A MALL Geometry of Interaction Based on Indexed Linear Logic

Masahiro Hamano

arXiv:1704.02711·cs.LO·June 22, 2023

A MALL Geometry of Interaction Based on Indexed Linear Logic

Masahiro Hamano

PDF

TL;DR

This paper develops a geometry of interaction model for MALL using indexed linear logic, enabling dynamic analysis of additive cut elimination with explicit indices.

Contribution

It extends the categorical GoI framework to handle additives in MALL through indexed logic, capturing additive features at a finer dynamic level.

Findings

01

Indexed logic effectively models additive cut elimination.

02

Execution formulas are invariant under cut elimination.

03

Indices diminish during execution, modeling erasure in proof reduction.

Abstract

We construct a geometry of interaction (GoI: dynamic modeling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli-Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi-Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard's original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation…

Equations138

(π_{[Δ], Γ})^{*} \subseteq ∣Γ∣

(π_{[Δ], Γ})^{*} \subseteq ∣Γ∣

R_{2} \circ R_{1} = {(g_{1}, g_{2}) ∣ \exists d \in D (g_{1}, d) \in R_{1} \land (d, g_{2}) \in R_{2}} \subseteq G_{1} \times G_{2}

R_{2} \circ R_{1} = {(g_{1}, g_{2}) ∣ \exists d \in D (g_{1}, d) \in R_{1} \land (d, g_{2}) \in R_{2}} \subseteq G_{1} \times G_{2}

sl (Δ) := {Δ_{S} ∣ S \subseteq {1, \dots, m}}

sl (Δ) := {Δ_{S} ∣ S \subseteq {1, \dots, m}}

[sl (Δ)] = S \subseteq {1, \dots, m} \sum ∣ Δ_{S} ∣,

[sl (Δ)] = S \subseteq {1, \dots, m} \sum ∣ Δ_{S} ∣,

[sl (Δ)] ≅ [sl (Δ_{1})] \times \dots \times [sl (Δ_{k})]

[sl (Δ)] ≅ [sl (Δ_{1})] \times \dots \times [sl (Δ_{k})]

∣ π_{[Δ], Γ} ∣ \subseteq [sl (Δ)] \times ∣Γ∣,

∣ π_{[Δ], Γ} ∣ \subseteq [sl (Δ)] \times ∣Γ∣,

∣ π_{[Δ_{1}, Δ_{2} A, A^{⊥}], Γ_{1}, Γ_{2}} ∣ :≅ ∣ π_{[Δ_{1}], Γ_{1}, A}^{1} ∣ \times ∣ π_{[Δ_{2}], A^{⊥}, Γ_{2}}^{2} ∣

∣ π_{[Δ_{1}, Δ_{2} A, A^{⊥}], Γ_{1}, Γ_{2}} ∣ :≅ ∣ π_{[Δ_{1}], Γ_{1}, A}^{1} ∣ \times ∣ π_{[Δ_{2}], A^{⊥}, Γ_{2}}^{2} ∣

\subseteq [sl (Δ_{1})] \times ∣ Γ_{1} ∣ \times ∣ A ∣ \times [sl (Δ_{2})] \times ∣ A^{⊥} ∣ \times ∣ Γ_{2} ∣

∣ π_{[Δ_{1}, Δ_{2}, Σ], Γ, A_{1} & A_{2}} ∣ :=

∣ π_{[Δ_{1}, Δ_{2}, Σ], Γ, A_{1} & A_{2}} ∣ :=

{(λ_{1}, (1, a_{1})) ∣ (λ_{1}, a_{1}) \in ∣ π_{[Δ_{1}, Δ], Γ, A_{1}}^{1} ∣} + {(λ_{2}, (2, a_{2})) ∣ (λ_{2}, a_{2}) \in ∣ π_{[Δ_{2}, Σ], Γ, A_{2}}^{2} ∣} \subseteq

([sl (Δ_{1}, Σ)] \times ∣Γ∣ \times {1} \times ∣ A_{1} ∣) + ([sl (Δ_{2}, Σ)] \times ∣Γ∣ \times {2} \times ∣ A_{2} ∣) \subseteq [sl (Δ_{1}, Δ_{2}, Σ)] \times ∣Γ∣ \times ∣ A_{1} & A_{2} ∣

⊢_{J} ([Δ], Γ) ⟨ ν ⟩ = ⊢_{J} [Δ ⟨⟨ δ ⟩⟩], Γ ⟨ γ ⟩

⊢_{J} ([Δ], Γ) ⟨ ν ⟩ = ⊢_{J} [Δ ⟨⟨ δ ⟩⟩], Γ ⟨ γ ⟩

ν \in ∣ π_{[Δ], Γ} ∣^{J} .

ν \in ∣ π_{[Δ], Γ} ∣^{J} .

⊢_{J} ([Δ], Γ) ⟨ ν ⟩ .

⊢_{J} ([Δ], Γ) ⟨ ν ⟩ .

\vdash_{J_{1}+J_{2}}[\Delta_{1},\Delta_{2},\Omega,\Delta_{3},(A_{1}\&A_{2}),(A_{2}^{\bot}\oplus A_{1}^{\bot})],\Gamma,\Xi\lx@proof@logical@and\vdash_{J_{1}+J_{2}}[\Delta_{1},\Delta_{2},\Omega],\Gamma,A_{1}\&A_{2}\left\{\begin{array}[]{c}\vdash_{J_{i}}[\Delta_{i},\Omega\!\upharpoonright_{J_{i}}],\Gamma\!\upharpoonright_{J_{i}},A_{i}\end{array}\right\}^{i=1,2}\vdash_{J_{1}+J_{2}}[\Delta_{3}],\Xi,A_{1}^{\bot}\oplus A_{2}^{\bot}\vdash_{J_{1}+J_{2}}[\Delta_{3}],\Xi,A_{1}^{\bot}

\vdash_{J_{1}+J_{2}}[\Delta_{1},\Delta_{2},\Omega,\Delta_{3},(A_{1}\&A_{2}),(A_{2}^{\bot}\oplus A_{1}^{\bot})],\Gamma,\Xi\lx@proof@logical@and\vdash_{J_{1}+J_{2}}[\Delta_{1},\Delta_{2},\Omega],\Gamma,A_{1}\&A_{2}\left\{\begin{array}[]{c}\vdash_{J_{i}}[\Delta_{i},\Omega\!\upharpoonright_{J_{i}}],\Gamma\!\upharpoonright_{J_{i}},A_{i}\end{array}\right\}^{i=1,2}\vdash_{J_{1}+J_{2}}[\Delta_{3}],\Xi,A_{1}^{\bot}\oplus A_{2}^{\bot}\vdash_{J_{1}+J_{2}}[\Delta_{3}],\Xi,A_{1}^{\bot}

\vspace{1.5ex}\blacktriangleright^{\textsc{\tiny$I$}}\hskip 113.81102pt\vdash_{J_{1}}[\Delta_{1},\Omega,\Delta_{3}\!\upharpoonright_{J_{1}},A_{1},A_{1}^{\bot}\!\upharpoonright_{J_{1}}],\Gamma,\Xi\lx@proof@logical@and\vdash_{J_{1}}[\Delta_{1},\Omega],\Gamma,A_{1}\vdash_{J_{1}}[\Delta_{3}\!\upharpoonright_{J_{1}}],\Xi\!\upharpoonright_{J_{1}},A_{1}^{\bot}\!\upharpoonright_{J_{1}}

\vspace{1.5ex}\blacktriangleright^{\textsc{\tiny$I$}}\hskip 113.81102pt\vdash_{J_{1}}[\Delta_{1},\Omega,\Delta_{3}\!\upharpoonright_{J_{1}},A_{1},A_{1}^{\bot}\!\upharpoonright_{J_{1}}],\Gamma,\Xi\lx@proof@logical@and\vdash_{J_{1}}[\Delta_{1},\Omega],\Gamma,A_{1}\vdash_{J_{1}}[\Delta_{3}\!\upharpoonright_{J_{1}}],\Xi\!\upharpoonright_{J_{1}},A_{1}^{\bot}\!\upharpoonright_{J_{1}}

ρ_{⊢_{J} ([Δ], Γ) ⟨ ν ⟩} \ignorespaces \ignorespaces \ignorespaces \ignorespaces

ρ_{⊢_{J} ([Δ], Γ) ⟨ ν ⟩} \ignorespaces \ignorespaces \ignorespaces \ignorespaces

⊢ [Δ, B^{⊥}, A], B, Γ \lx@proof@logical@and ⊢ B, B^{⊥} ⊢ [Δ], A, Γ

⊢ [Δ, B^{⊥}, A], B, Γ \lx@proof@logical@and ⊢ B, B^{⊥} ⊢ [Δ], A, Γ

⊢ [Δ], B, Γ

⊢ [Δ], B, Γ

J^{'} = {j \in J ∣ b_{j} = a_{j}}

J^{'} = {j \in J ∣ b_{j} = a_{j}}

J^{'} = J_{1} and ν^{'} = τ ↾_{J_{1}} \times λ^{'}

J^{'} = J_{1} and ν^{'} = τ ↾_{J_{1}} \times λ^{'}

λ^{'} \in ∣ π^{3} ∣^{J^{'}}

\vdash[\Xi,\Sigma_{1},\Sigma_{2},\Omega,A,A^{\bot}],\Delta,\Gamma,B_{1}\&B_{2}\lx@proof@logical@and\vdash[\Xi],\Delta,A\vdash[\Sigma_{1},\Sigma_{2},\Omega],A^{\bot},\Gamma,B_{1}\&B_{2}\left\{\begin{array}[]{c}\vdash[\Sigma_{i},\Omega],A^{\bot},\Gamma,B_{i}\end{array}\right\}^{i=1,2}

\vdash[\Xi,\Sigma_{1},\Sigma_{2},\Omega,A,A^{\bot}],\Delta,\Gamma,B_{1}\&B_{2}\lx@proof@logical@and\vdash[\Xi],\Delta,A\vdash[\Sigma_{1},\Sigma_{2},\Omega],A^{\bot},\Gamma,B_{1}\&B_{2}\left\{\begin{array}[]{c}\vdash[\Sigma_{i},\Omega],A^{\bot},\Gamma,B_{i}\end{array}\right\}^{i=1,2}

\vdash[\Sigma_{1},\Sigma_{2},A,A^{\bot},\,A,A^{\bot},\Xi,\Omega],\Delta,\Gamma,B_{1}\&B_{2}\left\{\begin{array}[]{c}\vdash[\Xi,\Sigma_{i},\Omega,A,A^{\bot}],\Delta,\Gamma,B_{i}\lx@proof@logical@and\vdash[\Xi],\Delta,A\vdash[\Sigma_{i},\Omega],A^{\bot},\Gamma,B_{i}\end{array}\right\}^{i=1,2}

\vdash[\Sigma_{1},\Sigma_{2},A,A^{\bot},\,A,A^{\bot},\Xi,\Omega],\Delta,\Gamma,B_{1}\&B_{2}\left\{\begin{array}[]{c}\vdash[\Xi,\Sigma_{i},\Omega,A,A^{\bot}],\Delta,\Gamma,B_{i}\lx@proof@logical@and\vdash[\Xi],\Delta,A\vdash[\Sigma_{i},\Omega],A^{\bot},\Gamma,B_{i}\end{array}\right\}^{i=1,2}

(C, \otimes, I, s_{U, U}, j : U \otimes U ⊲ U : k, 0_{U})

(C, \otimes, I, s_{U, U}, j : U \otimes U ⊲ U : k, 0_{U})

k \circ 0_{U} \circ j = 0_{U} \otimes 0_{U}

k \circ 0_{U} \circ j = 0_{U} \otimes 0_{U}

(s_{U, U})^{0} := 0_{U \otimes U}

(s_{U, U})^{0} := 0_{U \otimes U}

(s_{U, U})^{0} := (0_{U} \otimes 0_{U}) \circ s_{U, U} \circ (0_{U} \otimes 0_{U}) = 0_{U} \otimes 0_{U} = 0_{U \otimes U}

(s_{U, U})^{0} := (0_{U} \otimes 0_{U}) \circ s_{U, U} \circ (0_{U} \otimes 0_{U}) = 0_{U} \otimes 0_{U} = 0_{U \otimes U}

s

s

⊢_{{*}} ([Δ], Γ) ⟨ x ⟩ = ⊢_{{*}} [Δ ⟨⟨ x^{'} ⟩⟩], Γ ⟨ x^{''} ⟩

⊢_{{*}} ([Δ], Γ) ⟨ x ⟩ = ⊢_{{*}} [Δ ⟨⟨ x^{'} ⟩⟩], Γ ⟨ x^{''} ⟩

Δ

Δ

\vrule height=6.30554pt,depth=2.0pt,width=0.4pt \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt \vrule height=6.30554pt,depth=2.0pt,width=0.4pt \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt x \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt \vrule height=6.30554pt,depth=2.0pt,width=0.4pt \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt \vrule height=6.30554pt,depth=2.0pt,width=0.4pt_{π} : U^{ℓ} ⟶ U^{ℓ}

\vrule height=6.30554pt,depth=2.0pt,width=0.4pt \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt \vrule height=6.30554pt,depth=2.0pt,width=0.4pt \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt x \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt \vrule height=6.30554pt,depth=2.0pt,width=0.4pt \lower 2.0pt \vbox to8.30554pt \hrule width=1.0pt,height=0.0pt,depth=0.4pt \vfil \hrule width=1.0pt,height=0.4pt,depth=0.0pt \vrule height=6.30554pt,depth=2.0pt,width=0.4pt_{π} : U^{ℓ} ⟶ U^{ℓ}

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

A MALL Geometry of Interaction

Based on Indexed Linear Logic

Masahiro HAMANO

Institute of Information Science, Academia Sinica, Taiwan

[email protected]

Abstract

We construct a geometry of interaction (GoI: dynamic modeling of Gentzen-style cut elimination) for multiplicative–-additive linear logic ( ${\sf MALL}$ ) by employing Bucciarelli–Ehrhard indexed linear logic ${\sf MALL}(I)$ to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1 [Gir89]. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category ${\sf Rel}$ , but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, w.r.t symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

1 Introduction

The indexed multiplicative additive linear logic ${\sf MALL}(I)$ , introduced by Bucciarelli– Ehrhard [BucEhr00], is a conservative extension of Girard’s ${\sf MALL}$ in which all formulas and proofs come equipped with sets of indices. The usual ${\sf MALL}$ is stipulated to be the restriction to the empty set. The status of the indexed syntactical system is noteworthy as it stems from the denotational semantics of Rel, a simple, yet pivotal categorical model of ${\sf MALL}$ . With the enabling of an explicit notion of location in linear proof theory, the indices can enumerate the locations of formulas and proofs, corresponding to denotational interpretations of ${\sf MALL}$ . The notion of location becomes a requirement for the additives, although it is redundant for the multipicatives, for which the singleton $\{*\}$ suffices. To work with parallelism, which the additives bring intrinsically, different locations need to be handled rather than the sole location $*$ . In the context of parallelism, superpositions are known to typically arise under the syntactic additive $\&$ -rule. Indices allow one to deal with superpositions by identifying multiple occurrences of formulas in the different indices and by enlarging (or restricting) the indices.

The original motivation for indexed logic was to provide a bridge between a truth-valued semantics (for provability) for ${\sf MALL}(I)$ and the denotational semantics of (nonindexed) ${\sf MALL}$ . By means of this bridge, Bucciarelli–Ehrhard obtained a new kind of denotational completeness theorem in [BucEhr00] for ${\sf MALL}$ and later extended it to the exponential in [BucEhr01].

This paper investigates indexed ${\sf MALL}$ from the perspective of a dynamic semantics for cut elimination, a topic that—to the best of our knowledge—has remained untouched aside from the precursory work of Duchesne [Duc09] since the original work of Bucciarelli–Ehrhard [BucEhr00]. The dynamic semantics is the Girard project of Geometry of Interaction (GoI), whereby cut-elimination is modelled, using operator algebras [Gir89] and more generally traced monoidal categories [JSV96]. The GoI project was successful [Gir89, HS06] for ${\sf MLL}$ with the exponential, and inspired a new model of computation for $\beta$ reduction of $\lambda$ -calculus [DR95]. This paper aims to initiate an exploration of how to combine the two notions of location, which the indexed logic brings, and of dynamics, which GoI brings to cut-elimination. The combination is important in understanding additive cut-elimination. For this goal, we employ the indices to construct a GoI model for (non-indexed) ${\sf MALL}$ . We combine the Haghverdi–Scott categorical GoI situation [HS06] with the indices in such a way that that the original ${\sf MLL}$ GoI situation represents a collapse to the singleton index $\{*\}$ . The dynamics of cut-elimination is captured by a feedback mechanism determined by traces of morphisms interpreting proofs. We further augment the situation with two kinds of actions, identical and zero, over the symmetries interpreting the cut-rule. These two actions provide representations of matches and of mismatches among locations. These come into play during a Gentzen style cut-elimination procedure, in which one encounters noncommunication of individual proofs, due to the additive parallelism. Crucial instances of GoI situation such as ${\sf Rel}_{+}$ and ${\sf Hilb_{2}}$ [HS06] are directly lifted to our framework, the latter of which is the operator algebraic origin of the Girard project.

We study Girard’s execution formula [Gir89] in the general categorical setting of a traced symmetric monoidal category. The execution formula accommodates indices, and faithfully simulates ${\sf MALL}$ cut-elimination by a hybrid method relating the indexed syntax to the relational semantics. Each location in the relation interpreting a proof is first assigned an endomorphism on a reflexive object $U$ . The cut-rule before execution is interpreted as a tensor product of two premise morphisms, more loosely than their composition. This interpretation allows extraction of the dynamical meaning of the cut, which the usual categorical composition hides by virtue of its static approach. In the loose interpretation, there remain redundant indices when interpreting rules: however, they are shown to disappear, while running the Execution formula in terms of the categorical trace structure. The disappearance of indices is modelled by zero morphisms, which exist in the traced monoidal categories for GoI. Proof-theoretically, the zero morphisms allow us to interpret discarding subproofs specific to additive cut-elimination, and in the way of theory of indices, they provide a way to to interpret mismatches among locations. In traced monoidal categories, the zero morphisms are supposed to act partially on symmetries for cut-formulas, and also to act partially on retractions and co-retractions of the reflexive objects. The latter action arises via tracing along the zero morphisms which takes feedback into account along with the zero. We prove zero-convergence which means that execution formulas converge to zero when two proofs interact with mismatched locations. Thus the execution formulas terminate to the denotational interpretations of proofs, while diminishing sets of indices in order to recover the relational composition. This is realized by properly coupling indices to trace axioms, especially for “generalized yanking” and “dinaturality”. The former axiom directly designates that traces are primitive enough to retrieve the categorical composition in a monoidal category, and the latter axiom concerns the interaction of bidirectional flow of morphisms.

In contrast to the precursory work of Duchesne [Duc09] concerning both indices and GoI, the present paper accommodates the indices directly in GoI semantics in order to simulate (nonindexed) ${\sf MALL}$ cut elimination. The diminution of sets of indices is a typical dynamic aspect our GoI captures using the zero morphisms of our traced monoidal category. The precursor in [Duc09], on the contrary, first accommodates the index into the static category of ${\sf Rel}$ , using the semantic method of localisation, of which the indexed syntax provides a precise description. Then (nonindexed) dynamic GoI action over the indexed denotational semantics is investigated to characterise the static fix points as the denotational interpretation.

We prove two main result: (i) (Invariance of the execution formula during ${\sf MALL}$ normalization): The execution formula in our dynamic categorical modelling is shown to converge to the denotational interpretation of proofs in the static categorical model. This characterises the normalization of proofs by categorical invariants. (ii) (Diminution of indices while running the execution formula): The execution formula may converge to [math], making the redundant indices disappear. Part (i) is seen as a pointwise collection of invariants, as previously established for the multiplicatives [Gir89, HS06]. Part (ii) is specific to the additives: Proof-theoretically. it reflects erasure of subproofs as well as additive (co) contraction and superposition, in cut-elimination. Category-theoretically, it ensures that our categorical ingredient (the execution formula) is fine grained enough to retrieve a static monoidal category as well as a relational category handling indices.

The rest of this paper is organized as follows: Section 2 introduces a syntax ${\sf MALL}^{[\mathfrak{c}]}(I)$ for indexed ${\sf MALL}$ with a cut list as well as its relational counterpart ${\sf Rel}^{[\mathfrak{c}]}$ . A fundamental lemma is proved, which connects a provable ${\sf MALL}^{[\mathfrak{c}]}(I)$ sequent to an indexed subset of the interpretation of a ${\sf MALL}$ proof with cuts. In Section 3, ${\sf MALL}$ proof reduction for cut elimination is lifted to ${\sf MALL}(I)$ proof transformation with diminishing sets of indices. Section 4 concerns our ${\sf MALL}$ GoI interpretation by means of the indexed system in a traced symmetric monoidal category with zero morphism. Execution formulas are run indexwise, and the main theorem is proved.

2 ${\sf MALL}(I)$ with cut list and relational semantics

2.1 ${\sf MALL}(I)$ with cut list

(Inference rules of ${\sf MALL}^{[\mathfrak{c}]}$ with cut formulas)

We accommodate a stack to record cut formulas into the syntax of the multiplicative–-additive linear logic ${\sf MALL}$ . To stress this, the system is written as ${\sf MALL}^{[\mathfrak{c}]}$ . To accommodate the stack into the additive fragment, one has to work with superpositions that arise inside the stack as well as in the conclusion (outside the stack).

A ${\sf MALL}^{[\mathfrak{c}]}$ sequent $\vdash[\Delta],\Gamma$ with a cut list is a set $\Gamma$ of formula occurrences together with pairwise-dual formulas occurrences $\Delta$ inside the brackets. Each dual pair in $\Delta$ is written $A,A^{\perp}$ .

Sequents are proved using the following rules:

$\vdash A,A^{\perp}$

$\vdash[\Delta_{1},\Delta_{2}],\,\,\Gamma_{1},\Gamma_{2},A\otimes B\lx@proof@logical@and\vdash[\Delta_{1}],\,\,\Gamma_{1},A\vdash[\Delta_{2}],\,\,\Gamma_{2},B$

$\vdash[\Delta],\,\,\Gamma,A\parr B\vdash[\Delta],\,\,\Gamma,A,B$

$\vdash[\Delta_{1},\Delta_{2},A,A^{\perp}],\,\,\Gamma_{1},\Gamma_{2}\lx@proof@logical@and\vdash[\Delta_{1}],\,\,\Gamma_{1},A\vdash[\Delta_{2}],\,\,A^{\perp},\Gamma_{2}$

$\vdash[\Delta_{1},\Delta_{2},\Sigma],\,\,\Gamma,A_{1}\&A_{2}\lx@proof@logical@and\vdash[\Delta_{1},\Sigma],\,\,\Gamma,A_{1}\vdash[\Delta_{2},\Sigma],\,\,\Gamma,A_{2}$

$\vdash[\Delta],\,\,\Gamma,A_{1}\oplus A_{2}\vdash[\Delta],\,\,\Gamma,A_{1}$

$\vdash[\Delta],\,\,\Gamma,A_{1}\oplus A_{2}\vdash[\Delta],\,\,\Gamma,A_{2}$

Note: In the $\&$ -rule, not only is $\Gamma$ superposed in the conclusion, but so is $\Sigma$ in the stack. The superposition among cut formulas inside the stack causes the well-known additive (co-) contraction that arises in ${\sf MALL}$ cut elimination. The formulas occurrences $\Sigma$ is not deterministically chosen in the premises, so that $\Sigma$ in general is neither empty (i.e., never superimpose cuts) nor maximal (i.e., superimpose as many cuts as possible). Thus, the $\&$ -rule has several possible instances depending on the choice of $\Sigma$ . The exchange rule is eliminated under the assumption that formula occurrences are implicitly tracked by the premises and the conclusion of a rule.

We extend the above accommodation of cut lists to Bucciarelli–Ehrhard indexed system ${\sf MALL}(I)$ [BucEhr00]. To stress this, the system is written as ${\sf MALL}^{[\mathfrak{c}]}(I)$ . The extension stipulates that a set of indices is consistently associated with each formula (including cut formulas) and sequent (including cut lists).

We fix an index set $I$ , once and for all. Each formula $A$ of ${\sf MALL}(I)$ is associated with a set $d(A)\subseteq I$ , called the domain of $A$ .

( ${\sf MALL}(I)$ ** formulas and domains**)

Formulas in the domain $J$ are defined by the following grammar: $\bm{0}_{\emptyset}$ and $T_{\emptyset}$ are formulas of the domain $\emptyset$ . For any $J,K,L\subseteq I$ such that $K\cap L=\emptyset$ and $K\cup L=J$ ,

$\begin{aligned} X_{J}::=\quad\bm{1}_{J}\quad\arrowvert\quad\bot_{J}\quad\arrowvert\quad X_{J}\otimes X_{J}\quad\arrowvert\quad X_{J}\parr X_{J}\quad\arrowvert\quad X_{K}\oplus X_{L}\quad\arrowvert\quad X_{K}\&X_{L}.\end{aligned}$

For any ${\sf MALL}(I)$ formula $A$ with $d(A)=J$ , its negation $A^{\perp}$ with $d(A^{\perp})=J$ is defined using the De Morgan duality for the ${\sf MALL}$ formula.

(Restriction)

For a ${\sf MALL}(I)$ formula $A$ with $d(A)=J$ and $K\subseteq J$ , the restriction $A\!\upharpoonright_{K}$ of $A$ by $K$ is defined to be a ${\sf MALL}(I)$ formula in the domain $J\cap K$ as follows:

$\bm{0}_{\emptyset}\!\upharpoonright_{K}=\bm{0}_{\emptyset}$ and $T_{\emptyset}\!\upharpoonright_{K}=T_{\emptyset}$ . $\perp_{J}\!\upharpoonright_{K}=\perp_{J\cap K}$ and $\bm{1}_{J}\!\upharpoonright_{K}=\bm{1}_{J\cap K}$ $(A\otimes B)\!\upharpoonright_{K}=A\!\upharpoonright_{K}\!\otimes\;B\!\upharpoonright_{K}$

$(A\parr B)\!\upharpoonright_{K}=A\!\upharpoonright_{K}\!\parr\;B\!\upharpoonright_{K}$ $(A\oplus B)\!\upharpoonright_{K}=A\!\upharpoonright_{K}\!\oplus\;B\!\upharpoonright_{K}$ $(A\&B)\!\upharpoonright_{K}=A\!\upharpoonright_{K}\!\&\;B\!\upharpoonright_{K}$

It trivially follows that $(A^{\perp})\!\upharpoonright_{K}=(A\!\upharpoonright_{K})^{\perp}$ . If $\Gamma$ is a sequence of ${\sf MALL}(I)$ formulas $A_{1},\ldots,A_{n}$ of domains $J$ , we define $\Gamma\!\upharpoonright_{K}=A_{1}\!\upharpoonright_{K},\ldots,A_{n}\!\upharpoonright_{K}$ .

(Inference rules of ${\sf MALL}^{[\mathfrak{c}]}(I)$ with cut lists)

We augment ${\sf MALL}^{[\mathfrak{c}]}$ with indices. This makes it possible to accommodate a stack recording cut formulas to the original ${\sf MALL}(I)$ [BucEhr00]. Although this is straightforward for the multiplicative fragment, careful treatment is required for the listing of cut formulas in the additive fragment. Two kinds of sequences of formulas are considered in our ${\sf MALL}^{[\mathfrak{c}]}(I)$ -syntax: One is a sequence $\Xi$ whose all formulas occurrences have a same domain $J$ uniformly, which is denoted by $d(\Xi)=J$ . The other is a sequence $\Xi$ whose any formula occurrence has a domain contained in $J$ , which is denoted by $d(A)\subseteq J$ . Each sequent is of the form $\vdash_{J}[\Delta],\Gamma$ , in which $d(\Gamma)=J$ and $d(\Delta)\subseteq J$ with $d(A)=d(A^{\bot})\subseteq J$ for any pairwise-dual formulas $A$ and $A^{\bot}$ in $\Delta$ within the stack.

**Note: ** The uniformity requirement that all formulas in $\Gamma$ have the same domain $I$ does not apply to the stack $\Delta$ of the cut formulas. Formulas from different cuts have various domains contained in $J$ .

**Axioms and cut: **

$\vdash_{J}\bm{1}_{J}$

$\vdash_{\emptyset}\Gamma,T_{\emptyset}$

$\vdash_{J}[\Delta_{1},\Delta_{2},A,A^{\perp}],\,\,\Gamma_{1},\Gamma_{2}\lx@proof@logical@and\vdash_{J}[\Delta_{1}],\,\,\Gamma_{1},A\vdash_{J}[\Delta_{2}],\,\,A^{\perp},\Gamma_{2}$

Note that $d(A)=d(A^{\perp})=J$ for cut formulas $A$ and $A^{\perp}$ .

Multiplicative rules:

$\vdash_{J}[\Delta],\,\,\Gamma,\bot_{J}\vdash_{J}[\Delta],\,\,\Gamma$

$\vdash_{J}[\Delta_{1},\Delta_{2}\,],\,\,\Gamma_{1},\Gamma_{2},A\otimes B\lx@proof@logical@and\vdash_{J}[\Delta_{1}],\,\,\Gamma_{1},A\vdash_{J}[\Delta_{2}],\,\,\Gamma_{2},B$

$\vdash_{J}[\Delta],\,\,\Gamma,A\parr B\vdash_{J}[\Delta],\,\,\Gamma,A,B$

Additive rules:

$\vdash_{J_{1}+J_{2}}[\Delta_{1},\Delta_{2},\Sigma\,],\,\,\Gamma,A_{1}\&A_{2}\lx@proof@logical@and\vdash_{J_{1}}[\Delta_{1},\Sigma\!\upharpoonright_{J_{1}}\,],\,\,\Gamma\!\upharpoonright_{J_{1}},A_{1}\vdash_{J_{2}}[\Delta_{2},\Sigma\!\upharpoonright_{J_{2}}\,],\,\,\Gamma\!\upharpoonright_{J_{2}},A_{2}$

Note that the superposed context $\Gamma$ encompasses the whole domain $J=J_{1}+J_{2}$ , while the superposed context $\Sigma$ in the stack has a domain contained in $J$ .

$\vdash_{J}[\Delta],\,\,\Gamma,A_{1}\oplus A_{2}\vdash_{J}[\Delta],\,\,\Gamma,A_{1}$

$\vdash_{J}[\Delta],\,\,\Gamma,A_{1}\oplus A_{2}\vdash_{J}[\Delta],\,\,\Gamma,A_{2}$

Note $d(A_{3-i})=\emptyset$ in

each rule $\oplus_{i}$ ( $i=1,2$ ).

${\sf MALL}(I)$ has no propositional variables; the only atomic formulas are the constants. Then, the usual identity axiom is readily derived:

Lemma 2.1 (Identity)

$\vdash_{J}A,A^{\perp}$ * is provable for any ${\sf MALL}(I)$ formula $A$ of domain $J$ .*

Lemma 2.2 (Restricting provable sequents)

*If $\vdash_{J}[\Delta],\,\,\Gamma$ is provable, then so is

$\vdash_{J\cap K}[\Delta\!\upharpoonright_{K}],\,\,\Gamma\!\upharpoonright_{K}$ for any $K\subseteq I$ .*

For each inference rule of ${\sf MALL}^{[\mathfrak{c}]}(I)$ , if the conclusion sequent has the domain $\emptyset$ , then so does the premise sequent(s). Thus, the rules for sequents deriving the empty domain are identified with the rules of ${\sf MALL}^{[\mathfrak{c}]}$ . As a consequence, every ${\sf MALL}^{[\mathfrak{c}]}(I)$ -proof $\pi$ for $\vdash_{\emptyset}[\Delta],\Gamma$ contains only sequents of the empty domain. Hence $\pi$ is considered as a ${\sf MALL}$ -proof for $\vdash[\Delta],\Gamma$ . To sum up,

Lemma 2.3

${\sf MALL}^{[\mathfrak{c}]}(I)$ * is a conservative extension of ${\sf MALL}^{[\mathfrak{c}]}$ .*

Accordingly, in the sequel ${\sf MALL}^{[\mathfrak{c}]}$ is identified with ${\sf MALL}^{[\mathfrak{c}]}(\emptyset)$ .

2.2 ${\sf MALL}^{[\mathfrak{c}]}(I)$ and Relational Semantics ${\sf Rel}^{[\mathfrak{c}]}$

It is well known that the category ${\sf Rel}$ of sets and relations constitutes a denotational semantics of ${\sf MALL}$ , that is, the interpretation is invariant, $(\pi_{[\Delta],\Gamma})^{*}=(\pi^{\prime}_{[\Delta^{\prime}],\Gamma})^{*}$ , for any reduction $\pi_{[\Delta],\Gamma}\rhd\pi^{\prime}_{[\Delta^{\prime}],\Gamma}$ of ${\sf MALL}$ cut elimination. In particular, the denotation of $\pi$ is equal to that of a cut-free $\pi^{\prime}$ when $\Delta^{\prime}$ is empty. The cut rule is interpreted by relational composition in ${\sf Rel}$ , and this interpretation makes the semantics denotational.

Definition 2.4** (Denotational interpretation $(\pi_{[\Delta],\Gamma})^{*}$

in ${\sf Rel}$ )**

Every ${\sf MALL}$ proof $\pi_{[\Delta],\Gamma}$ of a sequent $\vdash[\Delta]\,\,\Gamma$ is interpreted as a subset of an associated set of the conclusion (without the cut list),

[TABLE]

Note that in the interpretation, the cut formulas $\Delta$ become invisible by virtue of the relational composition: Two relations $R_{1}\subseteq G_{1}\times D$ and $R_{2}\subseteq D\times G_{2}$ compose in ${\sf Rel}$

[TABLE]

The interpretation $\pi^{*}$ of (1) is known in [BucEhr00] as the relational interpretation of ${\sf MALL}$ proofs in a compact closed category ${\sf Rel}_{\times}$ with biproducts $+$ . The interpretation is specified as follows accordingly to the ${\sf MALL}$ rules, for which we refer to the above ${\sf MALL}^{[\mathfrak{c}]}$ -rules ridden of the cut-lists. First, every formula $A$ is interpreted as a set $|A|$ , and every sequence $\Gamma=A_{1},\ldots,A_{n}$ as $|\Gamma|=|A|\times\cdots\times|A_{n}|$ : When $A$ is $A_{1}\otimes A_{2}$ or $A_{1}\wp A_{2}$ , $|A|=|A_{1}|\times|A_{2}|$ , and when $A$ is $A_{1}\oplus A_{2}$ or $A_{1}\&A_{2}$ , $|A|=(\{1\}\times|A_{1}|)\cup(\{2\}\times|A_{2}|)$ .

Then

(axiom) $\pi^{*}=\{(a,a)\mid a\in|A|\}\subseteq|A,A^{\perp}|$

(cut-rule) $\pi^{*}=\{(\gamma_{1},\gamma_{2})\mid(\gamma_{1},a)\in\pi_{1}^{*}\,\mbox{and}\,(a,\gamma_{2})\in\pi_{2}^{*}\}\subseteq|\Gamma_{1},\Gamma_{2}|$

( $\&$ -rule) $\pi^{*}=\{(\gamma,(1,a))\mid(\gamma,a)\in\pi_{1}^{*}\}\cup\{(\gamma,(2,a)\mid(\gamma,a)\in\pi_{2}^{*}\}\subseteq|\Gamma,A_{1}\&A_{2}|$

( $\oplus_{i}$ -rule) $\pi^{*}=\{(\gamma,(i,a))\mid(\gamma,a)\in\pi_{i}^{*}\}\subseteq|\Gamma,A_{1}\oplus A_{2}|$

( $\otimes$ -rule) $\pi^{*}=\{(\gamma_{1},\gamma_{2},(a_{1},a_{2}))\mid(\gamma_{1},a_{1})\in\pi_{1}^{*}\,\mbox{and}\,(\gamma_{2},a_{2})\in\pi_{2}^{*}\}\subseteq|\Gamma,A_{1}\otimes A_{2}|$

( $\parr$ -rule) $\pi^{*}=\{(\gamma,(a_{1},a_{2}))\mid(\gamma,a_{1},a_{2})\in\pi^{{}^{\prime}*}\}\subseteq|\Gamma,A_{1}\parr A_{2}|$

Our aim in this paper is to investigate a dynamics of cuts hidden in such a static categorical composition. We begin by interpreting proofs in ${\sf Rel}$ but without performing cuts by means of relational composition. To stress this interpretation with the unexecuted cuts, the categorical framework is denoted by ${\sf Rel}^{[\mathfrak{c}]}$ , in which the cut list $[\Delta]$ is interpreted explicitly.

To deal with the additives in ${\sf Rel}^{[\mathfrak{c}]}$ , we have to work with a sublist and the set of all the sublists: Let $\Delta$ be $C_{1},C_{1}^{\bot},\ldots,C_{m},C_{m}^{\bot}$ . For a subset $S$ of $\{1,\ldots,m\}$ , let $\Delta_{S}$ denote the sublist $\ldots C_{i},C_{i}^{\bot}\ldots$ where $i$ ranges in $S$ . Then the set $\mathfrak{sl}(\Delta)$ of all the sublists (including $\Delta$ and $\Delta_{\emptyset}$ ) is defined;

[TABLE]

Consequently, we interpret (2) as an object in ${\sf Rel}$ in terms of the disjoint union of each interpretations of the sublists:

[TABLE]

in which $|\Delta_{S}|$ , for a nonempty sequence, is the usual interpretation of the sequence in ${\sf Rel}$ and $|\Delta_{\emptyset}|:=\{*\}=|\bot|$ . The disjoint sum $\Sigma$ is taken over different $S$ ’s.

In what follows in this paper, when $S$ is clear from context, a sublist $\Delta_{S}$ is often abbreviated by $\widehat{\Delta}$ , whose hat indicates a pairwise deletion of some cut formulas.

Lemma 2.5

If $\Delta=\Delta_{1},\ldots,\Delta_{k}$ such that $\Delta_{i}$ are lists of pairwise-dual formulas, then

[TABLE]

E.g., a particular choice of each $\Delta_{i}$ is $A_{i},A_{i}^{\perp}$ .

In what follows in Sections 2 and 3, $\cong$ denotes an iso modulo the symmetry of the set-theoretical cartesian product. In the sequel, the symmetry corresponds to the exchange of formula occurrences. As the exchange is always clear from the context and fixed, we use the terminologies $\subseteq_{\cong}$ and $\in_{\cong}$ consistently as follows: $A\subseteq_{\cong}B$ (resp. $a\in_{\cong}B$ ) means that $A$ is a subset (resp. a member) of $\sigma(B)$ where $\sigma$ is the exchange for $\cong$ .

Definition 2.6** (Interpretation $|\pi_{[\Delta],\Gamma}|$

of proofs with unexecuted cuts in ${\sf Rel}^{[\mathfrak{c}]}$ )**

Every ${\sf MALL}$ proof $\pi_{[\Delta],\Gamma}$ of a sequent $\vdash[\Delta],\,\,\Gamma$ is interpreted by

[TABLE]

which is defined inductively and in the same manner as in Definition 2.4, except for the cut rule to make the interpretation differ from the standard (1) in that $\Delta$ is visible without performing the relational composition.

*(cut rule) *

When $\pi$ is $\vdash[\Delta_{1},\Delta_{2},A,A^{\perp}],\,\,\Gamma_{1},\Gamma_{2}\lx@proof@logical@and\vdash[\Delta_{1}],\,\,\Gamma_{1},A\pi^{1}\vdash[\Delta_{2}],\,\,A^{\perp},\Gamma_{2}\pi^{2}$

[TABLE]

The symmetry used for the definition is the exchange between the conclusion of the cut and merging those of $\pi_{i}$ ’s. In obtaining the last inclusion, Lemma 2.5 is used because the two lists $\Delta_{1}$ and $\Delta_{2}$ are disjoint.

( $\&$ *-rule) *

When $\pi$ is $\vdash[\Delta_{1},\Delta_{2},\Sigma\,],\,\,\Gamma,A_{1}\&A_{2}\lx@proof@logical@and\vdash[\Delta_{1},\Sigma\,],\,\,\Gamma,A_{1}\pi^{1}\vdash[\Delta_{2},\Sigma\,],\,\,\Gamma,A_{2}\pi^{2}$

[TABLE]

For the last inclusion, the monotonicity, $[\mathfrak{sl}(\Delta_{i},\Sigma)]\subseteq[\mathfrak{sl}(\Delta_{1},\Delta_{2},\Sigma)]$ is used.

We extend Bucciarelli–Ehrhard translation of Definition 2.7 to Definition 2.8 to accommodate cut formulas inside the stack.

Definition 2.7** (Translation of indexed relation $\gamma$

to ${\sf MALL}(I)$ sequent $\Gamma\langle\gamma\rangle$ [BucEhr00])**

To any ${\sf MALL}$ formula $A$ and a family $a\in|A|^{J}$ , a formula $A\langle a\rangle$ of ${\sf MALL}(I)$ is associated, with domain $J$ so that $A\langle a\rangle\!\upharpoonright_{\emptyset}$ is $A$ . For a sequence $\Gamma=A_{1},\ldots,A_{n}$ of ${\sf MALL}$ formulas, every $\gamma\in|\Gamma|^{J}$ is written uniquely as $\gamma=\gamma^{1}\times\cdots\times\gamma^{n}$ with $\gamma^{m}\in|A_{m}|^{J}$ , and we set $\Gamma\langle\gamma\rangle=A_{1}\langle\gamma_{1}\rangle,\ldots,A_{n}\langle\gamma_{n}\rangle$ .

-For $A=\bm{0}$ or $A=T$ , if $J\not=\emptyset$ , we have $|A|^{J}=\emptyset$ , and $A\langle a\rangle$ is undefined. If $J=\emptyset$ , $|A|^{J}$ has exactly one element, namely, the empty family $\emptyset$ , and we set $\bm{0}\langle\emptyset\rangle=\bm{0}$ and $T\langle\emptyset\rangle=T$ .

-If $A=\bm{1}$ or $A=\>\perp$ , $a$ is the constant family, and we set $\bm{1}\langle(*)_{J}\rangle=\bm{1}_{J}$ and $\perp\langle(*)_{J}\rangle=\perp_{J}$ .

-If $A=B\otimes C$ , then $a=b\times c$ with $b\in|B|^{J}$ and $c\in|C|^{J}$ , and we set $A\langle a\rangle=B\langle b\rangle\otimes C\langle c\rangle$ which is a well-formed formula of ${\sf MALL}(I)$ of domain $J$ . Here $b\times c$ denotes the mediating morphism of the set-theoretical cartesian product.

Similarly, for $A=B\parr C$ , we set $A\langle a\rangle=B\langle b\rangle\parr C\langle c\rangle$ .

-If $A=B\oplus C$ , then $a=b+c$ with $b\in|B|^{K}$ and $c\in|C|^{L}$ and $K+L=J$ . Then we set $A\langle a\rangle=B\langle b\rangle\oplus C\langle c\rangle$ which is a well-formed formula of ${\sf MALL}(I)$ of domain $J$ .

Similarly for $A=B\;\&\;C$ , we set $A\langle a\rangle=B\langle b\rangle\,\&\,C\langle c\rangle$ .

(Notation)

Let $X$ be a set and $J=J_{1}+J_{2}$ . Every $x\in X^{J}$ yields the restrictions $x_{i}=x\!\upharpoonright_{J_{i}}\in X^{J_{i}}$ with $i=1,2$ . Conversely, the two restrictions allow us to recover $x$ . We write this as $x=x_{1}\raisebox{2.15277pt}{$ \frown $}x_{2}$ .

Definition 2.8** (Translation to ${\sf MALL}^{[\mathfrak{c}]}(I)$

sequent $\vdash_{J}[\Delta\langle\langle\delta\rangle\rangle],\,\Gamma\langle\gamma\rangle$ )**

Let $\Delta$ be a sequence of pairwise-dual ${\sf MALL}$ formulas and $\delta\in[\mathfrak{sl}(\Delta)]^{J}$ for some $J\subseteq I$ . Then the ${\sf MALL}(I)$ sequence $\Delta\langle\langle\delta\rangle\rangle$ of pairwise-dual formulas is associated such that $d(\Delta\langle\langle\delta\rangle\rangle)\subseteq J$ and $\Delta\langle\langle\delta\rangle\rangle\!\upharpoonright_{\emptyset}$ is $\Delta$ as follows.

First, we write $\Delta=\Delta_{1},\ldots,\Delta_{n}$ , where each $\Delta_{i}$ is a list of two dual formulas $A_{i}$ and $A_{i}^{\bot}$ . By Lemma 2.5, $\delta=\delta^{1}\times\cdots\times\delta^{n}$ , so $\delta^{i}\in[\mathfrak{sl}(\Delta_{i})]^{J}$ . Because $\mathfrak{sl}(\Delta_{i})=\{\Delta_{i},\Delta_{\emptyset}\}$ , we have $[\mathfrak{sl}(\Delta_{i})]=|A_{i},A_{i}^{\perp}|+\{*\}$ . Recall that $\{*\}$ interprets the empty list in (3). Thus every $\delta^{i}$ makes $J$ divide into $J=J_{i}+K_{i}$ to yield $\delta^{i}=\delta^{i}\!\upharpoonright_{J_{i}}\!\raisebox{2.15277pt}{$ \frown $}\,\,\delta^{i}\!\upharpoonright_{K_{i}}$ so that $\delta^{i}\!\upharpoonright_{J_{i}}\in|A_{i},A_{i}^{\perp}|^{J_{i}}$ and $\delta^{i}\!\upharpoonright_{K_{i}}\in|*|^{K_{i}}$ . (explicitly $J_{i}=\{x\mid\delta_{i}(x)\in|A_{i},A_{i}^{\perp}|\}$ and $K_{i}=\{x\mid\delta_{i}(x)=*\}$ .) Then, using the $\delta^{i}\!\upharpoonright_{J_{i}}\,$ , we define

$\begin{aligned} \Delta\langle\langle\delta\rangle\rangle=\Delta_{1}\langle\delta^{1}\!\upharpoonright_{J_{1}}\rangle,\,\ldots,\,\Delta_{n}\langle\delta^{n}\!\upharpoonright_{J_{n}}\rangle,\end{aligned}$ **

in which the two dual formulas in each $\Delta_{i}\langle\delta^{n}\!\upharpoonright_{J_{i}}\rangle$ have the same domain $J_{i}\subseteq J$ .

Then, by employing Definition 2.7, for a given ${\sf MALL}$ sequent $\vdash[\Delta],\Gamma$ , every $\nu\in(|\mathfrak{sl}(\Delta)|\times|\Gamma|)^{J}$ is associated with a ${\sf MALL}(I)$ sequent, for which we write $\nu=\delta\times\gamma$ , so that $\delta\in[\mathfrak{sl}(\Delta)]^{J}$ and $\gamma\in|\Gamma|^{J}$ :

[TABLE]

Here $\vdash_{J}([\Delta],\,\Gamma)\langle\nu\rangle$ restricted to $\emptyset$ is $\vdash[\Delta],\Gamma$ . Note that all the formulas in $\Gamma\langle\gamma\rangle$ have domain $J$ , while each formula in $\Delta\langle\langle\delta\rangle\rangle$ has a domain contained in $J$ . The $\Delta\langle\langle\delta\rangle\rangle$ ’s inside the stack become a list of pairwise-dual ${\sf MALL}(I)$ formulas in which each pair has the same domain.

The translations commute with restriction of indices, and Lemma 2.2 can be restated:

Lemma 2.9 (Restricting translation)

For any $\gamma\in|\Gamma|^{J}$ , it holds that $\Gamma\langle\gamma\!\upharpoonright_{J\cap K}\rangle=\Gamma\langle\gamma\rangle\!\upharpoonright_{K}$ .

-

For any $\delta\in[\mathfrak{sl}(\Delta)]^{J}$ , it holds that $\Delta\langle\langle\delta\!\upharpoonright_{J\cap K}\rangle\rangle=\Delta\langle\langle\delta\rangle\rangle\!\upharpoonright_{K}$ .

-

If $\vdash_{J}[\,\Delta\langle\langle\delta\rangle\rangle\,],\,\,\Gamma\langle\gamma\rangle$ is provable, then so is $\vdash_{J\cap K}[\,\Delta\langle\langle\delta\rangle\rangle\!\upharpoonright_{J\cap K}\,],\,\,\Gamma\langle\gamma\rangle\!\upharpoonright_{J\cap K}$ .

2.3 Fundamental lemma

Indexed linear logic arises essentially due to its tight connection to the relational semantics. The connection is realized by a fundamental lemma due to Bucciarelli & Ehrhard (proposition 20 of [BucEhr00]) establishing a correspondence between indexed sets in ${\sf Rel}$ and indexed sequents in ${\sf MALL}(I)$ . The former is semantic in ${\sf MALL}$ , while the latter is syntactic in ${\sf MALL}(I)$ . The fundamental lemma is shown to be preserved under our extended syntax and semantics, designed to accommodate cut formulas in ${\sf MALL}^{[\mathfrak{c}]}(I)$ and in ${\sf Rel}^{[\mathfrak{c}]}$ , respectively.

Proposition 2.10 (Fundamental lemma à la Bucciarelli–Ehrhard)

For $\nu\in(|\mathfrak{sl}(\Delta)|\times|\Gamma|)^{J}$ with $J\subseteq I$ , the following two statements are equivalent and induce a relationship $\rho\!\upharpoonright_{\emptyset}=\pi$ between $\pi$ of (i) and $\rho$ of (ii):

(i)

There exists a ${\sf MALL}^{[\mathfrak{c}]}$ proof $\pi$ such that

[TABLE] 2. (ii)

There exists a ${\sf MALL}^{[\mathfrak{c}]}(I)$ proof $\rho$ of the sequent

[TABLE]

** Proof**.

See lemmas B.1 and B.2 in the Appendix B.1. $\Box$

3 Lifting ${\sf MALL}$ reduction over indices

This section describes how our indexed syntax ${\sf MALL}^{[\mathfrak{c}]}(I)$ analyzes Gentzen-style reduction of cut elimination for nonindexed ${\sf MALL}$ . Every ${\sf MALL}$ reduction with cut elimination is shown to be lifted to a directed transformation between two ${\sf MALL}(I)$ proofs. These transformations diminish sets of the indices of proofs overall.

Definition 3.1** ( ${\sf MALL}^{[\mathfrak{c}]}(I)$ proof transformation $\blacktriangleright^{\textsc{\tiny$ I $}}$ with

diminishing sets of indices)**

A ${\sf MALL}^{[\mathfrak{c}]}(I)$ transformation $\blacktriangleright^{\textsc{\tiny$ I $}}$ with diminishing sets of indices, written as $\rho_{\,\vdash_{J}[\Delta],\,\Gamma}\blacktriangleright^{\textsc{\tiny$ I $}}\rho^{\prime}_{\,\vdash_{J^{\prime}}[\Delta^{\prime}],\,\Gamma}$ , is a transformation from one ${\sf MALL}^{[\mathfrak{c}]}(I)$ proof $\rho$ for $\vdash_{J}[\Delta],\,\Gamma$ to another, $\rho^{\prime}$ for $\vdash_{J^{\prime}}[\Delta^{\prime}],\,\Gamma$ with $J^{\prime}\subseteq J$ , satisfying the following condition:

-

*(Restriction to the empty domain) *

*Restricting the two ${\sf MALL}^{[\mathfrak{c}]}(I)$ proofs to $\emptyset$ gives rise to a ${\sf MALL}^{[\mathfrak{c}]}$ reduction $\pi_{[\Delta],\Gamma}\,\rhd\,\pi^{\prime}_{[\Delta^{\prime}],\Gamma}$ by cut-elimination. *

*Schematically, this can be written: * $\textstyle{\rho_{\,\vdash_{J}[\Delta],\,\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\!\upharpoonright_{\emptyset}}$$\textstyle{\blacktriangleright^{\textsc{\tiny$ I $}}}$$\textstyle{\rho^{\prime}_{\,\vdash_{J^{\prime}}[\Delta^{\prime}],\,\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\!\upharpoonright_{\emptyset}}$$\textstyle{\pi_{[\Delta],\Gamma}}$$\textstyle{\rhd}$$\textstyle{\pi^{\prime}_{[\Delta^{\prime}],\Gamma}}$ **

The transformation $\rho\,\blacktriangleright^{\textsc{\tiny$ I $}}\,\rho^{\prime}$ is called a lifting of $\pi\,\rhd\,\pi^{\prime}$ . The lifting simulates a ${\sf MALL}$ proof reduction for cut elimination in terms of ${\sf MALL}(I)$ proof transformation.

The lifting in Definition 3.1 is not unique for a given ${\sf MALL}^{[\mathfrak{c}]}$ reduction, as any subset $J^{\prime\prime}$ of $J^{\prime}$ , $\rho\,\blacktriangleright^{\textsc{\tiny$ I $}}\,\rho^{\prime}\!\upharpoonright_{J^{\prime\prime}}$ obviously becomes a lifting for $\rho$ and $\rho^{\prime}$ under this definition.

Example 3.2

The following is a ${\sf MALL}^{[\mathfrak{c}]}(I)$ reduction with diminishing sets of indices whose restriction to $\emptyset$ is a Gentzen reduction eliminating the pairwise-dual additive connectives $\&$ and $\oplus$ in the cut formulas:

[TABLE]

The sets of the indices are diminished from $J_{1}+J_{2}$ to $J_{1}$ as a result of erasing the subproof $\pi^{2}$ within the proof transformation.

Proposition 3.3 (Lifting to indexed transformation)

Let $\nu\in|\pi_{[\Delta],\Gamma}|^{J}$ and consider a ${\sf MALL}$ reduction $\pi_{[\Delta],\Gamma}\,\rhd\,\pi^{\prime}_{[\Delta^{\prime}],\Gamma}$ . Then there exist $J^{\prime}\subseteq J$ and $\nu^{\prime}\in|\pi_{[\Delta^{\prime}],\Gamma}|^{J^{\prime}}$ lifting the given reduction:

[TABLE]

$\rho$ and $\rho^{\prime}$ are ${\sf MALL}$ proofs ensured by the fundamental lemma (Proposition 2.10) for the sequents $\vdash_{J}([\Delta],\,\Gamma)\langle\nu\rangle$ and $\vdash_{J^{\prime}}([\Delta^{\prime}],\,\Gamma)\langle\nu^{\prime}\rangle$ , respectively. Hence, we can also denote the lifting by

$\begin{aligned} \nu\in|\pi_{[\Delta],\Gamma}|^{J}&\quad\blacktriangleright^{\textsc{\tiny$ I $}}\quad&\nu^{\prime}\in|\pi^{\prime}_{[\Delta^{\prime}],\Gamma}|^{J^{\prime}}.\end{aligned}$

Note: There is no straight connection between $\nu$ and $\nu^{\prime}$ such as the former is the restriction to the latter.

** Proof**.

For every kind of reduction $\rhd$ , we can directly construct $\nu^{\prime}$ together with $J^{\prime}$ . There are three crucial cases:

(Crucial case 1)

[TABLE]

(with $A$ and $B$ being different occurrences of the same formula) reduces to

[TABLE]

(identifying the occurrence of $A$ with $B$ ).

Let $\epsilon\in|\textrm{ax}_{B,B^{\perp}}|^{J}$ and $\tau\in|\pi^{\prime}_{[\Delta],A,\Gamma}|^{J}$ . Then, for each $j\in J$ , we have $\epsilon_{j}=(b_{j},b_{j})$ with $b_{j}\in|B|=|B^{\perp}|$ , and $\tau_{j}=(\delta_{j},a_{j},\lambda_{j})$ with $\delta_{j}\in[\mathfrak{sl}(\Delta)]$ , $a_{j}\in|A|$ and $\lambda_{j}\in|\Gamma|$ . Note that $\nu_{j}=(\delta_{j},b_{j},a_{j},b_{j},\lambda_{j})$ . We define

[TABLE]

(Crucial case 2)

This case is the ${\sf MALL}$ reduction arisen by Example 3.2 above, when restricting to the empty domain $\emptyset$ and identifying ${\sf MALL}(\emptyset)$ with ${\sf MALL}$ .

By $\pi$ ’s last rule, $\nu\cong\tau\times\lambda$ with $\tau\in|\&(\pi^{1},\pi^{2})|^{J}$ and $\lambda\in|\oplus_{1}\!(\pi^{3})|^{J}$ , where the conclusion of $\pi^{i}$ with $i=1,2$ is $\vdash[\Delta_{i},\Omega],\Gamma,A_{i}$ and that of $\pi^{3}$ is $\vdash[\Delta_{3}],\Xi,A_{1}^{\bot}$ . Then the $\&$ -rule of the left premise divides $J$ into $J=J_{1}+J_{2}$ . We define

[TABLE]

(Crucial case 3) Here $\pi$

[TABLE]

reduces to $\pi^{\prime}$

[TABLE]

Note that in the last $\&$ -rule of $\pi^{\prime}$ , $\Xi$ and $\Omega$ inside the stack are chosen to be superposed.

By $\pi$ ’s last rule, $\nu\cong\lambda\times\tau$ , so $\lambda\in|\rho_{[\Xi],\Delta,A}|^{J}$ and $\tau\in|\&(\pi^{1},\pi^{2})|^{J}$ . The last $\&$ -rule of the right premise divides $J$ into $J=J_{1}+J_{2}$ so that $\tau\cong\tau_{1}\raisebox{2.15277pt}{$ \frown $}\tau_{2}$ and $\tau_{i}\in|\pi^{i}_{[\Sigma_{i},\Omega],A^{\perp},\Gamma,B_{i}}|^{J_{i}}$ . Then $\lambda\!\upharpoonright_{J_{i}}\times\tau_{i}\in_{\cong}|\operatorname{cut}(\rho,\pi^{i})|^{J_{i}}$ . We define

$\begin{aligned} J^{\prime}=J&\quad\text{and}&\nu^{\prime}\cong(\lambda\!\upharpoonright_{J_{1}}\!\times\tau_{1})\raisebox{2.15277pt}{$ \frown $}(\lambda\!\upharpoonright_{J_{2}}\!\times\tau_{2}).\end{aligned}$

$\Box$

4 ${\sf MALL}$ GoI Interpretation

4.1 Execution formula with zero action on symmetries of cuts

4.1.1 Interpretation of indexed point

in ${\sf MALL}$ proof

Our categorical framework is a minimal part of the Haghverdi–Scott GoI situation [HS06] with a reflexive object $U$ in a traced symmetric monoidal category ${\cal C}$ with tensor unit $I$ . Ours in addition requires that ${\cal C}$ has zero morphisms, in particular, a zero endomorphism $0_{U}$ on $U$ :

[TABLE]

$s_{U,U}$ is a symmetry $U\otimes U\longrightarrow U\otimes U$ of tensor product. $j:\!U^{2}\lhd U\!:k$ denotes a pair of morphisms $j$ and $k$ respectively from $U^{2}$ to $U$ and the other way around. $j$ and $k$ are called respectively co-retraction and retraction for the reflexive $U$ when $k\circ j=U\otimes U=U^{2}$ . The $m$ -ary tensor folding $\underbrace{\star\otimes\cdots\otimes\star}_{m}$ is denoted by $\star^{m}$ both for object $\star$ or morphism $\star$ . The trace structure will be introduced later in (15).

We require the commutativity of the pair $(j,k)$ and the zero $0_{U}$ ;

[TABLE]

Indeed, (6) is equivalent to the two commutativity $j\circ(0_{U}\otimes\,0_{U})=0_{U}\circ j$ and $(0_{U}\otimes\,0_{U})\circ k=k\circ\,0_{U}$ .

Note: The zero morphism $0_{U}$ is absorbing with respect to composition, but not with respect to tensor. That is $f\otimes 0_{U}$ and $0_{U}\otimes f$ are not in general $0_{U^{2}}$ for any endomorphism $f$ on $U$ .

Lemma 4.1 (tensoring zero)

$0_{U}\otimes 0_{U}=0_{U^{2}}$ . More generally, $0_{U}^{m}=0_{U^{m}}$ for any natural number $m$ .

** Proof**.

The first assertion is derived by the condition (6). The general assertion is by iterating the condition $(k^{n}\otimes U)\circ(k\circ 0_{U}\circ j)\circ(j^{n}\otimes U)=0_{U}^{n+2}$ . $\Box$

The zero endomorphism $0_{U}$ acts on the symmetry $s$ as follows.

Definition 4.2 (zero-action $(s_{U,U})^{0}$ )

[math]* action on the symmetry $s_{U,U}$ on $U^{2}$ is defined to annihilate the symmetry $s_{U,U}$ to the zero endomorphism on $U^{2}$ .*

[TABLE]

Alternatively, the action is defined to be the following decomposition in terms of conjugation (both precomposing and composing):

[TABLE]

We abbreviate $s_{U,U}$ and $(s_{U,U})^{0}$ as $s$ and $s^{0}$ , respectively.

To avoid collapsing the categorical framework whose GoI interpretation becomes the degenerate zero, we assume

[TABLE]

This is a technical assumption for the main theorem (Theorem 4.16) of this paper to characterise the diminution of the index set in terms of the convergence to the zero, distinguishable from the other morphisms.

The zero morphism, which is required in our framework, exists in crucial examples of GoI situations: (i) ${\sf Rel}_{+}$ is ${\sf Rel}$ with the disjoint union $+$ of sets as $\otimes$ and a reflexive object $\mathbb{N}$ . The empty relation on $\mathbb{N}$ is the zero morphism. Furthermore $0_{\mathbb{N}}+0_{\mathbb{N}}=0_{\mathbb{N}+\mathbb{N}}$ sufficient to the condition (6). (ii) The monoidal subcategories ${\sf Pfn}$ and ${\sf PInj}$ of ${\sf Rel}_{+}$ , both in which resides the zero morphism. ${\sf PInj}$ is known to be equivalent to the original category ${\sf Hilb_{2}}$ of Hilbert spaces and partial isometries for Girard’s GoI 1 [Gir89].

Note: The above examples of GoI situations happen to be sum-style monoidal structures [HS11], whose $\otimes$ is given by the disjoint union. The style is known to capture the notion of feedback as data flow in terms of streams of tokens around graphical networks. However our categorical framework (5) in the present paper is the general one, hence does not assume that the monoidal product is sum-style.

The $i$ -th constituent $x_{i}$ of $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\pi_{[\Delta],\Gamma}|$ of Definition 2.6 corresponds, in terms of the membership relation, to the $i$ -th occurrence of formulas $\widehat{\Delta},\Gamma$ with a unique sublist $\widehat{\Delta}$ .

Lemma 4.3** (Tag of $x_{i}$ with $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\widehat{\Delta}|\times|\Gamma|$

for $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ )**

Every element $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\pi_{[\Delta],\Gamma}|$ interpreting $\pi$ in Definition 2.6 belongs to $|\widehat{\Delta}|\times|\Gamma|$ with a unique sublist $\widehat{\Delta}$ of $\Delta$ . That is, for $\bm{x}=(x_{1},\ldots,x_{\ell})$ , there exists a unique sublist $\widehat{\Delta}$ such that the $i$ -th constituent $x_{i}\in|A_{i}|$ for the $i$ -th formula $A_{i}$ in $\widehat{\Delta}$ (resp. in $\Gamma$ ) when $i\leq 2m$ (resp. $i>2m$ ), where $2m$ is the number of formulas in $\widehat{\Delta}$ . The formula $A_{i}$ is called the tag of the $i$ -th constituent $x_{i}$ of $\bm{x}$ in $|\pi|$ . Note the sublist $\widehat{\Delta}$ is determined not only by $\pi$ but also by $\bm{x}$ , as shown in the construction (8) in the following proof.

** Proof**.

As $|\pi_{[\Delta],\Gamma}|\cong|\pi_{[\Delta],\Gamma}|^{\{*\}}$ , by (4) in Definition 2.8 when $J$ is the singleton set $\{*\}$ , every $\bm{x}=(x_{1},\ldots,x_{\ell})$ factors $\bm{x}=\bm{x}^{\prime}\times\bm{x}^{\prime\prime}$ so that

[TABLE]

While all the formulas in the sequence $\Gamma\langle\bm{x}^{\prime\prime}\rangle$ of ${\sf MALL}(I)$ formulas has the domain $\{*\}$ , each formula in the sequence $\Delta\langle\langle\bm{x}^{\prime}\rangle\rangle$ has the domain either $\{*\}$ or $\emptyset$ . Thus the unique sublist $\widehat{\Delta}$ is determined by the following two steps: (i) Ridding $\Delta\langle\langle\bm{x}^{\prime}\rangle\rangle$ of all the formulas $D$ such that $d(D)=\emptyset$ , which yields the subsequence of $\Delta\langle\langle\bm{x}^{\prime}\rangle\rangle$ . (ii) $\widehat{\Delta}$ is defined to be the subsequence of (i) restricted to $\emptyset$ (i.e., forgetting the domain) in order to obtain non-indexed formulas.

[TABLE]

Note whenever a cut formula occurs in $\widehat{\Delta}$ , so does its dual formula, hence $\bm{x}^{\prime}=(x_{1},\ldots,x_{2m})$ for a natural number $m$ , then $\bm{x}^{\prime\prime}=(x_{2m+1},\ldots,x_{\ell})$ . By Definitions 2.7 and 2.8, the membership relation for the assertion of $x_{i}$ and the $i$ -th formula in $\widehat{\Delta},\Gamma$ follows. $\Box$

In what follows, we make permutations $(x_{\tau(1)},\ldots,x_{\tau(\ell)})$ among the constituents implicit so that $\bm{x}$ is up to the permutation. This is because the permutation corresponds to the exchange rule eliminated from our syntax. The permutations will be reflected by the symmetry of monoidal product of ${\cal C}$ , in the following interpretation $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , which we denote by $\cong$ .

Definition 4.4

Endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ on tensor folding $U$ ’s and tensor folding $\sigma_{\bm{x}}$ of symmetry $s$ and of zero $s^{0}$ **

•

*Every $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\pi_{[\Delta],\Gamma}|$ is interpreted as an endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}$ on the tensor product $U^{\ell}$ together with an endomorphism $\sigma_{\bm{x}}^{\pi}$ on a subfactor $U^{2m}$ of $U^{\ell}$ . The endomorphism $\sigma_{\bm{x}}^{\pi}$ interprets cut rules in $\pi_{[\Delta],\Gamma}$ and is an $m$ -ary tensor folding of morphisms which are either $s^{1}=s$ or $s^{0}=0$ (cf. Definition 4.2) both on $U^{2}$ : *

[TABLE]

To distinguish each $i$ -th component $U$ of $U^{\ell}$ , we label each component $U_{x_{i}}$ with $x_{i}$ by abuse of notation, since the label $x_{i}$ is always clearly specified to designate the $i$ -th constituent of $\bm{x}$ . Then $x_{i}\in|A_{i}|$ so that $A_{i}$ is the tag of $x_{i}$ . Under this labelling, $\eta(i)$ of $s^{\eta(i)}$ on $U_{a}\otimes U_{a^{\prime}}$ is defined to be the Kronecker delta $\delta_{a,a^{\prime}}$ , where the tags of $a$ and $a^{\prime}$ are pairwise dual formulas in $\widehat{\Delta}$ . That is,

[TABLE]

*We define $(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi},\sigma_{\bm{x}}^{\pi})$ by induction on the construction of the proof $\pi$ . *

•

We simultaneously define that a component $U_{x_{i}}$ such that the tag of $x_{i}$ is a formula in $\Gamma$ is contracted by the induction on $\pi$ 111 For the choice of $x_{i}$ (i.e., a choice of a formula (not in the cut-list $\Delta$ but) in $\Gamma$ ), the construction is free from cut.. Since $U_{x_{i}}$ appears both in the co-domain and in the domain of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , every contracted component in the domain (resp. co-domain) of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is a domain (resp. co-domain) of a unique retraction (resp. co-retraction), called associated retraction (resp. associated co-retraction) 222assoc-ret (resp. assoc-coret) for short. We simply say $i$ is contracted when so is $U_{x_{i}}$ .

In the definition, $\pi_{1}$ and $\pi_{2}$ denote the two premise proofs of the binary rules, and $\pi^{\prime}$ of the unary rules.

(Axiom)

$\bm{x}=(\bar{*},*)\in|A^{\perp},A|$ with $A=1$ and $A^{\perp}=\perp$ . We define $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}$ to be a symmetry $s_{U_{\bar{*}},U_{*}}$ on $U_{\bar{*}}\otimes U_{*}$ of ${\cal C}$ . Because $\pi$ is cut-free, $\sigma_{\bm{x}}^{\pi}$ is empty by definition.

$\bm{x}$ has no contracted component so that neither $U_{*}$ nor $U_{\bar{*}}$ are contracted.

(Cut rule)

$\bm{x}=(\bm{v}^{1},\bm{v}^{2},a,a^{\prime},\bm{w}^{1},\bm{w}^{2})\in|\widehat{\Delta_{1}}|\times|\widehat{\Delta_{2}}|\times|A|\times|A^{\perp}|\times|\Gamma_{1}|\times|\Gamma_{2}|$ , so $\bm{x}_{1}=(\bm{v}^{1},\bm{w}^{1},a)$ and $\bm{x}_{2}=(\bm{v}^{2},a^{\prime},\bm{w}^{2})$ belong respectively to $|\pi_{1}|$ and to $|\pi_{2}|$ .

We define

[TABLE]

That is, if $a=a^{\prime}$ (resp. $a\not=a^{\prime}$ ), then $\sigma_{\bm{x}}^{\pi}$ on $U_{a}\otimes U_{a^{\prime}}$ is $s$ (resp. $s^{0}$ ), and $\sigma_{\bm{x}}^{\pi}$ on the remaining components is $\sigma_{\bm{x}_{1}}\otimes\sigma_{\bm{x}_{2}}$ . Note the definition makes sense because $\sigma_{\bm{x}_{1}}^{\pi_{1}}\otimes\sigma_{\bm{x}_{2}}^{\pi_{2}}$ acts on the domain distinct both from $U_{a}$ and $U_{a^{\prime}}$ .

We say the cut (of the last rule of $\pi$ ) matches (resp. mismatches) in $\bm{x}$ if $a=a^{\prime}$ (resp. otherwise).

( $\parr$ -rule)

$\bm{x}=(\bm{v},(a,b))$ , so that $\bm{x^{\prime}}=(\bm{v},a,b)\in|\pi^{\prime}|$ . Note $A\parr B$ is the tag of $(a,b)$ in $\pi$ , while $A$ (resp. $B$ ) is the tag of $a$ (resp. $b$ ) in the premise. $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}$ is obtained directly from $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi^{\prime}}$ on $U^{\ell+1}=U^{\ell}\times U_{(a,b)}$ by the retraction $U_{a}\otimes U_{b}\lhd U_{(a,b)}$ . That is, $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}=\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi^{\prime}}^{(j,k)}=(U^{\ell-1}\otimes j)\,\circ\,\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi^{\prime}}\,\circ\,(U^{\ell-1}\otimes k)$ . We also define $\sigma_{\bm{x}}^{\pi}$ by $\sigma_{\bm{x^{\prime}}}^{\pi^{\prime}}$ .

( $\otimes$ -rule)

$\bm{x}=(\bm{v}^{1},\bm{v}^{2},\bm{w}^{1},\bm{w}^{2},(a,b))\in|\widehat{\Delta_{1}}|\times|\widehat{\Delta_{2}}|\times|\Gamma_{1}|\times|\Gamma_{2}|\times|A|\times|B|$ , so that $\bm{x}_{1}=(\bm{v}^{1},\bm{w}^{1},a)$ and $\bm{x}_{2}=(\bm{v}^{2},\bm{w}^{2},b)$ are respectively from $|\pi_{1}|$ and $|\pi_{2}|$ . Note $A\otimes B$ is the tag of $(a,b)$ in $\pi$ , while $A$ (resp. $B$ ) is the tag of $a$ (resp. $b$ ) in $\pi_{1}$ (resp. in $\pi_{2}$ ). The endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is obtained directly from $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}_{\pi_{1}}\otimes\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}_{\pi_{2}}$ on $U^{\ell+1}=U^{\ell}\times U_{(a,b)}$ by the retraction $U_{a}\otimes U_{b}\lhd U_{(a,b)}$ . That is, $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}\cong(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}_{\pi_{1}}\otimes\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}_{\pi_{2}})^{(j,k)}=(U^{\ell-1}\otimes j)\,\circ\,(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}_{\pi_{1}}\otimes\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}_{\pi_{2}})\,\circ\,(U^{\ell-1}\otimes k)$ . We also define $\sigma_{\bm{x}}^{\pi}$ by $\sigma_{\bm{x}_{1}}^{\pi_{1}}\otimes\sigma_{\bm{x}_{2}}^{\pi_{2}}$ .

In the above both multiplicatives rules ( $\otimes$ and $\parr$ ), the introduced $U_{(a,b)}$ in the domain (resp. co-domain) is a contracted component, and the assoc-ret (resp. assoc-coret) is $k:U_{(a,b)}\rhd U_{a}\otimes U_{b}$ (resp. $j:U_{a}\otimes U_{b}\lhd U_{(a,b)}$ ). Other contracted components are those of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle(\bm{v},a,b) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle(\bm{v},a,b) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle(\bm{v},a,b) $}\,\mathclose{\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle(\bm{v},a,b) $}\,\mathclose{\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}}\,}$ for $\parr$ and $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle(\bm{v}{1},\bm{w}{1},a) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle(\bm{v}{1},\bm{w}{1},a) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle(\bm{v}{1},\bm{w}{1},a) $}\,\mathclose{\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle(\bm{v}{1},\bm{w}{1},a) $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}}\,}$ and $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle(\bm{v}{2},\bm{w}{2},b) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle(\bm{v}{2},\bm{w}{2},b) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle(\bm{v}{2},\bm{w}{2},b) $}\,\mathclose{\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle(\bm{v}{2},\bm{w}{2},b) $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to9.03888pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.28888pt,width=0.4pt}}}\,}$ for $\otimes$ distinct from the components $U_{a}$ and $U_{b}$ . Note that $(\bm{v},a,b)\in|\mbox{premise of$ \parr $}|$ and $(\bm{v}_{1},\bm{w}_{1},a)\in|\mbox{left premise of$ \otimes $}|$ and $(\bm{v}_{2},\bm{w}_{2},b)\in|\mbox{right premise of$ \otimes $}|$ .

( $\&$ -rule)

$\bm{x}$ is either $(\bm{v},(1,a))$ or $(\bm{v},(2,a))$ , so that $(\bm{v},a)$ are either from $|\pi_{1}|$ or $|\pi_{2}|$ , respectively. Note $A_{1}\&A_{2}$ is the tag of $(i,a)$ in $\pi$ , while $A_{i}$ is the tag of $a$ in $\pi_{1}$ or in $\pi_{2}$ when $i=1$ or $i=2$ , respectively. We define $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}=\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}}\,}_{\pi_{i}}$ by relabelling the component $U_{a}$ either by $U_{(1,a)}$ or $U_{(2,a)}$ for the domain (equally for the codomain) of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}_{\pi}$ . We also define $\sigma_{\bm{x}}^{\pi}$ by $\sigma_{(\bm{v},a)}^{\pi_{i}}$ .

( $\oplus_{i}$ -rule) Same as $\&$ -rule but using the unique premise $\pi^{\prime}$ deterministically.

In the above both additives rules ( $\&$ and $\oplus_{i}$ ), contracted components are those of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}\hbox{\lower 4.5pt\hbox{\vbox to14.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=9.5pt,depth=4.5pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}\hbox{\lower 3.75pt\hbox{\vbox to11.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.25pt,depth=3.75pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle(\bm{v},a) $}\,\mathclose{\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}\hbox{\lower 3.25pt\hbox{\vbox to9.0pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.75pt,depth=3.25pt,width=0.4pt}}}\,}$ under the relabelling $U_{a}$ by $U_{(i,a)}$ for the component of the domain (equally of the codomain). Note that $(\bm{v},a)$ belongs to one of $|\mbox{left premise}|$ and $|\mbox{right premise}|$ in $\&$ -rule depending on $i=1$ or $i=2$ , and obviously to $|\mbox{the unique premise}|$ in $\oplus_{i}$ -rule.

In the sequel, the pair of Definition 4.4 is simply written $(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,},\sigma_{\bm{x}})$ by omitting $\pi$ , since the proof $\pi$ will be always specified clearly from the context.

(**Remark on Def 4.4 **) [The endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ as an I/O box]

The endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is seen as an input/output (I/O) box on the $(n+2m)$ -ary tensor folding of $U$ , whose inputs/outputs are the formulas occurring in $\Gamma,\widehat{\Delta}$ , in which $\Gamma$ contains $n$ occurrences of formulas, and a sublist $\widehat{\Delta}$ contains $2m$ occurrences of (pairwise dual) formulas. The formulas are the tags of $x_{i}$ s where $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\pi_{[\Delta],\Gamma}|$ .

The endomorphism $\sigma_{\bm{x}}$ is seen as a more special box consisting of $m$ -ary tensor folding of $\{s,s^{0}\}$ for the I/O formulas in the sublist $\widehat{\Delta}$ . See Figure 1 below for $(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,},\sigma_{\bm{x}})$ .

A characterisation of contracted component is derived:

Lemma 4.5

$U_{x_{i}}$ * is a contracted component of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ for $\bm{x}\in\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\pi $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\pi $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\pi $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\pi $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ if and only if $x_{i}$ ’s tag contains a multiplicative connective (i.e., $\otimes$ or $\parr$ ).*

** Proof**.

Straightforward accordingly to the inductive step of Definition 4.4, in which the retraction and the co-retraction are used only for multiplicatives-rules ( $\parr$ or $\otimes$ ) so that $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is constructed $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to9.8611pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.125pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x^{\prime}} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.26389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=2.0pt,width=0.4pt}}}\,}^{(j,k)}$ or $(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}\otimes\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,})^{(j,k)}$ , respectively. $\Box$

(labelling associated retractions $\rhd_{i}$ and co-retractions $\lhd_{i}$ )

Every associated retraction $\rhd$ (resp. associated co-retraction $\lhd$ ) is by Definition 4.4 uniquely labelled with a contracted $i$ such that the tag of $x_{i}$ is a formula in $\Gamma$ . The labelling is written $\rhd_{i}$ (resp. $\lhd_{i}$ ). In what follows, the labelling is made implicit except when an explicit labelling makes an explanation easier to understand.

(assoc-rets and assoc-corets in I/O box $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ )

When the endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is seen as the I/O box, the assoc-rets and the assoc-corets are those $\rhd$ ’s and $\lhd$ ’s whose domains and co-domains lie respectively among the inputs and the outputs of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ . By the construction of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , they lie pairwise in the inputs and the outputs. See Figure 1 for $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ depicting the occurrence of the assoc-rets $\rhd$ ’s and the assoc-corets $\lhd$ ’s.

(Convention omitting $\operatorname{Id}_{U}$ s) When an indicated occurrence of a contracted component $U$ is clear from the context, $\rhd\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ (resp. $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ\lhd$ ) is an abbreviation for the composition $(\operatorname{Id}_{U}\otimes\cdots\otimes\operatorname{Id}_{U}\otimes\rhd\otimes\operatorname{Id}_{U}\otimes\cdots\otimes\operatorname{Id}_{U})\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ (resp. $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ(\operatorname{Id}_{U}\otimes\cdots\otimes\operatorname{Id}_{U}\otimes\lhd\otimes\operatorname{Id}_{U}\otimes\cdots\otimes\operatorname{Id}_{U})$ , where the domain of $\rhd$ (resp. the codomain of $\lhd$ ) is the contracted $U$ . This abbreviation is generalised for plural indicated occurrences of contracted components $U_{1},\ldots U_{r}$ in $U^{\ell}$ as follows: $(\otimes^{r}\rhd)\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ (resp. $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ(\otimes^{r}\lhd)$ ) stands for the composition (resp. precomposition) to $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ by the morphism tensoring $\rhd$ (resp. $\lhd$ ) on the contracted components indicated and $\operatorname{Id}_{U}$ on the remaining components. Note because $r\leq\ell$ , the abbreviation is for omitting identities on $U$ s.

\opencutleft{cutout}00pt3

In the sequel, the two abbreviations are pictured as in the left hand respectively. Using a notation of the $r$ -ary tensor folding of $\rhd$ (resp. $\lhd$ ), it is also written by $\rhd^{r}\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ (resp. $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ\lhd^{r}$ ). This convention is equally employed when indicated occurrences of assoc-(co)rets are clear from the context.

Under this convention, for any contracted component in the co-domain (resp. domain) of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , it holds;

[TABLE]

That is, $\lhd\circ\rhd$ composes (resp. precomposes) with any retraction (resp. co-retraction) as the identity. In other word, $\lhd\circ\rhd$ is a projector on a contracted component in the codomain (resp. domain) by composition (resp. precomposition). Pictorially,

In Equation (11), the assoc-coret (resp. assoc-ret) occurs explicitly as the last composed $\lhd$ (resp. the first precomposed $\rhd$ ). Thus, the left most $\lhd$ (resp. right most $\rhd$ ) in (11) is seen labelled $\lhd_{i}$ (resp. $\rhd_{i}$ ) such that the tag of $x_{i}$ is a formula occurrence in $\Gamma$ , where $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\pi_{[\Delta],\Gamma}|$ . Since every contracted component occurs pairwise in the co-domain and the domain of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , the two equations can be written successively all at once;

[TABLE]

The co-domain and the domain $U^{\ell}$ of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ have in general several contracted components $U$ s labelled with $x_{i}$ s, where $i$ ranges in the set $\mathfrak{r}$ of the contraced $i$ s. Thus, for all the several pairs of contracted components in the domain and the co-domain of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , the parallel compositions and precompositions with $\lhd\circ\rhd$ s to each contracted components act as the identity on $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ :

[TABLE]

where $r$ is the cardinality of $\mathfrak{r}$ , hence is a number of the contracted components, and the third equality is by $(\lhd\circ\rhd)^{r}=\lhd^{r}\!\circ\rhd^{r}$ , as $(-)^{r}$ is the $r$ -ary tensor folding. Since the last composed $\lhd^{r}$ (resp. the first precomposed $\rhd^{r}$ ) in the rightmost expression of (12) are the explicit occurrences of the assoc-rets (resp. assoc-corets), the endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is written so that all the assoc-rets $\rhd^{r}$ and the assoc-corets $\lhd^{r}$ can be made explicit as follows:

[TABLE]

Roughly speaking, $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}^{\mathrm{o}}$ is $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ stripped of all the assoc-rets and assoc-corets. which is depicted in the following Figure 3:

4.1.2 The action $\epsilon_{\bm{x}}$ annihilating associated

(co)retractions

This subsection is concerned with defining the action $\epsilon_{\bm{x}}$ (Definition 4.8) over the associated retractions (resp. co-retractions) in Definition 4.4 above. The action $\epsilon_{\bm{x}}$ arises from $\sigma_{\bm{x}}$ of (9) when the feedback on the trace of ${\cal C}$ is taken into account, and annihilates, using the zero morphism $0_{U}$ , a certain class of retractions and co-retractions. This class is defined in Definition 4.8 in terms of zero input and output.

In what follows, we shall see how feedback stemming from Gentzen cut-elimination for a ${\sf MALL}$ proof $\pi$ acts on the assoc-rets and the assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ for $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ . The action is stipulated in terms of the zero morphism added in our framework. First, in a categorical framework of Girard’s GoI project, the feedback is modelled by the trace structure (cf. [HS06]) defined by the seven axioms below:

[TABLE]

There are three kinds of naturality axioms: naturality in $X$ and naturality in $Y$ , and dinaturality in $Z$ . The other axioms are vanishing I,II, superposing and yanking. See Appendix A.1 for the seven axioms: the three naturalities and the four axioms.

In our setting of Definition 4.4, the endomorphism $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is on $U^{n+2m}$ so that $n$ and $2m$ are the numbers of formulas respectively in $\Gamma$ and in a sublist $\widehat{\Delta}$ , and $\sigma_{\bm{x}}$ is on the subfactor $U^{2m}$ . Then the feedback is calculated by;

[TABLE]

See Figure 4.

Note that when $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ comes from a proof $\pi$ of the multiplicative fragment, the equation is exactly the GoI interpretation of the proof $\pi_{[\Delta],\Gamma}$ (cf. [HS06]). This is because in the multiplicative fragment, the index set $I$ becomes redundantly the singleton $\{*\}$ , thus $|\pi_{[\Delta],\Gamma}|=\{\bm{x}\}$ , whereby $\sigma_{\bm{x}}$ is a simple tensor folding of the symmetry $s$ (free of [math] morphism).

By the naturalities of traces, the assoc-corets (resp. the assoc-rets) of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ commute with ${\sf Tr}^{U^{2m}}_{U^{n},U^{n}}$ , hence taking a trace of (11) composed with $\operatorname{Id}\otimes\sigma_{\bm{x}}$ yields for any $i$ such that $i$ -th component of $\bm{x}=(x_{1},\ldots,x_{\ell})$ is contracted.

[TABLE]

Thus all the assoc-rets and assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ are written explicitly;

[TABLE]

where $\mathfrak{r}$ is the set of all contracted $i$ s and $r$ is the cardinality of $\mathfrak{r}$ .

Equation(17) is depicted in Figure 5, in which the dotted squares are the scopes of the traces and the shifting of the scopes are naturalities of the $\rhd$ s and the $\lhd$ s;

While inside the sole $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , the assoc-rets and the assoc-corets (written explicitly in (14)) do not interact with zero morphisms because the construction of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ of Definition 4.4 is free from the zero morphisms. Remember that the zero morphisms reside only in $\sigma_{\bm{x}}$ as subfactors (cf.(9)). However when they are put inside the context ${\sf Tr}^{U^{2m}}_{U^{n},U^{n}}\left((\operatorname{Id}\otimes\sigma_{\bm{x}})\circ-\right)$ (written explicitly in (17)), they may interact with zero morphisms arising from $\sigma_{\bm{x}}$ via the feedback of the trace. That is, the trace in a monoidal category takes feedback into account, hence makes the zeros stemming from $\sigma_{\bm{x}}$ interact with the assoc-rets and the assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ . This yields a certain action $\epsilon_{\bm{x}}$ on the assoc-(co)rets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ , as defined in Definition 4.8 below.

Definition 4.6

zero input (resp. output) of assoc-coret (resp. assoc-ret) w.r.t the interpretation $\sigma_{\bm{x}}$ of cuts**

(zero input of assoc-coret $\lhd$ ) An assoc-coret $\lhd$ of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is said to have zero input w.r.t $\sigma_{\bm{x}}$ when $\lhd$ decomposes in ${\sf ex}\left(\sigma,\bm{x}\right)$ either as $\lhd\circ(0_{U}\!\otimes U)$ or as $\lhd\circ(U\otimes 0_{U})$ .

(zero output of assoc-ret $\rhd$ ) An assoc-ret $\rhd$ of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is said to have zero output w.r.t $\sigma_{\bm{x}}$ when $\rhd$ decomposes in ${\sf ex}\left(\sigma,\bm{x}\right)$ either as $(0_{U}\!\otimes U)\circ\rhd$ or as $(U\otimes 0_{U})\circ\rhd$ .

Why do we use the terminology zero input (resp. output) ? The object $U\otimes U$ of $\lhd$ ’s domain (resp. $\rhd$ ’s co-domain) can be regarded as having two inputs (resp. outputs), one left component $U$ and the other right one. Then the decomposition in each case says that one of two inputs (resp. outputs) is zero.

Pictorially,

$\begin{array}[]{r}\stackrel{{\scriptstyle 0_{U}}}{{\longrightarrow}}\\ \stackrel{{\scriptstyle\text{\normalsize$ \longrightarrow $}}}{{\text{\scriptsize$ U $}}}\end{array}\!\!\!\raisebox{1.72218pt}{$ \lhd $}$

or

$\begin{array}[]{r}\stackrel{{\scriptstyle U}}{{\longrightarrow}}\\ \stackrel{{\scriptstyle\text{\normalsize$ \longrightarrow $}}}{{\text{\scriptsize$ 0_{U} $}}}\end{array}\!\!\!\raisebox{1.72218pt}{$ \lhd $}$

for the zero input and

$\raisebox{1.72218pt}{$ \rhd $}\!\!\!\begin{array}[]{l}\stackrel{{\scriptstyle 0_{U}}}{{\longrightarrow}}\\ \stackrel{{\scriptstyle\text{\normalsize$ \longrightarrow $}}}{{\text{\scriptsize$ U $}}}\end{array}$

or

$\raisebox{1.72218pt}{$ \rhd $}\!\!\!\begin{array}[]{l}\stackrel{{\scriptstyle U}}{{\longrightarrow}}\\ \stackrel{{\scriptstyle\text{\normalsize$ \longrightarrow $}}}{{\text{\scriptsize$ 0_{U} $}}}\end{array}$

for the zero output.

Note that Definition 4.6 is alternatively stated as follows: When the assoc-coret (resp. assoc-ret) is written explicitly as ${\sf ex}\left(\sigma,\bm{x}\right)=\lhd\circ{\sf g}$ (resp. ${\sf ex}\left(\sigma,\bm{x}\right)={\sf g}\circ\rhd$ ) (cf. (11)), the assoc-coret $\lhd$ (resp. assoc-ret $\rhd$ ) has zero input (resp. output) iff either $0_{U}\otimes U$ or $U\otimes 0_{U}$ acts trivially on ${\sf g}$ by composing (resp. precomposing) to the indicated component $U\otimes U$ .

Example 4.7

Let $\pi_{[A\&A,A^{\perp}\oplus A^{\perp}]\,\,A^{\perp},A\otimes B,B^{\perp}}$ be a proof obtained by a $\otimes$ -rule between $\pi_{1}$ of Section C (the first paragraph) and $ax_{B^{\perp},B}$ . Let $\bm{x}:=\nu_{2}\times(\bar{\star},\star)\in|\pi|$ , where $\nu_{2}$ is in Section C and $(\bar{\star},\star)\in|ax_{B^{\perp},B}|$ .

\opencutleft{cutout}

00pt7

Then $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ has the unique pair of assoc-ret and assoc-coret both interpreting the $\otimes$ -rule. See the left-hand dotted rectangle representing $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle x $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle x $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle x $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle x $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ with the assoc-ret and the assoc-coret. The pair of assoc-ret and the assoc-coret appears explicitly in the second and the third of the following equations (in which $\sigma_{\bm{x}}=s^{0}_{U,U}$ ):

$\begin{aligned} {\sf Tr}^{U^{2}}_{U^{3},U^{3}}\left((\operatorname{Id}\otimes\sigma_{\bm{x}})\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\right)=\lhd\circ{\sf Tr}^{U^{2}}_{U^{4},U^{4}}\left((\operatorname{Id}\otimes\sigma_{\bm{x}})\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}^{\mathrm{o}}\right)\circ\rhd=\lhd\circ(s_{U,U}^{0}\otimes s_{U,U})\circ\rhd\end{aligned}$ *, where $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}^{\mathrm{o}}$ is $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ without the assoc-ret and the assoc-coret.

See the above figure whose LHS and RHS are the first and the last equations, respectively, The assoc-coret (resp. assoc-ret) has zero input (resp. output) because the right picture depicts $\lhd$ (resp. $\rhd$ ) having a zero input (resp. output) from the northwest (resp. to the northeast). Hence $(0_{U}\otimes U)$ composes (resp. precomposes) to $s_{U,U}^{0}\otimes s_{U,U}$ trivially.*

Definition 4.8** (action $\epsilon_{\bm{x}}$ on

assoc-rets and assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ )**

The endomorphism $\sigma_{\bm{x}}$ of Definition 4.4 for $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ yields the following action $\epsilon_{\bm{x}}$ on the assoc-rets and the assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ . The action $\epsilon_{\bm{x}}$ acts on each assoc-ret and assoc-coret as either zero or the identity by (pre)composition on them, as follows:

[TABLE]

where zero actions $\rhd^{0}$ and $\lhd^{0}$ are defined respectively as follows:

[TABLE]

That is, the zero annihilates the pair of assoc-ret and assoc-coret $j:U\otimes U\lhd U:k$ to the pair of the zero morphisms $j^{0}:U\otimes U\lhd^{0}U:k^{0}$ , where $j^{0}=\lhd^{0}$ and $k^{0}=\rhd^{0}$ .

The action $\epsilon_{\bm{x}}$ of Definition 4.8 is by definition conjugate on the pairwise tensor foldings $(\otimes_{i\in\mathfrak{r}}\,\rhd_{i},\otimes_{i\in\mathfrak{r}}\,\lhd_{i})=(\rhd^{r},\lhd^{r})$ of the assoc-rets and the assoc-corets represented in (14), where $\mathfrak{r}=\{i\mid\mbox{$ i $is contracted with$ \bm{x}=(x_{1},\ldots,x_{\ell}) $}\}$ . Hence we define to formulate the action on $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ by conjugation:

[TABLE]

Pictorially,

[TABLE]

This action of $\epsilon_{\bm{x}}$ , by naturalities, extends to the action $\epsilon_{\bm{x}}$ on the corresponding retractions and co-retractions in (17):

[TABLE]

Note by (16) that the LHS of the first equation is ${\sf ex}\left(\sigma,\bm{x}\right)^{\epsilon_{\bm{x}}}$ . Recall that $r$ is the number of the assoc-rets $\rhd_{i}$ (equally the assoc-corets $\lhd_{i}$ ) of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ such that $i$ is contracted.

4.1.3 The Execution formula

Definition 4.9** (Execution formula ${\sf Ex}\left(\sigma,\bm{x}\right)$

for $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ )**

*For every $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ , the endomorphism ${\sf Ex}\left(\sigma,\bm{x}\right)$ is defined by *

[TABLE]

*where $(\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,},\sigma_{\bm{x}})$ is the pair of the endomorphism on $U^{n+2m}$ and on the subfactor $U^{2m}$ in Definition 4.4 and $\epsilon_{\bm{x}}$ is the action in Definition 4.8 on the assoc-rets and the assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ . The domains (resp. the co-domains) of the assoc-rets (resp. the assoc-corets) lie among the subfactor $U^{n}$ in the domain (resp. the co-domain) of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ . See Figure 6. *

Example 4.10

Let $\bm{x}$ be of Example 4.7. Since $\epsilon_{\bm{x}}$ acts as zero both on the unique assoc-ret $\rhd$ and on the unique assoc-coret $\lhd$ , $\begin{aligned} {\sf Ex}\left(\sigma,\bm{x}\right)=\lhd^{0}\circ(s_{U,U}^{0}\otimes s_{U,U})\circ\rhd^{0}=0_{U^{3},U^{3}}.\end{aligned}$

Finally, the execution formula is run point-wise for every enumerated set $\nu$ in interpretation of a proof in ${\sf Rel}^{[\mathfrak{c}]}$ .

Definition 4.11** (Execution formula ${\sf Ex}_{J}\left(\sigma,\nu\right)$

for $\nu\in|\pi_{[\Delta],\Gamma}|^{J}$ )**

Let $\pi_{[\Delta],\Gamma}$ be a ${\sf MALL}^{[\mathfrak{c}]}$ proof. For every $\nu\in|\pi_{[\Delta],\Gamma}|^{J}$ , ${\sf Ex}\left(\sigma,\nu\right)\in|\Gamma|^{J}$ is defined indexwise by:

[TABLE]

4.2 Zero Convergence of Execution Formula

This subsection concerns the main proposition (Proposition 4.15), which says that communicating two proofs via mismatched pair yields zero convergence of Ex. We start with the tracing zero lemma derivable from some trace axioms.

Lemma 4.12 (tracing zero)

For any natural number $n\geq 1$ ,

[TABLE]

** Proof**.

First by Lemma 4.1 $0_{U^{m}}=(0_{U})^{m}$ for any natural number $m$ . Then by superposing ${\sf Tr}^{U}_{U^{n+1},U^{n+1}}\left(0_{U^{n+2}}\right)=0_{U^{n}}\otimes{\sf Tr}^{U}_{U,U}\left(0_{U^{2}}\right)$ , it suffices to prove the assertion for $n=1$ . Second, observe the equation333In a more general setting, the natural iso $(b\otimes U)\circ s_{U,U}\circ(a\otimes U)\cong a\otimes b\cong(U\otimes a)\circ s_{U,U}\circ(U\otimes b)$ , for any endomorphisms $a$ and $b$ on $U$

[TABLE]

Thus ${\sf Tr}^{U}_{U,U}\left(0_{U^{2}}\right)=0_{U}\circ{\sf Tr}^{U}_{U,U}\left(s_{U,U}\right)\circ\,0_{U}=0_{U}\circ U\circ\,0_{U}$ , where the first equation is by naturalities and the second equation is by yanking. $\Box$

We prepare the following Lemma 4.14, which will directly entail the main Proposition 4.15.

Definition 4.13 (action $\delta_{\bm{x}}$ )

For $\bm{x}\in|\pi_{[\Delta],\Gamma}|$ , let us put $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ into the context $(\operatorname{Id}\otimes\,0_{U})\circ(-)\circ(\operatorname{Id}\otimes\,0_{U})$ , allowing interaction of the assoc-rets and the assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ with the two zeros $0_{U}$ in the context. Zero input (resp. zero output) of assoc-ret (resp. assoc-coret) in this context is defined in the same manner, yielding the action, say $\delta_{\bm{x}}$ , on the assoc-rets and the assoc-corets of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ same as in Definition 4.8 (but simpler without the feed back): That is, the morphism $\rhd^{\delta_{\bm{x}}}$ is defined to be $\rhd^{0}$ (resp. $\rhd$ ) if $\rhd$ decomposes in $(\operatorname{Id}\otimes\,0_{U})\circ\,\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ\,(\operatorname{Id}\otimes\,0_{U})$ either as $(0_{U}\otimes\,U)\circ\rhd$ or $(U\otimes\,0_{U})\circ\rhd$ (resp. otherwise). Symmetrically, $\lhd^{\delta_{\bm{x}}}$ is defined to be $\lhd^{0}$ (resp. $\lhd$ ) if $\lhd$ decomposes in $(\operatorname{Id}\otimes\,0_{U})\circ\,\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ\,(\operatorname{Id}\otimes\,0_{U})$ either as $\lhd\circ(0_{U}\otimes\,U)$ or $\lhd\circ(U\otimes\,0_{U})$ (resp. otherwise).

Lemma 4.14 (lemma for Prop 4.15)

[TABLE]

where ${\sf ex}\left(\sigma,\bm{x}\right)^{\delta_{\bm{x}}}$ is ${\sf ex}\left(\sigma,\bm{x}\right)$ of (16) whose $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}$ is replaced by $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x} $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}^{\delta_{\bm{x}}}$ using the action $\delta_{\bm{x}}$ of Definition 4.13.

See Figure 7 (upper-right) depicting the equation. The lemma holds up to the the permutations $\tau$ on $U^{n}$ so that the left ${\sf ex}\left(\sigma,\bm{x}\right)$ is read by $\tau^{-1}\circ\,{\sf ex}\left(\sigma,\bm{x}\right)\circ\,\tau$ . Hence the assertion is independent of the choice of $U$ for the $0_{U}$ . The choice is of one formula occurrence from $\Gamma$ , as each occurrence is interpreted by the distinct $U$ .

** Proof**.

Induction on the construction of $\pi$ for $\bm{x}$ in Definition 4.4. In the proof, Equation(20) in the proof of Lemma 4.12 is used. In the following, for $i=1,2$ , $\bm{x}_{i}$ are the premises of $\bm{x}$ (i.e., $\bm{x}_{1}=\bm{y}$ and $\bm{x}_{2}=\bm{z}$ in Definition 4.4), and ${\sf ex}_{i}$ denotes ${\sf ex}\left(\sigma,\bm{x}_{i}\right)$ .

(axiom)

$(\operatorname{Id}\otimes\,0_{U})\circ\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle ax $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle ax $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to8.30554pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle ax $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to7.01389pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=2.0pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle ax $}\,\mathclose{\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}\hbox{\lower 2.0pt\hbox{\vbox to6.15277pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=2.0pt,width=0.4pt}}}\,}\circ(\operatorname{Id}\otimes\,0_{U})=(\operatorname{Id}\otimes\,0_{U})\circ s_{U,U}\circ(\operatorname{Id}\otimes\,0_{U})=0_{U}\otimes 0_{U}$ .

( $\otimes$ -rule) (case 1) $U$ is introduced by the $\otimes$ -rule.

$\begin{aligned} &(\operatorname{Id_{1}}\otimes\,0_{U}\otimes\operatorname{Id_{2}})\circ\lhd\circ({\sf ex}_{1}\otimes{\sf ex}_{2})\circ\rhd\circ(\operatorname{Id_{1}}\otimes\,0_{U}\otimes\operatorname{Id_{2}})\\ &=\lhd\circ(((\operatorname{Id_{1}}\otimes\,0_{U})\circ\,{\sf ex_{1}}\circ(\operatorname{Id_{1}}\otimes\,0_{U}))\otimes((\operatorname{Id_{2}}\otimes\,0_{U})\circ\,{\sf ex_{2}}\circ(\operatorname{Id_{2}}\otimes\,0_{U}))\circ\rhd=0_{U^{n_{1}}}\otimes 0_{U^{n_{2}}}\end{aligned}$

The last equation is by I.H.’s on $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}$ and $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{2} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}$ .

( $\otimes$ -rule) (case 2) other than case 1:

In this case, the $U$ for the $0_{U}$ of $\operatorname{Id}\otimes 0_{U}$ is a factor from the (co)domain of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{i} $}\,\mathclose{\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{i} $}\,\mathclose{\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}\hbox{\lower 3.84666pt\hbox{\vbox to10.1522pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.84666pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.31905pt,width=0.4pt}}\hbox{\lower 3.31905pt\hbox{\vbox to8.33293pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.31905pt,width=0.4pt}}\hbox{\lower 3.31905pt\hbox{\vbox to8.33293pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{i} $}\,\mathclose{\hbox{\lower 3.31905pt\hbox{\vbox to8.33293pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.31905pt,width=0.4pt}}\hbox{\lower 3.31905pt\hbox{\vbox to8.33293pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.31905pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.31905pt,width=0.4pt}}\hbox{\lower 3.31905pt\hbox{\vbox to7.47182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.31905pt,width=0.4pt}}\hbox{\lower 3.31905pt\hbox{\vbox to7.47182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{i} $}\,\mathclose{\hbox{\lower 3.31905pt\hbox{\vbox to7.47182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.31905pt,width=0.4pt}}\hbox{\lower 3.31905pt\hbox{\vbox to7.47182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.31905pt,width=0.4pt}}}\,}$ . We assume without loss of generality that $i=1$ . Then, $(\operatorname{Id}\otimes\,0_{U})\circ\,{\sf ex_{1}}\circ(\operatorname{Id}\otimes\,0_{U})=0_{U^{n_{1}}}$ by I.H on $\bm{x}_{1}$ . This directly implies that the co-retraction and the retraction $(j,k)$ interpreting the $\otimes$ -rule are acted by $\delta$ as zero, denoted by $(j^{0},k^{0})$ , since $j$ ’s output and $k$ ’s input both on $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}$ are zeros by the I.H. Hence, when $(j,k)$ is written by $(\lhd,\rhd)$ ,

$\begin{aligned} \lhd^{0}\circ(0_{U^{n_{1}}}\otimes{\sf ex_{2}})\circ\rhd^{0}&=\lhd\circ(0_{U^{n_{1}}}\otimes(0_{U}\otimes\operatorname{Id})\circ\,{\sf ex_{2}}\circ(0_{U}\otimes\operatorname{Id}))\circ\rhd\\ &=\lhd\circ(0_{U^{n_{1}}}\otimes 0_{U^{n_{2}}})\circ\rhd=0_{U^{n}}\end{aligned}$

The first equation is by the assumption and the second equation is by I.H. on $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x_{2}} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x_{2}} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x_{2}} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x_{2}} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}$ .

(cut-rule)

By the rule,

[TABLE]

We assume without loss of generality that the $U$ for the $0_{U}$ of the $\operatorname{Id}\otimes\,0_{U}$ is a factor from the (co)domain of $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{1} $}\,\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}}\,}$ . Then, LHS of the assertion is equal to

$\begin{array}[]{llr}&(\operatorname{Id}\otimes\,0_{U})\circ{\sf Tr}^{U^{2}}_{U^{n},U^{n}}\left((\operatorname{Id}_{1}\otimes s_{U,U}\otimes\operatorname{Id}_{2})\circ({\sf ex}_{1}^{\delta}\otimes{\sf ex}_{2}^{\delta})\right)\circ(\operatorname{Id}\otimes\,0_{U})\\ &={\sf Tr}^{U^{2}}_{U^{n},U^{n}}\left((\operatorname{Id}\otimes\,0_{U})\circ((\operatorname{Id_{1}}\otimes s_{U,U}\otimes\operatorname{Id_{2}})\circ({\sf ex}_{1}^{\delta}\otimes{\sf ex}_{2}^{\delta}))\circ(\operatorname{Id}\otimes\,0_{U})\right)&\text{naturalities}\\ &={\sf Tr}^{U^{2}}_{U^{n},U^{n}}\left((\operatorname{Id_{1}}\otimes s_{U,U}\otimes\operatorname{Id_{2}})\circ((\operatorname{Id}\otimes\,0_{U})\circ\,{\sf ex}_{1}^{\delta}\,\circ(\operatorname{Id}\otimes\,0_{U}))\otimes{\sf ex}_{2}^{\delta})\right)&\text{by the asm.}\\ &={\sf Tr}^{U^{2}}_{U^{n},U^{n}}\left((\operatorname{Id_{1}}\otimes s_{U,U}\otimes\operatorname{Id_{2}})\circ(0_{U^{n_{1}+1}}\otimes{\sf ex}_{2}^{\delta})\right)&\hfil\text{I.H. on$ \mathchoice{,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\displaystyle\bm{x_{1}}$ },\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}},}{,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\textstyle\bm{x_{1}}$ },\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}},}{,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\scriptstyle\bm{x_{1}}$ },\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}},}{,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\scriptscriptstyle\bm{x_{1}}$ },\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}},} $}\\ &={\sf Tr}^{U^{2}}_{U^{n},U^{n}}\left((\operatorname{Id_{1}}\otimes(0_{U}\otimes U)\circ s_{U,U}\circ(0_{U}\otimes U)\otimes\operatorname{Id_{2}})\circ(0_{U^{n_{1}+1}}\otimes{\sf ex}_{2}^{\delta})\right)&\text{ dinaturality}\end{array}$

$\begin{array}[]{llr}&={\sf Tr}^{U}_{U^{n_{1}},U^{n_{1}}}\left(0_{U^{n_{1}+1}}\right)\otimes{\sf Tr}^{U}_{U^{n_{2}},U^{n_{2}}}\left((0_{U}\otimes\operatorname{Id})\circ\,{\sf ex}_{2}^{\delta}\right)&\text{(\ref{prps}) and superposing}\\ &={\sf Tr}^{U}_{U^{n_{1}},U^{n_{1}}}\left(0_{U^{n_{1}+1}}\right)\otimes{\sf Tr}^{U}_{U^{n_{2}},U^{n_{2}}}\left((0_{U}\otimes\operatorname{Id})\circ\,{\sf ex}_{2}^{\delta}\circ(0_{U}\otimes\operatorname{Id})\right)&\text{dinaturality}\\ &=0_{U^{n_{1}}}\otimes 0_{U^{n_{2}}}&\hfil\text{I.H. on$ \mathchoice{,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\displaystyle\bm{x}_{2}$ },\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}},}{,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\textstyle\bm{x}_{2}$ },\mathclose{\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}\hbox{\lower 3.80444pt\hbox{\vbox to10.10999pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=3.80444pt,width=0.4pt}}},}{,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\scriptstyle\bm{x}_{2}$ },\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to8.30276pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=3.28888pt,width=0.4pt}}},}{,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}},\hbox{ $\scriptscriptstyle\bm{x}_{2}$ },\mathclose{\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}\hbox{\lower 3.28888pt\hbox{\vbox to7.44165pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=3.28888pt,width=0.4pt}}},} $}\end{array}$

The first dinaturality is via the decomposition $0_{U^{n_{1}+1}}=(U^{n_{1}}\otimes 0_{U})\circ\,0_{U^{n_{1}}}\circ\,(U^{n_{1}}\otimes 0_{U})$ and the second dinaturality is via the decomposition $0_{U}=0_{U}\circ\,0_{U}$ .

( $\parr$ -rule and additives)

Direct from the construction.

See Figure 8 for a pictorial proof depicting the above rewriting in each case. $\Box$

Proposition 4.15** (Mismatch gives rise to

zero convergence of Ex)**

For two ${\sf MALL}$ proofs $\pi^{i}_{[\Delta_{i}],\Gamma_{i},A_{i}}$ with $i=1,2$ , let $\bm{x}_{i}=\lambda_{i}\times(\gamma_{i},a_{i})\in|\pi^{i}_{[\Delta_{i}],\Gamma_{i},A_{i}}|$ so that $a_{i}\in|A_{i}|$ with $a_{1}\not=a_{2}$ and $A_{1}$ and $A_{2}$ are pairwise dual formulas. Then

[TABLE]

Note that the LHS of the assertion is, by Definition 4.9, the following, in which $\epsilon$ is the action arising from $\sigma_{a_{1},a_{2}}=s_{U_{a_{1}},U_{a_{2}}}^{0}$ of Definition 4.8:

[TABLE]

In the above $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\bm{x}{j} $}\,\mathclose{\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\bm{x}{j} $}\,\mathclose{\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to11.24109pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.30554pt,depth=4.93555pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.01389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.1107pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.1107pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\bm{x}{j} $}\,\mathclose{\hbox{\lower 4.09682pt\hbox{\vbox to9.1107pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.1107pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.01389pt,depth=4.09682pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=4.15277pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to8.24959pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to8.24959pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\bm{x}{j} $}\,\mathclose{\hbox{\lower 4.09682pt\hbox{\vbox to8.24959pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to8.24959pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=4.15277pt,depth=4.09682pt,width=0.4pt}}}\,}$ is an endomorphism on $U^{2(m_{j}+1)+n_{j}}$ with the subfactor $U^{2m_{j}}$ for $\sigma_{\bm{x}_{j}}$ .

Before the proof of Proposition 4.15, let us observe a general equation derivable from certain trace axioms (dinaturality and yanking), where $f:X\otimes U\longrightarrow Y\otimes U$ and $0_{U,I}$ (resp. $0_{I,U}$ ) is the zero morphism from $U$ to the tensor unit $I$ (resp. the other way around).

[TABLE]

Note first that the zero morphisms $0_{U,I}$ and $0_{I,U}$ above are derivable from $0_{U}$ using the trace:

$\begin{aligned} 0_{U,I}={\sf Tr}^{U}_{U,I}\left(U\otimes U\stackrel{{\scriptstyle 0_{U}\otimes 0_{U}}}{{\longrightarrow}}U\otimes U\stackrel{{\scriptstyle j}}{{\longrightarrow}}U\cong I\otimes U\right)\\ 0_{I,U}={\sf Tr}^{U}_{U,I}\left(I\otimes U\cong U\stackrel{{\scriptstyle k}}{{\longrightarrow}}U\otimes U\stackrel{{\scriptstyle 0_{U}\otimes 0_{U}}}{{\longrightarrow}}U\otimes U\right)\end{aligned}$

See Figure 7 (lower-right) depicting Equation (24).

(proof of (24))

By the decomposition $0_{U}=0_{I,U}\circ\,0_{U,I}$ , the LHS is ${\sf Tr}^{I}_{X\otimes I,Y\otimes I}\left((\operatorname{Id}\otimes\,0_{U,I})\circ f\circ(\operatorname{Id}\otimes\,0_{I,U})\right)$ , by dinaturality, which is equal to the RHS by vanishing. (end of proof of (24))

Finally we go to:

** Proof**.

[Proof of Proposition 4.15] We prove the following instance of the proposition using Equation (21), where $n=n_{1}+n_{2}$ , since $\sigma_{a_{1},a_{2}}=s^{0}=0_{U}^{2}$ :

[TABLE]

On the other hand, Lemma 4.14 and (24) say for all $i=1,2$

[TABLE]

Since the two actions $\epsilon$ and $\delta$ coincide again by (24), the formula (29) becomes equal to $0_{U^{n_{1}}}\otimes 0_{U^{n_{2}}}=0_{U^{n}}$ . $\Box$

4.3 Main Theorem

This section concerns the main theorem of this paper.

Theorem 4.16 (Ex is invariant and diminishes sets of indices)

Let $\nu\in|\pi_{[\Delta],\Gamma}|^{J}\blacktriangleright^{\textsc{\tiny$ I $}}\nu^{\prime}\in|\pi^{\prime}_{[\Delta^{\prime}],\Gamma}|^{J^{\prime}}$ be any ${\sf MALL}^{[\mathfrak{c}]}(I)$ proof transformation. Then

(i)

[TABLE] 2. (ii)

In particular, when $\pi^{\prime}$ is cut-free so that $\Delta^{\prime}$ is empty, then

[TABLE]

** Proof**.

We prove (i) according to the cases of Proposition 3.3, since (ii) follows directly from (i) as follows: For a cut-free $\pi^{\prime}$ , $\sigma_{\nu^{\prime}}$ is empty, hence ${\sf Ex}\left(\sigma,\nu_{j}\right)=\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}}\,}$ , where $\mathchoice{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \displaystyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \textstyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}\hbox{\lower 4.93555pt\hbox{\vbox to12.79665pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=7.8611pt,depth=4.93555pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptstyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to10.22182pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=6.125pt,depth=4.09682pt,width=0.4pt}}}\,}{\,\mathopen{\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}}\,\hbox{$ \scriptscriptstyle\nu^{\prime}{j} $}\,\mathclose{\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}\hbox{\lower 4.09682pt\hbox{\vbox to9.3607pt{\hrule width=1.0pt,height=0.0pt,depth=0.4pt\vfil\hrule width=1.0pt,height=0.4pt,depth=0.0pt}}}\hbox{\hbox{\vrule height=5.26389pt,depth=4.09682pt,width=0.4pt}}}\,}\not=0$ holds directly both from the construction of Definition 4.4 and from the non-collapsing assumption (7).

The invariance of (i) is direct by the yanking axiom in case 1, and by induction on the proof $\pi$ in cases 2 and 3: This proof method directly comes as an instance of the known method in the symmetric traced monoidal category modelling multiplicative GoI [HS06]. Thus we prove the zero convergence for the diminution of $J$ . The following crucial cases are those of the proof of Proposition 3.3.

(Crucial case 1)

Each instance of $\nu$ at $j\in J\setminus J^{\prime}$ is $\nu_{j}=(\delta_{j},b_{j},a_{j},b_{j},\lambda_{j})$ , so that $a_{j}\not=b_{j}$ since $j\not\in J^{\prime}$ ,

then ${\sf Ex}\left(\sigma,\nu_{j}\right)=0$ by Proposition 4.15.

(Crucial case 2)

$J=J_{1}+J_{2}$ diminishes into $J_{1}$ . Each instance of $\tau$ at $j_{2}\in J_{2}$ is $\tau_{j_{2}}=(\omega,\delta_{2},\gamma,(2,a_{2}))\in|\&(\pi^{1},\pi^{2})|$ . Thus each instance of $\nu$ at $j_{2}\in J_{2}$ is $\nu_{j_{2}}=(\omega,\delta_{2},\delta_{3},(2,a_{2}),(1,a_{1}),\gamma,\xi)\in|\pi|$ . Since $(2,a_{2})\not=(1,a_{1})$ , we have ${\sf Ex}\left(\sigma,\nu_{j_{2}}\right)=0$ by Proposition 4.15.

(Crucial case 3)

$J$ does not diminish in this case. $\Box$

Appendix C is read as an elucidating example of Theorem 4.16.

Conclusion and Future Work

This paper offers two main contributions:

(i)

Presenting an indexed ${\sf MALL}$ system for stacking cut formulas and its relational counterpart to simulate ${\sf MALL}$ proof reduction of cut elimination. 2. (ii)

Constructing an execution formula for the interpretation of ${\sf MALL}$ proofs equipped with indices. The ${\sf MALL}$ proof reduction is characterised by the convergence of the execution formula to the denotational interpretation. Furthermore, the zero convergence of the execution formula characterises the diminution of indices, which is specific to additive cut elimination.

Our explicit use of indexed-syntactical manipulations directly overcomes known difficulties in additive GoI. We hope that this paper, from the perspective of indexed linear logic, will shed light on an approachable understanding of the preceding literature on additive GoI, from precursory ones [Gir95, Duc09] to more recent developments [Gir11, Sei16].

We discuss some future directions.

For a genuine ${\sf MALL}$ GoI without bypassing via indexed logic, a syntax-free counterpart is required to replace the indices. We construct such a genuine GoI [Ham17] using an algebraic ingredient: a scalar extension of Girard’s $*$ -algebra of partial isometries over a boolean polynomial semi-ring. The genuine GoI may help us connect our syntactic manipulation of indices to Girard’s semantic use of clauses for predicates in the precursory ${\sf MALL}$ GoI [Gir95].

In a syntactic direction, the status of Gentzen cut elimination for ${\sf MALL}(I)$ remains open since the present paper only concerns lifting the image to the indices of ${\sf MALL}$ cut-reduction. The status will complement the reduction-free cut elimination, known to be derivable from the Fundamental lemma 2.10 (cf. [BucEhr00, BucEhr01, HamTak08]).

Extending the present paper to the exponentials is challenging to use Bucciarelli–Ehrhard ${\sf LL}(I)$ [BucEhr01] for modelling GoI. This will involve extending our methodology of a traced monoidal category with a zero morphism to the whole GoI situation [HS06, HS11], compatibly with the multisets interpretation of the exponential connective in ${\sf Rel}$ . The explicit accommodation of the indices to GoI will give a novel approach to the (non indexed) GoI modelling for the exponentials.

Acknowledgment We wish to thank the referees for detailed and very helpful comments that have greatly improved the presentation.

Appendix A Axioms of Traced Monoidal Category

Definition A.1** (Trace axioms of the family

${\sf Tr}^{Z}_{X,Y}$ of (15) [JSV96, HS06])**

*(Natural in $X$ ) *

$\begin{aligned} {\sf Tr}^{Z}_{X,Y}\left(f\right)\circ g={\sf Tr}^{Z}_{X^{\prime},Y}\left(f\circ(g\otimes Z)\right)&&\mbox{where}\quad f:X\otimes Z\longrightarrow Y\otimes Z\quad\mbox{and}\quad g:X^{\prime}\rightarrow X\end{aligned}$ ** 2. 2.

*(Natural in $Y$ ) *

$\begin{aligned} g\circ{\sf Tr}^{Z}_{X,Y}\left(f\right)={\sf Tr}^{Z}_{X^{\prime},Y}\left((g\otimes Z)\circ f\right)&&\mbox{where}\quad f:X\otimes Z\longrightarrow Y\otimes Z\quad\mbox{and}\quad g:Y\rightarrow Y^{\prime}\end{aligned}$ ** 3. 3.

*(Dinatural in $Z$ ) *

$\begin{aligned} {\sf Tr}^{Z}_{X,Y}\left((Y\otimes g)\circ f\right)={\sf Tr}^{Z^{\prime}}_{X,Y}\left(f\circ(X\otimes g)\right)&&\mbox{where}\quad f:X\otimes Z\longrightarrow Y\otimes Z\quad\mbox{and}\quad g:Z^{\prime}\rightarrow Z\end{aligned}$ ** 4. 4.

*(Vanishing I ) *

$\begin{aligned} {\sf Tr}^{I}_{X,Y}\left(f\otimes I\right)=f&&\mbox{where}\quad f:X\longrightarrow Y\end{aligned}$ ** 5. 5.

*(Vanishing II ) *

$\begin{aligned} {\sf Tr}^{Z\otimes W}_{X,Y}\left(f\right)={\sf Tr}^{Z}_{X,Y}\left({\sf Tr}^{W}_{X\otimes Z,Y\otimes Z}\left(f\right)\right)&&\mbox{where}\quad f:X\otimes Z\otimes W\longrightarrow Y\otimes Z\otimes W\end{aligned}$ ** 6. 6.

*(Superposing) *

$\begin{aligned} g\otimes{\sf Tr}^{Z}_{X,Y}\left(f\right)={\sf Tr}^{Z}_{W\otimes X,V\otimes Y}\left(g\otimes f\right)&&\mbox{where}\quad f:X\otimes Z\longrightarrow Y\otimes Z\quad\mbox{and}\quad g:W\longrightarrow V\end{aligned}$ ** 7. 7.

*(Yanking) *

$\begin{aligned} {\sf Tr}^{X}_{X,X}\left(s_{X,X}\right)=X&&\mbox{for the symmetry}\quad s_{X,X}:X\otimes X\longrightarrow X\otimes X\end{aligned}$ **

Lemma A.2 (Generalized Yanking [HS11])

Let $s_{Z,Y}$ denote the symmetry from $Z\otimes Y$ to $Y\otimes Z$ .

[TABLE]

** Proof**.

$LHS\stackrel{{\scriptstyle nat}}{{=}}{\sf Tr}^{Z}_{Z,Y}\left(s_{Z,Y}\circ(Z\otimes g)\right)\circ f\stackrel{{\scriptstyle dinat}}{{=}}{\sf Tr}^{Y}_{Z,Y}\left((Y\otimes g)\circ s_{Z,Y}\right)\circ f$ . Inside the trace $(Y\otimes g)\circ s_{Z,Y}=s_{Y,Y}\circ(g\otimes Y)$ , thus ${\sf Tr}^{Y}_{Z,Y}\left(s_{Y,Y}\circ(g\otimes Y)\right)\stackrel{{\scriptstyle nat}}{{=}}{\sf Tr}^{Y}_{Y,Y}\left(s_{Y,Y}\right)\circ g\stackrel{{\scriptstyle yank}}{{=}}g$ $\Box$

Appendix B Omitted Proofs

B.1 Proof for Proposition 2.10 (Fundamental Lemma)

Lemma B.1 ((i) implies (ii))

Let $\pi_{[\Delta],\Gamma}$ be a proof of a sequent $\vdash[\Delta],\,\,\Gamma$ in ${\sf MALL}^{[\mathfrak{c}]}$ . Let $\delta\times\gamma\in|\pi_{[\Delta],\Gamma}|^{J}$ (for some $J\subseteq I$ ) with $\delta\in[\mathfrak{sl}(\Delta)]^{J}$ and $\gamma\in|\Gamma|^{J}$ . The sequent $\vdash_{J}[\Delta\langle\langle\delta\rangle\rangle],\,\Gamma\langle\gamma\rangle$ has a proof $\rho$ in ${\sf MALL}^{[\mathfrak{c}]}(I)$ such that $\rho\!\upharpoonright_{\emptyset}=\pi$ .

** Proof**.

By construction on the ${\sf MALL}$ proof $\pi$ . The proof figures are referred in Definition 2.6.

(cut rule)

$\delta\times\gamma\cong\delta_{1}\times\delta_{2}\times\gamma_{1}\times\gamma_{2}$ with $\delta_{1}\times\gamma_{1}\in\pi^{1}_{[\Delta_{1}],\Gamma_{1},A}$ and $\delta_{2}\times\gamma_{2}\in\pi^{2}_{[\Delta_{2}],A^{\perp},\Gamma_{2}}$ . By I.H.s on $(\delta_{i}\times\gamma_{i})\,$ s, there are ${\sf MALL}(I)$ -proofs of the sequents $\vdash_{J}[\Delta_{1}\langle\langle\delta_{1}\rangle\rangle],\,\Gamma_{1}\langle\gamma_{1}^{{}^{\prime}}\rangle,A\langle\gamma_{1}^{{}^{\prime\prime}}\rangle$ and $\vdash_{J}[\Delta_{2}\langle\langle\delta_{2}\rangle\rangle],\,A^{\perp}\langle\gamma_{2}^{{}^{\prime}}\rangle,\Gamma_{2}\langle\gamma_{2}^{{}^{\prime\prime}}\rangle$ with $\gamma_{i}=\gamma_{i}^{{}^{\prime}}\times\gamma_{i}^{{}^{\prime\prime}}$ . Note that $A\langle\gamma_{1}^{{}^{\prime\prime}}\rangle$ and $A^{\perp}\langle\gamma_{2}^{{}^{\prime}}\rangle$ are dual formulas since they have the same domain $J$ . Hence the cut between the dual formulas is applied to prove $\vdash_{J}[\,\,\Delta_{1}\langle\langle\delta_{1}\rangle\rangle,\Delta_{2}\langle\langle\delta_{2}\rangle\rangle,A\langle\gamma_{1}^{{}^{\prime\prime}}\rangle,A^{\perp}\langle\gamma_{2}^{{}^{\prime}}\rangle\,\,],\,\Gamma_{1}\langle\gamma_{1}^{{}^{\prime}}\rangle,\Gamma_{2}\langle\gamma_{2}^{{}^{\prime\prime}}\rangle$ . The assertion follows since $\Delta_{1}\langle\langle\delta_{1}\rangle\rangle\times\Delta_{2}\langle\langle\delta_{2}\rangle\rangle=(\Delta_{1},\Delta_{2})\langle\langle\delta_{1}\times\delta_{2}\rangle\rangle$ .

( $\&$ -rule)

Let $\nu=|\pi_{[\Delta],\Gamma}|$ . Then $\nu=\{(x_{1},z,y,(1,a_{1}))\mid(x_{1},z,y,a_{1})\in\nu_{1}\}\,+\,\{(x_{2},z,y,(2,a_{2}))\mid(x_{2},z,y,a_{2})\in\nu_{2}\}\,\cong\,\nu_{1}\raisebox{2.15277pt}{$ \frown $}\nu_{2}$ with $\gamma_{i}\in|\pi^{i}_{[\Delta_{i},\Delta],\Gamma,A_{i}}|^{J_{1}}$ and $J=J_{1}+J_{2}$ . By I.H.s on $\pi^{i}\,$ s, there are ${\sf MALL}(I)$ -proofs of $\vdash_{J_{i}}[\,\Delta_{i}\langle\langle\delta_{i}^{{}^{\prime}}\rangle\rangle,\Delta\langle\langle\delta_{i}^{{}^{\prime\prime}}\rangle\rangle\,],\,\Gamma\langle\gamma_{i}^{{}^{\prime}}\rangle,A_{i}\langle\gamma_{i}^{{}^{\prime\prime}}\rangle$ with $\delta_{i}=\delta_{i}^{{}^{\prime}}\times\delta_{i}^{{}^{\prime\prime}}$ and $\gamma_{i}=\gamma_{i}^{{}^{\prime}}\times\gamma_{i}^{{}^{\prime\prime}}$ . Because $\Gamma\langle\gamma_{1}^{{}^{\prime}}\raisebox{2.15277pt}{$ \frown $}\gamma_{2}^{{}^{\prime}}\rangle\!\upharpoonright_{J_{i}}=\Gamma\langle\gamma_{i}^{{}^{\prime}}\rangle$ and $\Delta\langle\langle\delta_{1}^{{}^{\prime\prime}}\raisebox{2.15277pt}{$ \frown $}\delta_{2}^{{}^{\prime\prime}}\rangle\rangle\!\upharpoonright_{J_{i}}=\Delta\langle\langle\delta_{i}^{{}^{\prime\prime}}\rangle\rangle$ by Lemma 2.2, the $\&$ -rule is applied to prove

$\vdash_{J_{1}+J_{2}}[\,\,\Delta_{1}\langle\langle\delta_{1}^{{}^{\prime}}\rangle\rangle,\Delta_{2}\langle\langle\delta_{2}^{{}^{\prime}}\rangle\rangle,\Delta\langle\langle\delta_{1}^{{}^{\prime\prime}}\raisebox{2.15277pt}{$ \frown $}\delta_{2}^{{}^{\prime\prime}}\rangle\rangle\,\,],\,\Gamma\langle\gamma_{1}^{{}^{\prime}}\raisebox{2.15277pt}{$ \frown $}\gamma_{2}^{{}^{\prime}}\rangle,\,A_{1}\langle\gamma_{1}^{{}^{\prime\prime}}\rangle\,\&\,A_{2}\langle\gamma_{2}^{{}^{\prime\prime}}\rangle$ .

$\Box$

Lemma B.2 ((ii) implies (i))

Let $\vdash[\Delta],\,\,\Gamma$ be a sequent of ${\sf MALL}^{[\mathfrak{c}]}$ . Let $\nu\in(\mathfrak{sl}(\Delta)\times\Gamma)^{J}$ (for some $J\subseteq I$ ) and let $\rho$ be a proof of $\vdash_{J}([\Delta],\,\Gamma)\langle\nu\rangle$ in ${\sf MALL}^{[\mathfrak{c}]}(I)$ . Then $\nu\in|(\rho\!\upharpoonright_{\emptyset})_{[\Delta],\Gamma}|^{J}$

** Proof**.

By the construction on the ${\sf MALL}(I)$ proof $\rho$ .

(cut rule)

$\rho$ is $\vdash_{J}[\Delta_{1},\Delta_{2},A,A^{\perp}],\,\,\Gamma_{1},\Gamma_{2}\lx@proof@logical@and\vdash_{J}[\Delta_{1}],\,\,\Gamma_{1},A\rho^{1}\vdash_{J}[\Delta_{2}],\,\,A^{\perp},\Gamma_{2}\rho^{2}$ .

The conclusion is written $\vdash_{J}([\Delta_{1},\Delta_{2},A,A^{\perp}],\,\Gamma_{1},\Gamma_{2})\langle\nu\rangle$ . From the construction, $\nu\cong\nu_{1}\times\nu_{2}$ so that the conclusions of $\rho^{1}$ and $\rho_{2}$ are respectively $\vdash_{J}([\Delta_{1}],\,\Gamma_{1},A)\langle\nu_{1}\rangle$ and $\vdash_{J}([\Delta_{2}],\,A^{\perp},\Gamma_{2})\langle\nu_{2}\rangle$ . By I.H.s on $\rho^{i}$ s, $\nu_{1}\in|(\rho^{1}\!\upharpoonright_{\emptyset})_{[\Delta_{1}],\Gamma_{1},A}|^{J}$ and $\nu_{2}\in|(\rho^{2}\!\upharpoonright_{\emptyset})_{[\Delta_{2}],A^{\perp},\Gamma_{2}}|^{J}$ . The assertion follows since

$|(\rho^{1}\!\upharpoonright_{\emptyset})_{[\Delta_{1}],\Gamma_{1},A}|^{J}\times|(\rho^{2}\!\upharpoonright_{\emptyset})_{[\Delta_{2}],A^{\perp},\Gamma_{2}}|^{J}\cong|(\rho\!\upharpoonright_{\emptyset})_{[\Delta_{1},\Delta_{2},A,A^{\perp}],\Gamma_{1},\Gamma_{2}}|^{J}.$

( $\&$ -rule)

$\rho$ is $\vdash_{J_{1}+J_{2}}[\Delta_{1},\Delta_{2},\Sigma\,],\,\,\Gamma,A_{1}\&A_{2}\lx@proof@logical@and\vdash_{J_{1}}[\Delta_{1},\Sigma\,],\,\,\Gamma,A_{1}\rho^{1}\vdash_{J_{2}}[\Delta_{2},\Sigma\,],\,\,\Gamma,A_{2}\rho^{2}$

$\nu$ is of the form $\nu_{1}\raisebox{2.15277pt}{$ \frown $}\nu_{2}$ so that the conclusions of $\rho_{i}\,$ s are $\vdash_{J_{i}}([\Delta_{i},\Sigma],\,\Gamma,A_{i})\langle\nu_{i}\rangle$ . By I.H.s on $\rho^{i}$ s, $\nu_{i}\in|(\rho^{i}\!\upharpoonright_{\emptyset})_{[\Delta_{i},\Sigma],\Gamma_{1},A_{i}}|^{J_{i}}$ . The assertion follows since

$|(\rho^{1}\!\upharpoonright_{\emptyset})_{[\Delta_{1},\Sigma],\Gamma,A_{1}}|^{J_{1}}\raisebox{2.15277pt}{$ \frown $}|(\rho^{1}\!\upharpoonright_{\emptyset})_{[\Delta_{2},\Sigma],\Gamma,A_{2}}|^{J_{2}}\cong|(\rho\!\upharpoonright_{\emptyset})_{[\Delta_{1},\Delta_{2},\Sigma],\Gamma,A_{1}\&A_{2}}|^{J_{1}+J_{2}}$

$\Box$

Appendix C Indices and Additive Cut Elimination

This appendix elucidates the fundamental idea of the paper. The appendix may read as a prologue of the paper by readers yet familiar with ${\sf MLL}$ GoI interpretation on a reflexive object in a traced monoidal category.

Consider a sequence $\pi_{1}\rhd\pi_{2}\rhd\pi_{3}$ of cut eliminations for proofs in the additive fragment of ${\sf MALL}$ . In our sequent notation, pairwise cut formulas, if present, are stored inside a stack $[\cdot]$ in a sequent. The first reduction, intrinsic to the additives, eliminates a $\&$ in a cut, whereby the subproof $\textrm{ax}_{2}$ is pruned. The second reduction eliminates a redundant cut against an axiom:

$\vdash[A\&A,A^{\perp}\oplus A^{\perp}]\,\,A^{\perp},A\lx@proof@logical@and\vdash A^{\perp},A\&A\lx@proof@logical@and\vdash A^{\perp},A\vdash A^{\perp},A\vdash A^{\perp}\oplus A^{\perp},A\vdash A^{\perp},A$

$\rhd$

$\vdash[A,A^{\perp}]\,\,A^{\perp},A\lx@proof@logical@and\vdash A^{\perp},A\vdash A^{\perp},A$

$\rhd$

$\vdash A^{\perp}A$

Step 1 (Interpretation $|\pi|$ in ${\sf Rel}$ with unperformed cuts and indices for additives)

We begin by interpreting proofs in ${\sf Rel}$ but without relational composition. For this, the cut rule is interpreted same as the tensor rule. This interpretation is consistent with the syntactic convention, starting from Girard’s GoI 1, which puts the cut formulas into a stack.

For simplicity and in accordance with the fact that the multiplicative dual elements 1 and $\perp$ are interpreted in ${\sf Rel}$ as the singleton set, we take $A=1$ and, dually, $A^{\perp}=\>\perp$ , so that $|1|=|\!\perp\!|$ is the singleton, whose unique element is denoted $*$ or $\bar{*}$ (obviously, $*=\bar{*}$ ), depending on whether it comes from $|A|$ or $|A^{\perp}|$ , respectively.

An axiom is interpreted in ${\sf Rel}$ by the diagonal, so that $|\textrm{ax}_{i}|=\{(\bar{*},*)\}\subseteq|A^{\perp}|\times|A|$ . The proof $\pi_{2}$ is interpreted as $|\textrm{cut}(\textrm{ax}_{1},\textrm{ax}_{2})|=\{(\bar{*},*,\bar{*},*)\}\subseteq|A^{\perp}|\times\,[\,|A|\times|A^{\perp}|\,]\,\times|A|$ , in which the pair $(*,\bar{*})$ in the cut slot from $[\,|A|\times|A^{\perp}|\,]$ remains explicit, rather than being hidden by relational composition through $*=\bar{*}$ . Note that both interpretations $|\pi_{2}|$ and $|\pi_{3}|$ are singletons. More generally, it is straightforward to see that any proof in the multiplicatives can be interpreted by a singleton whenever literals are interpreted by singletons. However, this is not the case for the additives. When interpreting $\pi_{1}$ with additive rules, singletons prove insufficient, and this is where the indices become necessary: The left and right premises of $\pi_{1}$ are interpreted respectively by $|\&(\textrm{ax}_{1},\textrm{ax}_{2})|=\{(\bar{*},(1,*)),(\bar{*},(2,*))\}\subseteq|A^{\perp}|\times(|A|+|A|)$ and $|\oplus_{1}\!(\textrm{ax}_{3})|=\{((1,\bar{*}),*)\}\subseteq(|A^{\perp}|+|A^{\perp}|)\times|A|$ . A set of indices $J=\{1,2\}$ is employed to describe these two interpretations: the first yields $\delta\in|\&(\textrm{ax}_{1},\textrm{ax}_{2})|^{J}$ so that $\delta_{1}=(\bar{*},(1,*))$ and $\delta_{2}=(\bar{*},(2,*))$ , and the second yields $\tau\in|\oplus_{1}\!(\textrm{ax}_{3})|^{J}$ , so $\tau_{1}=\tau_{2}=((1,\bar{*}),*)$ .

Then $\pi_{1}$ is interpreted by $\nu$ :

$\begin{aligned} \nu:=\delta\times\tau\in|\textrm{cut}(\&(\textrm{ax}_{1},\textrm{ax}_{2}),\oplus_{1}(\textrm{ax}_{3}))|^{J}\subseteq(|A^{\perp}|\times[\,(|A|\!+\!|A|)\times(|A^{\perp}|\!+\!|A^{\perp}|)\,]\times|A|)^{J},\end{aligned}$

and therefore $(\delta\times\tau)_{1}=(\delta_{1},\tau_{1})=(\bar{*},(1,*),(1,\bar{*}),*)$ and $(\delta\times\tau)_{2}=(\delta_{2},\tau_{2})=(\bar{*},(2,*),(1,\bar{*}),*)$ , where $\delta\times\tau$ denotes the mediating morphism of the set-theoretical cartesian product. Summing up, $|\pi_{1}|=\{v_{j}\mid j\in J=\{1,2\}\}$ , where $v_{1}=(\bar{*},(1,*),(1,\bar{*}),*)$ and $v_{2}=(\bar{*},(2,*),(1,\bar{*}),*)$ .

Step 2 ( ${\sf Ex}_{J}\left(\sigma,\nu\right)$ for $\nu\in|\pi|^{J}$ : Executing cuts using trace structures)

In addition to step 1, our GoI interpretation runs an execution formula for $|\pi_{i}|$ to perform cut elimination against the unperformed cut formulas, syntactically in the stack and semantically in the noncompositional interpretation.

Each point in $|\pi|$ is interpreted as an endomorphism on a certain tensor folding of a reflexive object $U$ in a traced monoidal category ${\cal C}$ with a zero morphism on $U$ . The object $U$ uniformly interprets each element in the interpretation of the conclusion of $\pi$ ; e.g., in $|\pi_{1}|$ , $U$ has the elements $\bar{*}$ , $*$ , $(1,\bar{*})$ , $(1,*)$ , and $(2,*)$ . In the following, these points are identified with their interpretation $U$ .

For the most primitive case, e.g., for $|\pi_{3}|$ , each diagonal point $(\bar{*},*)$ interpreting the axiom is interpreted as a symmetry of $\cal{C}$ :

[TABLE]

The unique point of $|\pi_{2}|$ is interpreted by the endomorphism ${\sf Ex}\left(\sigma_{\textrm{cut}},|\pi_{2}|\right)$ on $U_{\bar{*}}\otimes U_{*}$ , in which $\sigma_{\textrm{cut}}$ , as the interpretation of the cut, is the symmetry $s_{*,\bar{*}}$ acting on the cut formulas:

[TABLE]

Note that the symmetries $s$ ’s occurring in (31) interpret respective axioms.

This is equal to (30) in ${\cal C}$ by the trace axioms. The adjacent diagrams illustrate (30) and (31), where the equality is found in the diagram for (31) by chasing arrows with respect to both composition and feedback.

The GoI interpretation of $\pi_{1}$ is that for the indexed $\nu\in|\pi_{1}|^{J}$ in Step 1, which is defined pointwise (for $j\in J=\{1,2\}$ ) at $\nu_{1}$ and $\nu_{2}$ , in which $\sigma_{\nu_{1}}$ and $\sigma_{\nu_{2}}$ are stipulated respectively by $\sigma_{\textrm{cut}}$ and 0, where $\sigma_{\textrm{cut}}$ is a symmetry $s_{(1,*),(1,\bar{*})}$ for the cut formulas while 0 is the zero morphism $0_{U^{2}}$ resulting by zero action on a symmetry $s_{(2,*),(1,\bar{*})}$ :

[TABLE]

Here (32) is equivalent to (31), while (33) reduces to 0 in ${\cal C}$ by virtue of the trace axioms with zero morphisms. The next two diagrams, for (32) and (33), illustrate that (33) yields a zero morphism because chasing any arrow results in passing through 0.

At $j=2$ ,

${\sf Ex}\left(\sigma_{\nu_{2}},\nu_{2}\right)=0$ , so we delete the index 2, reducing $J$ into the singleton $\{1\}$ . For index 1, ${\sf Ex}\left(\sigma_{\nu_{1}},\nu_{1}\right)$ is identical to the symmetry $(\ref{ex3})$ of $|\pi_{3}|$ and, hence, to the denotational interpretation.

Bibliography13

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[Buc Ehr 00] A. Bucciarelli and T. Ehrhard, On phase semantics and denotational semantics in multiplicative-additive linear logic. Ann. Pure Appl. Logic 102(3) 247-282 (2000)
2[Buc Ehr 01] A. Bucciarelli, T. Ehrhard, On phase semantics and denotational semantics: the exponentials. Ann. Pure Appl. Logic 109(3): 205-241 (2001)
3[DR 95] V. Danos and L. Regnier, Proof-nets and the Hilbert space. In J.-Y. Girard, Y. Lafont, and L. Regnier, eds. Advances in Linear Logic, Cambridge University Press, (1995) 307–328
4[Duc 09] E. Duchesne, La Localisation en Logique : Géométrie de l’Interaction et Sémantique Dénotationnelle. Thèse de doctorat, Université Aix-Marseille II, 2009.
5[Gir 89] J-Y. Girard, Geometry of Interaction I: Interpretation of System F, in: Logic Colloquium ’88 , North-Holland, 1989, pp. 221-260.
6[Gir 95] J-Y. Girard, Geometry of Interaction III: Accommodating the Additives, in: Advances in Linear Logic, LNS 222 , CUP, 1995, 329–389.
7[Gir 11] J-Y. Girard, Geometry of Interaction V: Logic in the Hyperfinite Factor, Theor. Comput. Sci. Vol. 412 No. 20 (2011) pp. 1860-1883
8[HS 06] E. Haghverdi and P. Scott, A Categorical Model for the Geometry of Interaction, Theor. Comput. Sci. Vol. 350 (2-3), (2006), pp. 252-274.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

A MALL Geometry of Interaction

Abstract

1 Introduction

2 MALL(I){\sf MALL}(I)MALL(I) with cut list and relational semantics

2.1 MALL(I){\sf MALL}(I)MALL(I) with cut list

Lemma 2.1** (Identity)**

Lemma 2.2** (Restricting provable sequents)**

Lemma 2.3

2.2 MALL[c](I){\sf MALL}^{[\mathfrak{c}]}(I)MALL[c](I) and Relational Semantics Rel[c]{\sf Rel}^{[\mathfrak{c}]}Rel[c]

Definition 2.4** (Denotational interpretation (π[Δ],Γ)∗(\pi_{[\Delta],\Gamma})^{*}(π[Δ],Γ​)∗

Lemma 2.5

Definition 2.6** (Interpretation ∣π[Δ],Γ∣|\pi_{[\Delta],\Gamma}|∣π[Δ],Γ​∣

Definition 2.7** (Translation of indexed relation γ\gammaγ

Definition 2.8** (Translation to MALL[c](I){\sf MALL}^{[\mathfrak{c}]}(I)MALL[c](I)

Lemma 2.9** (Restricting translation)**

2.3 Fundamental lemma

Proposition 2.10** (Fundamental lemma à la Bucciarelli–Ehrhard)**

3 Lifting MALL{\sf MALL}MALL reduction over indices

Definition 3.1** (MALL[c](I){\sf MALL}^{[\mathfrak{c}]}(I)MALL[c](I) proof transformation \blacktriangleright^{\textsc{\tinyI}} with

Example 3.2

Proposition 3.3** (Lifting to indexed transformation)**

4 MALL{\sf MALL}MALL GoI Interpretation

4.1 Execution formula with zero action on symmetries of cuts

4.1.1 Interpretation of indexed point

Lemma 4.1** (tensoring zero)**

Definition 4.2** (zero-action (sU,U)0(s_{U,U})^{0}(sU,U​)0)**

Lemma 4.3** (Tag of xix_{i}xi​ with x=(x1,…,xℓ)∈∣Δ^∣×∣Γ∣\bm{x}=(x_{1},\ldots,x_{\ell})\in|\widehat{\Delta}|\times|\Gamma|x=(x1​,…,xℓ​)∈∣Δ∣×∣Γ∣

Definition 4.4

Lemma 4.5

4.1.2 The action ϵx\epsilon_{\bm{x}}ϵx​ annihilating associated

Definition 4.6

Example 4.7

Definition 4.8** (action ϵx\epsilon_{\bm{x}}ϵx​ on

4.1.3 The Execution formula

Definition 4.9** (Execution formula Ex(σ,x){\sf Ex}\left(\sigma,\bm{x}\right)Ex(σ,x)

Example 4.10

Definition 4.11** (Execution formula ExJ(σ,ν){\sf Ex}_{J}\left(\sigma,\nu\right)ExJ​(σ,ν)

4.2 Zero Convergence of Execution Formula

Lemma 4.12** (tracing zero)**

Definition 4.13** (action δx\delta_{\bm{x}}δx​)**

Lemma 4.14** (lemma for Prop 4.15)**

Proposition 4.15** (Mismatch gives rise to

4.3 Main Theorem

Theorem 4.16** (Ex is invariant and diminishes sets of indices)**

Appendix A Axioms of Traced Monoidal Category

Definition A.1** (Trace axioms of the family

Lemma A.2** (Generalized Yanking [HS11])**

Appendix B Omitted Proofs

B.1 Proof for Proposition 2.10 (Fundamental Lemma)

Lemma B.1** ((i) implies (ii))**

Lemma B.2** ((ii) implies (i))**

Appendix C Indices and Additive Cut Elimination

2 ${\sf MALL}(I)$ with cut list and relational semantics

2.1 ${\sf MALL}(I)$ with cut list

Lemma 2.1 (Identity)

Lemma 2.2 (Restricting provable sequents)

2.2 ${\sf MALL}^{[\mathfrak{c}]}(I)$ and Relational Semantics ${\sf Rel}^{[\mathfrak{c}]}$

Definition 2.4** (Denotational interpretation $(\pi_{[\Delta],\Gamma})^{*}$

Definition 2.6** (Interpretation $|\pi_{[\Delta],\Gamma}|$

Definition 2.7** (Translation of indexed relation $\gamma$

Definition 2.8** (Translation to ${\sf MALL}^{[\mathfrak{c}]}(I)$

Lemma 2.9 (Restricting translation)

Proposition 2.10 (Fundamental lemma à la Bucciarelli–Ehrhard)

3 Lifting ${\sf MALL}$ reduction over indices

Definition 3.1** ( ${\sf MALL}^{[\mathfrak{c}]}(I)$ proof transformation $\blacktriangleright^{\textsc{\tiny$ I $}}$ with

Proposition 3.3 (Lifting to indexed transformation)

4 ${\sf MALL}$ GoI Interpretation

Lemma 4.1 (tensoring zero)

Definition 4.2 (zero-action $(s_{U,U})^{0}$ )

Lemma 4.3** (Tag of $x_{i}$ with $\bm{x}=(x_{1},\ldots,x_{\ell})\in|\widehat{\Delta}|\times|\Gamma|$

4.1.2 The action $\epsilon_{\bm{x}}$ annihilating associated

Definition 4.8** (action $\epsilon_{\bm{x}}$ on

Definition 4.9** (Execution formula ${\sf Ex}\left(\sigma,\bm{x}\right)$

Definition 4.11** (Execution formula ${\sf Ex}_{J}\left(\sigma,\nu\right)$

Lemma 4.12 (tracing zero)

Definition 4.13 (action $\delta_{\bm{x}}$ )

Lemma 4.14 (lemma for Prop 4.15)

Theorem 4.16 (Ex is invariant and diminishes sets of indices)

Lemma A.2 (Generalized Yanking [HS11])

Lemma B.1 ((i) implies (ii))

Lemma B.2 ((ii) implies (i))