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Article

# The Logic of Pseudo-Uninorms and Their Residua

by
SanMin Wang
Faculty of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
Symmetry 2019, 11(3), 368; https://doi.org/10.3390/sym11030368
Submission received: 15 February 2019 / Revised: 6 March 2019 / Accepted: 8 March 2019 / Published: 12 March 2019
(This article belongs to the Special Issue Mathematical Fuzzy Logic and Fuzzy Set Theory)

## Abstract

:
Our method for density elimination is generalized to the non-commutative substructural logic $GpsUL *$. Then, the standard completeness of $HpsUL *$ follows as a lemma by virtue of previous work by Metcalfe and Montagna. This result shows that $HpsUL *$ is the logic of pseudo-uninorms and their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.
MSC:
03B50; 03F05; 03B52; 03B47

## 1. Introduction

Prof. Metcalfe, Olivetti and Gabbay conjectured that the Hilbert system $HpsUL$ is the logic of pseudo-uninorms and their residua in 2009 in [1]. It is not the case, as shown by Prof. Wang and Zhao in [2], although $HpsUL$ is the logic of bounded representable residuated lattices. We constructed the system $HpsUL *$ by adding the weakly commutativity rule
$( W C M ) ⊢ ( A ⇝ t ) → ( A → t )$
to $HpsUL$ and conjectured that it is the logic of residuated pseudo-uninorms and their residua in 2013 in [3].
In this paper, we prove the conjecture by showing that the density elimination holds for the hypersequent system $GpsUL *$ corresponding to $HpsUL *$. Then, the standard completeness of $HpsUL *$ follows as a lemma by virtue of previous work by Metcalfe and Montagna [4]. That is, $HpsUL *$ is complete with respect to algebras whose lattice reduct is the real unit interval $[ 0 , 1 ]$. Thus, $HpsUL *$ is a kind of substructural fuzzy logic [4], and potentially has certain applications to fuzzy inferences and expert Systems [5,6,7,8]. Our result also shows that that $HpsUL *$ is an axiomatization for the variety of residuated lattices generated by all dense residuated chains. Thus, we have also answered the question posed by Prof. Metcalfe and Tsinakis in [9] in 2017.
In proving the density elimination for $GpsUL *$, we have to overcome several difficulties as follows. Firstly, cut-elimination doesn’t holds for $GpsUL *$. Note that $( W C M )$ and the density rule $( D )$ are formulated as
$G | Γ , Δ ⇒ t G | Δ , Γ ⇒ t , G | Π ⇒ p | Γ , p , Δ ⇒ B G | Γ , Π , Δ ⇒ B$
in $GpsUL *$, respectively. Consider the following derivation fragment.
By the induction hypothesis of the proof of cut-elimination, we get that $G 1 | G 2 | Γ 1 , Γ 2 , Δ 2 , Δ 1 ⇒ A$ from $G 2 | Γ 2 , Δ 2 ⇒ t$ and $G 1 | Γ 1 , t , Δ 1 ⇒ A$ by $( C U T )$. However, we can’t deduce $G 1 | G 2 | Γ 1 , Δ 2 , Γ 2 , Δ 1 ⇒ A$ from $G 1 | G 2 | Γ 1 , Γ 2 , Δ 2 , Δ 1 ⇒ A$ by $( W C M )$. We overcome this difficulty by introducing the following weakly cut rule into $GpsUL *$
$G 1 | Γ , t , Δ ⇒ A G 2 | Π ⇒ t G 1 | G 2 | Γ , Π , Δ ⇒ A ( W C T ) .$
Secondly, the proof of the density elimination for $GpsUL *$ becomes troublesome even for some simple cases in $GUL$ [4]. Consider the following derivation fragment
Here, the major problem is how to extend $( D )$ such that it is applicable to $G 2 | Γ 2 , Π 2 ′ , p , Π 2 ″ , Σ 2 ⇒ p$. By replacing p with t, we get $G 2 | Γ 2 , Π 2 ′ , t , Π 2 ″ , Σ 2 ⇒ t$. However, there exists no derivation of $G 1 | G 2 | Γ 1 , Π 2 ′ , Γ 2 , Π 1 , Σ 2 , Π 2 ″ , Σ 1 ⇒ A 1$ from $G 2 | Γ 2 , Π 2 ′ , Π 2 ″ , Σ 2 ⇒ t$ and $G 1 | Γ 1 , Π 1 , Σ 1 ⇒ A 1$. Notice that $Γ 2 , Π 2 ′$ and $Π 2 ″ , Σ 2$ in $G 2 | Γ 2 , Π 2 ′ , p , Π 2 ″ , Σ 2 ⇒ p$ are commutated simultaneously in $G 1 | G 2 | Γ 1 , Π 2 ′ , Γ 2 , Π 1 , Σ 2 , Π 2 ″ , Σ 1 ⇒ A 1$, which we can’t obtain by $( W C M )$. It seems that $( W C M )$ can’t be strengthened further in order to solve this difficulty. We overcome this difficulty by introducing a restricted subsystem $GpsUL Ω$ of $GpsUL *$. $GpsUL Ω$ is a generalization of $GIUL Ω$, which we introduced in [10] in order to solve a longstanding open problem, i.e., the standard completeness of $IUL$. Two new manipulations, which we call the derivation-splitting operation and derivation-splicing operation, are introduced in $GpsUL Ω$.
The third difficulty we encounter is that the conditions of applying the restricted external contraction rule $( E C Ω )$ become more complex in $GpsUL Ω$ because new derivation-splitting operations make the conclusion of the generalized density rule to be a set of hypersequents rather than one hypersequent. We continue to apply derivation-grafting operations in the separation algorithm of the multiple branches of $GIUL Ω$ in [10], but we have to introduce a new construction method for $GpsUL Ω$ by induction on the height of the complete set of maximal $( p E C )$-nodes rather than on the number of branches.
The structure of this paper is as follows. In Section 2, we present two hypersequent calculi $GpsUL *$ and $GpsUL Ω$, and prove that Cut-elimination does not hold for $GpsUL *$. Because of the absence of the commutativity rule, we have to introduce two novel operations, i.e., the derivation-splitting operation and derivation-splicing operation, in $GpsUL Ω$ in Section 3, and then we present a suitable definition of the generalized density rule $( D )$ for $GpsUL Ω$. In Section 4, we adapt the old main algorithm in the system $GIUL Ω$ to the new system $GpsUL Ω$. In Section 5, we propose two directions for future research.

## 2. $GpsUL$, $GpsUL *$ and $GpsUL Ω$

Definition 1.
([1]) $GpsUL$ consists of the following initial sequents and rules:
Initial sequents
$A ⇒ A ( I D ) ⇒ t ( t r ) Γ , ⊥ , Δ ⇒ A ( ⊥ l ) Γ ⇒ ⊤ ( ⊤ r ) ,$
Structural Rules
$G | Γ ⇒ A | Γ ⇒ A G | Γ ⇒ A ( E C ) G G | Γ ⇒ A ( E W ) ,$
$G 1 | Γ 1 , Π 1 , Δ 1 ⇒ A 1 G 2 | Γ 2 , Π 2 , Δ 2 ⇒ A 2 G 1 | G 2 | Γ 1 , Π 2 , Δ 1 ⇒ A 1 | Γ 2 , Π 1 , Δ 2 ⇒ A 2 ( C O M ) ,$
Logical Rules
$G 1 | Γ ⇒ A G 2 | Δ ⇒ B G 1 | G 2 | Γ , Δ ⇒ A ⊙ B ( ⊙ r ) G 1 | Γ , B , Δ ⇒ C G 2 | Π ⇒ A G 1 | G 2 | Γ , Π , A → B , Δ ⇒ C ( → l ) G 1 | Π ⇒ A G 2 | Γ , B , Δ ⇒ C G 1 | G 2 | Γ , A ⇝ B , Π , Δ ⇒ C ( ⇝ l ) G 1 | Γ , A , Δ ⇒ C G 2 | Γ , B , Δ ⇒ C G 1 | G 2 | Γ , A ∨ B , Δ ⇒ C ( ∨ l ) G 1 | Γ ⇒ A G 2 | Γ ⇒ B G 1 | G 2 | Γ ⇒ A ∧ B ( ∧ l ) G | Γ , A , Δ ⇒ C G | Γ , A ∧ B , Δ ⇒ C ( ∧ r r ) G | Γ , Δ ⇒ A G | Γ , t , Δ ⇒ A ( t l ) G | Γ , A , B , Δ ⇒ C G | Γ , A ⊙ B , Δ ⇒ C ( ⊙ l ) G | A , Γ ⇒ B G | Γ ⇒ A → B ( → r ) G | Γ , A ⇒ B G | Γ ⇒ A ⇝ B ( ⇝ r ) G | Γ ⇒ A G | Γ ⇒ A ∨ B ( ∨ r r ) G | Γ ⇒ B G | Γ ⇒ A ∨ B ( ∨ r l ) G | Γ , B , Δ ⇒ C G | Γ , A ∧ B , Δ ⇒ C ( ∧ r l ) .$
Cut Rule
$G 1 | Γ , A , Δ ⇒ B G 2 | Π ⇒ A G 1 | G 2 | Γ , Π , Δ ⇒ B ( C U T ) .$
Definition 2.
([3]) $GpsUL *$ is $GpsUL$ plus the weakly commutativity rule
$G | Γ , Δ ⇒ t G | Δ , Γ ⇒ t ( W C M ) .$
Definition 3.
$GpsUL * D$ is $GpsUL *$ plus the density rule $G | Π ⇒ p | Γ , p , Δ ⇒ B G | Γ , Π , Δ ⇒ B ( D )$.
Lemma 1.
$G ≡ B ∨ ( ( D → B ) ⊙ C ⊙ ( C → D ) ⊙ A → A )$ is not a theorem in $HpsUL$.
Proof.
Let $A = ( { 0 , 1 , 2 , 3 } , ∧ , ∨ , ⊙ , → , ⇝ , 2 , 0 , 3 )$ be an algebra, where $x ∧ y = min ( x , y )$, $x ∨ y = max ( x , y )$ for all $x , y ∈ { 0 , 1 , 2 , 3 }$, and the binary operations ⊙, → and ⇝ are defined by the following tables (see [2]).
By easy calculation, we get that $A$ is a linearly ordered $HpsUL$-algebra, where 0 and 3 are the least and the greatest element of $A$, respectively, and 2 is its unit. Let $v ( A ) = v ( B ) = v ( C ) = v ( D ) = 1$. Then, $v ( G ) = 1 ∨ ( 3 ⊙ 1 ⊙ 3 ⊙ 1 → 1 ) = 1 < 2$. Hence, G is not a tautology in $HpsUL$. Therefore, it is not a theorem in $HpsUL$ by Theorem 9.27 in [1]. □
Theorem 1.
Cut-elimination doesn’t hold for $GpsUL *$.
Proof.
$G ≡ ⇒ B ∨ ( ( D → B ) ⊙ C ⊙ ( C → D ) ⊙ A → A )$ is provable in $GpsUL *$, as shown in Figure 1.
Suppose that G has a cut-free proof $ρ$. Then, there exists no occurrence of t in $ρ$ by its subformula property. Thus, there exists no application of $( W C M )$ in $ρ$. Hence, G is a theorem of $GpsUL$, which contradicts Lemma 1. □
Remark 1.
Following the construction given in the proof of Theorem 53 in [4], $( C U T )$ in Figure 1 is eliminated by the following derivation, as shown in Figure 2. However, the application of $( W C M )$ in ρ is invalid, which illustrates the reason why the cut-elimination theorem doesn’t hold in $GpsUL *$.
Definition 4.
$GpsUL * *$ is constructed by replacing $( C U T )$ in $GpsUL *$ with
$G 1 | Γ , t , Δ ⇒ A G 2 | Π ⇒ t G 1 | G 2 | Γ , Π , Δ ⇒ A ( W C T ) .$
We call it the weakly cut rule and denote it by $( W C T )$.
Theorem 2.
If $⊢ GpsUL * G$, then $⊢ GpsUL * * G$.
Proof.
It is proved by a procedure similar to that of Theorem 53 in [4] and omitted. □
Definition 5.
( [10]) $GpsUL Ω$ is a restricted subsystem of $GpsUL *$ such that
(i) p is designated as the unique eigenvariable by which we mean that it is not used to build up any formula containing logical connectives and is only used as a sequent-formula.
(ii) Each occurrence of p in a hypersequent is assigned one unique identification number i in $GpsUL Ω$ and written as $p i$. Initial sequent $p ⇒ p$ of $GpsUL *$ has the form $p i ⇒ p i$ in $GpsUL Ω$. p doesn’t occur in $A , Γ$ or Δ for each initial sequent $Γ , ⊥ , Δ ⇒ A$ or $Γ ⇒ ⊤$ in $GpsUL Ω$.
(iii) Each sequent S of the form $Γ 0 , p , Γ 1 , ⋯ , Γ λ − 1 , p , Γ λ ⇒ A$ in $GpsUL *$ has the form $Γ 0 , p i 1 , Γ 1 , ⋯ , Γ λ − 1 , p i λ , Γ λ ⇒ A$ in $GpsUL Ω$, where p does not occur in $Γ k$ for all $0 ⩽ k ⩽ λ$ and, $i k ≠ i l$ for all $1 ⩽ k < l ⩽ λ$. Define $v l ( S ) = { i 1 , ⋯ , i λ }$, $v r ( S ) = { j 1 }$ if A is an eigenvariable with the identification number $j 1$ and, $v r ( S ) = ∅$ if A isn’t an eigenvariable.
Let G be a hypersequent of $GpsUL Ω$ in the form $S 1 | ⋯ | S n$ then $v l ( S k ) ⋂ v l ( S l ) = ∅$ and $v r ( S k ) ⋂ v r ( S l ) = ∅$ for all $1 ⩽ k < l ⩽ n$. Define $v l ( G ) = ⋃ k = 1 n v l ( S k )$, $v r ( G ) = ⋃ k = 1 n v r ( S k )$.
(iv) A hypersequent G of $GpsUL Ω$ is called closed if $v l ( G ) = v r ( G )$. Two hypersequents $G ′$ and $G ″$ of $GpsUL Ω$ are called disjoint if $v l ( G ′ ) ⋂ v l ( G ″ ) = ∅$, $v l ( G ′ ) ⋂ v r ( G ″ ) = ∅$, $v r ( G ′ ) ⋂ v l ( G ″ ) = ∅$ and $v r ( G ′ ) ⋂ v r ( G ″ ) = ∅$. $G ″$ is a copy of $G ′$ if they are disjoint and there exist two bijections $σ l : v l ( G ′ ) → v l ( G ″ )$ and $σ r : v r ( G ′ ) → v r ( G ″ )$ such that $G ″$ can be obtained by applying $σ l$ to antecedents of sequents in $G ′$ and $σ r$ to succedents of sequents in $G ′$.
(v) A hypersequent $G | G 1 | G 2$ can be contracted as $G | G 1$ in $GpsUL Ω$ under certain conditions given in Construction 3, which we called the constraint external contraction rule and denote by $G ′ | G 1 | G 2 G ′ | G 1 ( E C Ω )$.
(vi) $( E W )$ is forbidden in $GpsUL Ω$ and $( E C )$ and $( C U T )$ are replaced with $( E C Ω )$ and $( W C T )$, respectively.
(vii) Two rules $( ∧ r )$ and $( ∨ l )$ of $GL$ are replaced with $G 1 | Γ 1 ⇒ A G 2 | Γ 2 ⇒ B G 1 | G 2 | Γ 1 ⇒ A ∧ B | Γ 2 ⇒ A ∧ B ( ∧ r w )$ and $G 1 | Γ 1 , A , Δ 1 ⇒ C 1 G 2 | Γ 2 , B , Δ 2 ⇒ C 2 G 1 | G 2 | Γ 1 , A ∨ B , Δ 1 ⇒ C 1 | Γ 2 , A ∨ B , Δ 2 ⇒ C 2 ( ∨ l w )$ in $GpsUL Ω$, respectively.
(viii) $G 1 | S 1$ and $G 2 | S 2$ are closed and disjoint for each two-premise inference rule $G 1 | S 1 G 2 | S 2 G 1 | G 2 | H ′ ( I I )$ of $GpsUL Ω$ and, $G ′ | S ′$ is closed for each one-premise inference rule $G ′ | S ′ G ′ | S ″ ( I )$.
Proposition 1.
Let $G ′ | S ′ G ′ | S ″ ( I )$ and $G 1 | S 1 G 2 | S 2 G 1 | G 2 | H ′ ( I I )$ be inference rules of $GpsUL Ω .$ Then, $v l ( G ′ | S ″ ) = v r ( G ′ | S ″ ) = v r ( G ′ | S ′ ) = v l ( G ′ | S ′ )$ and $v l ( G 1 | G 2 | H ′ ) = v l ( G 1 | S 1 ) ⋃ v l ( G 2 | S 2 ) =$ $v r ( G 1 | G 2 | H ′ ) = v r ( G 1 | S 1 ) ⋃ v r ( G 2 | S 2 )$.
Proof.
Although $( W C T )$ makes t’s in its premises disappear in its conclusion; it has no effect on identification numbers of the eigenvariable p in a hypersequent because t is a constant in $GpsUL Ω$ and is distinguished from propositional variables. □
Definition 6.
Let G be a closed hypersequent of $GpsUL Ω$ and $S ∈ G$. $[ S ] G : = ⋂ { H : S ∈ H ⊆ G , v l ( H ) = v r ( H ) }$ is called a minimal closed unit of G.

## 3. The Generalized Density Rule $( D )$ for $GpsUL Ω$

In this section, $GL Ω cf$ is $G ps UL Ω$ without $( E C Ω )$. Generally, $A , B , C , ⋯$, denote a formula other than an eigenvariable $p i$.
Construction 1.Given a proof $τ *$ of $H ≡ G | Γ , p j , Δ ⇒ p j$ in $GL Ω cf$, let , where $H 0 ≡ p j ⇒ p j$, $H n ≡ H$. By $Γ k , p j , Δ k ⇒ p j$, we denote the sequent containing $p j$ in $H k$. Then, $Γ 0 = ∅$, $Δ 0 = ∅$, $Γ n = Γ$ and $Δ n = Δ$. Hypersequents , and their proofs , are constructed inductively for all $0 ⩽ k ⩽ n$ in the following such that , , and .
(i) , and are built up with $⇒ t$.
(ii) Let $G ′ | S ′ G ″ | S ″ G ′ | G ″ | H ′ ( I I )$ (or $G ′ | S ′ G ′ | S ″ ( I ) )$ be in $τ *$, $H k = G ′ | S ′$ and $H k + 1 = G ′ | G ″ | H ′$ (accordingly $H k + 1 = G ′ | S ″$ for $( I ) )$ for some $0 ⩽ k ⩽ n − 1$. There are three cases to be considered.
Case 1.
$S ′ = Γ k , p j , Δ k ⇒ p j$. If all focus formula(s) of $S ′$ is (are) contained in $Γ k$,
(accordingly for $( I )$) and, is constructed by combining the derivation and (accordingly for $( I ) )$ and, is constructed by combining and . The case of all focus formula(s) of $S ′$ contained in $Δ k$ is dealt with by a procedure dual to above and omitted.
Case 2.
. (accordingly for $( I ) )$, and is constructed by combining the derivation and (accordingly for $( I ) )$ and, is constructed by combining and .
Case 3.
. It is dealt with by a procedure dual to Case 2 and omitted.
Definition 7.
The manipulation described in Construction 1 is called the derivation-splitting operation when it is applied to a derivation and the splitting operation when applied to a hypersequent.
Corollary 1.
Let $⊢ GL Ω cf G | Γ , p 1 , Δ ⇒ p 1$. Then, there exist two hypersequents $G 1$ and $G 2$ such that $G = G 1 ⋃ G 2$, $G 1 ⋂ G 2 = ∅$, $⊢ GL Ω cf G 1 | Γ ⇒ t$ and $⊢ GL Ω cf G 2 | Δ ⇒ t$.
Construction 2.
Given a proof $τ *$ of $H ≡ G | Π ⇒ p j | Γ , p j , Δ ⇒ A$ in $GL Ω cf$, let $T h τ * ( p j ⇒ p j ) = ( H 0 , ⋯ , H n )$, where $H 0 ≡ p j ⇒ p j$ and $H n ≡ H$. Then, there exists $1 ⩽ m ⩽ n$ such that $H m$ is in the form $G ′ | Π ′ ⇒ p j | Γ ′ , p j , Δ ′ ⇒ A ′$ and $H m − 1$ is in the form $G ″ | Γ ″ , p j , Δ ″ ⇒ p j$. A proof of $G | Γ , Π , Δ ⇒ A$ in $GL Ω cf$ is constructed by induction on $n − m$ as follows:
• For the base step, let $n − m = 0$. Then, $H n − 1 ≡ G ′ | Π ′ , Γ ′ , p j , Δ ′ , Π ‴ ⇒ p j G ″ | Γ ″ , Π ″ , Δ ″ ⇒ A H n ≡ G ′ | G ″ | Π ′ , Π ″ , Π ‴ ⇒ p j | Γ ″ , Γ ′ , p j , Δ ′ , Δ ″ ⇒ A ( C O M ) ∈ τ *$, where $G ′ | G ″ = G$ and $Π ′ , Π ″ , Π ‴ = Π$ and $Γ ″ , Γ ′ = Γ$ and $Δ ′ , Δ ″ = Δ$. It follows from Corollary 1 that there exist $G 1 ′$ and $G 2 ′$ such that $G ′ = G 1 ′ ⋃ G 2 ′$, $G 1 ′ ⋂ G 2 ′ = ∅$, $⊢ GL Ω cf G 1 ′ | Π ′ , Γ ′ ⇒ t$ and $⊢ GL Ω cf G 2 ′ | Δ ′ , Π ‴ ⇒ t$. Then, $G | Γ , Π , Δ ⇒ A$ is proved as follows:
$G ″ | Γ ″ , Π ″ , Δ ″ ⇒ A G ″ | Γ ″ , t , Π ″ , Δ ″ ⇒ A ( t l ) G 1 ′ | Π ′ , Γ ′ ⇒ t G 1 ′ | Γ ′ , Π ′ ⇒ t ( W C M ) G ″ | G 1 ′ | Γ ″ , Γ ′ , Π ′ , Π ″ , Δ ″ ⇒ A ( W C T ) G ″ | G 1 ′ | Γ ″ , Γ ′ , Π ′ , Π ″ , t , Δ ″ ⇒ A ( t l ) G 2 ′ | Δ ′ , Π ‴ ⇒ t G 2 ′ | Π ‴ , Δ ′ ⇒ t ( W C M ) G ″ | G 1 ′ | G 2 ′ | Γ ″ , Γ ′ , Π ′ , Π ″ , Π ‴ , Δ ′ , Δ ″ ⇒ A ( W C T ) .$
• For the induction step, let $n − m > 0$. Then, it is treated using applications of the induction hypothesis to the premise followed by an application of the relevant rule. For example, let $H n − 1 = G ′ | Π ⇒ p j | Σ ′ , Γ ″ , p j , Δ ″ , Σ ‴ ⇒ A ′ G ″ | Γ ′ , Σ ″ , Δ ′ ⇒ A H n = G ′ | Π ⇒ p j | Σ ′ , Σ ″ , Σ ‴ ⇒ A ′ | G ″ | Γ ′ , Γ ″ , p j , Δ ″ , Δ ′ ⇒ A ( C O M ) ∈ τ *$, where $G ′ | G ″ | Σ ′ , Σ ″ , Σ ‴ ⇒ A ′ = G$ and $Γ ′ , Γ ″ = Γ$ and $Δ ″ , Δ ′ = Δ$. By the induction hypothesis, we obtain a derivation of $G | Γ , Π , Δ ⇒ A$:
$G ′ | Σ ′ , Γ ″ , Π , Δ ″ , Σ ‴ ⇒ A ′ G ″ | Γ ′ , Σ ″ , Δ ′ ⇒ A G ′ | Σ ′ , Σ ″ , Σ ‴ ⇒ A ′ | G ″ | Γ ′ , Γ ″ , Π , Δ ″ , Δ ′ ⇒ A ( C O M ) .$
Definition 8.
The manipulation described in Construction 2 is called the derivation-splicing operation when it is applied to a derivation and the splicing operation when applied to a hypersequent.
Corollary 2.
If $⊢ GL Ω cf G | Π ⇒ p j | Γ , p j , Δ ⇒ A$, then $⊢ GL Ω cf G | Γ , Π , Δ ⇒ A$.
Definition 9.
(i) Let $⊢ GL Ω cf H ≡ G | Γ , p j , Δ ⇒ p j$. Define $H j − = G 1 | Γ ⇒ t$, $H j + = G 2 | Δ ⇒ t$ and $D j ( H ) = { G 1 | Γ ⇒ t , G 2 | Δ ⇒ t }$, where $G 1$ and $G 2$ are determined by Corollary 1.
(ii) Let $⊢ GL Ω cf H ≡ G | Π ⇒ p j | Γ , p j , Δ ⇒ A$. Define $D j ( H ) = { G | Γ , Π , Δ ⇒ A } = H j$.
(iii) Let $⊢ GL Ω cf G$. $D j ( G ) = { G }$ if $p j$ does not occur in G.
(iv) Let $⊢ GL Ω cf G i$ for all $1 ⩽ i ⩽ n$. Define $D j ( { G 1 , ⋯ , G n } ) = D j ( G 1 ) ⋃ ⋯ ⋃ D j ( G n )$.
(v) Let $⊢ GL Ω cf G$ and $K = { 1 , ⋯ , n } ⊆ v ( G )$. Define $D K ( G ) = D n ( ⋯ D 2 ( D 1 ( G ) ) ⋯ )$. Especially, define $D ( G ) = D v l ( G ) ( G )$.
Theorem 3.
Let $⊢ GL Ω cf G$. Then, $⊢ GL Ω cf H$ for all $H ∈ D ( G )$.
Proof.
Immediately from Corollaries 1, 2 and Definition 9. □
Lemma 2.
Let $G ′$ be a minimal closed unit of $G | G ′$. Then, $G ′$ has the form $Γ ⇒ A | Γ i 2 ⇒ p i 2 | ⋯ | Γ i n ⇒ p i n$ if there exists one sequent $Γ ⇒ A ∈ G ′$ such that A is not an eigenvariable otherwise $G ′$ has the form $Γ i 1 ⇒ p i 1 | ⋯ | Γ i n ⇒ p i n$.
Proof.
Define $G 1 = Γ ⇒ A$ in Construction 5.2 in [10]. Then, $∅ = v r ( G 1 ) ⊆ v l ( G 1 )$. Suppose that $G k$ is constructed such that $v r ( G k ) ⊆ v l ( G k )$. If $v l ( G k ) = v r ( G k )$, the procedure terminates and $n : = k$; otherwise, $v l ( G k ) ∖ v r ( G k ) ≠ ∅$ and define $i k + 1$ to be an identification number in $v l ( G k ) ∖ v r ( G k )$. Then, there exists $Γ i k + 1 ⇒ p i k + 1 ∈ G ∖ G k$ by $v l ( G ) = v r ( G )$ and define $G k + 1 = G k | Γ i k + 1 ⇒ p i k + 1 .$ Thus, $v r ( G k + 1 ) = v r ( G k ) ⋃ { i k + 1 } ⊆ v l ( G k ) ⊆ v l ( G k + 1 )$. Hence, there exists a sequence $i 2 , ⋯ , i n$ of identification numberssuch that $v r ( G k ) ⊆ v l ( G k )$ for all $1 ⩽ k ⩽ n$, where $G 1 = Γ ⇒ A$, $G k = Γ ⇒ A | Γ i 2 ⇒ p i 2 | ⋯ | Γ i k ⇒ p i k$ for all $2 ⩽ k ⩽ n$. Therefore, $G ′$ has the form $Γ ⇒ A | Γ i 2 ⇒ p i 2 | ⋯ | Γ i n ⇒ p i n$. □
Definition 10.
Let $G ′$ be a minimal closed unit of $G | G ′$. $G ′$ is a splicing unit if it has the form $Γ ⇒ A | Γ i 2 ⇒ p i 2 | ⋯ | Γ i n ⇒ p i n$. $G ′$ is a splitting unit if it has the form $Γ i 1 ⇒ p i 1 | ⋯ | Γ i n ⇒ p i n$.
Lemma 3.
Let $G ′$ be a splicing unit of $G | G ′$ in the form $Γ ⇒ A | Γ i 2 ⇒ p i 2 | ⋯ | Γ i n ⇒ p i n$ and $K = { i 2 , ⋯ , i n }$. Then, .
Proof.
By the construction in the proof of Lemma 2, $i k ∈ v l ( G k − 1 )$ for all $2 ⩽ k ⩽ n$. Then, $p i 2 ∈ Γ$ and $D i 2 ( G | G ′ ) = G | Γ [ Γ i 2 ] ⇒ A | Γ i 3 ⇒ p i 3 | ⋯ | Γ i n ⇒ p i n$, where $Γ [ Γ i 2 ]$ is obtained by replacing $p i 2$ in $Γ$ with $Γ i 2$. Then, $p i 3 ∈ Γ [ Γ i 2 ]$. Repeatedly, we get $D i 2 ⋯ i n ( G | G ′ ) = D K ( G | G ′ ) = G | Γ [ Γ i 2 ] ⋯ [ Γ i n ] ⇒ A$. □
This shows that $D K ( G | G ′ )$ is constructed by repeatedly applying splicing operations.
Definition 11.
Let $G ′$ be a minimal closed unit of $G | G ′$. Define $V G ′ = v ( G ′ )$, $E G ′ = { ( i , j ) | Γ , p i , Δ ⇒ p j ∈ G ′ }$ and, j is called the child node of i for all $( i , j ) ∈ E G ′$. We call $Ω G ′ = ( V G ′ , E G ′ )$ the Ω-graph of $G ′$.
Let $G ′$ be a splitting unit of $G | G ′$ in the form $Γ 1 ⇒ p 1 | ⋯ | Γ n ⇒ p n$. Then, each node of $Ω G ′$ has one and only one child node. Thus, there exists one cycle in $Ω G ′$ by . Assume that, without loss of generality, $( 1 , 2 ) , ( 2 , 3 ) , ⋯ , ( i , 1 )$ is the cycle of $Ω G ′$. Then, $p 1 ∈ Γ 2$, $p 2 ∈ Γ 3$, $⋯ , p i − 1 ∈ Γ i$ and $p i ∈ Γ 1$. Thus, $D i ⋯ 2 ( G | G ′ ) = G | Γ 1 [ Γ i ] [ Γ i − 1 ] ⋯ [ Γ 2 ] ⇒ p 1$ is in the form $G | Γ ′ , p 1 , Δ ′ ⇒ p 1$. By a suitable permutation $σ$ of $i + 1 , ⋯ , n$, we get $D i ⋯ 2 σ ( i + 1 ⋯ n ) ( G | G ′ ) = G | Γ 1 [ Γ i ] [ Γ i − 1 ] ⋯ [ Γ 2 ] [ Γ σ ( i + 1 ) ] ⋯ [ Γ σ ( n ) ] ⇒ p 1 = G | Γ , p 1 ,$ $Δ ⇒ p 1$. This process also shows that there exists only one cycle in $Ω G ′$. Then, we introduce the following definition.
Definition 12.
(i) $Γ j ⇒ p j$ is called a splitting sequent of $G ′$ and $p j$ its corresponding splitting variable for all $1 ⩽ j ⩽ i$.
(ii) Let $K = { 1 , 2 , ⋯ , n }$ and $D 1 ( G | Γ , p 1 , Δ ⇒ p 1 ) = { G 1 | Γ ⇒ t , G 2 | Δ ⇒ t }$. Define , and .
Lemma 4.
If $G ′$ be a splitting unit of $G | G ′$, $K = v ( G ′ )$ and k be a splitting variable of $G ′$. Then, $D K ∖ { k } ( G | G ′ )$ is constructed by repeatedly applying splicing operations and only the last operation $D k$ is a splitting operation.
Construction 3
(The constrained external contraction rule).Let , and be two copies of a minimal closed unit $S H$, where we put two copies into ${ } 1$ and ${ } 2$ in order to distinguish them. For any splitting unit , or , where . Then, is constructed by cutting off and some sequents in $G ′$ as follows.
(i) If and are two splicing units, then $G ″ : = G ′$;
(ii) If and are two splitting units and, k, $k ′$ their splitting variables, respectively, , , , , $D K ⋃ K ′ ( H ) = { G 1 ′ | Γ ⇒ t | Γ ⇒ t , G 2 ′ | Δ ⇒ t , G 2 ″ | Δ ⇒ t }$ or $D K ⋃ K ′ ( H ) = { G 1 ′ | Δ ⇒ t | Δ ⇒ t , G 2 ′ | Γ ⇒ t , G 2 ″ | Γ ⇒ t }$, where $G 1 ′ ⋃ G 2 ′ ⋃ G 2 ″ = G ′$ and $G 2 ″$ is a copy of $G 2 ′$. Then, $G ″ : = G ′ ∖ G 2 ″$.
The above operation is called the constrained external contraction rule, denoted by $〈 E C Ω * 〉$ and written as .
Lemma 5.
If $⊢ GL Ω cf H$ as above, then $⊢ GpsUL Ω H ′$ for all .

## 4. Density Elimination for $GpsUL *$

In this section, we adapt the separation algorithm of branches in [10] to $GpsUL *$ and prove the following theorem.
Theorem 4.
Density elimination holds for $GpsUL *$.
The proof of Theorem 4 runs as follows. It is sufficient to prove that the following strong density rule
is admissible in $GpsUL *$, where $n , m ⩾ 1$, p does not occur in $G ′ , Γ i , Δ i , A i , Π j$ for all $1 ⩽ i ⩽ n$, $1 ⩽ j ⩽ m$.
Let $τ$ be a proof of $G 0$ in $GpsUL * *$ by Theorem 2. Starting with $τ$, we construct a proof $τ *$ of $G | G *$ in $GL Ω cf$ by a preprocessing of $τ$ described in Section 4 in [10].
In Step 1 of preprocessing of $τ$, a proof $τ ′$ is constructed by replacing inductively all applications of $( ∧ r )$ and $( ∨ l )$ in $τ$ with $( ∧ r w )$ and $( ∨ l w )$ followed by an application of $( E C )$, respectively. In Step 2, a proof $τ ″$ is constructed by converting all $G i ‴ | { S i c } m i ′ G i ‴ | S i c ( E C * ) ∈ τ ′$ into $G i ″ | { S i c } m i ′ G i ″ | { S i c } m i ′ ( I D Ω )$, where $G i ‴ ⊆ G i ″$. In Step 3, a proof $τ ‴$ is constructed by converting $G ′ G ′ | S ′ ( E W ) ∈ τ ″$ into $G ″ G ″ ( I D Ω )$, where $G ″ ⊆ G ′$. In Step 4, a proof $τ ″ ″$ is constructed by replacing some $G ′ | Γ ′ , p , Δ ′ ⇒ A ′ ∈ τ ‴$ (or $G ′ | Γ ′ ⇒ p ∈ τ ‴$) with $G ′ | Γ ′ , ⊤ , Δ ′ ⇒ A ′$ (or $G ′ | Γ ′ ⇒ ⊥ )$. In Step 5, a proof $τ *$ is constructed by assigning the unique identification number to each occurrence of p in $τ ″ ″$. Let $H i c = G i ′ | { S i c } m i$ denote the unique node of $τ *$ such that $H i c ⩽ G i ″ | { S i c } m i$ and $S i c$ is the focus sequent of $H i c$ in $τ *$. We call $H i c$, $S i c$ the i-th $( p E C )$-node of $τ *$ and $( p E C )$-sequent, respectively. If we ignore the replacements from Step 4, each sequent of G is a copy of some sequent of $G 0$ and each sequent of $G *$ is a copy of some contraction sequent in $τ ′$.
Now, starting with $G | G *$ and its proof $τ *$, we construct a proof $τ ☆$ of $G ☆$ in $GpsUL Ω$ such that each sequent of $G ☆$ is a copy of some sequent of G. Then, $⊢ GpsUL Ω D ( G ☆ )$ by Theorem 3 and Lemma 5. Then, $⊢ GpsUL * D 0 ( G 0 )$ by Lemma 9.1 in [10].
In [10], $G ☆$ is constructed by eliminating $( p E C )$-sequents in $G | G *$ one by one. In order to control the process, we introduce the set $I = { H i 1 c , ⋯ , H i m c }$ of maximal $( p E C )$-nodes of $τ *$ (see Definition 13) and the set $I$ of the branches relative to I and construct $G I ☆$ such that $G I ☆$ doesn’t contain the contraction sequents lower than any node in I, i.e., $S j c ∈ G I ☆$ implies $H j c | | H i c$ for all $H i c ∈ I$. The procedure is called the separation algorithm of branches in [10].
The problem we encounter in $GpsUL Ω$ is that Lemma 7.11 of [10] doesn’t hold because new derivation-splitting operations make the conclusion of $( D )$-rule to be a set of hypersequents rather than one hypersequent. Then, $G ‡ m q ′$ generally can’t be contracted to $G ‡$ in Step 2 of Stage 1 in the main algorithm in [10] and ${ G I l ∖ r ☆ } m q ′$ can’t be contracted to $G I l ∖ r ☆$ in Step 2 of Stage 2. Furthermore, we sometimes can’t construct some branches to I in $GpsUL Ω$ before we construct $τ I ☆$. Therefore, we have to introduce a new induction strategy for $GpsUL Ω$ and don’t perform the induction on the number of branches. First, we give some primary definitions and lemmas.
Definition 13.
A $( p E C )$-node $H i c$ is maximal if no other $( p E C )$-node is higher than $H i c$. Define $I 0$ to be the set of maximal $( p E C )$-nodes in $τ *$. A nonempty subset I of $I 0$ is complete if I contains all maximal $( p E C )$-nodes higher than or equal to the intersection node $H I V$ of I. Define $H I V = H i c$ if $I = { H i c }$, i.e., the intersection node of a single node is itself.
Proposition 2.
(i) $H i c ∥ H j c$ for all $i ≠ j$, $H i c , H j c ∈ I 0$.
(ii) Let I be complete and $H j c ⩾ H I V$. Then, $H j c ⩽ H i c$ for some $H i c ∈ I$.
(iii) $I 0$ is complete and ${ H i c }$ is complete for all $H i c ∈ I 0$.
(iv) If $I ⊆ I 0$ is complete and $I > 1$, then $I l$ and $I r$ are complete, where $I l$ and $I r$ denote the sets of all maximal $( p E C )$-nodes in the left subtree and right subtree of $τ * ( H I V )$, respectively.
(v) If $I 1 , I 2 ⊆ I 0$ are complete, then $I 1 ⊆ I 2$, $I 2 ⊆ I 1$ or $I 1 ⋂ I 2 = ∅$.
Proof.
Only (v) is proved as follows. $I 1 ⊆ I 2$, $I 2 ⊆ I 1$ or $I 1 ⋂ I 2 = ∅$ holds by $H I 2 V ⩽ H I 1 V$, $H I 1 V ⩽ H I 2 V$ or $H I 2 V ∥ H I 1 V$, respectively. □
Definition 14.
A labeled binary tree ρ is constructed inductively by the following operations:
(i) The root of ρ is labeled by $I 0$ and leaves labeled ${ H i c } ⊆ I 0$.
(ii) If an inner node is labeled by I, then its parent nodes are labeled by $I l$ and $I r$, where $I l$ and $I r$ are defined in Proposition 2(iv).
Definition 15.
We define the height $o ( I )$ of $I ∈ ρ$ by letting $o ( I ) = 1$ for each leave $I ∈ ρ$ and, $o ( I ) = max { o ( I l ) , o ( I r ) } + 1$ for any non-leaf node.
Note that in Lemma 7.11 in [10] only uniqueness of $G H 1 : G 2 ☆ ( J ) | S 2 ^$ in $G H i k c ☆$ doesn’t hold in $GpsUL Ω$ and the following lemma holds in $GpsUL Ω$.
Lemma 6.
Let $G 1 | S 1 G 2 | S 2 H 1 ≡ G 1 | G 2 | H ″ ( I I ) ∈ τ *$, $τ G b | S j c * ∈ τ H i c ☆$, . Then, $H ″$ is separable in $τ H i c ☆ ( J )$ and there are some copies of $G H 1 : G 2 ☆ ( J ) | S 2 ^$ in $G H i c ☆$.
Lemma 7.
(New main algorithm for GpsULΩ)Let I be a complete subset of $I 0$ and $I ¯ = { H i c : H i c ⩽ H j c f o r s o m e H j c ∈ I }$. Then, there exists one close hypersequent $G I ☆ ⊆ c G | G *$ and its derivation $τ I ☆$ in $GpsUL Ω$ such that
(i) $τ I ☆$ is constructed by initial hypersequent , the fully constraint contraction rules of the form and elimination rule of the form
where $1 ⩽ w ⩽ | I | , H j k c ↭ H j l c$ for all $1 ⩽ k < l ⩽ w$, , $I j = { S j 1 c , S j 2 c , ⋯ , S j w c }$, $I j = { G b 1 | S j 1 c , G b 2 | S j 2 c , ⋯ , G b w | S j w c }$, $G b k | S j k c$ is closed for all $1 ⩽ k ⩽ w$. Then, $H i c ¬ ⩽ H j c$ for each $S j c ∈ G I j *$ and $H i c ∈ I$.
(ii) For all $H ∈ τ ¯ I ☆$, let
where $τ ¯ I ☆$ is the skeleton of $τ I ☆$, which is defined by Definition 7.13 [10]. Then, for some $1 ⩽ k ⩽ w$ in $τ I j *$;
(iii) Letting $H ∈ τ ¯ I ☆$ and $G | G * < ∂ τ I ☆ H ⩽ H I V$, then $G H I V : H ☆ ( J ) ∈ τ I ☆$ and it is built up by applying the separation algorithm along $H I V$ to H, and is an upper hypersequent of either if it is applicable, or , otherwise.
(iv) $S j c ∈ G I ☆$ implies $H j c ∥ H i c$ for all $H i c ∈ I$ and, $S j c ∈ G I j *$ for some $τ I j * ∈ τ I ☆$.
Proof.
$τ I ☆$ is constructed by induction on $o ( I )$. For the base case, let $o ( I ) = 1$; then, $τ I ☆$ is built up by Construction 7.3 and 7.7 in [10]. For the induction case, suppose that $o ( I ) ⩾ 2$, $τ I l ☆$ and $τ I r ☆$ are constructed such that Claims from (i) to (iv) hold.
Let $G ′ | S ′ G ″ | S ″ G ′ | G ″ | H ′ ( I I ) ∈ τ *$, where $G ′ | G ″ | H ′ = H I V$. Then, $I l$ and $I r$ occur in the left subtree $τ * ( G ′ | S ′ )$ and right subtree $τ * ( G ″ | S ″ )$ of $τ * ( H I V )$, respectively. Here, almost all manipulations of the new main algorithm are the same as those of the old main algorithm. There are some caveats that need to be considered.
Firstly, all leaves are replaced with $τ I r ☆$ in Step 3 at Stage 1 in the old main algorithm and are replaced with $τ I l ☆$ in Step 3 at Stage 2. Secondly, we abandon the definitions of branch to I and Notation 8.1 in [10] and then the symbol $I$ of the set of branches, which occur in $τ I ☆$ in [10], is replaced with I in the new algorithm. We call the new algorithm the separation algorithm along I. We also replace $Ω$ in $τ I Ω$ with ☆. Thirdly, under the new requirement that I is complete, we prove the following property.
Property (A)$G I l ☆$ contains at most one copy of $G H I V : G ″ ☆ ( J ) | S ″ ^$.
Proof. Suppose that there exist two copies and of $G H I V : G ″ ☆ ( J ) | S ″ ^$ in $G I l ☆$, and we put them into ${ } 1$ and ${ } 2$ in order to distinguish them. Let $S G I l ☆$ be a splitting unit of $G I l ☆$ and S its splitting sequent. Then, . Thus, S is a $( p E C )$-sequent and has the form $S i c$ by $S G I l ☆ ⊆ c G | G *$. Then, $[ S ] G I l ☆ = [ S i c ] G I l ☆$, $H i c ∥ H j c$ for all $H j c ∈ I l$ and $S i c ∈ G I j l *$ for some $τ I j l * ∈ τ I l ☆$ by Claim (iv). Since $I l$ is complete and $G ′ | S ′ ⩽ H I l V$, then $H i c ∥ G ′ | S ′$.
Let $τ I j l *$ be in the form where $G 1 | S 1 ⩽ G ′ | S ′$, $G 2 | S 2 ⩽ H i c$, $G 1 | G 2 | H ″$ is the intersection node of $H i c$ and $G ′ | S ′$, as shown in Figure 3. Then, by $G 1 | S 1 ⩽ G ′ | S ′ ⩽ H I l V$ and $S i c ∈ G I j l *$. Since $S 2$ is separable in $G I l ☆$ by $G ′ | S ′ ⩽ H I l V$, then $S i c ∈ G 2 | S 2$ and $S i c$ is not $S 2$.
Before proceeding to prove Property (A), we present the following property of .
Property (B) The set of splitting sequents of is equal to that of .
Proof.
Let $G 1 ′ | S 1 ′ G 2 ′ | S 2 ′ H 1 ′ ≡ G 1 ′ | G 2 ′ | H ‴ ( I I ) ∈ τ *$, $G 1 ′ | S 1 ′ ⩽ H 1$ and . Then, $S 1 ′$ and $S 2 ′$ are separable in $G I l ☆$. Thus, $G H 1 ′ : G 2 ′ ☆ ( J ) | S 2 ′ ^ ⊆ G I l ☆$ is closed. Hence, $G H 1 : G 2 ☆ ( J ) | S 2 ^ − ⋃ G 2 ′ | S 2 ′ G H 1 ′ : G 2 ′ ☆ ( J ) | S 2 ′ ^$ is closed, where $G 2 ′ | S 2 ′$ in $⋃ G 2 ′ | S 2 ′$ runs over all $I I ∈ τ *$ above such that $G H 1 ′ : G 2 ′ ☆ ( J ) | S 2 ′ ^ ⊆ G H 1 : G 2 ☆ ( J ) | S 2 ^$. Therefore, $v ( G H 1 : G 2 ☆ ( J ) | S 2 ^ − ⋃ G 2 ′ | S 2 ′ G H 1 ′ : G 2 ′ ☆ ( J ) | S 2 ′ ^ ) = v ( G 2 | S 2 )$, ${ S j c : S j c ∈ G 2 | S 2 , H j c ⩾ G 2 | S 2 } = { S j c : S j c ∈ G H 1 : G 2 ☆ ( J ) | S 2 ^ − ⋃ G 2 ′ | S 2 ′ G H 1 ′ : G 2 ′ ( J ) | S 2 ′ ^ }$ and . Then, the set of splitting sequents of is equal to that of since each splitting sequent is a $( p E C )$-sequent by and $S ‴ ∈ c G | G *$. This completes the proof of Property (B). □
We therefore assume that, without loss of generality, $S i c$ is in the form $Γ , p k , Δ ⇒ p k$ by Property (B), Lemma 5 and the observation that each derivation-splicing operation is local. There are two cases to be considered in the following.
Case 1.
for all $τ G b | S j c * ∈ τ H I V : G ″ ☆ ( J )$, $G 1 | S 1 ⩽ H j c ⩽ H I V$. Then, $G H 1 : G 2 ☆ ( J ) ⋂ G H I V : G ″ ☆ ( J ) = ∅$. We assume that, without loss of generality, , . Then, since $S = Γ , p k , Δ ⇒ p k$ isn’t a focus sequent at all nodes from $G 2 | S 2$ to $G I l ☆$ in $τ I l ☆$ and, $H j c ⩽ H 1$ or $H j c | | G 1 | S 1$ for all $S j c ∈ G 2 ′$ by Lemma 6.7 in [10]. Thus, . Therefore, because $S G I l ☆ ⊆ G H 2 : G 2 ☆ ( J ) | S 2 ^$, and . This shows that any splitting unit $S G I l ☆$ outside $G H I V : G ″ ☆ ( J ) | S ″ ^$ in $G I l ☆$ doesn’t take two copies of $G H I V : G ″ ☆ ( J ) | S ″ ^$ apart, i.e., the case of and doesn’t happen.
Case 2.
for some $τ G b | S j c * ∈ τ H I V : G ″ ☆ ( J )$, $G 1 | S 1 ⩽ H j c ⩽ H I V$. Then, . Thus, $G H 1 : G 2 ☆ ( J ) | S 2 ^ ⊆ G H I V : G ″ ☆ ( J ) | S ″ ^$. Hence, . The case of $S i c ∈ G ″$ is tackled with the same procedure as the following. Let . Then, there exists a copy of $S G I l ☆$ in and let $Γ , p k ′ , Δ ⇒ p k ′$ be its splitting sequent. We put two splitting units into ${ } k$ and ${ } k ′$ in order to distinguish them. Then, and . We assume that, without loss of generality, , . Then, . Thus, by . Then, , , where we put two copies of $Δ ⇒ t$ into ${ } k$ and ${ } k ′$ in order to distinguish them. Then, , , and is a copy of . Then, could be cut off of one of them because they are the two same sets of hypersequents in $D ( G I l ☆ )$. Meanwhile, two copies of $Δ ⇒ t$ in can’t be taken apart by any splitting unit outside $G H I V : G ″ ☆ ( J ) | S ″ ^$ in $G I l ☆$ for the reason as shown in Case 1 and thus could be contracted into one by $( E C )$ in $D ( G I l ☆ )$. Therefore, two copies and of $G H I V : G ″ ☆ ( J ) | S ″ ^$ can be contracted into one in $G I l ☆$ by . This completes the proof of Property (A). □
With Property (A), all manipulations in the old main algorithm in [10] work well. This completes the construction of $τ I ☆$ and the proof of Theorem 4. □
Theorem 5.
The standard completeness holds for $HpsUL *$.
Proof.
Let $⟷ i$ denote the i-th logical link of iff in the following. $⊧ K A$ means that $v ( A ) ⩾ t$ for every algebra $A$ in $K$ and valuation v on $A$. Let $psUL *$, $LIN ( psUL * )$, $psUL * D$ and $[ 0 , 1 ] psUL *$ denote the classes of all $psUL *$-algebras, $psUL *$-chain, dense $psUL *$-chain and standard $psUL *$-algebras (i.e., their lattice reducts are $[ 0 , 1 ]$), respectively. We have an inference sequence, as shown in Figure 4.
Links from 1 to 4 show Jenei and Montagna’s algebraic method to prove standard completeness and, currently, it seems hopeless to build up link 3 (see [11,12,13,14]). Links from $1 ∘$ to $4 ∘$ show Metcalfe and Montagna’s proof-theoretical method. Density elimination is at Link $2 ∘$ in Figure 4 and other links are proved by standard procedures with minor revisions and omitted (see [1,4,15,16,17]). □

## 5. Future Works

Generally, for any existing fuzzy logic system, we can consider its corresponding non-commutative system, just as $HpsUL$ is obtained by removing the commutativity of the strong conjunctive connective ⊙ in $UL$. Therefore, we can consider the corresponding non-commutative systems of many systems. A natural question is whether the method of the density elimination proposed in this paper can be generalized to these systems. It has often been the case in the past that metamathematical methods have corresponding algebraic analogues. The method proposed in this paper is essentially proof-theoretic. A natural problem is whether there is an algebraic proof corresponding to our proof-theoretic one.

## Funding

This research was funded by the National Foundation of Natural Sciences of China (Grant No: 61379018, 61662044, 11571013, and 11671358).

## Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. A proof $τ$ of G.
Figure 1. A proof $τ$ of G.
Figure 2. A possible cut-free proof ρ of G.
Figure 2. A possible cut-free proof ρ of G.
Figure 3. A fragment of $τ I l ☆$.
Figure 3. A fragment of $τ I l ☆$.
Figure 4. Two ways to prove standard completeness.
Figure 4. Two ways to prove standard completeness.

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Wang, S. The Logic of Pseudo-Uninorms and Their Residua. Symmetry 2019, 11, 368. https://doi.org/10.3390/sym11030368

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Wang S. The Logic of Pseudo-Uninorms and Their Residua. Symmetry. 2019; 11(3):368. https://doi.org/10.3390/sym11030368

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Wang, SanMin. 2019. "The Logic of Pseudo-Uninorms and Their Residua" Symmetry 11, no. 3: 368. https://doi.org/10.3390/sym11030368

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