diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzedpm" "b/data_all_eng_slimpj/shuffled/split2/finalzzedpm"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzedpm"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nFor many interpretations  of the modal operators -- e.g., for deontic, epistemic,  game-theoretic, and high-probability interpretations -- it is necessary to adopt logics that are weaker than the normal ones; e.g., deontic paradoxes, see \\cite{G13,M06},  are one of the main motivations for adopting a non-normal  deontic logic. Non-normal  logics, see \\cite{C80} for naming conventions,  are quite well understood from a semantic point of view  by means of neighbourhood semantics  \\cite{H09,P17}.\n  Nevertheless, until recent years their proof theory has been rather limited  since it was mostly confined to Hilbert-style axiomatic systems. This situation seems to be rather unsatisfactory since it is difficult to find derivations in axiomatic systems. When the aim is to find derivations and to analyse their structural properties, sequent calculi are to be preferred to  axiomatic systems. Recently  different kinds of sequent calculi for non-normal logics have been proposed: Gentzen-style calculi  \\cite{I05,I11,L00,O15}; labelled \\cite{GM14,CO18} and display \\cite{P19} calculi based on  translations into normal modal logics; labelled calculi based on the internalisation of neighbourhood  \\cite{N17,NO19} and  bi-neighbourhood  \\cite{DON} semantics;   and, finally, \\mbox{linear nested sequents  \\cite{L17}.}\n \n  \n This paper, which extends the results  presented in \\cite{O15},  concentrates on Gentzen-style calculi since they are better suited than labelled calculi, display calculi, and  nested sequents to give  decision procedures (computationally well-behaved) and constructive proofs of interpolation theorems. We consider cut- and contraction-free G3-style sequent calculi for all the logics in the cube of non-normal modalities and for their  extensions with the deontic axioms $D^\\Diamond:=\\Box A\\supset\\Diamond A$ and $D^\\bot:=\\neg\\Box\\bot$.  The calculi we present have the subformula property and  allow for a straightforward decision procedure by a terminating loop-free proof search. Moreover, with the  exception of the calculi for {\\bf EC(N)} and its deontic extensions, they are \\emph{standard} \\cite{G16} -- i.e., each operator is handled by a finite number of rules with a finite number of premisses -- and they admit of a Maehara-style constructive proof of Craig's interpolation  theorem.\n \n This work improves on previous ones  on Gentzen-style calculi for non-normal logics  in that we  prove cut admissibility   for non-normal modal and deontic logics, and not only for the modal ones  \\cite{L00,I05,I11}. Moreover, we prove height-preserving admissibility of weakening and contraction, whereas neither weakening nor contraction is admissible in \\cite{L00,I05} and weakening but not contraction is admissible in \\cite{I11}. The admissibility of contraction is a major improvement since, as it is well known, contraction can be as bad as cut for  proof search: we may continue to duplicate some formula forever  and, therefore,  we need a (computationally expensive) loop-checker to ensure termination.  Proof search  procedures based on contraction-free calculi terminate because the height of derivations is bounded by a number depending on the complexity of the end-sequent and, therefore, we avoid the need of  loop-checkers.  To illustrate, the introduction  of contraction-free calculi has allowed to give computationally optimal decision procedures for propositional intuitionistic logic  ({$\\mathbf{IL_p}$) \\cite{H93} and for the normal modal  logics {\\bf K} and {\\bf T} \\cite{B97,H95}.   The existence of a loop-free terminating decision procedure has also allowed  to give a constructive proof of uniform interpolation for $\\mathbf{IL_p}$ \\cite{P92} as well as for  {\\bf K} and {\\bf T} \\cite{B07}.  The cut- and contraction-free calculi for non-normal logics considered here are such that the height of each derivation is bounded by the weight of its end-sequent and, therefore, we easily obtain a polynomial space upper complexity bound for proof search. This upper bound  is  optimal for the  logics  having $C$  as theorem (the satisfiability problem for non-normal modal logics without $C$   is in  {\\sc NP}, see \\cite{V89}).\n \n  Moreover,  the introduction of well-behaved calculi for non-normal deontic logics is interesting since  proof analysis can be applied to  the deontic paradoxes \\cite{M06} that are one of the central topics of deontic reasoning. We illustrate this in Section  \\ref{forrester} by considering Forrester's Paradox \\cite{F84} and by showing that proof analysis cast  doubts on the widespread opinion \\cite{M06,P17,T97} that Forrester's argument provides evidence against rule $RM$ (see Table \\ref{rulesinf}).\nIf   Forrester's argument  is formalized as in \\cite{M06} then  it does not compel us to adopt a deontic logic weaker than {\\bf KD}. If, instead, it is formalised as in \\cite{T97} then it forces the adoption of a logic where $RM$ fails, but the formal derivation  differs substantially from Forrester's informal argument. \n \n It is also given a constructive proof of interpolation for all logics having a standard calculus. To our knowledge in the literature there is no other constructive  study of interpolation in non-normal  logics.  In \\cite[Chap(s). 3.8 and 6.6]{F83} a constructive proof of Craig's (and Lyndon's)  interpolation theorem is given for the modal logics {\\bf K} and {\\bf R}, and for some of their extensions, including the deontic ones, but the proof makes use of model-theoretic notions. A proof of interpolation by the  Maehara-technique for {\\bf KD} is given in \\cite{V93}. For a thorough study of  interpolation in modal logics we refer the reader to \\cite{G05}. A model-theoretic proof of interpolation for {\\bf E} is given in \\cite{H09}, and a coalgebraic proof  of (uniform) interpolation for all the logics considered here, as well as all other rank-1 modal logics (see below), is given in \\cite{P13}. As it is explained in Example \\ref{prob}, we have not been able to prove interpolation for calculi containing the non-standard rule $LR$-$C$ (see Table \\ref{Modal rules}) and, as far as we know, it is still an open problem whether it is possible to give a constructive proof of interpolation for these logics.\n \n \\paragraph{Related Work.}\\label{related}\n The modal rules of inference  presented in Table \\ref{Modal rules} are obtained from the rules presented in \\cite{L00} by adding weakening contexts to the conclusion of the rules. This minor modification, used also in \\cite{I11,P13,PS10} for several modal rules, allows us to shift from set-based sequents to multiset-based ones  and to prove not only that cut is admissible, as it is done in \\cite{I05,I11,L00}, but also that weakening and contraction are height-preserving admissible. Given that implicit contraction is not eliminable from set-based sequents, the decision procedure for non-normal logics given in \\cite{L00} is based on  a model-theoretic inversion technique so that it is possible to define a  procedure that  outputs a derivation for all valid sequents and a finite   countermodel  for all invalid ones. One weakness of this decision  procedure is that  it does not respect the subformula property for logics without rule $RM$ (the procedure  adds instances of the excluded middle).\n \n\n \nThe paper \\cite{I05} considers multiset-based calculi for the non-normal logic  {\\bf M(N)} and for its extensions with axioms $D^\\Diamond,T, 4, 5$, and $B$. Nevertheless,  neither weakening nor contraction is eliminable  because there are no weakening contexts in the conclusion of the modal rules. In \\cite{I11} multiset-based sequent calculi for the non-normal  logic {\\bf E(N)} and  for its extensions with  axioms $D^\\Diamond,T$, 4, 5, and $B$ are given. The rules $LR$-$E$ and $R$-$N$ are  as in Table \\ref{Modal rules}, but the deontic axiom  $D^\\Diamond$ is expressed  by the following  rule:\n\n$$\n\\infer[\\infrule D\\text{-}2]{\\Box A,\\Box B,\\Gamma\\Rightarrow\\Delta}{A,B\\Rightarrow&(\\Rightarrow A,B)}\n$$\nwhere the right premiss  is present when we are working over $LR$-$E$ and  it has to be omitted when we work over $LR$-$M$. In the calculi in \\cite{I05,I11} weakening and contraction are taken as primitive rules and not as admissible ones as in the present approach. Even if it is easy to show that weakening is eliminable from the calculi in \\cite{I11}, contraction cannot be eliminated because rule \\emph{D-2} has exactly two principal formulas and, therefore, it is not possible to permute contraction up with respect to instances of  rule \\emph{D-2} (see Theorem \\ref{contr}). \nThe presence of a non-eliminable rule of contraction makes the elimination of cut more problematic: in most cases we cannot  eliminate the cut directly, but we have to consider the rule known as multicut \\cite[p. 88]{NP01}.  Moreover, cut is not eliminable from the calculus given in \\cite{I11} for the deontic  logic {\\bf END}. The formula  $D^\\bot:= \\neg\\Box\\bot$ is a theorem of this logic, but it can be derived only with a non-eliminable instance of  cut  as in:\n\n$$\n\\infer[\\infrule R\\neg]{\\Rightarrow \\neg\\Box\\bot}{\n\\infer[\\infrule Cut]{\\Box\\bot\\Rightarrow}{\n\\infer[\\infrule R\\mbox{-}N]{\\Rightarrow\\Box\\top}{\\Rightarrow\\top}&\n\\infer[\\infrule D\\mbox{-}2]{\\Box\\top,\\Box\\bot\\Rightarrow}{\\bot,\\top\\Rightarrow&\\Rightarrow\\bot,\\top}}}\n$$ \n\nFinally, it is worth noticing that all the non-normal logics we consider here are  \\emph{rank-1} logics in the sense of \\cite{P13,PS10,PS09} -- i.e., logics whose modal axioms are propositional combinations of formulas of shape $\\Box\\phi$, where  $\\phi$ is purely propositional -- and the calculi we give for the  modal logics {\\bf E}, {\\bf M}, {\\bf K} and {\\bf KD} are explicitly considered in \\cite{P13,PS09}.  Thus, they are part of the family of modal coalgebraic logics  \\cite{P13,PS10,PS09} and most of the results in this paper can be seen as  instances of general results that hold for rank-1 (coalgebraic) logics. If, in particular, we consider cut-elimination for coalgebraic logics \\cite{PS10} then all our calculi  absorb congruence  and Theorem \\ref{contr} and case 3 of Theorem \\ref{cut} show that they absorb  contraction and cut. Hence, \\cite[Thm. 5.7]{PS10} entails  that cut and contraction are   admissible in these calculi; moreover, \\cite[Props. 5.8 and 5.11]{PS10} entail that they are one-step cut free complete w.r.t. coalgebraic semantics. This latter result gives a semantic proof of cut admissibility  in the calculi considered here. Analogously, if we consider decidability, the polynomial space upper bound we find in Section \\ref{decision} coincides with that found in \\cite{PS09} for rank-1 modal logics. \n\n\n\\paragraph{Synopsis. }\nSection \\ref{secaxiom} summarizes the basic notions of axiomatic systems and of neighbourhood semantics for  non-normal  logics.  Section \\ref{seccalculi} presents G3-style  sequent calculi for these logics and then shows that weakening and contraction are height-preserving admissible and that cut is (syntactically) admissible.  Section \\ref{secdecax} describes a  terminating proof-search decision procedure for all calculi, it shows that each calculus is equivalent to the  corresponding axiomatic system, and  it applies proof search to Forrester's paradox.  Finally, Section \\ref{secinterpol} gives a Maehara-style    constructive proof of Craig's interpolation theorem for the logics having a standard calculus.\n\\section{Non-normal Logics}\\label{secaxiom}\n\\subsection{Axiomatic Systems}\nWe introduce, following \\cite{C80},  the basic notions  of non-normal  logics. Given a countable set of propositional variables $\\{p_n\\,|\\,n\\in \\mathbb{N}\\}$, the formulas of the modal language $\\mathcal{L}$  are generated by:\n\n$$\nA::= \\;p_n\\;|\\;\\bot\\;|\\;A\\wedge A\\;|\\;A\\lor A\\;|\\;A\\supset A\\;|\\;\\Box A\n$$\nWe remark that $\\bot$ is a 0-ary logical symbol. This will be extremely important in the proof of Craig's interpolation theorem. As usual $\\neg A$ is a shorthand for $A\\supset\\bot$, $\\top$ for $\\bot\\supset\\bot$, $A \\leftrightarrow B$  for $(A\\supset B)\\wedge(B\\supset A)$, and $\\Diamond A$ for $\\neg\\Box\\neg A$. We follow the usual conventions for parentheses. \n\n\n\n\\begin{table}\n\\caption{ Rules of inference}\\label{rulesinf}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hline\\hline\\noalign{\\smallskip}\n\\infer[\\infrule RE]{\\Box A\\leftrightarrow\\Box B}{A\\leftrightarrow B}\n&$\\quad$&\n\\infer[\\infrule RM]{\\Box A\\supset\\Box B}{A\\supset B}\n\\\\\\noalign{\\smallskip\\smallskip\\smallskip}\n\\infer[\\infrule RR,\\; n\\geq 1]{(\\Box A_1\\wedge\\dots\\wedge\\Box A_n)\\supset\\Box B}{(A_1\\wedge\\dots\\wedge A_n)\\supset B}\n&$\\quad$&\n\\infer[\\infrule RK,\\; n\\geq 0]{(\\Box A_1\\wedge\\dots\\wedge\\Box A_n)\\supset\\Box B}{(A_1\\wedge\\dots\\wedge A_n)\\supset B}\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\caption{ Axioms}\\label{axioms}\n\\begin{center}\n\\begin{tabular}{cclccclcccl}\n\\hline\\hline\\noalign{\\smallskip}\n$M$)&$\\quad$&$\\Box (A\\wedge B)\\supset(\\Box A\\wedge\\Box B)$&${}$\\qquad{}\\qquad{}\\qquad&$C$)&$\\quad$&$(\\Box A\\wedge\\Box B)\\supset\\Box(A\\wedge B)$ \\\\\\noalign{\\smallskip\\smallskip\\smallskip}$N$)&$\\quad$&$\\Box \\top$&${}$\\qquad{}\\qquad{}\\qquad&\n$D^\\bot$)&$\\quad$&$\\neg\\Box\\bot$ \\\\\\noalign{\\smallskip\\smallskip\\smallskip}$D^\\Diamond$)&$\\quad$&$ \\Box A\\supset\\Diamond A$\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nLet {\\bf L}  be the logic containing  all $\\mathcal{L}$-instances of propositional tautologies as axioms, and   modus ponens ($MP$) as inference rule.  The minimal non-normal modal logic {\\bf E} is the logic {\\bf L} plus the rule $RE$ of Table \\ref{rulesinf}. We will consider all the logics that are obtained by extending {\\bf E} with some set of axioms from Table \\ref{axioms}. We will denote the logics according to the axioms that define them, e.g. {\\bf EC} is the logic  {\\bf E}$\\,\\oplus \\,C$, and {\\bf EMD$^\\bot$} is {\\bf E}$\\,\\oplus\\, M\\oplus D^\\bot$.  By {\\bf X} we denote any of these logics and we  write \\mbox{{\\bf X} $\\vdash A$} whenever $A$ is a theorem of {\\bf X}. We will call \\emph{modal} the logics containing neither $D^\\bot$ nor $D^\\Diamond$, and \\emph{deontic} those containing at least one of them. We have followed the usual naming conventions for the modal axioms, but we have introduced new  conventions for the deontic ones: $D^\\bot$ is usually called either  $CON$ or $P$ and $D^\\Diamond$ is usually called $D$, cf. \\cite{G00,G13,M06}.\n \n It is also possible to give an equivalent  rule-based axiomatization of some of these logics. In particular, the logic {\\bf EM}, also called {\\bf M}, can be axiomatixed as {\\bf L} plus the rule $RM$ of Table \\ref{rulesinf}.  The logic  {\\bf EMC}, also called {\\bf R}, can be axiomatized as {\\bf L} plus the rule  $RR$ of Table \\ref{rulesinf}. Finally,  the logic {\\bf EMCN}, i.e. the smallest normal modal logic {\\bf K},  can be axiomatized as {\\bf L} plus the rule $RK$ of Table \\ref{rulesinf}. These rule-based axiomatizations will be useful later on since they simplify the proof of the equivalence between  axiomatic systems and  sequent calculi (Theorem \\ref{comp}).\n\n\n\n\n\n\n\nThe following proposition states the well-known relations between the theorems of non-normal modal  logics. For a proof  the reader is referred to \\cite{C80}.\n\n\n\\begin{proposition} For any formula $A\\in \\mathcal{L}$ we have that {\\bf E} $\\vdash A$ implies {\\bf M}  $\\vdash A$; {\\bf M} $\\vdash A$   implies {\\bf R} $\\vdash A$; {\\bf R} $\\vdash A$  implies {\\bf K} $\\vdash A$. Analogously for the logics containing axiom $N$ and\/or axiom $C$.\\end{proposition}\n\n\nAxiom $D^\\bot$ is {\\bf K}-equivalent to $D^\\Diamond$, but the correctness of $D^\\Diamond$ has been a big issue in the literature on deontic logic. This fact urges the study of  logics weaker than {\\bf KD}, where $D^\\bot$ and $D^\\Diamond$ are no more equivalent \\cite{C80}. \nThe deontic  formulas $D^\\bot$ and $D^\\Diamond$ have the following relations in the logics we are considering.\n\\begin{proposition} $D^\\bot$ and $D^\\Diamond$ are  independent  in {\\bf E}; $D^\\bot$ is derivable from $D^\\Diamond$ in  non-normal logics containing at least one of the axioms  $M$  and $N$; $D^\\Diamond$ is derivable from $D^\\bot$ in non-normal logics containing  axiom $C$.\n\\end{proposition}\nIn Figure  \\ref{cube} the reader finds the lattice of non-normal modal logics, see \\cite[p. 237]{C80}, and in Figure \\ref{cubed}  the lattice of non-normal deontic logics. \n \\begin{figure}[t\n\\begin{center}\n\\begin{tikzpicture}\n\\node at (1,-2)  {EM={\\bf M}};\n\\node at (4,-4)  {{\\bf E}};\n\\node at (4,-2)  {{\\bf EC}};\n\\node at (7,-2)  {{\\bf EN}};\n\\node at (1,0)  {EMC={\\bf R}};\n\\node at (4,2)  {EMCN={\\bf K}};\n\\node at (4,0)  {{\\bf EMN}};\n\\node at (7,0)  {{\\bf ECN}};\n\n\\draw (3.8,-3.8) -- (1,-2.2);\n\\draw (4,-3.8) -- (4,-2.2); \n\\draw (4.2,-3.8) -- (7,-2.2);\n\n\\draw (1,-1.8) -- (1,-0.2);\n\\draw (4,0.2) -- (4,1.8); \n\\draw (7,-1.8) -- (7,-0.2);\n\n\\draw (1.2,0.2) -- (3.8,1.8);\n\\draw (6.8,0.2) -- (4.2,1.8);\n\\draw (1.2,-1.8) -- (3.8,-0.2);\n\\draw (6.8,-1.8) -- (4.2,-0.2);\n\n\\draw (3.8,-1.8) -- (1.2,-0.2);\n\\draw (4.2,-1.8) -- (6.8,-0.2);\n\\end{tikzpicture}\n\\caption{Lattice of non-normal modal logics}\\label{cube}\n\\end{center}\n\\end{figure}\n\n \\begin{figure}[t]\n\\begin{center}\n\\scalebox{0.80000}{\\begin{tikzpicture}\n\\filldraw  [black] (5,-4.2) circle (4pt) \n\t\t     (6.7,-4.2) circle (4pt)\n\t\t     (5,-3) circle (4pt);\n\\node[below] at (4.8,-4.3) {\\small{\\bf ED$^\\bot$}};\n\\node[below] at (6.7,-4.3)  {\\small{\\bf ED$^\\Diamond$}};\n\\node[left] at (4.9,-3) {\\small{ ED$^\\bot$D$^\\Diamond$}=};\n\\node[left] at (4.9,-3.3) {\\small{\\bf ED}};\n\\draw (5,-4.2) -- (1,-1.7)\n\\draw (5,-3) -- (1,-0.5)\n\\draw (5,-4.2) -- (5,-3)\n\\draw (6.7,-4.2) -- (5,-3)\n\\draw  (5,-3) -- (5,-0.5)\n\\draw (6.7,-4.2) -- (6.7,-1.6)\n\\draw (5,-4.2) -- (9,-1.7)\n\\draw (5,-3) -- (9,-0.5)\n\n\n\n\\filldraw  [black] (6.7,-1.5) circle (4pt) \n\t\t     (5,-0.5) circle (4pt);\n\\node[left] at (6.5,-1.5)  {\\small{\\bf ECD$^\\Diamond$}};\n\\node[left] at (4.9,-0.5)  {\\small{ ECD$^\\bot$=}};\n\\node[left] at (4.9,-0.8)  {\\small{ \\bf ECD}};\n\\draw (5,-0.5) -- (9,2)\n\\draw (6.7,-1.5)-- (5,-0.5)\n\\draw (5,-0.5) -- (1,2)\n\n\n\\filldraw  [black] (9,-1.7) circle (4pt) \n\t\t   \n\t\t     (9,-0.5) circle (4pt);\n\\node[right] at (9.2,-1.7)  {\\small{\\bf END$^\\bot$}};\n\\node[right] at (9.1,-0.5)  {\\small{ END$^\\Diamond$}= {\\bf END}};\n\\draw (9,-0.5) -- (9,2)\n\\draw (9,-1.7) -- (9,-0.5)\n\\draw (9,-1.7) -- (5,0.8)\n\\draw (9,-0.5) -- (5,2)\n\n\n\\filldraw  [black] (1,2) circle (4pt)  \n\\node[left] at (0.9,2)  {\\small{{ RD$^\\bot$}= {RD$^\\Diamond$}= {\\bf RD}}};\n\\draw (1,2) -- (5,4.5);\n\n\\filldraw  [black] (5,4.5) circle (4pt)  \n\\node[above] at (5,4.6)  {\\small{{KD$^\\bot$}= {KD$^\\Diamond$}= {\\bf KD}}};\n\n\\filldraw  [black] (1,-1.7) circle (4pt) \n\t\t     (1,-0.5) circle (4pt);\n\\node[left] at (0.9,-1.7)  {\\small{\\bf MD$^\\bot$}};\n\\node[left] at (0.9,-0.5) {\\small{ MD$^\\Diamond$}= {\\bf MD}};\n\\draw (1,-1.7) -- (5,0.8)\n\\draw (1,-0.5) -- (5,2)\n\\draw (1,-1.7) -- (1,-0.5)\n\t\t     (5,2) circle (4pt);\n\\draw (1,-0.5) -- (1,2)\n\n\\filldraw  [black] (5,0.8) circle (4pt) \n\t\t    (5,2) circle (4pt); \n\\node[right] at (5.2,0.8)  {\\small{\\bf MND$^\\bot$}};\n\\node[right] at (5.1,2)  {\\small{ MND$^\\Diamond$}= {\\bf MND}};\n\\draw (5,0.8) -- (5,2);\n\\draw (5,2) -- (5,4.5);\n\n\\filldraw  [black]\n\t\t     (9,2) circle (4pt);\n\\node[right] at (9.1,2)  {\\small{ECND$^\\bot$} = {ECND$^\\Diamond$}= {\\bf ECND}};\n\\draw (9,2) -- (5,4.5)\n\\end{tikzpicture}}\n\\caption{Lattice of non-normal deontic logics}\\label{cubed}\n\\end{center}\n\\end{figure\n\n\n\n\\subsection{Semantics}\\label{semantics}\nThe most widely known semantics for non-normal logics is neighbourhood semantics. We sketch its main tenets following \\cite{C80}, where neighbourhood models are called \\emph{minimal models}. \n\n\\begin{definition}A \\emph{neighbourhood model} is a triple $\\mathcal{M}:=\\langle W,\\, N,\\,P\\rangle$, where $W$ is a non-empty set of possible worlds; $N:W\\longrightarrow 2^{2^W}$ is a neighbourhood function that associates to each possible world $w$ a set $N(w)$ of subsets   of $W$; and $P$ gives a truth value to each propositional variable at each world. \\end{definition}\n\n\\noindent The definition of truth of a formula $A$ at a world $w$ of a neighbourhood model $\\mathcal{M}$ -- $\\models_w^\\mathcal{M}A$ -- is the standard one for the classical connectives with the addition of\n\n$$\n\\models_w^\\mathcal{M}\\Box A\\qquad \\textnormal{iff}\\qquad || A||^\\mathcal{M}\\in N(w)\n$$\nwhere $||A||^\\mathcal{M}$ is the truth set of $A$ -- i.e., $||A||^\\mathcal{M}=\\{w\\,|\\,\\models_w^\\mathcal{M} A\\}$. We say that a formula $A$  is \\emph{valid} in a class $\\mathcal{C}$ of neighbourhood models iff it is true in every world of every $\\mathcal{M}\\in\\mathcal{C}$.\n\nIn order to give  soundness and completeness results for non-normal modal and deontic  logics with respect to (classes of) neighbourhood models, we introduce the following definition.\n\\begin{definition} Let $\\mathcal{M}=\\langle W,\\, N,\\,P\\rangle$ be a neighbourhood model, $X,Y \\in 2^W$, and $w\\in W$, we say that: \n\\begin{itemize}\n\\item $\\mathcal{M}$ is \\emph{supplemented}    if $X\\cap Y\\in N(w)$ imples $X\\in N(w)$ and $Y\\in N(w)$;\n\\item$\\mathcal{M}$ is \\emph{closed under finite intersection}  if $X\\in N(w)$ and $Y\\in N(w)$ imply $X\\cap Y\\in N(w)$;\n\\item $\\mathcal{M}$ \\emph{contains the unit} if   $W\\in N(w)$;\n\\item $\\mathcal{M}$ is \\emph{non-blind}  if $X\\in N(w)$ implies $X\\neq \\emptyset$;\n\\item $\\mathcal{M}$ is \\emph{complement-free} if $X\\in N(w)$ implies $W-X\\not\\in N(w)$.            \n\\end{itemize}\\end{definition}\n\n\n\\begin{proposition}\\label{corrax} We have the following correspondence results between $\\mathcal{L}$-formulas and the properties of the neighbourhood function defined above:\n\\begin{itemize}\n\\item Axiom $M$ corresponds to supplementation;\n\\item Axiom $C$ corresponds to closure under finite intersection;\n\\item Axiom $N$ corresponds to containment of the unit;\n\\item Axiom $D^\\bot$ corresponds to non-blindness;\n\\item Axiom $D^\\Diamond$ corresponds to complement-freeness.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{theorem}\\label{compax} {\\bf E} is sound and complete with respect to the class of all neighbourhood models. Any logic {\\bf X} which is obtained by extending {\\bf E} with some axioms from Table \\ref{axioms} is sound and complete with respect to the class of all neighbourhood models which satisfies all the properties corresponding to the axioms of {\\bf X}.\n\\end{theorem}\n\nSee \\cite{C80} for the proof of Proposition \\ref{corrax} and of Theorem \\ref{compax}.\n\n\\section{Sequent Calculi}\\label{seccalculi}\n\n We introduce  sequent calculi for non-normal logics that extend the multiset-based  sequent calculus  {\\bf G3cp} \\cite{NP01,NP11,T00} for classical propositional logic  -- see Table \\ref{G3cp} -- by adding some modal and deontic rules from Table \\ref{Modal rules}.   In particular, we  consider the modal sequent calculi given in Table \\ref{modcalculi}, which will be shown to capture the modal logics of Figure \\ref{cube}, and their deontic extensions given in Table \\ref{deoncalculi}, which will be shown to capture all deontic logics of Figure \\ref{cubed}.\nWe adopt the following notational conventions: we use {\\bf G3X} to denote a generic calculus from either Table \\ref{modcalculi} or Table \\ref{deoncalculi}, and we use {\\bf G3Y(Z)}  to denote both {\\bf G3Y} and {\\bf GRYZ}.  All the rules in Tables \\ref{G3cp} and  \\ref{Modal rules} but  $LR$-$C$ and $L$-$D^{\\Diamond_C}$ are standard rules in the sense of \\cite{G16}: each of them is a  single rule with a fixed number of premisses; $LR$-$C$ and $L$-$D^{\\Diamond_C}$, instead, stand for a recursively enumerable set of rules with a variable number of premisses.   \n\n For an introduction to {\\bf G3cp} and  the relevant notions, the reader is referred to \\cite[Chapter 3]{NP01}.\n   We sketch here the main notions that will be used in this paper.   A \\emph{sequent} is an expression  $\\Gamma\\Rightarrow \\Delta$, where $\\Gamma$ and $\\Delta$ are finite, possibly empty, multisets of formulas. If $\\Pi$ is the (possibly empty) multiset $A_1,\\dots,A_m$ then $\\Box\\Pi$ is  the (possibly empty) multiset $\\Box A_1,\\dots,\\Box A_m$. A \\emph{derivation} of a sequent $\\Gamma\\Rightarrow\\Delta$ in {\\bf G3X} is an upward growing  tree of sequents having $\\Gamma\\Rightarrow\\Delta$ as root, initial sequents or instances of rule $L\\bot$ as leaves, and such that each non-initial node is the conclusion of an instance of one  rule of {\\bf G3X} whose premisses are its children. In the rules in Tables \\ref{G3cp} and \\ref{Modal rules}, the multisets $\\Gamma$ and $\\Delta$ are called \\emph{contexts}, the other formulas occurring in the conclusion (premiss(es), resp.) are called \\emph{principal} (\\emph{active}). In a sequent the \\emph{antecedent} (\\emph{succedent}) is the multiset occurring to the left (right) of the sequent arrow $\\Rightarrow$.  As for {\\bf G3cp}, a sequent $\\Gamma\\Rightarrow\\Delta$ has the following  \\emph{denotational interpretation}: the conjunction of the formulas in $\\Gamma$ implies the disjunction of the formulas in $\\Delta$.\n \nAs measures for inductive proofs we  use the weight of a formula and the height of a derivation. The \\emph{weight} of a formula $A$, $w(A)$, is defined inductively as follows: $w(\\bot)=w(p_i)=0$; $w(\\Box A)=w(A)+1$; $w(A\\circ B)=w(A)+w(B)+1$ (where    $\\circ$ is one of the binary connectives $\\wedge,\\,\\lor,\\,\\supset$). The \\emph{weight} of a sequent is the sum of the weight of the formulas occurring in that sequent.  The \\emph{height} of a derivation is the length of its longest branch minus one. A rule of inference is said to be (\\emph{height-preserving}) \\emph{admissible} in {\\bf G3X} if, whenever its premisses are derivable in {\\bf G3X}, then also its conclusion is derivable (with at most the same derivation height) in {\\bf G3X}. The \\emph{modal depth} of a formula (sequent) is the maximal number of nested modal operators occurring in it(s members).\n  \n\n   \\begin{table}\n\\caption{ The sequent calculus {\\bf G3cp}}\\label{G3cp}\n\\begin{center}\n\\scalebox{0.85000}{\\begin{tabular}{cccc}\n\\hline\\hline\\noalign{\\smallskip}\n Initial sequents: &$\\qquad p_n,\\Gamma\\Rightarrow\\Delta,p_n$&& $p_n$ propositional variable \\\\\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  Propositional rules:&\n  \\infer[\\infrule L\\wedge]{A\\wedge B,\\g\\To\\d}{A,B,\\g\\To\\d}\n\n&$\\qquad$&\n\\infer[\\infrule R\\wedge]{\\g\\To\\d,A\\wedge B}{\\g\\To\\d,A\\quad&\\g\\To\\d,B}\n\n\\\\\\noalign{\\smallskip\\smallskip}\n\\infer[\\infrule L\\bot]{\\bot,\\g\\To\\d}{}&\n\\infer[\\infrule L\\lor]{A\\lor B,\\g\\To\\d}{A,\\g\\To\\d\\quad&B,\\g\\To\\d}\n\n&&\n\\infer[\\infrule R\\lor]{\\g\\To\\d,A\\lor B}{\\g\\To\\d,A,B}\n\n\\\\\\noalign{\\smallskip\\smallskip}&\n\\infer[\\infrule L\\supset]{A\\supset B,\\g\\To\\d}{\\g\\To\\d,A\\quad&B,\\g\\To\\d}\n\n&&\n\\infer[\\infrule R\\supset]{\\g\\To\\d,A\\supset B}{A,\\g\\To\\d,B}\n\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}}\n\\end{center}\n\\caption{Modal and deontic rules}\\label{Modal rules}\n\\begin{center}\n\\scalebox{0.87000}{\\begin{tabular}{llll}\n\\hline\\hline\\noalign{\\smallskip}\n\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\g\\To\\d,\\Box B}{A\\Rightarrow B\\quad&B\\Rightarrow A}\n&\n\\infer[\\infrule LR\\mbox{-}M ]{\\Box A,\\g\\To\\d,\\Box B}{A\\Rightarrow B}\n&\n\\multicolumn{2}{c}{\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box \\Pi,\\g\\To\\d,\\Box B}{A,\\Pi\\Rightarrow B}}\n\n\\\\\\noalign{\\smallskip\\smallskip}\n\\multicolumn{2}{c}{\\infer[\\infrule LR\\mbox{-}C ]{\\Box A_1,\\dots,\\Box A_n,\\g\\To\\d,\\Box B}{A_1,\\dots,A_n\\Rightarrow B&B\\Rightarrow A_1&{}^{\\dots}&B\\Rightarrow A_n}\n}\n&\n\\infer[\\infrule LR\\mbox{-}K]{\\Box\\Pi,\\g\\To\\d,\\Box B}{\\Pi\\Rightarrow B}&\n\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box B}{\\Rightarrow B}\n\\\\\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\end{tabular}}\n\n\\scalebox{0.8700}{\\begin{tabular}{llllll}\n\n\n&\n\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box A,\\g\\To\\d}{A\\Rightarrow}\n&\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}, \\,|\\Pi|\\leq 2]{\\Box\\Pi,\\g\\To\\d}{\\Pi\\Rightarrow&\\Rightarrow\\Pi}\n&\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_M}, \\,|\\Pi|\\leq 2]{\\Box\\Pi,\\g\\To\\d}{\\Pi\\Rightarrow}\n\\\\\\noalign{\\smallskip\\smallskip}\n\\phantom{aaaaaaaa}&\n\\multicolumn{2}{c}{\\infer[\\infrule L\\mbox{-}D^{\\Diamond_C}]{\\Box \\Pi,\\Box\\Sigma,\\g\\To\\d}{\\Pi,\\Sigma\\Rightarrow&\\{ \\Rightarrow A, B|\\; A\\in\\Pi, B\\in \\Sigma \\}}\n}\n&\n\\infer[\\infrule L\\mbox{-}D^*]{\\Box\\Pi,\\g\\To\\d}{\\Pi\\Rightarrow}&\\phantom{aaaaaaa}\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}}\\end{center}\n \\caption{Modal  sequent calculi (\\checkmark= rule of the calculus, $\\star$ = admissible rule, $-$ = neither)}\\label{modcalculi}\n \n \\begin{center}\n\\scalebox{0.66000}{ \\begin{tabular}{r|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}  \n\\noalign{\\smallskip} \\hline\\hline\\noalign{\\smallskip\\smallskip}\n&{\\bf G3E}&{\\bf G3EN}&{\\bf G3M}& {\\bf G3MN}&{\\bf G3C}&{\\bf G3CN}& {\\bf G3R}&{\\bf G3K}\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$E$&\\checkmark&\\checkmark&$\\star$&$\\star$&$\\star$&$\\star$&$\\star$&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$M$&$-$&$-$&\\checkmark&\\checkmark&$-$&$-$&$\\star$&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$C$&$-$&$-$&$-$&$-$&$\\checkmark$&$\\checkmark$&$\\star$&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$R$&$-$&$-$&$-$&$-$&$-$&$-$&\\checkmark&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$K$&$-$&$-$&$-$&$-$&$-$&$-$&$-$&\\checkmark\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$R$-$N$&$-$&$\\checkmark$&$-$&$\\checkmark$&$-$&$\\checkmark$&$-$&$\\star$\\\\\\noalign{\\smallskip}\\hline\\hline\n  \\end{tabular}}\\end{center}\n\\caption{Deontic sequent calculi (\\checkmark= rule of the calculus, $\\star$ = admissible rule, $-$ = neither)}\\label{deoncalculi}\n\\begin{center}\n\\scalebox{0.66000}{\\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|}\n\n\\hline\\hline\\noalign{\\smallskip\\smallskip\\smallskip}\n  &{\\bf G3E(N)D$^\\bot$}&{\\bf G3ED$^\\Diamond$}&{\\bf G3E(N)D}&{\\bf G3M(N)D$^\\bot$}&{\\bf G3M(N)D}&{\\bf G3CD$^\\Diamond$}&{\\bf G3C(N)D}&{\\bf G3RD}&{\\bf G3KD}\\\\\n  \n     \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^\\bot$&\\checkmark&$-$&\\checkmark&\\checkmark&$\\star$&$-$&$\\star$&$\\star$&$\\star$\\\\\n  \n   \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^{\\Diamond_E}$&$-$&\\checkmark&\\checkmark&$-$&$\\star$&$\\star$&$\\star$&$\\star$&$\\star$\\\\\n  \n    \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^{\\Diamond_M}$&$-$&$-$&$-$&$-$&\\checkmark&$\\star$&$\\star$&$\\star$&$\\star$\\\\\n  \n   \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^{\\Diamond_C}$&$-$&$-$&$-$&$-$&$-$&\\checkmark&$\\star$&$\\star$&$\\star$\\\\\n  \n   \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^*$&$-$&$-$&$-$&$-$&$-$&$-$&\\checkmark&\\checkmark&\\checkmark\n   \n\\\\\n\\noalign{\\smallskip}\\hline\\hlin\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Structural rules of inference}\nWe are now going to prove that the calculi {\\bf G3X} have the same good structural properties of {\\bf G3cp}: weakening and contraction are height-preserving admissible  and  cut is admissible. All proofs are extension of those for {\\bf G3cp}, see  \\cite[Chapter 3]{NP01}; in most cases, the modal rules have to be treated differently from the propositional ones  because of the presence of empty contexts in the premiss(es) of the modal ones. We adopt the following notational convention: given a derivation tree $\\mathcal{D}_k$, the derivation tree of the $n$-th leftmost premiss of its last step is  denoted by $\\mathcal{D}_{kn}$. We begin by showing that the restriction to atomic initial sequents, which is needed to have the propositional rules invertible, is not limitative in that initial  sequents with arbitrary principal  formula are derivable in {\\bf G3X}.\n\n\\begin{proposition}\\label{genax} Every instance of $A,\\Gamma\\Rightarrow\\Delta, A$ is derivable in {\\bf G3X}.\\end{proposition}\n\n\\begin{proof} By induction on the weight of $A$. If $w(A)=0$ -- i.e., $A$ is atomic or $\\bot$ -- then we have an instance of an initial sequent or of a conclusion of $L\\bot$ and there is nothing to prove. If $w(A)\\geq1$,  we  argue by cases according to the construction of $A$. In each case we apply, root-first, the appropriate rule(s) in order to obtain sequents where some proper subformula of $A$ occurs both in the antecedent and in the succedent. The claim then holds by the inductive hypothesis (IH). To illustrate, if $A\\equiv\\Box B$ and we are in {\\bf G3M(ND)}, we have:\n\n$$\n\\infer[\\infrule LR\\mbox{-}M]{\\Box B,\\g\\To\\d,\\Box B}{\\infer[\\infrule IH]{B\\Rightarrow B}{}}\n$$\n  \\end{proof}\n\n\\begin{theorem}\\label{weak} The left and right rules of weakening are height-preserving admissible in {\\bf G3X}\\\n\n$$\n\\infer[\\infrule LW]{A,\\g\\To\\d}{\\g\\To\\d}\n\\qquad\n\\infer[\\infrule RW]{\\g\\To\\d,A}{\\g\\To\\d}\n$$\n  \\end{theorem}\n\n\\begin{proof}The proof is a straightforward induction on the height of the derivation $\\mathcal{D}$ of $\\Gamma\\Rightarrow \\Delta$.\n If the last step of $\\mathcal{D}$ is by a propositional rule, we have to apply the same rule to the weakened premiss(es), which are derivable by IH, see \\cite[Thm. 2.3.4]{NP01}. If it is by a modal or deontic rule, we proceed by adding $A$ to the appropriate weakening context of the conclusion of that rule  instance. To illustrate, if the last rule is $LR$-$E$, we transform \n \n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box B,\\g\\To\\d,\\Box C}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_1}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_2}  }\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}E]{\\Box B,A,\\g\\To\\d,\\Box C}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_1}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_2}  }\n$$\n\n{}\\end{proof}\n\n\n\n\\noindent Before considering contraction, we recall some facts that will be useful later on. \n\\begin{lemma} In {\\bf G3X} the rules$\\quad$\n\\infer[\\infrule L\\neg]{\\neg A,\\g\\To\\d}{\\g\\To\\d,A}\nand$\\quad$\n\\infer[\\infrule R\\neg]{\\g\\To\\d,\\neg A}{A,\\g\\To\\d}\nare admissible.\n\\end{lemma}\n\\begin{proof}\nWe have the following derivations (the step by $RW$ is admissible thanks to Theorem \\ref{weak}):\n\n$$\\infer[\\infrule L\\supset]{A\\supset\\bot,\\g\\To\\d}{\\g\\To\\d,A&\\infer[\\infrule L\\bot]{\\bot,\\g\\To\\d}{}}\n\\qquad\\qquad\n\\infer[\\infrule R\\supset]{\\g\\To\\d,A\\supset\\bot}{\\infer[\\infrule RW]{A,\\g\\To\\d,\\bot}{A,\\g\\To\\d}}\n$$\n\\end{proof}\n\n\\begin{lemma}\\label{inv}All propositional rules are height-preserving invertible in {\\bf G3X}, that is the derivability of (a possible instance of) a conclusion of a propositional rule entails the derivability, with at most the same derivation height, of its premiss(es).\\end{lemma}\n\n\\begin{proof} We have only to extend the proof  for {\\bf G3cp}, see  \\cite[Thm. 3.1.1]{NP01}, with new cases for the modal and deontic rules. If $A\\circ B$ occurs in the antecedent (succedent) of the conclusion of an instance of a  modal or deontic rule then it must be a member of the weakening context $\\Gamma$ ($\\Delta$)  of this rule instance, and we have only to change the weakening context according to the rule we are inverting.\\end{proof}\n\n\\begin{theorem}\\label{contr}The left and right rules of contraction are height-preserving admissible in {\\bf G3X}\\vspace{0.3cm}\n\n$$\n\\infer[\\infrule LC]{A,\\g\\To\\d}{A,A,\\g\\To\\d}\n\\qquad\n\\infer[\\infrule RC]{\\g\\To\\d,A}{\\g\\To\\d,A,A}\n$$\n \\end{theorem}\n \n \\begin{proof} The proof is by simultaneous  induction on the height  of the derivation $\\mathcal{D}$  of the  premiss for left and right  contraction. The base case is straightforward. For the inductive steps, we have different strategies according to whether the last step in $\\mathcal{D}$ is by a propositional  rule or not. If the last step in $\\mathcal{D}$ is  by a propositional rule, we have two subcases: if the contraction formula is not principal in that step, we apply the inductive hypothesis and then the rule. Else  we start by using the height-preserving invertibility --  Lemma \\ref{inv} -- of that rule, and then we apply the inductive hypothesis and the rule, see \\cite[Thm. 3.2.2]{NP01} for details. \n \n If the last step in $\\mathcal{D}$ is by a modal or deontic rule, we have two subcases: either (the last step is by one  of $LR$-$C$, $LR$-$R$, $LR$-$K$, $L$-$D^{\\Diamond_E}$, $L$-$D^{\\Diamond_M}$, $L$-$D^{\\Diamond_C}$ and $L$-$D^*$ and) both occurrences of the contraction formula $A$ of $LC$ are principal in the last step or  some  instance of the contraction formula  is    introduced in the appropriate  weakening context of the conclusion. In the first subcase, we apply the inductive hypothesis to the premiss and then the rule. An interesting example is when the last step in $\\mathcal{D}$ is by $L$-$D^{\\Diamond_E}$. We transform\n \n$$\n\\infer[\\infrule LC]{\\Box B,\\Gamma\\Rightarrow\\Delta}{\\infer[\\infrule L\\mbox{-}D^{\\Diamond}]{\\Box B,\\Box B,\\Gamma\\Rightarrow\\Delta}{\\deduce{B,B\\Rightarrow\\quad}{\\vdots\\;\\mathcal{D}_1}&\\deduce{\\Rightarrow B,B}{\\vdots\\;\\mathcal{D}_2}}}\n\\qquad\\textrm{ into }\\qquad\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond}]{\\Box B,\\Gamma\\Rightarrow\\Delta}{\\deduce{B\\Rightarrow\\qquad}{\\vdots\\;IH(\\mathcal{D}_1)}&\\deduce{\\Rightarrow B\\qquad}{\\vdots\\;IH(\\mathcal{D}_2)}}\n$$\n \n\\noindent where $IH(\\mathcal{D}_1)$ is obtained  by applying the inductive hypothesis for the left rule of contraction to $\\mathcal{D}_1$ and $IH(\\mathcal{D}_2)$ is obtained by applying the inductive hypothesis for the right rule of contraction to $\\mathcal{D}_2$.\n\nIn the second subcase,  we apply an instance of the same   modal or deontic rule which introduces one less occurrence of $A$ in the appropriate context of the conclusion. Let's consider $RC$.  If the last step is by $LR$-$M$ and no  instance of $A$ is principal in the last rule, we transform\\vspace{0.3cm}\n$$\n\\infer[\\infrule  RC]{\\Box B,\\Gamma'\\Rightarrow\\Delta',A,\\Box C}{ \\infer[\\infrule LR\\mbox{-}M]{\\Box B,\\Gamma'\\Rightarrow\\Delta',A,A,\\Box C}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_1}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}M]{\\Box B,\\Gamma'\\Rightarrow\\Delta',A,\\Box C}{\\deduce{B\\Rightarrow C\\quad}{\\vdots\\;\\mathcal{D}_1}}\n$$\n\n\\end{proof}\n \n \\begin{theorem}\\label{cut}The rule of cut is admissible in {\\bf G3X}\n$$\n\\infer[\\infrule Cut]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma}{\\deduce{\\g\\To\\d,D}{\\vdots\\;\\mathcal{D}_1}&\\deduce{D,\\Pi\\Rightarrow\\Sigma}{\\vdots\\;\\mathcal{D}_2}}\n$$ \n\\end{theorem}\n \n \\begin{proof}\n We consider an uppermost application of $Cut$ and we show that either it is eliminable, or it can be permuted upward in the derivation until we reach sequents where it is eliminable. The proofs, one for each calculus, are by induction on the weight of the cut formula $D$ with a sub-induction on the sum of the heights of the derivations of the two premisses (cut-height for shortness). The proof can be organized in 3 exhaustive cases:\n \n \\begin{enumerate}\n \\item At least one of the premisses of cut  is an initial sequent or a conclusion of $L\\bot$;\n\\item The cut formula in not principal in the last step of at least one of the two premisses;\n \\item The cut formula is principal in both premisses.\n \\end{enumerate}\n \n\n\n\\medskip\n\\noindent {\\bf{\\large\\textbullet}$\\quad$ Case (1).}$\\quad$ Same as for {\\bf G3cp}, see \\cite[Thm. 3.2.3]{NP01} for the details.\n \\medskip\n \n\n \\medskip\n\\noindent {\\bf{\\large\\textbullet}$\\quad$ Case (2).}$\\quad$ We have many subcases according to the last rule applied in the derivation  ($\\mathcal{D}^\\star$)  of the  premiss where the cut formula is not principal. For the propositional rules, we refer the reader to \\cite[Thm. 3.2.3]{NP01}, where it is given a procedure that allows to reduce the cut-height. If the last rule applied in $\\mathcal{D}^\\star$ is a modal or deontic one, we  can transform the derivation into a cut-free one because the conclusion of \\mbox{\\it{Cut}} is derivable  by replacing the last step of  $\\mathcal{D}^\\star$ with  the appropriate instance of the same modal or deontic rule. We present explicitly only the cases where the last step of the left premiss is by $LR$-$E$ and $L$-$D^\\bot$ and the cut formula is not principal in it, all other transformations being similar.\n \\medskip\n\n \n\n\n\\noindent $\\mathbf{LR\\textrm{-}E}:\\quad$ \\, If the left premiss  is by rule $LR$-$E$  (and $\\Gamma\\equiv \\Box A,\\Gamma'$ and $\\Delta\\equiv \\Delta',\\Box B$), we  transform\n\n$$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta',\\Box B,\\Sigma}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta',\\Box B,D}{\\deduce{A\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{B\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\deduce{D,\\Pi\\Rightarrow\\Sigma}{\\vdots\\;\\mathcal{D}_2}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}E ]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta',\\Box B,\\Sigma}{\\deduce{A\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{B\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}\n$$\n\n\n\n\\noindent $\\mathbf{L}${\\bf-}$\\mathbf{D^\\bot}:\\quad$ \\, If the left premiss  is by rule $L$-$D^\\bot$, we  transform\n\n$$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box A,\\Gamma'\\Rightarrow\\Delta,D}{\\deduce{A\\Rightarrow }{\\vdots\\;\\mathcal{D}_{11}}}&\\deduce{D,\\Pi\\Rightarrow\\Sigma}{\\vdots\\;\\mathcal{D}_2}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule L\\mbox{-}D^\\bot ]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\deduce{A\\Rightarrow }{\\vdots\\;\\mathcal{D}_{11}}}\n$$\n\n\n\n\n \n\n\n\\medskip\n\n\\noindent {\\bf {\\large\\textbullet}$\\quad$ Case (3).}$\\quad$ If the cut formula $D$  is principal in both premisses, we have cases according to the principal operator of $D$. In each case we have a procedure that allows to reduce the weight of the cut formula, possibly increasing the cut-height. For the propositional cases, which are the same for all the logics considered here, see \\cite[Thm. 3.2.3]{NP01}.\n\n If $D\\equiv\\Box C$, we consider the different logics one by one, without repeating the common cases. \\\\\n\n\\noindent\\textbullet$\\quad${\\bf G3E(ND).}$\\quad$ Both premisses are by rule $LR$-$E$, we have\n\n$$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule LR\\mbox{-}E]{\\Box C,\\Pi\\Rightarrow\\Sigma',\\Box B}{\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}&\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{22}}}}\n$$\n\n\\noindent and we transform it into the following derivation that has two  cuts with cut formulas of lesser weight, which are admissible by IH.\n\n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{ A\\Rightarrow B}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}&\\infer[\\infrule Cut]{B\\Rightarrow A}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{22}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}}\n$$\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3EN(D).}$\\quad$ Left premiss by $R$-$N$ and right one by $LR$-$E$. We transfor\n\n$$\n\\infer[\\infrule Cut]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box A}{\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule LR\\mbox{-}E]{\\Box C,\\Pi\\Rightarrow\\Sigma',\\Box A}{\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{21}}&\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{22}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule R\\mbox{-}N ]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box A}{\\infer[\\infrule Cut]{\\Rightarrow A}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\n\\noindent \\textbullet$\\quad${\\bf G3E(N)D$^\\bot$.}$\\quad$\nLeft premiss is by $LR$-$E$, and  right one by $L$-$D^\\bot$. We transform \n\n $$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box C,\\Pi\\Rightarrow\\Sigma}{\\deduce{C\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule L\\mbox{-}D^\\bot ]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{A\\Rightarrow }{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow }{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\\noindent \\textbullet$\\quad${\\bf G3E(N)D$^\\Diamond$.}$\\quad$\nLeft premiss is by $LR$-$E$, and  right one by $L$-$D^{\\Diamond_E}$. We transform ($|\\Xi|\\leq 1$\n\n{\\small $$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}]{\\Box C,\\Box\\Xi,\\Pi'\\Rightarrow\\Sigma}{\\deduce{C,\\Xi\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}&\\deduce{\\Rightarrow C,\\Xi}{\\vdots\\;\\mathcal{D}_{22}}}}\n\\text{into} \n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}]{\\Box A,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{ \\Rightarrow\\Xi,A}{\\deduce{\\Rightarrow \\Xi, C}{\\vdots\\;\\mathcal{D}_{22}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule Cut]{A,\\Xi\\Rightarrow }{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,\\Xi\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n$$}\n\n\\noindent\\textbullet$\\quad${\\bf G3E(N)D.}$\\quad$ Left premiss by $LR$-$E$ and right one by   $L$-$D^\\bot$ or $L$-$D^{\\Diamond_E}$. Same as  above.\\vspace{0.3cm}\n\n\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3END$^\\bot$.}$\\quad$ Left premiss by $R$-$N$ and right one by $L$-$D^\\bot$. We transform\n\n $$\n\\infer[\\infrule Cut]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box C,\\Pi\\Rightarrow\\Sigma}{\\deduce{C\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer=[\\infrule LWs\\mbox{ and }RWs ]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{\\phantom{C}\\Rightarrow^{} }{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow }{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3END.}$\\quad$ Left premiss by $R$-$N$ and right one by $L$-$D^{\\Diamond_E}$. We transform ($|\\Xi|\\leq 1$)\n\n$$\n\\infer[\\infrule Cut]{\\Box \\Xi,\\Gamma,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\n\\infer[\\infrule R\\mbox{-}N]{\\Gamma\\Rightarrow\\Delta,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\n\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\Box C,\\Box \\Xi,\\Pi'\\Rightarrow\\Sigma}{\\deduce{C,\\Xi}{\\vdots\\;\\mathcal{D}_{21}}\\Rightarrow&\\Rightarrow \\deduce{C,\\Xi}{\\vdots\\;\\mathcal{D}_{22}}}}\n\\qquad\\text{into}\\qquad\n\\infer[\\infrule (\\star)]{\\Box \\Xi,\\Gamma,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{\\Xi\\Rightarrow}{\n\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\n\\deduce{C,\\Xi\\Rightarrow}{\\vdots\\;\\mathcal{D}_{21}}}}\n$$\nwhere $(\\star)$ is an instance of $L$-$D^\\bot$ if $|\\Xi|=1$, else ($|\\Xi|=0$ and) it is some instances\\mbox{ of $LW$ and $RW$.} \n\n\\noindent \\textbullet$\\quad${\\bf G3M(ND).}$\\quad$\n Both premisses are by rule $LR$-$M$, we transfor\n $$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule LR\\mbox{-}M]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule LR\\mbox{-}M]{\\Box C,\\Pi\\Rightarrow\\Sigma',\\Box B}{\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}M]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{A\\Rightarrow B }{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow B}{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3MN(D).}$\\quad$ Left premiss by $R$-$N$ and right one by $LR$-$M$. Similar to the case with left premiss by $R$-$N$ and right one by $LR$-$E$.\\vspace{0.3cm}\n\n\\noindent \\textbullet$\\quad${\\bf G3M(N)D$^\\bot$} and {\\bf G3M(N)D.}$\\quad$\nLeft premiss is by $LR$-$M$, and  right one by  $L$-$D^\\bot$ or $L$-$D^{\\Diamond_M}$. Similar to the  case with left premiss by $LR$-$E$ and right by $L$-$D^\\bot$ or $L$-$D^{\\Diamond_E}$, respectively.\\vspace{0.3cm}\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3MND$^\\bot$} and {\\bf G3MND.}$\\quad$ The cases with left premiss by $R$-$N$ and right one by  a deontic rule are like the analogous ones we have already considered.\n\n\n\\medskip\n\\noindent \\textbullet$\\quad${\\bf G3C(ND).} Both premisses are by rule $LR$-$C$. Let us agree to use $\\Lambda$ to denote the non-empty  multiset $A_1,\\dots,A_n$, and $\\Xi$ for the (possibly empty) multiset $B_2,\\dots B_m$.  The derivation\n\n$$\n\\infer[\\infrule Cut]{\\Box\\Lambda,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box E}{\\infer[\\infrule LR\\mbox{-}C]{\\Box\\Lambda,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{\\Lambda\\Rightarrow C}{\\vdots \\;\\mathcal{D}_{11}}& \\deduce{C\\Rightarrow A_1}{\\vdots \\;\\mathcal{D}_{A_1}}&{}^{\\dots}&\\deduce{C\\Rightarrow A_n}{\\vdots \\;\\mathcal{D}_{A_n}}}&\n\\infer[\\infrule LR\\mbox{-}C]{\\Box C,\\Box\\Xi,\\Pi'\\Rightarrow\\Sigma',\\Box E}{\\deduce{C,\\Xi\\Rightarrow E}{\\vdots \\;\\mathcal{D}_{21}}&\\deduce{ E\\Rightarrow C}{\\vdots \\;\\mathcal{D}_{C}}&{}^{\\dots}& \\deduce{E\\Rightarrow B_m}{\\vdots \\;\\mathcal{D}_{B_m}}}}\n$$\n\n\\noindent is transformed into the following derivation having  $n+1$ cuts on formulas of lesser weight\n\\noindent\\scalebox{0.8000}{$$\n\\infer[\\infrule LR\\mbox{-}C]{\\Box\\Lambda,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box E}{\\infer[\\infrule Cut]{\\Lambda,\\Xi\\Rightarrow E}{\\deduce{\\Lambda\\Rightarrow C}{\\vdots\\:\\mathcal{D}_{11}}&\\deduce{C,\\Xi\\Rightarrow E}{\\vdots\\;\\mathcal{D}_{21}}}&\n\\infer[\\infrule{Cut}\\;\\dots]{E\\Rightarrow A_1}{\\deduce{E\\Rightarrow C}{\\vdots\\:\\mathcal{D}_{C}}&\\deduce{C\\Rightarrow A_1}{\\vdots\\;\\mathcal{D}_{A_n}}}&\n\\infer[\\infrule Cut]{E\\Rightarrow A_n}{\\deduce{E\\Rightarrow C}{\\vdots\\:\\mathcal{D}_{C}}&\\deduce{C\\Rightarrow A_n}{\\vdots\\;\\mathcal{D}_{A_n}}}&\n \\deduce{E\\Rightarrow B_1}{\\vdots \\;\\mathcal{D}_{B_1}}\n &{}^{\\dots}& \\deduce{E\\Rightarrow B_m}{\\vdots \\;\\mathcal{D}_{B_m}}\n }\n$$}\\vspace{0.3cm}\n\n\\noindent \\textbullet$\\quad${\\bf G3CN(D).} Left premiss by $R$-$N$ and right premiss by $LR$-$C$. We have\n\n$$\n\\infer[\\infrule Cut]{\\Gamma,\\Box A_1,\\dots,\\Box A_n,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box B}{\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\n\\infer[\\infrule LR\\mbox{-}C]{\\Box C,\\Box A_1,\\dots,\\Box A_n,\\Pi'\\Rightarrow\\Sigma',\\Box B}{\\deduce{C,A_1,\\dots,A_n\\Rightarrow B}{\\vdots \\;\\mathcal{D}_{21}}&\\deduce{ B\\Rightarrow C}{\\vdots \\;\\mathcal{D}_{C}}&{}^{\\dots}& \\deduce{B\\Rightarrow A_n}{\\vdots \\;\\mathcal{D}_{A_n}}}}\n$$\n where $A_1,\\dots,A_n$ (and thus also $\\Box A_1,\\dots,\\Box A_n$)  may or may not be the empty multiset. If $A_1,\\dots,A_n$ is not empty, we transform it  into the following derivation having one cut with cut formula of lesser weigh\n \n$$\n\\infer[\\infrule LR\\mbox{-}C]{\\Gamma,\\Box A_1,\\dots\\Box A_n,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box B}{\n\\infer[\\infrule Cut]{A_1,\\dots,A_n\\Rightarrow B}{\\deduce{\\Rightarrow C\\quad}{\\vdots\\:\\mathcal{D}_{11}}&\\deduce{C,A_1,\\dots, A_n\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}&\n \\deduce{B\\Rightarrow A_1}{\\vdots \\;\\mathcal{D}_{A_1}}\n &{}^{\\dots}& \\deduce{B\\Rightarrow A_n}{\\vdots \\;\\mathcal{D}_{A_n}}\n }\n$$\nIf, instead, $A_1,\\dots,A_n$ is empty, we transform it into\n\n$$\n\\infer[\\infrule R\\mbox{-}N]{\\Gamma,\\Pi'\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{\\Rightarrow B}{\\deduce{\\Rightarrow C\\qquad}{\\vdots\\:\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}}\n$$\n\n\\noindent \\textbullet$\\quad${\\bf G3CD$^{\\Diamond}$.} Left premiss by $LR$-$C$ and right premiss by $L$-$D^{\\Diamond_C}$. We transform  (we assume $\\Xi= A_1,\\dots A_k$, $\\Theta= C, B_2,\\dots, B_m$ and $\\Lambda= D_1,\\dots, D_n$) \n\n$$\n\\infer[\\infrule Cut ]{\\Box\\Xi,\\Box B_1,\\dots, \\Box B_m,\\Box \\Lambda,\\Gamma',\\Pi'\\Rightarrow\\Delta,\\Sigma}{\n\\infer[LR\\mbox{-}C]{\\Box\\Xi,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{\\{ C\\Rightarrow A_i\\,|\\, A_i\\in\\Xi\\}}{\\vdots\\;\\mathcal{D}_{1A_i}}}&\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_C}]{\\Box C,\\Box B_2,\\dots,\\Box B_m,\\Box\\Lambda,\\Pi'\\Rightarrow\\Sigma}{\n\\deduce{\\Theta,\\Lambda\\Rightarrow}{\\vdots\\;\\mathcal{D}_{21}}&\n\\deduce{\\{ \\Rightarrow E,D_j\\,|\\, E\\in\\Theta\\text{ and } D_j\\in\\ \\Lambda\\}}{\\vdots\\;\\mathcal{D}_{\\Theta_{i}\\Lambda_j}}}}\n$$\ninto the following derivation having $1+(k\\times n)$ cuts on formulas of lesser weight\n\n$$\\scalebox{0.830}{\n\\infer[\\infrule{ L\\mbox{-}D^{\\Diamond_C}}]{\\Box\\Xi,\\Box B_1,\\dots, \\Box B_m,\\Box \\Lambda,\\Gamma',\\Pi'\\Rightarrow\\Delta,\\Sigma}{\n\\infer[\\infrule Cut]{\\Xi,B_2,\\dots, B_m,\\Lambda\\Rightarrow}{\n\\deduce{\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,B_2,\\dots,B_m,\\Lambda\\Rightarrow}{\\vdots\\;\\mathcal{D}_{21}}}\n&\n\\infer[\\infrule Cut]{\\{\\Rightarrow A_i,D_j| A_i\\in\\Xi,\\, D_j\\in\\Lambda\\}}{\\deduce{\\Rightarrow D_j,C}{\\vdots\\;\\mathcal{D}_{\\Theta_1,\\Lambda_j}}&\\deduce{C\\Rightarrow A_i}{\\vdots\\;\\mathcal{D}_{1A_i}}}\n&\n\\deduce{\\{\\Rightarrow B_i,D_j| B_i\\in\\Theta-C,\\, D_j\\in \\Lambda\\}}{\\vdots\\;\\mathcal{D}_{\\Theta_i\\Lambda_j}}\n}\n}$$\n\n\n\n\n\n\n\n\n\n\n\\noindent \\textbullet$\\quad${\\bf G3C(N)D.}$\\quad$ Left premiss by $LR$-$C$ and right one by $L$-$D^*$. It is straightforward to transform the derivation into another one having one cut with cut formula of lesser weight. \\medskip\n\n\n\n\n\\noindent \\textbullet$\\quad${\\bf G3R(D).}$\\quad$ Both premisses are by rule $LR$-$R$, we transform\n\n\\noindent\\scalebox{0.88000}{ $$\n\\infer[\\infrule Cut]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box\\Xi,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule LR\\mbox{-}R]{\\Box C,\\Box\\Psi,\\Pi'\\Rightarrow\\Sigma',\\Box B}{\\deduce{C,\\Psi\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}}\n\\quad\\mbox{into}\\quad\n\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{A,\\Xi,\\Psi\\Rightarrow B }{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,\\Psi\\Rightarrow B}{\\vdots\\,\\mathcal{D}_{21}}}}\n$$}\n\\medskip\n\n\\noindent\\noindent \\textbullet$\\quad${\\bf G3RD$^\\star$.}$\\quad$Left premiss is by $LR$-$R$, and  right one by $L$-$D^\\star$, we transform\n\n\\noindent$$\n\\infer[\\infrule Cut]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box\\Xi,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule L\\mbox{-}D^*]{\\Box C,\\Box\\Psi,\\Pi'\\Rightarrow\\Sigma}{\\deduce{C,\\Psi\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n\\quad\\mbox{into}\\quad\n\\infer[\\infrule L\\mbox{-}D^*]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{A,\\Xi,\\Psi\\Rightarrow  }{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,\\Psi\\Rightarrow }{\\vdots\\,\\mathcal{D}_{21}}}}\n$$ \n\n\n\\noindent \\textbullet$\\quad${\\bf G3K(D).}$\\quad$\nThe new cases with respect to {\\bf G3R(D)} are those with left premiss by an instance of $LR$-$K$ that has no principal formula in the antecedent. These cases can be treated like cases with left premiss by $R$-$N$.\\end{proof}\n\n\n\n\n\n\\section{Decidability and syntactic completeness}\\label{secdecax}\n\\subsection{Decision procedure for {\\bf G3X}}\\label{decision}\nEach calculus {\\bf G3X} has the strong subformula property since all  active formulas of  each rule  in Tables \\ref{G3cp} and \\ref{Modal rules} are  proper subformulas of the the principal formulas and no formula  disappears in moving from premiss(es) to conclusion. As usual, this gives us a syntactic proof of consistency.\n\\begin{proposition}\\label{cons}\\\n\\begin{enumerate}\n\\item Each  premiss of each rule of {\\bf G3X} has smaller weight than its conclusion;\n\\item Each premiss of each modal or deontic rule of {\\bf G3X} has smaller modal depth than its conclusion;\n\\item The calculus {\\bf G3X} has the subformula property:  a {\\bf G3X}-derivation of a sequent $\\mathcal{S}$ contains only sequents composed of subformulas of $\\mathcal{S}$;\n\\item The empty sequent is not {\\bf G3X}-derivable.\n\\end{enumerate}\\end{proposition} \nWe also have an effective method to decide the derivability of a sequent in {\\bf G3X}: we start from the desired sequent $\\Gamma\\Rightarrow\\Delta$ and we construct all possible {\\bf G3X}-derivation trees until either we find a tree where each leaf is an initial sequent or a conclusion of $L\\bot$ -- we have found a {\\bf G3X}-derivation of $\\Gamma\\Rightarrow\\Delta$ -- or we have checked all possible {\\bf G3X}-derivations   and we have found none --  $\\Gamma\\Rightarrow\\Delta$ is not {\\bf G3X}-derivable. \n\nMore in details, we present  a depth-first procedure that tests {\\bf G3X}-derivability in polynomial space. As it is usual in decision procedures involving non-invertible rules, we have trees involving two kinds of branching. Application of a rule with more than one premiss produce an \\emph{AND-branching} point, where all branches have to be derivable to obtain a derivation. Application of a non-invertible rule to a sequent that can be the conclusion of different instances of non-invertible rules produces an \\emph{OR-branching} point, where only one branch need be derivable to obtain a derivation.\nIn the procedure below we will assume that, given a calculus {\\bf G3X} and given a sequent $\\Box\\Pi,\\Gamma^p\\Rightarrow\\Delta^p,\\Box\\Sigma$ (where $\\Gamma^p$ and $\\Delta^p$ are multisets of propositional variables), there is some fixed way of ordering the finite (see below) set of instances of modal and deontic rules of {\\bf G3X} (\\emph{{\\bf X}-instances}, for shortness) having that sequent as conclusion. Moreover, we will represent the root of branches above an OR-branching point  by nodes of shape $\\Box_i$, where $\\Box_i$ is the name of the $i$-th {\\bf X}-instance applied (in the order of  all {\\bf X}-instances having that conclusion).  To illustrate, if we are in {\\bf G3EN} and we have to consider  $\\Box A,\\Box B,\\Gamma^p\\Rightarrow\\Delta^p,\\Box C$ then we obtain (fixing one way of ordering the three {\\bf X}-instances having that sequent as conclusion):\n \n$$\n\\infer{\\Box A,\\Box B,\\Gamma^p\\Rightarrow\\Delta^p,\\Box C}{\n\\infer[\\qquad]{LR\\mbox{-}E}{A\\Rightarrow C& C\\Rightarrow A}&\n\\infer[\\qquad]{LR\\mbox{-}E}{B\\Rightarrow C&C\\Rightarrow B}&\n\\infer{R\\mbox{-}N}{\\Rightarrow B}}\n$$\nwhere the lowermost sequent is an  OR-branching point  and the two nodes   $LR$-$E_1$ and $LR$-$E_2$ are AND-branching points. Finally, Given an AND(OR)-branching point \n\n$$\\infer{\\mathcal{S}}{\\mathcal{S}_1&\\dots&\\mathcal{S}_n}$$ we say that the branch above $\\mathcal{S}_i$ is an  \\emph{unexplored AND(OR)-branch} if no one  of its nodes has already been active.\n\n\n\\begin{definition}[{\\bf G3X}-decision procedure]\\label{decisiontree}\\\n\\begin{description}\n\\item[Stage 1.] We write the one node tree $\n\\Gamma\\Rightarrow\\Delta$\nand we label $\\Gamma\\Rightarrow\\Delta$ as active.\n\\item[Stage n+1.] Let $\\mathcal{T}_n$ be the tree constructed at stage $n$, let $\\mathcal{S}\\equiv\\Pi\\Rightarrow\\Sigma$ be its active sequent, and let $\\mathcal{B}$ be the branch  going from   the root  of $\\mathcal{T}_n$ to $\\mathcal{S}$. \n\\begin{description}\n\\item[Closed.] If $\\mathcal{S}$ is such that $p\\in\\Pi\\cap\\Sigma$ (for some propositional variable $p$) or $\\bot\\in\\Pi$, then we label $\\mathcal{S}$ as closed and\n\\begin{description}\n\\item[Derivable.] If $\\mathcal{B}$ contains no  unexplored AND-branch, the procedure ends and  \\mbox{$\\Gamma\\Rightarrow\\Delta$} is {\\bf G3X}-derivable;\n\\item[AND-backtrack.]If, instead, $\\mathcal{B}$ contains  unexplored AND-branches, we choose the topmost one and we label as active its leftmost unexplored leaf. {\\bf Else}\n\\end{description}\n\\item[Propositional.] if $\\mathcal{S}$ can be the conclusion of some instances $\\circ_1,\\dots,\\circ_m$ of the invertible propositional rules, we extend  $\\mathcal{B}$ by applying  one of such instances: \n\n$$\\infer[\\infrule{\\circ_i\\quad 1\\leq i\\leq m}]{\\mathcal{S}}{\\mathcal{S}_1&(\\mathcal{S}_2)}$$ \\noindent where, if  $\\mathcal{S}_2$, if present,  $\\mathcal{S}$ is an AND-branching point. {\\bf Else}\n\\item[Modal.] If $\\mathcal{S}$ can be the conclusion of the following  canonically ordered list of   {\\bf X}-instances:\n\n\n$$\n\\infer[\\infrule \\Box_1]{\\mathcal{S}}{\\mathcal{S}_1^1&\\dots&\\mathcal{S}^1_k }\n\\qquad\\deduce[\\dots]{\\phantom{a}}{}\\qquad\n\\infer[\\infrule \\Box_m]{\\mathcal{S}}{\\mathcal{S}_1^m&\\dots&\\mathcal{S}^m_l }\n$$ \nthen we extend  $\\mathcal{B}$ as follows:\n\n\n$$\n\\infer{\\mathcal{S}}{\\infer{\\Box_1\\phantom{^1}}{\\mathcal{S}_1^1&\\dots&\\mathcal{S}^1_k }&\\deduce[\\dots]{\\phantom{a}}{}&\\infer{\\Box_m\\phantom{^1}}{\\mathcal{S}_1^m&\\dots&\\mathcal{S}^m_l}\n}\n$$\nwhere, if $m\\geq 2$,  $\\mathcal{S}$ is  OR-branching  and, if $\\Box_i$ is a rule with more than one premiss, $\\Box_i$ is AND-branching. Moreover, we label $\\mathcal{S}_1^1$ as active. {\\bf Else} \n\\item[Open.] No rule of {\\bf G3x} can be applied to $\\mathcal{S}$, then  we label $\\mathcal{S}$ as open and\n\\begin{description}\n\\item[Underivable.] If $\\mathcal{B}$ contains no unexplored  OR-branch, the procedure ends and  \\mbox{$\\Gamma\\Rightarrow\\Delta$} is not  {\\bf G3X}-derivable;\n\\item[OR-backtrack.]If, instead, $\\mathcal{B}$ contains unexplored OR-branches, we choose the topmost one and we label as active its leftmost unexplored leaf. \n\\end{description}\n\\end{description}\n\\end{description}\n\\end{definition}\n \nTermination  can be shown as follows. Proposition \\ref{cons}.1 entails that the height of each branch of the tree $\\mathcal{T}$  constructed in a {\\bf G3X}-decision procedure for a sequent $\\Gamma\\Rightarrow\\Delta$  is bounded by the weight of  $\\Gamma\\Rightarrow\\Delta$  (in particular, given Proposition \\ref{cons}.2, the number of OR-branching points occurring in a branch is bounded by the modal depth of $\\Gamma\\Rightarrow\\Delta$). Moreover, $\\mathcal{T}$ is  finitary branching  since all rules of {\\bf G3X} are finitary branching rules, and since each sequent can be the conclusion of a finite number  $k$ of  {\\bf X}-instances (for each {\\bf G3X} $k$ is bounded by a function of $|\\Gamma|$ and $|\\Delta|$)}.   Hence, after a finite number of stages we are either in case {\\bf Derivable} or in case {\\bf Underivable} and, in both cases, the procedure ends. In the first case we can easily extract a {\\bf G3X}-derivation of $\\Gamma\\Rightarrow\\Delta$ from $\\mathcal{T}$ (we just have to delete  all unexplored branches as well as all underivable sub-trees   above an OR-branching point). In the latter case, thanks to Proposition \\ref{cons}.3, we know that (modulo the order of the invertible propositional rules)  we have explored the whole search space for a {\\bf G3X}-derivation of $\\Gamma\\Rightarrow\\Delta$ and we have found none. \n\n\n\nWe  prove that it is possible to test {\\bf G3X}-derivability in polynomial space by showing how it is possible to store only the active node together with a stack containing  information sufficient to reconstruct unexplored branches. For the propositional part of the calculi, we proceed as in \\cite{B97,H93,H95}: each entry of the stack is a triple containing the name of the rule applied, an index recording which of its premisses  is active, and its principal formula. For the  {\\bf X}-instances two complications arise: we need to record which OR-branches are unexplored yet, and we have to keep track of the weakening contexts of the conclusion in the premisses of {\\bf X}-instances. The first problem has already been  solved by having assumed that the {\\bf X}-instances applicable to a given sequent have a fixed canonical order.  The second problem is solved by adding a numerical superscript to the formulas occurring in a sequent and by imposing that:\\\\\n- All formulas in the end-sequent have 1 as superscript;\\\\\n - The superscript $k$ of the principal formulas of rules and of initial sequents are maximal in that sequent;\\\\- Active formulas of {\\bf X}-instances (propositional rules) have $k+2$ ($k$, respectively) as superscript;\\\\-  \n  Contexts are copied in the premisses of each rule.\\\\ By doing so, the contexts of the conclusion are copied in the premisses in each rule of {\\bf G3X},  but they cannot be principal in the trees above the premisses of the  {\\bf X}-instances because their superscript is never maximal therein.  It is immediate to see the the superscripts occurring in a derivation are bounded by (twice) the modal depth of the end-sequent.\n  \n  Instances of all modal and deontic rules in Table \\ref{Modal rules} but $LR$-$C$ and $L$-$D^{\\Diamond_C}$ are such that there is no need to  record their principal formulas in the stack entry: they are the boxed version of the formulas  having maximal superscript in the active premiss; moreover, the name of the rule and the number of the premiss  allow to reconstruct the position of the principal formulas (for the right premiss of  $LR$-$E$ and $L$-$D^{\\Diamond_E}$, we have to switch the two formulas). In instances of rules $LR$-$C$ and $L$-$D^{\\Diamond_C}$, instead,  this  doesn't hold since in all premisses but the leftmost one  there is no subformula of some principal formulas. We can overcome this problem by copying in each premiss all principal formulas having no  active subformula in that premiss and by adding one to  their superscript. We also keep fixed the position of all formulas (modulo the swapping of the two active formulas). To illustrate, one such instance is:\n  \n  $$ \\scalebox{0.9}{\\infer[\\infrule LR\\mbox{-}C]{\\Box A^k_1,\\Box A^k_2,\\Gamma\\Rightarrow\\Delta,\\Box B^k}{A^{k+2}_1,A^{k+2}_2,\\Gamma\\Rightarrow\\Delta, B^{k+2}\\quad&\n B^{k+2},\\Box A^{k+1}_2,\\Gamma\\Rightarrow\\Delta, A_1^{k+2}\n \\quad&\n \\Box A^{k+1}_1,B^{k+2},\\Gamma\\Rightarrow\\Delta, A_2^{k+2}\n }\n}$$\n In this way, given  the name of the modal or deontic rule applied, any premiss of this rule instance, and its position among the premisses of this rule, we can reconstruct both the conclusion of this rule instance and  its position  in the fixed order of {\\bf X}-instances concluding that sequent (thus we know which OR-branches are unexplored yet). In doing so, we use the hp-admissibility of contraction to ensure that no formula has more than one occurrence in the antecedent or in the succedent of the conclusion of {\\bf X}-instances (otherwise we might be unable to reconstruct which of two identical {\\bf X}-instances we are considering). Hence, for {\\bf X}-instances each stack entry records the name of the rule applied and an index recording which premiss we are considering. \n \n\nThe decision procedure is like in Definition \\ref{decisiontree}. The only novelty is that at each stage, instead of storing the full tree constructed so far, we store only the active node and the stack, we push an entry  in the stack and, if we are in a backtracking case, we pop stack entries (and we  use them  to reconstruct the corresponding active sequent) until we reach an entry recording unexplored branches of the appropriate kind, if any occurs. \n\n\\begin{theorem}\n{\\bf G3X}-derivability is decidable in $\\mathcal{O}(n\\, \\log{} n)$-{\\sc space}, where $n$ is the weight of the end-sequent.\n\\end{theorem}\n\n\\begin{proof}\nWe have already argued that proof search terminates.\nAs   in \\cite{B97,H93,H95}, Proposition \\ref{cons}.1 entails that the  stack depth is bounded by $\\mathcal{O}(n)$ and, by storing the principal formulas of propositional rules as indexes into the end-sequent,  each entry requires $\\mathcal{O}(\\log{}n)$ space. Hence we have an $\\mathcal{O}(n\\, \\log{} n)$  space bound for the stack. Moreover, the active sequent contains at most $\\mathcal{O}(n)$ subformulas of the end-sequent and their numerical superscripts. Each such subformula requires $\\mathcal{O}(\\log{}n)$ space since it can be recorded as an index into the end-sequent;  its numerical superscript requires $\\mathcal{O}(\\log{}n)$ too since there are at most $\\mathcal{O}(n)$ superscripts. Hence also the active sequent requires $\\mathcal{O}(n\\log{}n)$ space\n\\end{proof}\n\n\n\\subsection{Equivalence with the axiomatic systems}\nIt is now time to show that the sequent calculi introduced are equivalent to the non-normal logics of Section \\ref{secaxiom}.  We write {\\bf G3X} $\\vdash \\Gamma\\Rightarrow\\Delta$ if the sequent $\\Gamma\\Rightarrow\\Delta$ is  derivable in {\\bf G3X}, and we say that $A$ is derivable in {\\bf G3X} whenever {\\bf G3X} $\\vdash \\;\\Rightarrow A$. We begin by proving the following \n\n\\begin{lemma}\\label{ax} All the axioms of the  axiomatic system {\\bf X} are derivable in {\\bf G3X}.\\end{lemma}\n\n\\begin{proof} A straightforward application of the rules of the appropriate sequent calculus, possibly using Proposition \\ref{genax}.  As an example, we show that the deontic axiom $D^\\bot$ is derivable by means of rule $L$-$D^\\bot$ and that axiom  $C$ is derivable by means of  $LR$-$C$.\n\n$$\n\\infer[\\infrule R\\neg]{\\Rightarrow\\neg\\Box\\bot}{\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box\\bot\\Rightarrow}{\\infer[\\infrule L\\bot]{\\bot\\Rightarrow}{}}}\n\\qquad\n\\infer[\\infrule R\\supset]{\\Rightarrow \\Box A\\wedge\\Box B\\supset\\Box(A\\wedge B)}{\\infer[\\infrule L\\wedge]{\\Box A\\wedge \\Box B\\Rightarrow \\Box (A\\wedge B)}{\\infer[\\infrule LR\\mbox{-}C]{\\Box A,\\Box B\\Rightarrow \\Box(A\\wedge B)}{\\infer[\\infrule R\\wedge]{A,B\\Rightarrow A\\wedge B}{\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow A}{}&\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow B}{}}&\n\\infer[\\infrule L\\wedge  ]{A\\wedge B\\Rightarrow A}{\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow A}{}}&\n\\infer[\\infrule L\\wedge]{A\\wedge B\\Rightarrow B}{\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow B}{}}}}}\n$$\n\n\\end{proof}\n\nNext we prove  the equivalence of the sequent calculi for non-normal logics with the corresponding axiomatic systems in the sense that  a sequent $\\Gamma\\Rightarrow\\Delta$ is derivable in {\\bf G3X} if and only if  its characteristic formula $\\bigwedge\\Gamma\\supset\\bigvee \\Delta$ is derivable in {\\bf X} (where the empty antecedent stands for $\\top$ and the empty succedent for $\\bot$).  As a consequence  each calculus is sound and complete with respect to the appropriate class of neighbourhood models (see  Section \\ref{semantics}).\n\n\\begin{theorem}\\label{comp} Derivability in  the sequent system {\\bf G3X} and in the axiomatic system {\\bf X} are equivalent, i.e.\n\n\\begin{center}\n{\\bf G3X} $\\vdash\\;\\Gamma\\Rightarrow\\Delta \\qquad$iff$\\qquad${\\bf X} $\\vdash\\bigwedge\\Gamma\\supset\\bigvee\\Delta$\n\\end{center}\\end{theorem}\n\n\\begin{proof}\nTo prove the right-to-left implication, we argue by induction on the height of the  axiomatic derivation in  {\\bf X}. The base case is covered by Lemma \\ref{ax}. For the inductive steps, the case of $MP$ follows by the admissibility of Cut and the invertibility of rule $R\\supset$.  If the last step is by $RE$, then $\\Gamma=\\emptyset$ and  $\\Delta$ is $\\Box C\\leftrightarrow \\Box D$. We know that (in {\\bf X}) we have derived $\\Box C\\leftrightarrow\\Box D$ from $C\\leftrightarrow D$. Remember that $C\\leftrightarrow D$ is defined as $(C\\supset D)\\wedge (D\\supset C)$.  Thus we assume, by inductive hypothesis (IH) , that {\\bf G3ED} $\\vdash\\;  \\Rightarrow C\\supset D\\wedge D\\supset C$. From this, by invertibility of  $R\\wedge$ and $R\\supset$ (Lemma \\ref{inv}), we obtain that {\\bf G3ED} $\\vdash\\;  C\\Rightarrow D$ and {\\bf G3ED} $\\vdash\\;  D\\Rightarrow C$.    We can thus  proceed as follows \\vspace{0.3cm}\n\n$$\n\\infer[\\infrule R\\wedge]{\\Rightarrow (\\Box C\\supset \\Box D)\\wedge(\\Box D\\supset \\Box C)}{\\infer[\\infrule R\\supset]{\\Rightarrow \\Box C\\supset\\Box D}{\\infer[\\infrule LR\\mbox{-}E]{\\Box C\\Rightarrow \\Box D}{\\infer[IH+\\ref{inv}]{C\\Rightarrow D}{}&\\infer[IH+\\ref{inv}]{D\\Rightarrow C}{}}\n}&\n\\infer[\\infrule R\\supset]{\\Rightarrow \\Box D\\supset\\Box C}{\\infer[\\infrule LR\\mbox{-}E]{\\Box D\\Rightarrow \\Box C}{\\infer[IH+\\ref{inv}]{D\\Rightarrow C}{}&\\infer[IH+\\ref{inv}]{C\\Rightarrow D}{}}}}\n$$\n\n\nFor the converse implication, we assume {\\bf G3X} $\\vdash \\Gamma\\Rightarrow\\Delta$, and show, by induction on the height of the derivation in sequent calculus, that {\\bf X} \\mbox{$\\vdash \\bigwedge \\Gamma\\supset\\bigvee\\Delta$.} If the derivation has height 0,  we have an initial sequent -- so $\\Gamma\\cap\\Delta\\neq\\emptyset$ -- or an instance on $L\\bot$ -- thus $\\bot\\in\\Gamma$. In both cases the claim holds. If the height is $n+1$, we consider the last rule applied in the derivation. If it is a propositional one, the proof is straightforward. If it is a modal rule, we argue by cases. \n\n\n\nIf the last step of a derivation in {\\bf G3E(ND)} is by  $LR$-$E$, we have derived $\\Box C,\\Gamma'\\Rightarrow\\Delta',\\Box D$ from $C\\Rightarrow D$ and $D\\Rightarrow C$. By IH and propositional reasoning, {\\bf ED} $\\vdash C\\leftrightarrow D$, thus {\\bf ED} $\\vdash \\Box C\\supset \\Box D$. By some propositional steps we conclude {\\bf ED} $\\vdash (\\Box C\\wedge\\bigwedge\\Gamma')\\supset (\\bigvee\\Delta'\\lor\\Box D).$  The cases of $LR$-$M$,  $LR$-$R$, and $LR$-$K$  can be treated in a similar manner (thanks, respectively, to the rule $RM$, $RR$,  and $RK$ from Table \\ref{rulesinf}).\n\nIf we are in {\\bf G3C(ND)}, suppose the last step is the following instance of $LR$-$C$:\n$$\n\\infer[\\infrule LR\\mbox{-}C]{\\Box C_1,\\dots,\\Box C_k,\\Gamma'\\Rightarrow \\Delta',\\Box D}{C_1,\\dots C_k\\Rightarrow D&D\\Rightarrow C_1&\\dots& D\\Rightarrow C_k}\n$$\nBy IH, we have  that  {\\bf C(ND)} $\\vdash D\\supset C_i$ for all $i\\leq k$, and, by propositional reasoning, we have that  {\\bf C(ND)} $\\vdash D\\supset C_1\\wedge\\dots\\wedge C_k$. We also know, by IH, that  {\\bf C(ND)} $\\vdash C_1\\wedge\\dots\\wedge C_k\\supset D$. By applying $RE$ to these two theorems we get that \n\\begin{equation}\\label{3}\n\\mathbf{ C(ND)} \\vdash \\Box(C_1\\wedge\\dots\\wedge C_k)\\supset \\Box D\n\\end{equation}\n By using axiom $C$ and propositional reasoning, we know that \n \\begin{equation}\\label{4}\n \\mathbf{ C(ND)} \\vdash \\Box C_1\\wedge\\dots\\wedge\\Box C_k\\supset\\Box(C_1\\wedge\\dots\\wedge C_k)\n \\end{equation}\n By applying transitivity to (\\ref{4}) and (\\ref{3}) and some propositional steps, we conclude that \n $$\n  \\mathbf{ C(ND)} \\vdash (\\Box C_1\\wedge\\dots\\wedge\\Box C_k\\wedge \\bigwedge\\Gamma')\\supset (\\bigvee\\Delta'\\lor\\Box D)\n  $$\n  \n  Let's now consider rule $L$-$D^\\bot$.  Suppose we are in {\\bf G3XD$^\\bot$}  and  we have derived  $\\Box C,\\Gamma'\\Rightarrow\\Delta$ from $C\\Rightarrow$. By IH,  {\\bf XD$^\\bot$} $\\vdash C\\supset\\bot$, and we know that {\\bf xD$^\\bot$} $\\vdash \\bot \\supset C$. Thus by $RE$ (or $RM$), we get {\\bf XD$^\\bot$} $\\vdash \\Box C\\supset \\Box \\bot$. By  contraposing it and then applying a $MP$ with the axiom $D^\\bot$, we get that {\\bf XD$^\\bot$} $\\vdash \\neg\\Box C$. By some easy propositional steps we  conclude  {\\bf XD$^\\bot$} $\\vdash ( \\Box C\\wedge\\bigwedge\\Gamma')\\supset \\bigvee\\Delta$. The case $R$-$N$ is similar.\n\nLet's consider rules $L$-$D^{\\Diamond_E}$. Suppose we are in {\\bf G3ED$^\\Diamond$}  and we have derived \\mbox{$\\Box A,\\Box B,\\Gamma'\\Rightarrow\\Delta$} from the premisses $A,B\\Rightarrow$ and $\\Rightarrow A,B$. By induction we get that {\\bf ED$^\\Diamond$}$\\vdash A\\wedge B\\supset\\bot$ and {\\bf ED$^\\Diamond$}$\\vdash A\\lor B$. Hence, {\\bf ED$^\\Diamond$}$\\vdash B\\supset \\neg A$ and {\\bf ED$^\\Diamond$}$\\vdash\\neg A\\supset B$. By applying $RE$ we get that  \n\n$$ \\mathbf{ED^\\Diamond}\\vdash \\Box B\\supset\\Box \\neg A$$\n which, thanks to  axiom $D^\\Diamond$, entails that  \n \n $$\\mathbf{ ED^\\Diamond}\\vdash \\Box B\\supset\\neg\\Box A$$ \n  By some propositional steps we conclude   \n  \n $$\\mathbf{ ED^\\Diamond}\\vdash (\\Box A\\wedge\\Box B\\wedge \\bigwedge\\Gamma')\\supset\\bigvee\\Delta$$\n   Notice that, thanks to Proposition \\ref{cons}.4 and Theorem \\ref{cut},  we can assume that instances of rule $L$-$D^\\Diamond$ always  have   two principal formulas. Otherwise the calculus would prove the empty sequent (we will also assume that neither $\\Pi$ nor $\\Sigma$ is empty in instances of rule $L$-$D^{\\Diamond_C}$).\n   \n    The case of $L$-$D^{\\Diamond_M}$ is analogous to that of $L$-$D^\\bot$ for instances with one principal formula and to that of $L$-$D^{\\Diamond_E}$ for instances with two principal formulas.\n   \n\nLet's consider rule $L$-$D^{\\Diamond_C}$. Suppose we have a {\\bf G3CD$^\\Diamond$}-derivation whose last step is:\n$$\n\\infer{\\Box\\Pi,\\Box\\Sigma,\\Gamma'\\Rightarrow \\Delta'}{\\Pi,\\Sigma\\Rightarrow &\\{\\Rightarrow A,B| \\,A\\in\\Pi\\text{ and }B\\in \\Sigma\\}}\n$$\nBy induction and by some easy propositional steps we   know that {\\bf ECD$^\\Diamond$} $\\vdash \\bigwedge\\Pi\\leftrightarrow\\neg\\bigwedge\\Sigma$. By rule $RE$ we derive {\\bf ECD$^\\Diamond$} $\\vdash\\Box\\bigwedge\\Pi\\supset\\Box\\neg\\bigwedge\\Sigma$, which, thanks to axiom $D^\\Diamond$, entails that {\\bf ECD$^\\Diamond$} $\\vdash\\Box\\bigwedge\\Pi\\supset\\neg\\Box\\bigwedge\\Sigma$. By transitivity with two (generalized) instances of axiom $C$ we obtain {\\bf ECD$^\\Diamond$} $\\vdash \\bigwedge\\Box\\Pi\\supset \\neg\\bigwedge\\Box\\Sigma$. By some easy  propositional steps we conclude that  \\mbox{{\\bf ECD$^\\Diamond$} $\\vdash (\\bigwedge\\Box\\Pi\\wedge\\bigwedge\\Box\\Sigma\\wedge\\bigwedge\\Gamma')\\supset\\bigvee\\Delta$.}\n\nThe admissibility of $L$-$D^*$ in {\\bf EC(N)D}, {\\bf RD}, and {\\bf KD} is similar to that of $LR$-$C$:  in (\\ref{3}) we replace $\\Box D$ with $\\Box \\bot$ and then we use theorem $D^\\bot$ to transform it into $\\bot$.\n\\end{proof}\nBy combining this and Theorem \\ref{compax} we have the following result.\n\\begin{corollary}\nThe calculus {\\bf G3X} is sound and complete with respect to the class of all neighbourhood models for {\\bf X}.\n\\end{corollary}\n\n\n\\subsection{Forrester's Paradox}\\label{forrester}\n  As an application of our decision procedure, we use it to analyse two formal reconstructions of Forrester's paradox \\cite{F84}, which   is one of the many paradoxes that endanger the normal deontic logic {\\bf KD} \\cite{M06}. Forrester's informal argument goes as follows:\n\n\\begin{quote}\n\nConsider the following three statements:\n\\begin{enumerate}\n\\item Jones murders Smith.\n\\item Jones ought not murder Smith.\n\\item  If Jones murders Smith, then Jones ought to murder Smith gently.\n\\end{enumerate}\nIntuitively, these sentences appear to be consistent. However 1 and 3 together imply that\n\\begin{itemize}\n\\item[4.] Jones ought to murder Smith gently.\n\\end{itemize}\nAlso we accept the following conditional:\n\\begin{itemize}\n\\item[5.]  If Jones murders Smith gently, then Jones  murder Smith.\n\\end{itemize}\nOf course, this is \\emph{not} a logical validity but, rather, a fact about the world we live in. Now, if we assume that the monotonicity rule is valid, then statement 5 entails\n\\begin{itemize}\n\\item[6.]  If Jones ought to murder Smith gently, then Jones  ought to murder Smith.\n\\end{itemize}\nAnd so, statements 4 and 6 together imply\n\\begin{itemize}\n\\item[7.] Jones ought to murder Smith.\n\\end{itemize}\nBut [given the validity of $D^\\Diamond$] this contradicts statement 2. The above argument suggests  that classical deontic logic should \\emph{not} validate the monotonicity rule [$RM$] \\cite[p. 16]{P17}  \n\\end{quote} \n \n\n We  show that Forrester's paradox is not a valid argument in deontic logics by presenting, in Figure \\ref{fig}, a failed {\\bf G3KD}-proof search  of the sequent that expresses it\n\\begin{equation}\ng\\supset m,m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow\n\\end{equation}\n where $m$ stands for 'John \\underline{m}urders Smith' and $g$ for `John murders Smith \\underline{g}ently' \\cite[pp. 87--91]{M06}.\nNote that, by Theorem \\ref{comp}, if Forrester's paradox  is not {\\bf G3KD}-derivable, then it is not valid  in all the weaker  deontic logics we have considered.\n\n\\begin{figure}\n$$\\infer[\\infrule L\\supset]{g\\supset m,m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow}{\n\\infer[\\infrule L\\supset]{m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow g}{\n\\deduce{m,\\Box \\neg m\\Rightarrow g,m^{\\phantom{a}}}{\\textnormal{closed}}&\n\\infer{\\Box g,\\Box\\neg m,m\\Rightarrow g^{\\phantom{a}}}{\n\\infer{L\\mbox{-}D^\\star}{\n\\infer[\\infrule L\\neg]{g,\\neg m\\Rightarrow}{\\deduce{g\\Rightarrow m}{\\textnormal{open}}}}\n&\n\\infer{L\\mbox{-}D^\\star}{\\deduce{g\\Rightarrow }{\\textnormal{open}}}\n&\n\\infer{L\\mbox{-}D^\\star}{\\infer[\\infrule L\\neg]{\\neg m\\Rightarrow}{\\deduce{\\Rightarrow m}{\\textnormal{open}}}}}\n}&\n\\deduce{m,m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow^{\\phantom{A}}}{\\vdots}}$$\n\\caption{Failed {\\bf G3KD}-proof search of Forrester's paradox \\cite{M06}}\\label{fig}\n\\end{figure}\n\n To make our failed proof search into a derivation of Forrester's paradox, we would have to add (to {\\bf G3MD$^\\Diamond$} or stronger calculi) a non-logical axiom $\\Rightarrow g\\supset m$, and to have cut as a primitive -- and  ineliminable -- rule of inference.  An  Hilbert-style axiomatization of Forrester's argument -- e.g., \\cite[p. 88]{M06} -- hides this cut with a non-logical axiom in the  step where   $\\Box g\\supset\\Box m$ is derived from $ g\\supset m$,  by one of $RM$, $R R$ or $RK$. This step -- i.e., the step from 5 to 6 in the informal argument above -- is not acceptable because none of these rules allows to infer its conclusion when the premiss is  an assumption  and not a  theorem. We have here an instance of the same  problem that has led many authors to conclude that the deduction theorem fails in modal logics, conclusion that has been  shown to be wrong in \\cite{NH12}.\n \n An alternative formulation of Forrester's argument is given in  \\cite{T97}, where the sentence `John murders Smith gently' is expressed by the complex formula $g\\wedge m$ instead of by the atomic $g$. In this case Forrester's argument becomes valid whenever the monotonicity rule is valid as it shown in Figure \\ref{figfig}. Nevertheless, whereas it was an essential ingredient of the informal version, under this formalization premiss 5 becomes dispensable. Hence it is disputable that this is an acceptable way of formalising Forrester's argument.\n \\begin{figure}\n$$ \\infer[\\infrule L\\supset]{m\\supset\\Box(m\\wedge g),\\Box\\neg m,m\\Rightarrow}{\n \\deduce{\\Box\\neg m,m\\Rightarrow m^{\\phantom{a}}}{\\textnormal{closed}}&\n\\infer{\\Box(g\\wedge m),\\Box\\neg m,m\\Rightarrow}{ \\infer{L\\mbox{-}D^\\star}{\n\\infer[\\infrule L\\wedge]{g\\wedge m,\\neg m\\Rightarrow}{\n\\infer[\\infrule L\\neg]{g, m,\\neg m\\Rightarrow}{\n\\deduce{g,m\\Rightarrow m}{\\textnormal{closed}}}}}&\n  \\infer{L\\mbox{-}D^\\star}{g\\wedge m\\Rightarrow}&\n   \\infer{L\\mbox{-}D^\\star}{\\neg m\\Rightarrow}}\n  }$$\n \\caption{ Succesfull {\\bf G3MD}-proof search for the alternative version of  Forrester's paradox \\cite{T97}}\\label{figfig}\n \\end{figure}\n  \n This is not the place to discuss at length  the correctness of formal representation of Forrester's argument  and their implications for deontic logics. We just wanted to illustrate how  the calculi {\\bf G3XD} can be  used to analyse formal representations of the  deontic paradoxes. If Forrester's argument  is formalised as in \\cite{M06} then  it does not force to adopt a deontic logic weaker than {\\bf KD}. If, instead, it is formalised as in \\cite{T97} then it forces the adoption of a logic where $RM$ fails, but the formal derivation  differs substantially from Forrester's informal argument \\cite{F84}.\n \n\\section{Craig's Interpolation Theorem}\\label{secinterpol}\nIn this section we use Maehara's \\cite{M60,M61} technique to prove Craig's interpolation theorem for each modal or deontic logic {\\bf X} which has  $C$  as theorem only if it has also $M$  (Example \\ref{prob} illustrates the problem with the non-standard rule $LR$-$C$).\n\n\\begin{theorem}[Craig's interpolation theorem] \\label{Craig}\nLet $A\\supset B$ be a theorem of a  logic {\\bf X}  that differs from {\\bf EC(N)} and its deontic extensions {\\bf EC(N)D} and {\\bf ECD$^\\Diamond$}, then \nthere is a formula $I$, which contains  propositional variables common to $A$ and $B$ only, such that both $A\\supset I$ and $I\\supset B$ are theorems of {\\bf X}.\n\n\\end{theorem}\n\n\\noindent In order to prove this theorem, we use the following notions.\n\n\\begin{definition}\n\nA \\emph{partition} of a sequent $\\Gamma\\Rightarrow \\Delta$  is any pair of sequents \\\\$\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>$ such that $\\Gamma_1,\\Gamma_2=\\Gamma$ and $\\Delta_1,\\Delta_2=\\Delta$.\\\\\nA \\emph{{\\bf G3X}-interpolant of a partition} $\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>$ is any formula $I$  such that:\n\\begin{enumerate}  \n\\item All propositional variables in $I$ are    in $(\\Gamma_1\\cup\\Delta_1)\\cap(\\Gamma_2\\cup\\Delta_2)$;\n \\item {\\bf G3X} $\\vdash\\Gamma_1\\Rightarrow\\Delta_1,I$ and {\\bf G3X} $\\vdash I,\\Gamma_2\\Rightarrow\\Delta_2$.\n \\end{enumerate}\n\n\\end{definition}\n\nIf $I$ is a {\\bf G3X}-interpolant of the partition $\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>$, we write\n\n$$(\\textrm{{\\bf G3X}}\\vdash)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{I}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\n$$\n\n\\noindent where one or more of the multisets $\\Gamma_1,\\Gamma_2,\\Delta_1,\\Delta_2$ may be empty. When the set of propositional variables in $(\\Gamma_1\\cup\\Delta_1)\\cap(\\Gamma_2\\cup\\Delta_2)$ is empty, the {\\bf X}-interpolant has to be constructed from $\\bot$ (and $\\top$).\nThe proof of  Theorem \\ref{Craig} is by the following lemma, originally due to Maehara \\cite{M60,M61} for (an extension of) classical logic.\n\n\\begin{lemma}[Maehara's lemma]\\label{Maehara} If {\\bf G3X} $\\vdash \\Gamma\\Rightarrow\\Delta$ and $LR$-$C$ (and $L$-$D^{\\Diamond_C}$) is not a  rule of {\\bf G3X} (see Tables \\ref{modcalculi} and \\ref{deoncalculi}), every partition   of $\\Gamma\\Rightarrow\\Delta$ has a {\\bf G3X}-interpolant.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is  by induction on the height of the derivation $\\mathcal{D}$ of $\\Gamma\\Rightarrow\\Delta$. We have to show that each partition of an initial sequent (or of a conclusion of a 0-premiss rule) has a  {\\bf G3X}-interpolant and that for each rule of {\\bf G3X} (but $LR$-$C$ and $L$-$D^{\\Diamond_C}$) we have an effective procedure that outputs a {\\bf G3X}-interpolant for any partition of its conclusion from the interpolant(s) of suitable partition(s) of its premiss(es). The proof is modular and, hence, we can consider the modal rules without having to reconsider them in the different calculi.\n\nFor the base case of initial sequents with $p$ principal formula, we have four possible partitions, whose interpolants are:\\begin{center}\\begin{tabular}{ccc}\n$(1)\\;\\<p,\\Gamma_1'\\Rightarrow\\Delta_1',p\\;\\stackrel{\\bot}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(2)\\;\\<p,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{p}{||}\\;\\Gamma_2\\Rightarrow\\Delta'_2,p\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1',p\\;\\stackrel{\\neg p}{||}\\;p,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(4)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\top}{||}\\;p,\\Gamma'_2\\Rightarrow\\Delta'_2,p\\>$\n\\end{tabular}\\end{center} \n\n\\noindent and for the base case of rule $L\\bot$, we have:\n\\begin{center}\\begin{tabular}{ccc}\n$(1)\\;\\<\\bot,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\bot}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(2)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\top}{||}\\;\\bot,\\Gamma'_2\\Rightarrow\\Delta_2,\\>$\n\\end{tabular}\\end{center}\n\n For the proof of (some of) the propositional  cases the reader is referred to \\cite[pp. 117-118]{T00}. Thus, we have only to prove that all the modal and deontic  rules of Table \\ref{Modal rules} (modulo  $LR$-$C$ and $L$-$D^{\\Diamond_C}$)  behave as desired.\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-E}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n A\\Rightarrow B\\qquad B\\Rightarrow A\n \\justifies\n \\Box A,\\Gamma\\Rightarrow\\Delta,\\Box B\n \\using LR\\textrm{-}E\n  \\end{prooftree} \n  \\end{center}\n  \\noindent we have  four kinds of partitions of the conclusion:\n\n\\begin{center}\\begin{tabular}{ccc}\n$(1)\\;\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(2)\\;\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta'_2,\\Box B\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(4)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>$\n\\end{tabular}\\end{center}\n\n\\noindent In each  case we have to choose  partitions of the premisses that permit to construct a {\\bf G3E(ND)}-interpolant for the partition under consideration.\n\n In case {\\bf(1)} we have \n $$\n\\framebox{ \\infer[\\infrule LR\\mbox{-}E]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{C}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>}{ \\<A\\Rightarrow B\\;\\stackrel{C}{||}\\;\\Rightarrow\\>&\\<B\\Rightarrow A\\;\\stackrel{D}{||}\\;\\Rightarrow\\>}\n} $$\n  \n\\noindent   This can be shown as follows. By  IH there is some $C$ ($D$) that is a {\\bf G3E(ND)}-interpolant of the given partition of the left (right) premiss.  Thus both $C$ and $D$ contains only propositional variables common to $A$ and $B$; and (i) $\\vdash A\\Rightarrow B,C\\;$ (ii) $\\vdash C\\Rightarrow\\;$ (iii) $\\vdash B\\Rightarrow A,D\\;$ and (iv) $\\vdash D\\Rightarrow\\;$. Since the common language of the partitions of the premisses is empty,   no propositional variable can occur in $C$ nor in $D$. Here is a proof that $C$ is a {\\bf G3E(ND)}-interpolant of the partition under consideration (the sequents $A\\Rightarrow B$ and $B\\Rightarrow A$ are derivable since they are the premisses of the instance of $LR$-$E$ we are considering):\n  \n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'_1\\Rightarrow\\Delta_1',\\Box B,C}{\\infer{A\\Rightarrow B}{}&\\infer{B\\Rightarrow A}{}}\n\\qquad\\qquad\n\\infer[\\infrule LWs+RWs]{C,\\Gamma_2\\Rightarrow\\Delta_2}{\\infer[\\infruler{(ii)}]{C\\Rightarrow}{}}\n$$\n\n\n\n\\noindent In case {\\bf(2)} we have\n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}E]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Gamma_2\\Rightarrow\\Delta'_2,\\Box B\\>}{ \\<A\\Rightarrow \\;\\stackrel{C}{||}\\;\\Rightarrow B\\> &  \\<B\\Rightarrow \\;\\stackrel{D}{||}\\;\\Rightarrow A\\>}\n}$$\n  \n\\noindent By IH it holds that  some $C$ and $D$ are {\\bf G3E(ND)}-interpolants of the given partitions of the premisses.  Thus, (i) $\\vdash A\\Rightarrow C\\;$ (ii) $\\vdash C\\Rightarrow B\\;$ (iii) $\\vdash B\\Rightarrow D\\;$ (iv) $\\vdash D\\Rightarrow A\\;$ and (v)  all propositional variables in $C\\cup D$ are in $A\\cap B$.\n Here is a proof that $\\Box C$ is a {\\bf G3E(ND)}-interpolant of the given partition (the language condition is satisfied thanks to (v)\\,):\n \n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1,\\Box C}{\\infer[\\infruler{(i)}]{A\\Rightarrow C}{}&\n\\infer[\\infrule Cut]{C\\Rightarrow A}{\\infer[\\infrule Cut]{C\\Rightarrow D}{\\infer[\\infruler{(ii)}]{C\\Rightarrow B}{}&\\infer[\\infruler{(iii)}]{B\\Rightarrow D}{}}&\n\\infer[\\infruler{(iv)}]{D\\Rightarrow A}{}}}\n$$\n\n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box C,\\Gamma_2\\Rightarrow\\Delta_2',\\Box B}{\\infer[\\infruler{(ii)}]{C\\Rightarrow B}{}&\\infer[\\infrule Cut]{B\\Rightarrow C}{\\infer[\\infruler{(iii)}]{B\\Rightarrow D}{}&\\infer[\\infrule Cut]{D\\Rightarrow C}{\\infer[\\infruler{(iv)}]{D\\Rightarrow A}{}&\\infer[\\infruler{(i)}]{A\\Rightarrow C}{}}}}\n$$\n\n\\noindent  In case {\\bf(3)} we have \n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}E]{\\<\\Gamma_1\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta_2\\>}{ \\<\\Rightarrow B\\;\\stackrel{C}{||}\\;A\\Rightarrow \\> &  \\<\\Rightarrow A\\;\\stackrel{D}{||}\\;B\\Rightarrow \\>}\n}$$\n\\noindent  By IH, there are $C$ and $D$ that are {\\bf G3E(ND)}-interpolants of the partitions of the premisses. Thus (i) $\\vdash \\Rightarrow B,C\\;$ (ii) $\\vdash C,A\\Rightarrow\\;$ (iii) $\\vdash \\Rightarrow A,D\\;$ and (iv) $\\vdash D,B\\Rightarrow\\;$. We prove that $\\Diamond C$  is a {\\bf G3E(ND)}-interpolant of the (given partition of the) conclusion as follows:\n\n$$\n\\infer[\\infrule R\\neg]{\\Gamma_1\\Rightarrow\\Delta_1',\\Box B,\\neg\\Box\\neg C}{\\infer[\\infrule LR\\mbox{-}E]{\\Box\\neg C,\\Gamma_1\\Rightarrow\\Delta_1',\\Box B}{\n\\infer[\\infrule L\\neg]{\\neg C\\Rightarrow B}{\\infer[\\infruler{(i)}]{\\Rightarrow B,C}{}}&\n\\infer[\\infrule R\\neg]{B\\Rightarrow\\neg C}{\\infer[\\infrule Cut]{B,C\\Rightarrow}{\n\\infer[\\infrule Cut]{C\\Rightarrow D}{\\infer[\\infruler{(iii)}]{\\Rightarrow D,A}{}&\\infer[\\infruler{(ii)}]{A,C\\Rightarrow}{}}&\n\\infer[\\infruler{(iv)}]{D,B\\Rightarrow}{}}}}}\n$$\n\n$$\n\\infer[\\infrule L\\neg]{\\neg\\Box\\neg C,\\Box A,\\Gamma_2'\\Rightarrow\\Delta_2}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma_2'\\Rightarrow\\Delta_2,\\Box\\neg C}{\\infer[\\infrule R\\neg]{A\\Rightarrow \\neg C}{\\infer[\\infruler{(ii)}]{C,A\\Rightarrow}{}}&\n\\infer[\\infrule L\\neg]{\\neg C\\Rightarrow A}{\\infer[\\infrule Cut]{\\Rightarrow A,C}{\n\\infer[\\infruler{(iii)}]{\\Rightarrow A,D}{}&\n\\infer[\\infrule Cut]{D\\Rightarrow C}{\\infer[\\infruler{(ii)}]{\\Rightarrow C,B}{}&\\infer[\\infruler{(iv)}]{B,D\\Rightarrow}{}}}}}}\n$$\n\\noindent  In case {\\bf(4)} we have \n\n$$\n\\framebox{\n\\infer[\\infrule LR-E]{\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>}{\\<\\Rightarrow \\;\\stackrel{C}{||}\\;A\\Rightarrow B\\>&\\<\\Rightarrow \\;\\stackrel{D}{||}\\;B\\Rightarrow A \\>}\n}\n$$\n \\noindent By IH, there are  {\\bf G3E(ND)}-interpolants $C$ and $D$  of the partitions of the premisses. Thus (i) $\\vdash \\Rightarrow C\\;$ (ii) $\\vdash C,A\\Rightarrow B\\;$ (iii) $\\vdash \\Rightarrow D\\;$ and (iv) $\\vdash D,B\\Rightarrow A\\;$.  Since the common language of the partitions of the premisses is empty,  no propositional variable  occurs in $C$ nor in $D$. We  show that $C$ is a {\\bf G3E(ND)}-interpolant of the partition under consideration as follows (as in case {\\bf (1)}, $A\\Rightarrow B$ and $B\\Rightarrow A$, being the premisses of the instance of $LR$-$E$ under consideration, are derivable):\n \n$$\n\\infer[\\infrule LWs+RWs]{\\Gamma_1\\Rightarrow\\Delta_1,C}{\\infer[\\infruler{(i)} ]{\\Rightarrow C}{}}\n\\qquad\n\\infer[\\infrule LR\\mbox{-}E]{C,\\Box A,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B}{\\infer{A\\Rightarrow B}{}&\\infer{B\\Rightarrow A}{}}\n$$\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-M}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule LR\\mbox{-}M]{\\Box A,\\Gamma\\Rightarrow\\Delta,\\Box B}{A\\Rightarrow B}\n$$\n\n\\noindent we give directly the {\\bf G3M(ND)}-interpolants of the possible  partitions  of the conclusion (and of the appropriate partition of the  premiss). The proofs are parallel to those for $LR$-$E$.\n\n$$\n\\framebox{ \\infer[\\infrule LR\\mbox{-}M]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{C}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>}{\\<A\\Rightarrow B\\;\\stackrel{C}{||}\\;\\Rightarrow\\> } \n\\qquad\n \\infer[\\infrule LR\\mbox{-}M]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Gamma_2\\Rightarrow\\Delta'_2,\\Box B\\>}{\\<A\\Rightarrow \\;\\stackrel{C}{||}\\;\\Rightarrow B\\>}\n }\n$$\n$$\\framebox{\n\\infer[\\infrule LR\\mbox{-}M]{\\<\\Gamma_1\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta_2\\>}{\\<\\Rightarrow B\\;\\stackrel{C}{||}\\;A\\Rightarrow \\>} \n\\qquad\n\\infer[\\infrule LR\\mbox{-}M]{\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>}{ \\<\\Rightarrow \\;\\stackrel{C}{||}\\;A\\Rightarrow B\\> }\n}$$\n  \n  \n\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-R}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box \\Pi,\\Gamma\\Rightarrow\\Delta,\\Box B}{A,\\Pi\\Rightarrow B}\n$$\n  \\noindent we have  four kinds of partitions of the conclusion:\n\n\n\\begin{center}\\begin{tabular}{ll}\n$(1)\\qquad$&$\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n$(2)$&$\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1\\;||\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)$&$\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Box A,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n$(4)$&$\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;||\\;\\Box A,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>$\n\\end{tabular}\\end{center}\n\n\n In case {\\bf(1)} we have two subcases according to whether $\\Pi_2$ is empty or not. If it is not empty we have \n\n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow \\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\; \\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2\\> }{\\<A,\\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\; \\Pi_2\\Rightarrow\\;\\> }\n}$$\n  \n\\noindent  By IH, there is  a {\\bf G3R(D$^\\star$)}-interpolant $C$ of the chosen partition of the premiss. Thus (i) $\\vdash A,\\Pi_1\\Rightarrow B,C$ and (ii) $\\vdash C,\\Pi_2\\Rightarrow$, and we have the following derivations\n\n$$\n\\infer[\\infrule R\\neg]{\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B,\\neg\\Box\\neg C}{\\infer[\\infrule LR\\mbox{-}R]{\\Box\\neg C,\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B}{\\infer[\\infrule L\\neg]{\\neg C,A,\\Pi_1\\Rightarrow B}{\\infer[\\infruler{(i)}]{A,\\Pi_1\\Rightarrow B,C}{}}}}\n\\qquad\n\\infer[\\infrule L\\neg]{\\neg\\Box\\neg C,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2}{\\infer[\\infrule LR\\mbox{-}R]{\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2,\\Box \\neg C}{\\infer[\\infrule R\\neg]{\\Pi_2\\Rightarrow \\neg C}{\\infer[\\infruler{(ii)}]{C,\\Pi_2\\Rightarrow}{}}}}\n$$\n\n\\noindent When $\\Pi_2$ (and $\\Box \\Pi_2$)  is empty we cannot proceed as above since we cannot apply $LR$-$R$  in the right derivation. But in this case, reasoning like in case {\\bf (1)} for rule $LR$-$E$, we can show that\n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow \\Delta_1',\\Box B\\;\\stackrel{C}{||}\\; \\Gamma_2'\\Rightarrow\\Delta_2\\> }{\\<A,\\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\; \\Rightarrow\\;\\> }\n}$\n\n\n\n\\noindent Cases {\\bf(2)} and {\\bf(3)} are similar to  the corresponding cases for rule $LR$-$E$:\n{\\footnotesize$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box A,\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>}{\\< A,\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow B\\>}\\quad\n\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta'_1\\,\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box A,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>}{\\<\\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\; A,\\Pi_2\\Rightarrow\\;\\>}\n}$$}\n  \n\n\n In case {\\bf(4)} we have two subcases according to whether $\\Pi_1$ is empty or not:\n{\\small$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||} \\Box A,\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B\\>}{\\<\\;\\Rightarrow\\;\\stackrel{C}{||}A,\\Pi_2\\Rightarrow B\\>}\\quad\n\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||} \\Box A,\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B\\>}{\\<\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}A,\\Pi_2\\Rightarrow B\\>}\n}$$ }\nThe proofs are similar to those for case {\\bf (1)}.\n  \n\n\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-K}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule LR\\mbox{-}K]{\\Box \\Pi,\\Gamma\\Rightarrow\\Delta,\\Box B}{\\Pi\\Rightarrow B}\n$$\n\\noindent we give directly the {\\bf G3K(D)}-interpolants of the two  possible  partitions  of the conclusion:\n\n{\\small$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}K]{\\<\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B\\>}{\\<\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow B\\>}\\qquad\n\\infer[\\infrule LR\\mbox{-}K]{\\<\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2\\>}{\\< \\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow\\;\\>}\n}$$}\nThe proofs are, respectively, parallel to those for cases {\\bf (2)} and {\\bf(3)} of $LR$-$E$ (when $\\Pi=\\emptyset$, we can proceed as for rule $R$-$N$ and use $C$ instead of $\\Box C$ and of $\\Diamond C$, respectively).\n  \n\n\n  \n\n\n\\noindent\\textbullet$\\quad$ {\\bf L-D}$^\\bot)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n A\\Rightarrow \n \\justifies\n \\Box A,\\Gamma\\Rightarrow\\Delta\n \\using \\infrule{L\\mbox{-}D^\\bot}\n  \\end{prooftree} \n  \\end{center}\n  \\noindent we have  two kinds of partitions of the conclusion, whose {\\bf G3XD$^\\bot$}-interpolants are, respectively:\n\n \\begin{center}\\framebox{ \\begin{prooftree}\n\\<A\\Rightarrow\\;\\stackrel{C}{||}\\;\\Rightarrow\\;\\>\n \\justifies\n\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\n \\using \\infrule{L\\mbox{-}D^\\bot}\n  \\end{prooftree}\\qquad\n   \\begin{prooftree}\n\\<\\;\\Rightarrow\\;\\stackrel{C}{||}\\; A\\Rightarrow\\;\\>\n \\justifies\n\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\; \\Box A,\\Gamma_2'\\Rightarrow\\Delta_2\\>\n \\using \\infrule{L\\mbox{-}D^\\bot}\n  \\end{prooftree} }\n  \\end{center}\n  \n\n\\noindent\\textbullet$\\quad$ {\\bf L-D$^\\Diamond$}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}]{\\Box A,\\Box B,\\Gamma\\Rightarrow \\Delta}{A,B\\Rightarrow\\qquad&\\Rightarrow A,B }\\qquad\n\\textnormal{or}\\qquad\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_M}]{\\Box A,\\Box B,\\Gamma\\Rightarrow \\Delta}{A,B\\Rightarrow}\n$$\n  \\noindent we have  three kinds of partitions of the conclusion:\n\n\n\\begin{center}\\begin{tabular}{ll}\n$(1)\\qquad$&$\\<\\Box A,\\Box B,\\Gamma'_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n$(2)$&$\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Box A,\\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)$&$\\<\\Box A,\\Gamma'_1\\Rightarrow\\Delta_1\\;||\\;\\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n\\end{tabular}\\end{center}\n\n In cases {\\bf (1)} and {\\bf (2)} we have, respectively (omitting the right premiss for $L$-$D^{\\Diamond_M}$):\n \n $$\n\\framebox{\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\<\\Box A,\\Box B,\\Gamma_1'\\Rightarrow \\Delta_1\\;\\stackrel{ C}{||}\\; \\Gamma_2\\Rightarrow\\Delta_2\\> }{\\<A,B\\Rightarrow \\;\\stackrel{C}{||}\\; \\Rightarrow\\;\\> \\qquad \\<\\Rightarrow \\;\\stackrel{D}{||}\\; \\Rightarrow \\;A,B\\>}\n\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\<\\Gamma_1\\Rightarrow \\Delta_1\\;\\stackrel{ C}{||}\\; \\Box A,\\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\> }{\\<A,B\\Rightarrow \\;\\stackrel{C}{||}\\; \\Rightarrow\\;\\> \\qquad \\<\\Rightarrow \\;\\stackrel{D}{||}\\; \\Rightarrow \\;A,B\\>}\n}$$\n\nFinally, in case {\\bf (3)} we have:\n\n $$\n\\framebox{\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\<\\Box A,\\Gamma_1'\\Rightarrow \\Delta_1\\;\\stackrel{ \\Box C}{||}\\; \\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\> }{\\<A\\Rightarrow \\;\\stackrel{C}{||}\\; B \\Rightarrow\\;\\> \\qquad \\<\\Rightarrow \\; A\\stackrel{D}{||}\\; \\Rightarrow \\;B\\>}\n}$$\nBy IH, we can assume that $C$ is an interpolant of the partition of the left premiss and $D$ of the right one.\nWe have the following {\\bf G3YD$^\\Diamond$}-derivations ({\\bf Y} $\\in\\{$ {\\bf E,M}$\\}$):\n\n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1,\\Box C}{\\infer[\\infrule IH]{A\\Rightarrow C}{}&\n\\infer[\\infrule Cut]{C\\Rightarrow A}{\\infer[\\infrule IH]{\\Rightarrow A,D}{}&\n\\infer[\\infrule Cut]{D,C\\Rightarrow}{\n\\infer[\\infrule IH]{D\\Rightarrow B}{}&\\infer[\\infrule IH]{B,C\\Rightarrow}{}\n}}}\n$$\n\n$$\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_{E}}]{\\Box C,\\Box B,\\Gamma_2'\\Rightarrow\\Delta_2}{\n\\infer[\\infrule IH]{C,B\\Rightarrow}{}&\\infer[\\infrule Cut]{\\Rightarrow C,B}{\n\\infer[\\infrule Cut]{\\Rightarrow C,D}{\n\\infer[\\infrule IH]{\\Rightarrow D,A}{}&\n\\infer[\\infrule IH]{A\\Rightarrow C}{}}&\n\\infer[\\infrule IH]{D\\Rightarrow B}{}}\n}\n$$\nIt is also immediate to notice that  $\\Box C$ satisfies the language condition for being a {\\bf G3YD$^\\Diamond$}-interpolant of the conclusion since, by IH, we know that each propositional variable occurring in $C$ occurs   in $A\\cap B$.\n\n\\noindent\\textbullet$\\quad$ {\\bf L-D$^\\star$}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n \\Pi\\Rightarrow \n \\justifies\n \\Box\\Pi, \\Gamma\\Rightarrow\\Delta\n \\using \\infrule{L\\mbox{-}D^\\star}\n  \\end{prooftree} \n  \\end{center}\n  \\noindent we have  the following kind of  partition:\n\\quad$\\< \\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>$\n\n\nIf $\\Pi_1$ is not empty we have: \n\\begin{center} \\framebox{\\begin{prooftree}\n\\<\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow\\;\\>\n \\justifies\n\\< \\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\n \\using  \\infrule{L\\mbox{-}D^\\star}\n  \\end{prooftree} }\n  \\end{center}\n\n\\noindent By IH, there is some  $C$ that is  an interpolant of the premiss. It holds  that $\\vdash \\Pi_1\\Rightarrow C$ and $\\vdash C,\\Pi_2\\Rightarrow\\;$. We  show that $\\Box C$ is a {\\bf G3YD}-interpolant ({\\bf Y} $\\in\\{${\\bf R,K}$\\}$) of the partition of the conclusion as follows:\n\n$$\n\\infer[\\infrule LR\\mbox{-}Y]{\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1,\\Box C}{\\infer[\\infruler{IH}]{\\Pi_1\\Rightarrow C}{}}\n\\qquad\n\\infer[\\infrule L\\mbox{-}D^*]{\\Box C,\\Box \\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2}{\\infer[\\infruler{IH}]{C,\\Pi_2\\Rightarrow}{}}\n$$\nIf, instead, $\\Pi_1$ is empty then $\\Pi_2$ cannot be empty and we have\n\\begin{center} \\framebox{\\begin{prooftree}\n\\<\\;\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow\\;\\>\n \\justifies\n\\< \\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Diamond C}{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\n \\using \\infrule{ L\\mbox{-}D^\\star}\n  \\end{prooftree} }\n  \\end{center} \n  \n\\noindent   By IH there is a formula $C$, containing no propositional variable, such that  $\\vdash \\;\\Rightarrow C$ and $\\vdash C,\\Pi_2\\Rightarrow\\;$ . Thus, {\\bf G3YD} $\\vdash\\Gamma_1\\Rightarrow\\Delta_1,\\Diamond C$ ($L$-$D^*$ makes $\\Rightarrow\\Diamond C$ derivable from $\\Rightarrow C$) and  {\\bf G3YD} $\\vdash\\Diamond\\top,\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2$ ($LR$-$Y$ makes  $\\Diamond C,\\Box\\Pi_2\\Rightarrow$ derivable from $C,\\Pi_2\\Rightarrow$ when $\\Pi_2\\neq \\emptyset$).\n \n \n\n \\noindent\\textbullet$\\quad$ {\\bf R-N}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n \\Rightarrow A \n \\justifies\n\\Gamma\\Rightarrow\\Delta,\\Box A\n \\using \\infrule{R\\mbox{-}N}\n  \\end{prooftree} \n  \\end{center}\n\n\\noindent The interpolants for the two possible partitions are\n\n\\noindent\\framebox{ \\begin{tabular}{cccc}\n $(1)\\;$& \\begin{prooftree}\n\\<\\;\\Rightarrow A\\stackrel{\\bot}{||}\\;\\Rightarrow\\;\\>\n \\justifies\n\\< \\Gamma_1\\Rightarrow\\Delta_1',\\Box A\\;\\stackrel{\\bot}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\n \\using \\infrule{R\\mbox{-}N}\\quad\n  \\end{prooftree} &\n \n  \n\n$(2)\\;$& \\begin{prooftree}\n\\<\\;\\Rightarrow \\;\\stackrel{\\top}{||}\\;\\Rightarrow A\\>\n \\justifies\n\\< \\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\top}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2',\\Box A\\>\n \\using \\infrule{R\\mbox{-}N}\n  \\end{prooftree} \n  \\end{tabular}}\\vspace{0.3cm}\n \n \n\n\\noindent This completes the proof.{}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{Craig}]\nAssume that $A\\supset B$ is a theorem of {\\bf X}. By Theorem \\ref{comp} and Lemma \\ref{inv} we have that {\\bf G3X} $\\vdash A\\Rightarrow B$. By Lemma \\ref{Maehara}  (taking $A$ as $\\Gamma_1$ and  $B$ as $\\Delta_2$ and $\\Gamma_2,\\Delta_1$ empty) and Theorem \\ref{comp} there exists  a formula $I$ that is an interpolant of $A\\supset B$ -- i.e. $I$ is such such that  all propositional variables occurring in $I$, if any, occur in both  $A$ and $B$ and such that  $A\\supset I$ and $I\\supset B$ are theorems of {\\bf X}.{}\n\\end{proof}\n\nObserve that the proof is constructive in that  Lemma \\ref{Maehara} gives a procedure to  extract  an interpolant for $A\\supset B$ from a given derivation of $A\\Rightarrow B$. Furthermore the proof is purely proof-theoretic in that it makes no use of model-theoretic notions.\n\nCraig's theorem is often -- e.g., in \\cite{M61} for an extension of  classical   logic -- stated in the following stronger  version:\n\\begin{quote}\nIf $A\\supset B$ is a theorem of the logic {\\bf X}, then \n\\begin{enumerate}\n\\item If $A$ and $B$ share some propositional variable, there is a formula $I$, which contains  propositional variables common to $A$ and $B$ only, such that both $A\\supset I$ and $I\\supset B$ are theorems of {\\bf X};\n\\item Else, either $\\neg A$  or $B$ is a theorem of {\\bf X}.\n\\end{enumerate}\n\\end{quote}\n\n\\noindent But the second condition  doesn't hold for   modal and deontic logics where at least one of $N:=\\Box\\top$ and $D^\\bot:=\\Diamond\\top$ is not a theorem.\nTo illustrate,  it holds that  $ \\Box\\top\\supset\\Box\\top$ is a theorem of {\\bf E} and its interpolant is $\\Box\\bot$ (see Figure \\ref{fig}), but neither  $\\neg \\Box\\top$ nor $\\Box\\top$ is a theorem of {\\bf E}. Analogously, we have that $\\Box\\bot\\supset\\Box\\bot$ is a theorem of  {\\bf E} and its interpolant is $\\Box\\bot$ (see Figure \\ref{fig}), but neither $\\neg \\Box\\bot$ nor $\\Box\\bot$ is a  theorem of {\\bf E}. These counterexamples work in all extensions of {\\bf E} that don't have both $N$ and $D^\\bot$ as theorems: to prove the stronger version of Craig's theorem we need $N$ and $D^\\bot$, respectively.\n\n\\begin{figure}\n\\scalebox{0.9900}{\\infer[\\infrule LR\\mbox{-}E]{\\< \\;\\Box\\top\\Rightarrow\\;\\stackrel{\\Box\\top}{||}\\;\\Rightarrow\\Box\\top\\;\\>}{\n\\< \\;\\top\\Rightarrow\\;\\stackrel{\\top}{||}\\;\\Rightarrow\\top\\;\\>& \\< \\;\\top\\Rightarrow\\;\\stackrel{\\top}{||}\\;\\Rightarrow\\top\\;\\>}\\qquad\n\\infer[\\infrule LR\\mbox{-}E]{\\< \\;\\Box\\bot\\Rightarrow\\;\\stackrel{\\Box\\bot}{||}\\;\\Rightarrow\\Box\\bot\\;\\>}{\n\\< \\;\\bot\\Rightarrow\\;\\stackrel{\\bot}{||}\\;\\Rightarrow\\bot\\;\\>& \\< \\;\\bot\\Rightarrow\\;\\stackrel{\\bot}{||}\\;\\Rightarrow\\bot\\;\\>}\n\n\n\\caption{Construction of an {\\bf ED}-interpolant for   $\\Box\\top\\supset\\Box\\top$ and for $\\Box\\bot\\supset\\Box\\bot$}\\label{fig}\n\\end{figure}\n\n\nAmong the deontic logics considered here, the stronger version of Craig's theorem holds only for {\\bf END$^{\\bot(\\Diamond)}$},  {\\bf MND$^{\\bot(\\Diamond)}$}, and {\\bf KD}, as shown by the following \n\n\\begin{corollary}\\label{cor} Let {\\bf XD} be one of {\\bf END$^{\\bot(\\Diamond)}$},  {\\bf MND$^{\\bot(\\Diamond)}$}, and {\\bf KD}. If $A\\supset B$ is a theorem of {\\bf XD} and $A$ and $B$ share no propositional variable, then either $\\neg A$ or $B$ is a theorem of {\\bf XD}.\n\\end{corollary}\n\n\\begin{proof}\nSuppose that {\\bf XD} $\\vdash A\\supset B$ and that $A$ and $B$ share no propositional variable, then the interpolant $I$  is constructed from $\\bot $ and $\\top$ by means of classical and deontic operators.  Whenever  $D^\\bot$ and $N$ are theorems of {\\bf XD}, we have that $\\Diamond\\top\\leftrightarrow \\top$, $\\Box\\top\\leftrightarrow \\top$,  $\\Diamond\\bot\\leftrightarrow \\bot$, and $\\Box\\bot\\leftrightarrow \\bot$ are theorems of {\\bf XD}.  Hence, the interpolant $I$ is (equivalent to) either $\\bot$ or $\\top$. In the first case {\\bf XD} $\\vdash\\neg A$ and in the second one {\\bf XD} $\\vdash B$.{}\n\\end{proof}\n\n\\noindent As noted in \\cite[p. 298]{F83}, Corollary \\ref{cor} is a Halld\\'en-completeness  result. A logic {\\bf X} is \\emph{Halld\\'en-complete} if, for every formulas $A$ and $B$ that share no propositional variable, {\\bf X} $\\vdash A\\lor B$ if and only if {\\bf X} $\\vdash A$ or {\\bf X} $\\vdash B$. All the modal and deontic logics considered here, being based on classical logic, are such that $A\\supset B$ is equivalent to $\\neg A \\lor B$. Thus  the deontic logics considered in Corollary \\ref{cor} are Halld\\'en-complete, whereas all other non-normal logics for which we have proved interpolation  are Halld\\'en-incomplete since they don't satisfy Corollary \\ref{cor}.\n\n\n\\begin{example}[Maehara's lemma and rule $LR$-$C$]\\label{prob}\n We have not been able to prove Maehara's Lemma \\ref{Maehara} for rule $LR$-$C$ because of the cases where the principal formulas of the antecedent are splitted in the two elements of the partition. In particular, if we have two principal formulas in the antecedent, the problematic partitions  are (omitting the weakening contexts):\n \\begin{center}\n (1)\\quad $\\<\\Box A_1\\Rightarrow\\;||\\;\\Box A_2\\Rightarrow \\Box B\\>$\\qquad\\qquad\n (2)\\quad $\\<\\Box A_1\\Rightarrow\\Box B\\;||\\;\\Box A_2\\Rightarrow\\>$\n \\end{center}\nTo illustrate, an interpolant of the first partition would be a formula $I$ such that:\n\n$$\n (i)\\quad \\vdash \\Box A_1\\Rightarrow I\\qquad (ii)\\quad\\vdash I,\\Box A_2\\Rightarrow\\Box B\\qquad(iii)\\quad p\\in I\\textnormal{ only if }p\\in (A_1)\\cap(A_2,B)\n$$\nBut we have not been able to find partitions of the premisses allowing to find such  $I$. More in details, for the first premiss it is natural to   consider the partition $\\< A_1\\Rightarrow\\;\\stackrel{C}{||}\\; A_2\\Rightarrow  B\\>$ in order to find an $I$ that satisfies $(iii)$.  But, for any combination of the partitions of the other two premisses that is compatible with $(iii)$, we can prove  that   $(ii)$ is satisfied (by $\\Box C$) but we have not been able to prove that also $(i)$ is satisfied.\n\\end{example}\n\n\\section{Conclusion}\\label{conc}\nWe presented cut- and contraction-free sequent calculi for non-normal modal and deontic logics. We have proved that these calculi have good structural properties in that weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Moreover, we have shown that these calculi allow for a terminating decision procedure whose complexity is in {\\sc Pspace}. Finally, we have given a constructive proof of Craig's interpolation property for all the logics that do not contain  rule $LR$-$C$. As far as we know, it is still an open problem whether it is possible to give a constructive proof of interpolation for these logics. Another open question is whether the calculi given here can be used to give a constructive proof of  the uniform interpolation property for non-normal logics as it is done in \\cite{P92} for $\\mathbf{IL_p}$ and in \\cite{B07} for  {\\bf K} and {\\bf T}. \n\n\n\n\n\n\n\\vspace{0.3cm}\n\n\\noindent {\\bf Thanks.} Thanks are due to Tiziano Dalmonte, Simone Martini,  and two anonymous referees for many helpful suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sxn0intrdxn}\nMRI is a household name as a diagnostic tool in the medical field\n\\cite{vp,tey},\nwith an impressive resume in many other fields, including the\nstudy of materials \\cite{callaghan,blumch,bjj,rugar},\ncorrosion of metals, monitoring batteries and supercapacitors\n\\cite{britton2010,ilott2014}.\n\nHowever, historically, MRI of bulk metals has been very rare, limited to\nspecialized cells using r.f. gradients (with limited control) \\cite{si0grld},\ninstead of the magnetic field gradients employed in conventional MRI.\nIn other studies involving bulk metals, the MRI targeted the surrounding\ndielectric (electrolyte in electrochemical and fuel cells with metallic\nelectrodes, or tissues with embedded metallic implants)\n\\cite{si0pines,si0camacho,si0bennet,si0olsen,si0alm,si0viano,si0shafiei,si0graf,\nsi0koch,si0jfz,si0garwood2011,si0garwood}.\nNotwithstanding, these studies addressed issues that can cause distortions and\nlimit sensitivity of the MRI images, such as bulk magnetic susceptibility (BMS) \neffects and eddy currents (produced on bulk metal surface due to\ngradient switching).\n\nThe dearth of mainstream bulk metal MRI is rooted in unique challenges posed by \nthe physics of propagation of r.f. EM fields in bulk metals\n(MRI employs r.f. pulses to generate the MR signal leading to the images).\nAll along, it has been known that the incident r.f. field decays rapidly and\nexponentially inside the metallic conductor (Fig.\\ref{figSI0dltaEff}), a\nphenomenon known as skin effect \\cite{jackson,si0griffiths,si0ulaby}.\nThe characteristic length of decay ($\\delta$, the {\\em skin depth}), typically\nof the order of several microns (Eq.(\\ref{eq:eq0sknDpth})),\ncharacterizes the limited r.f. penetration into the metal. This in turn, results\nin attenuated MR signal intensity for bulk metals \\cite{si0pines,rangeet}.\nTurning the tables, Bhattacharyya et. al.,\\cite{rangeet} exploited the skin\neffect to separate and quantify bulk and non-bulk metal NMR signals in Li ion\nbatteries to monitor the growth of dendritic metallic structures.\nSubsequently, for bulk metal MRI, yet another impediment was correctly diagnosed\n\\cite{chandrashekar}.\nIt was found that the orientation of the bulk metal surface, relative to \\mbox{$\\bf B_1$}\n(the r.f. magnetic field),\ncritically affected the outcome.\nUsing optimal alignment of the bulk metal (electrodes), relative to \\mbox{$\\bf B_1$},\nrecent studies have successfully demonstrated and highlighted bulk metal MRI\nalbeit, primarily applied to batteries and electrochemical cells\n\\cite{chandrashekar,ilott,romanenko,britton2014,hjc2015}.\n\nThough unanticipated at the time, the recent bulk metal MRI findings eased the\nimplementation of MRI of liquid electrolyte, by helping mitigate adverse effects\ndue to the metal in the vicinity of lithium, zinc and titanium electrodes,\nyielding fresh insights\n\\cite{romanenko,britton2014,si0britton2013,si0klett,si0furo}.\nSimilar benefits may be expected to accrue for MRI based radiology of soft\ntissues with embedded metallic implants (pacemeakers, prosthetics, dental\nimplants, etc.).\n\\\\\n\n\n\nHere, we present several key findings on bulk metal MRI and CSI:\n\\begin{itemize}\n\\item\nDuring a systematic noninvasive thickness measurement of bulk metal strips\nby MRI \\cite{romanenko}\n(section S\\ref{sxnSI0thcknss}),\nwe come across unexpected regions of intensity, and assign them to two mutually \northogonal pairs of faces of the strip.\n\n\\item\nTo explain the peculiar ratios of intensities from these different regions in\nbulk metal MRI and CSI, we derive formulae from first principles, unveling\na surprising underlying reason: \n{\\em differing effective elemental volumes for these different regions}.\n\n\\item\nIn the process, the images enable a visualization of a virtual EM vacuum\ninside the bulk metal via an {\\em MRI tunnel}.\n\n\\item\nAdditionally, we demonstrate that the bulk metal CSI distinguishes different\nfaces (surfaces) of a metal block according to their distinct NMR chemical\nshifts.\n\n\\end{itemize}\n\nWe attained these by employing three {\\em phantoms} (samples)\n{\\bf P0, P1, P3}, depicted schematically in Fig.\\ref{fig0phntms} and described\nin Methods section \\ref{subsxn0mthdsPhntms}.\nAll phantoms were derived from the same stock of 0.75 $mm$ thick lithium (Li)\nstrip. Phantom P3 is a super strip composed of three Li strips pressed together,\nforming an {\\em effective} single strip three times thicker than the\nindividual strips in phantoms P0 and P1.\n\nThe setup of phantoms, r.f. coil and the gradient assembly ensures that the\nimaging directions $x,y,z$ fulfill the condition that \n$x \\parallel a \\parallel \\mbox{$\\bf B_1$}$ and  $z \\parallel \\mbox{$\\bf B_0$}$\n(the static main magnetic field), with the possibility to reorient the phantom\nabout the $x$-axis; $a,b,c$  are the three sides of the strips.\n\nSince all MRI and CSI images to follow were acquired with the given phantom's\n$bc$ faces {\\em normal} ($\\perp$, {\\em perpendicular}) to \\mbox{$\\bf B_1$}, these images\nbear the imprint of having no contribution from these faces to the magnetic\nresonance (MR) signal \\cite{chandrashekar,ilott,romanenko,britton2014}, since\n\\mbox{$\\bf B_1$}\\ penetration into the metal is maximal when it is {\\em parallel}\n($\\parallel$) to the metal surface, and minimal when $\\perp$ metal surface\n\\cite{jackson,chandrashekar,ilott}.\n\nFor details on the MRI experiments, including the nomenclature, the reader is\nreferred to \nMethods section \\ref{subsxn0mthdsMri}.\n\n\n\\section{MRI}\n\\label{sxn0mri}\nFig.\\ref{fig0xy0yz} furnishes stackplots (intensity along the vertical axis) of\n\\mbox{$^7$Li} 2d MRI (without slice selection) of phantom P3.\nPanel (a) displays MRI($xy$). Panel (b) displays MRI($yz$).\n\nIt is straightforward to infer that the {\\em walls} of high intensity regions in\neither image emanate from $ac$ faces of the P3 strip\n\\cite{chandrashekar,ilott,romanenko}, as we did while measuring the\nthickness of metal strips (Figs.\\ref{fig0twoStrps} and \\ref{fig0xy},\nsection S\\ref{sxnSI0thcknss}).\nIn either image, contributions along the non imaged axis sum up to yield the\nhigh intensity walls.\n\nHowever, the unexpected intensity between the two $ac$ faces of the super strip,\nin both the images, is perplexing.\nThe 2d MRI($xy$) in Fig.\\ref{fig0xy0yz}a exhibits a low intensity {\\em plateau}\nspanning the walls.\nThe 2d MRI($yz$) in Fig.\\ref{fig0xy0yz}b displays low intensity {\\em ridges}  \nbridging the walls.\n\\\\\n\n\\subsection{Visualizing a virtual eletromagnetic vacuum by MRI}\nTo understand better these unexpected regions of intensity, we acquired\n\\mbox{$^7$Li} 3d MRI($xyz$) of phantom P3, shown in Fig.\\ref{fig0xyz}.\nIn addition to the $ac$ faces (separated along $y$),\nthe $ab$ faces (separated along $z$) are revealed for the first time.\n\nAs noted earlier, $bc$ faces (being $\\perp$ to \\mbox{$\\bf B_1$}) are absent.\nThe hollow region in MRI($xyz$) arises from the skin depth phenomenon\n\\cite{rangeet,chandrashekar,ilott,jackson,si0griffiths,si0ulaby},\nrestricting the EM fields to effectively access only a limited\nsubsurface underneath the $ac$ and $ab$ faces\n(section S\\ref{sxnSI0subSurface} and Fig.\\ref{figSI0dltaEff}).\nThe presence of faces $\\parallel$ \\mbox{$\\bf B_1$}, coupled with the conspicuous absence\nof faces $\\perp$ \\mbox{$\\bf B_1$}, in combination with the hollow region, imparts the 3d\nimage an appearance of an {\\em MRI tunnel}, supplying a compelling visualization\nof a virtual {\\em EM vacuum} in the interior of a metallic conductor\n(hitherto depicted only schematically in literature\n(for e.g. Ref.\\cite{rangeet})).\n\\\\\n\nWith the aid of 3d MRI in Fig.\\ref{fig0xyz}, the intensity regions in\n2d MRI($xy$) and 2d MRI($yz$) images of Fig.\\ref{fig0xy0yz}, can be easily\ninterpreted as simply regions resulting respectively from projections\nalong $z$ and $x$ axes of the 3d image.\nIt is convincingly clear that the intensity between $ac$ faces\n(either the plateau or the ridges), is due to the pair of $ab$ faces\nof the superstrip P3.\n\nYet, the basis for the relative intensity values remains elusive at\nthis stage.\n\nFor the 2d MRI($xy$) in Fig.\\ref{fig0xy0yz}a, it can be argued that,\nfor the $ac$ face the entire length of side $c=$7 $mm$\n(Fig.\\ref{fig0phntms}) contributes to the signal,\nwhile for the $ab$ face, only a subsurface depth\n$ \\mbox{$\\delta_{\\text{eff}}$} \\approx$ 9.49 $\\mu m$ contributes\n(Eq.(\\ref{eq:eq0sknDpth}), Eq.(\\ref{eq:eq0dltaEff}),\nsection S\\ref{sxnSI0subSurface} and Fig.\\ref{figSI0dltaEff}).\nThis would lead to a ratio of the corresponding intensities, $S_{ac}\/S_{ab}$,\nto be of the order of $c\/(2 \\mbox{$\\delta_{\\text{eff}}$}) \\approx 368 $ (Fig.\\ref{fig0xy0yz0sim}a),\nin obvious and jarring disagreement with the observed ratio\n(of maxima of $S_{ac}$ and $S_{ab}$) of 6.6.\n\nFor the 2d MRI($yz$) in Fig.\\ref{fig0xy0yz}b, the expected intensity pattern in \na stack plot would be one with equal intensities from $ab$ and $ac$ faces,\nsince they share the same side, $a$, along $x$ (non imaged) axis\n(Fig.\\ref{fig0xy0yz0sim}b).\nThis again, is in stark contrast with the observed ratio\n(of maxima of $S_{ac}$ and $S_{ab}$) of 10.\n\nFor the MRI($xyz$), naively, uniform intensity would be expected from both $ab$ \nand $ac$ faces, resulting in a ratio of unity.  Instead, the observed ratio\n(of maxima of $S_{ac}$ and $S_{ab}$) is found to be 3.8.\n \nThus, the MRI images bear peculiar intensity ratios from comfortably identified \n(from 3d MRI) regions of the bulk metal.\nWe will return to this topic later.\n\\\\\n\n\\section{CSI}\n\\label{sxn0csi}\nThe \\mbox{$^7$Li} NMR spectrum of phantom P3 (Fig.\\ref{fig0csi} inset)\ncontains two distinct peaks in the Knight shift region for metallic \\mbox{$^7$Li}\n(see Methods section \\ref{subsxn0mthdsMri}),\ncentered at $\\delta_1$= 256.4 and $\\delta_2$= 266.3 ppm.\nAt first sight it might seem odd that a metallic strip, of regular geometry and\nuniform density, that is entirely composed of identical Li atoms, gives\nrise to two NMR peaks instead of the expected single peak.\n\nTo gain additional insight as to the spatial distribution of the Li metal\nspecies with different NMR shifts, we turn to CSI, which combines an NMR\nchemical shift (CS) dimension with one or more imaging(I) dimensions\n\\cite{callaghan,haacke,si0spnglr,si0kwf}.\n\nThe 2d CSI($y$) shown in Fig.\\ref{figSI0csi} comprises of two bands separated\nalong $y$ located at $\\delta_2$, along the CS dimension,\nwhile a low intensity band spans them along $y$ at a CS of $\\delta_1$, strongly\nhinting at, the two bands (at $\\delta_2$) being associated with $ac$ faces.\n\nThis observation called for adding an additional\nimaging dimension along $z$, leading to 3d CSI($yz$), which is realized in\nFig.\\ref{fig0csi},\nwhere $y$ and $z$ are the imaging dimensions, accompanied by the CS dimension.\nThe bands separated along $z$, occur at $\\delta_1$.\nThe bands separated along $y$, occur at $\\delta_2$.\nIn conjunction with the 3d MRI image in Fig.\\ref{fig0xyz},\nit is evident that the pairs of bands at $\\delta_1$ and $\\delta_2$ arise from\n$ab$ and $ac$ faces respectively, completing the spatio-chemical assignment.\n\nThese assignments readily carry over to 2d CSI($y$) in Fig.\\ref{figSI0csi},\nwith the pair of bands at $\\delta_2$ and the low ridge spanning them at \n$\\delta_1$, being respectively identified with $ac$ and $ab$ faces.  Similarly,\nin the NMR spectrum, the short and tall peaks\nrespectively at $\\delta_{1,2}$ are assigned to $ab$ and $ac$ faces, consistent\nwith the reported \\cite{rangeet,ilott} experiments and simulations.\n\nThat different types of faces of the bulk metal strip suffer different NMR\n(Knight) shifts according to their orientations {\\em relative} to \\mbox{$\\bf B_0$},\nis consistent with previous observations and simulations \\cite{rangeet,ilott},\nand has been traced to bulk magnetic susceptibility (BMS) effect\n\\cite{rangeet,chandrashekar,ilott,hjc2015,hoffman,lina,hjc}.\n\nInterestingly, the 3d CSI sheds new light on previous bulk metal NMR studies.\nFor instance, in an earlier study \\cite{rangeet}, a similar shift\ndifference between NMR peaks was observed at $\\parallel$ and $\\perp$\norientations (relative to \\mbox{$\\bf B_0$}) of the major faces of a thinner metal strip,\nby carrying out two {\\em separate} experiments.\nHere, phantom P3 furnishes these two orientations in a {\\em single} experiment, \nvia $ac$ and $ab$ faces (Fig.\\ref{fig0phntms}).\nThe present work provides physical insight into another previous \\cite{ilott}\nobservation. It was found that the intensity of NMR peak arising from $ab$\nfaces, unlike that from the $ac$ faces, was invariant under rotation about\n$z \\parallel c \\parallel B_0$ axis.\nOur 3d MRI (Fig.\\ref{fig0xyz}) and 3d CSI (Fig.\\ref{fig0csi}), directly\ndemonstrate that such a rotation leaves the orientation of \\mbox{$\\bf B_1$}\\ relative to\n$ab$ face (but not the $ac$ face) the same (signal intensity from a given face\ndepends on its orientation relative to \\mbox{$\\bf B_1$} \\cite{chandrashekar,ilott}).\nNote that the shifts themselves remain unshifted since they depend on the\norientation of the faces relative to \\mbox{$\\bf B_0$}, which does not change under this\nrotation ($ac$ and $bc$ faces remain $\\parallel \\mbox{$\\bf B_0$}$, whilst $ab$ faces\nremain $\\perp \\mbox{$\\bf B_0$}$).\n\nThus, bulk metal CSI supplies direct\nevidence, that the bulk metal chemical (Knight) shifts resulting from BMS are\ncorrelated with the differing orientations (relative to \\mbox{$\\bf B_0$}) of different parts \nof the bulk metal.\\\\\n\nLike for MRI, the basis for the ratio of intensities from the $ac$ and $ab$\nfaces ($S_{ac}\/S_{ab} \\approx$ 2.8), is not immediately intuitively obvious\nand will be explored next.\n\n\n\n\\section{Intensity ratio formulae for bulk metal MRI and CSI}\n\\label{sxn0formulae}\nThe peculiar intensity ratios in MRI and CSI, of signals $S_{ab}$ and $S_{ac}$,\narising respectively from $ab$ and $ac$ faces of phantom P3\n(Fig.\\ref{fig0xy0yz}, sections \\ref{sxn0mri} and \\ref{sxn0csi}),\ncould be due to gradient switching involved in the MRI experiments (the\nresultant eddy currents could be different for $ab$ and $ac$ faces).\nHowever, as shown in section S\\ref{sxnSI0mri2dNtnst}, this can be ruled out on\nthe basis of 2d MRI($yz$) and MRI($zy$) at mutually orthogonal orientations\n(related by a rotation about $x \\parallel a \\parallel \\mbox{$\\bf B_1$}$),\nshown in Fig.\\ref{fig0p3hrzntl}.\n\nAnd yet, it is possible to derive, from elementary considerations\nand first principles, and arrive at expressions for the {\\em ratios} of MRI\nand CSI intensities from $ab$ and $ac$ faces.\\\\\n\nFor the 2d MRI($xy$), in Fig.\\ref{fig0xy0yz}a, the signal intensity from the\n$ab$ faces can be written as (see Eq.(\\ref{eq:eq1dltaEff}))\n\\begin{equation}\n S_{ab} (x,y)\\propto dx\\ dy\\ \\int dz =  dx\\ dy\\ 2\\delta_{\\text{eff}}\n\\label{eq:eq0sabxy}\n\\end{equation}\nwith  $dx\\ dy\\ dz$ denoting {\\em elemental} volume of the metal,\nand \\mbox{$\\delta_{\\text{eff}}$}\\ is the {\\em effective} subsurface {\\em depth} that would account\nfor the MR signal in the {\\em absence} of \\mbox{$\\bf B_1$}\\ decay\n(see Eq.(\\ref{eq:eq0dltaEff})\nand Fig.\\ref{figSI0dltaEff}).\nAbove, the integral over $z$, is replaced by \\mbox{$\\delta_{\\text{eff}}$}\\, underneath the two $ab$\nfaces separated along $z$.\n\nSimilarly, for the signal intensity from {\\em either} of the $ac$ faces,\n\\begin{equation}\n S_{ac} (x,y)\\propto dx\\ dy\\ \\int dz = dx\\ \\delta_{\\text{eff}}\\ c\n\\label{eq:eq0sacxy}\n\\end{equation}\nsince the subsurface now is $\\perp y$.\n\nEq.(\\ref{eq:eq0sabxy}) and Eq.(\\ref{eq:eq0sacxy}), reveal {\\em differing\neffective elemental volumes}(voxels) underneath these faces:\n\\begin{equation}\ndV_{\\text{eff}}^{\\text{ab}} = dx\\ dy\\ \\delta_{\\text{eff}}\n\\label{eq:eq0dVeffab}\n\\end{equation}\n\\begin{equation}\ndV_{\\text{eff}}^{\\text{ac}} = dx\\ \\delta_{\\text{eff}}\\ dz \n\\label{eq:eq0dVeffac}\n\\end{equation}\nFrom Eq.(\\ref{eq:eq0sabxy}) and Eq.(\\ref{eq:eq0sacxy}),\n\\begin{equation}\n \\frac{S_{ac}}{S_{ab}} =\\frac{c}{2\\Delta y}\n\\label{eq:eq0ratioxy}\n\\end{equation}\nwhere we have replaced $dy$ by $\\Delta y$, the resolution along\n$y\\ \\parallel b$.\nConsulting the Methods section \\ref{subsxn0mthdsMri} and Fig.\\ref{fig0phntms},\n$c=7$ $mm$, $\\Delta y$=0.25 $mm$ and Eq.(\\ref{eq:eq0ratioxy}) yields a\ncalculated ratio of $S_{ac}\/S_{ab}$= 14\n(as illustrated in Fig.\\ref{fig0xy0yz0drvd}a),\nwithin an order of magnitude of the observed ratio (section \\ref{sxn0mri},\nFig.\\ref{fig0xy0yz}a) and a 25 fold improvement relative to the expected ratio\n(Fig.\\ref{fig0xy0yz0sim}a).\n\nAlso, Eq.(\\ref{eq:eq0ratioxy}) reveals that $S_{ac}\/S_{ab}$ increases with\nincreasing resolution along $y$, as shown in the three images in\nFig.\\ref{figSI0td20td40td80},\nwith relative resolutions increasing by factors of 1, 2 and 4,\nyielding calculated  $S_{ac} \/ S_{ab}$ ratios of 7, 14 and 28 respectively.\nThe corresponding observed ratios (of maxima of $S_{ac}$ and $S_{ab}$) of\n3.3, 6.6, and 11.6, are within an order of magnitude of the calculated values.\nRemarkably, these observed ratios themselves increase by factors of 1, 2 and\n3.5, mimicking closely the corresponding factors of resolution increase.\n\\\\\n\nContinuing in the same vein, for the 2d MRI($yz$), in Fig.\\ref{fig0xy0yz}b,\n\\begin{equation}\n S_{ab} (y,z) \\propto dy\\ dz \\int dx = a\\ dy\\ \\delta_{\\text{eff}}\n\\label{eq:eq0sabyz}\n\\end{equation}\nwhile,\n\\begin{equation}\n S_{ac} (y,z) \\propto dy\\ dz \\int dx = a\\ \\delta_{\\text{eff}}\\ dz\n\\label{eq:eq0sacyz}\n\\end{equation}\nleading to\n\\begin{equation}\n \\frac{S_{ac}}{S_{ab}}= \\frac{\\Delta z}{\\Delta y}\n\\label{eq:eq0ratioyz}\n\\end{equation}\nonce again, replacing $dy,\\ dz$ by $\\Delta y,\\ \\Delta z$, the respective\nresolutions along $y,\\ z$.\nUsing the values of $\\Delta y, \\Delta z$= 0.0357, 1 $mm$\nin Eq.(\\ref{eq:eq0ratioyz}), ensues\na calculated ratio of $S_{ac} \/ S_{ab} \\approx$ 28\n(as illustrated in Fig.\\ref{fig0xy0yz0drvd}b),\nwithin an order of magnitude of the observed ratio (section \\ref{sxn0mri},\nFig.\\ref{fig0xy0yz}b), and 3.5 fold better than the expected ratio\n(Fig.\\ref{fig0xy0yz0sim}b). More importantly, the expected intensity pattern is \neven {\\em qualitatively} (visually) different from the experiment, unlike the\nderived pattern.\n\n\nSimilarly, for the 2d MRI($zy$) in Fig.\\ref{fig0p3hrzntl}b, of phantom P3 in\n{\\em horizontal} orientation, it can be easily shown that, \n\\begin{equation}\n \\frac{S_{ac}}{S_{ab}}= \\frac{\\Delta y}{\\Delta z}\n\\label{eq:eq0ratiozy}\n\\end{equation}\nUsing the values of $\\Delta z, \\Delta y$= 0.0357, 1 $mm$\nin Eq.(\\ref{eq:eq0ratiozy}), results in a calculated ratio of\n$S_{ac} \/ S_{ab} \\approx$ 28, within an order of magnitude of the observed\nratio (section S\\ref{sxnSI0mri2dNtnst}).\\\\\n\nProceeding along the same lines, for the MRI($xyz$) in Fig.\\ref{fig0xyz},\n\\begin{equation}\n S_{ab} (x,y,z) \\propto dx\\ dy\\ dz =  dx\\ dy\\ \\delta_{\\text{eff}}\n\\label{eq:eq0sabxyz}\n\\end{equation}\nwhile,\n\\begin{equation}\n S_{ac} (x,y,z) \\propto dx\\ dy\\ dz =  dx\\ \\delta_{\\text{eff}}\\ dz\n\\label{eq:eq0sacxyz}\n\\end{equation}\nAs usual by now, replacing $dy,\\ dz$ by $\\Delta y,\\ \\Delta z$, the respective\nresolutions along $y,\\ z$, we obtain again Eq.(\\ref{eq:eq0ratioyz}).\nConsulting the Methods section \\ref{subsxn0mthdsMri}, \n$\\Delta y, \\Delta z$= 0.25, 1 $mm$, respectively.  Using these values in \nEq.(\\ref{eq:eq0ratioyz}) yields a calculated ratio of $S_{ac} \/ S_{ab}$= 4,\nwithin an order of magnitude of the observed ratio\n(section \\ref{sxn0mri}).\n\nFig.\\ref{fig0xyzSlc25} shows (in stack plot representation, with vertical axis\ndenoting intensity) $xy$ slices (along $z$) from the 3d MRI($xyz$).\nThe central slice contains no signal between the walls of intensity (from $ac$\nfaces) as expected.\nHowever, the slice from the top $ab$ face exhibits a {\\em plateau} of intensity\nbetween the $ac$ faces, visually demonstrating that $S_{ab} \\neq S_{ac}$ in the \nnon-hollow regions of the 3d image, in place of the expected uniform intensity.\n\\\\\n\nSimilarly, for the 3d CSI($yz$) in Fig.\\ref{fig0csi}, it can be shown that\nthe ratio $S_{ac} \/ S_{ab}$ is given by Eq.(\\ref{eq:eq0ratioyz}), which along\nwith the relevant experimental parameters for this image,\nyields a calculated value of 2, within an order of magnitude of the observed\nratio (of maxima of $S_{ac}$ and $S_{ab}$) of 2.8.\n\nOn the other hand, for the 2d CSI($y$), in Fig.\\ref{figSI0csi}, it can be shown \nthat the ratio $S_{ac} \/ S_{ab}$ is given by Eq.(\\ref{eq:eq0ratioxy}), from\nwhich we obtain a calculated value of 14, using the experimental parameters in\nsection \\ref{subsxn0mthdsMri}.  The measured ratio (of maxima of $S_{ac}$ and\n$S_{ab}$) of 9.5, is again within an order of magnitude of the calculated value.\n\\\\\n\nThus, the $S_{ac} \/ S_{ab}$ ratios calculated from\nEqs.(\\ref{eq:eq0ratioxy}), (\\ref{eq:eq0ratioyz}) and (\\ref{eq:eq0ratiozy}),\nagree with the observed values within an order of magnitude\nfor 2d MRI($xy$), 2d MRI($yz$), 2d MRI($zy$), 3d MRI($xyz$), 3d CSI($yz$) and\n2d CSI($y$).\nIn fact, discrepancies between observed and derived $S_{ac}\/S_{ab}$ ratios range\nonly by factors of 0.7 to 2.8 across various MRI and CSI images\n(see Table.\\ref{table:tbl0ratio}).\nMore importantly, the derived patterns {\\em resemble} the observed patterns,\nunlike the expected patterns, which differ even visually from the observed\npatterns (for e.g., see\nFigs.\\ref{fig0xy0yz}, \\ref{fig0xy0yz0sim} and \\ref{fig0xy0yz0drvd}).\n\n\\begin{table}[h]\n \\caption{\n  The observed, derived (from\n  Eqs.(\\ref{eq:eq0ratioxy}), (\\ref{eq:eq0ratioyz}), (\\ref{eq:eq0ratiozy})),\n  and expected (from skin depth arguments alone) $S_{ac} \/ S_{ab}$ ratios.\n }\n \\begin{tabular}{|c| c c c|}\n  \\hline\n    & & $S_{ac} \/ S_{ab}$    &  \\\\\n  \\hline\n   Experiment & Observed & Derived & Expected \\\\\n\\hline\nMRI($xy$) & & & \\\\\nFig.\\ref{figSI0td20td40td80}a &  3.3 &  7 & 368 \\\\\nFig.\\ref{figSI0td20td40td80}b &  6.6 & 14 & 368 \\\\\nFig.\\ref{figSI0td20td40td80}c & 11.6 & 28 & 368 \\\\\n & & & \\\\\nMRI($yz$)    & 10  & 28  &   1 \\\\\nMRI($zy$)    & 10  & 28  &   1 \\\\\nMRI($xyz$)   & 3.8 &  4  &   1 \\\\\n2d CSI($y$)  & 9.5 & 14  & 368 \\\\\n3d CSI($yz$) & 2.8 &  2  &   1 \\\\\n\\hline\n \\end{tabular}\n \\label{table:tbl0ratio}\n\\end{table}\n\nIn summary, the formulae unveil the underlying reason for the significant\ndeparture of observed $S_{ac}\/S_{ab}$ from expected values:\n{\\em differing effective elemental volumes underneath these faces},\nas revealed by Eqs.(\\ref{eq:eq0dVeffab}) and (\\ref{eq:eq0dVeffac}).\nThe derived patterns bear closer resemblance to\nexperiment, than what is expected from skin depth consideratons alone, or from\nconventional specifications of the voxel= $\\Delta x \\Delta y \\Delta z$ (see for\ne.g.  Figs.\\ref{fig0xy0yz}, \\ref{fig0xy0yz0sim} and \\ref{fig0xy0yz0drvd}).\n\nOn a practical note, these formulae can guide experimental strategies to\nrelatively enhance MRI and CSI signals from different regions of the bulk metal.\n\n\n\\section{Conclusions}\n\\label{sxn0cnclusn}\nIn conclusion, the unexpected findings presented here may impact bulk metal MRI \nand CSI studies in general,\nvia fresh insights\nfor data collection, analysis and interpretation.\nThe bulk metal MRI and CSI\n(correlating different bulk metal surfaces with distinct chemical shifts)\nresults in this study have the noninvasive diagnostic potential in other fields\nsuch as\nstructure  of metals and alloys \\cite{rdrgz,flyn},\nmetallurgy (metal fatigue, fracture, strain)\n\\cite{mtllurg,bppag,mvrkks},\ncatalysis\n\\cite{zhong,yuan},\nbulk metal surface science and surface chemistry\n\\cite{tlptr,whttn,rgg},\nmetallic medical implants, dielectric MRI in the vicinity of bulk metals\netc.\n(section \\ref{sxn0intrdxn}).\n\nThe findings may also lead to as yet unforeseen applications\n(section \\ref{sxn0intrdxn}) since,\n(i) they are of a fundamental nature,\n(ii) there are no inherent limitations to the approach employed\n(scalability, different metals, systems other than batteries, etc., are all\npossible),\n(iii) the study utilizes only standard MRI tools (hardware, pulse sequences,\ndata acquisition and processing), ensuring ease of implementation and\nreproducibility. Thus it is likely to benefit from advances made in the\nmainstream (medical) MRI field.\n\n\n\n\n\\section{Methods}\n\\label{sxn0mthds}\n\\subsection{Phantoms}\n\\label{subsxn0mthdsPhntms}\nAll phantoms were assembled and sealed in an argon filled glove box.\nAll three phantoms, P0, P1 and P3, shown in Fig.\\ref{fig0phntms} \n(of dimensions $a \\times b \\times c$),\nwere derived from a (0.75 $mm$ thick) stock Li strip (Alfa Aesar 99.9\\%).\nThe Li strips were mounted on a 2.3 $mm$ thick teflon strip and the resulting\nsandwich bound together with Kapton tape.\nEach phantom was placed in a flat bottom glass tube (9.75 $mm$ inner diameter\n(I.D.), 11.5 $mm$ outer diameter (O.D.) and  5 $cm$ long), with\nthe longest side, $a$, $\\parallel$ to the tube axis and to the axis of the\nhome built horizontal loop gap resonator (LGR)  r.f. coil (32 $mm$ long, 15 $mm$\nO.D.), thus guaranteeing $\\mbox{$\\bf B_1$} \\parallel a$.\nThe phantom containing glass tubes were wrapped with Scotch tape\nto snugly fit into the r.f. coil.\n\nIn  Fig.\\ref{fig0phntms}, $x,y,z$ specify the imaging (gradient) directions,\nwith $z \\parallel \\mbox{$\\bf B_0$}$ (the main magnetic field).\nFor our horizontal LGR r.f. coil (the MR resonator) and the gradient assembly\nsystem, $\\mbox{$\\bf B_1$} \\parallel x$, resulting in $\\mbox{$\\bf B_1$} \\parallel x \\parallel a$.\n\\begin{itemize}\n\\item {\\bf P0}: Pair of Li strips separated by a teflon strip;\nfor each Li strip,\n$a \\times b \\times c=$ 20 x 0.75 x 7 $mm^3$.\n\\item\n{\\bf P1}: Single Li strip.\n$a \\times b \\times c=$ 15 x 0.75 x 7 $mm^3$.\n\\item\n{\\bf P3}: Three  Li strips pressed together to yield a single\ncomposite super strip.\n$a \\times b \\times c=$ 15 x 2.25 x 7 $mm^3$.\n\\end{itemize}\n\n\n\\subsection{MRI and CSI}\n\\label{subsxn0mthdsMri}\nMagnetic resonance experiments were conducted on a $B_0$=21$T$ magnet\n(corresponding to \\mbox{$^7$Li} Larmor frequency of 350 MHz)\noperating under Bruker Avance III system with Topspin spectrometer control and\ndata acquisition, and equipped with a triple axes ($x,y,z$) gradient amplifier\nassembly, using a multinulcear MRI probe\n(for a triple axes 63 $mm$ I.D. gradient stack by Resonance Research Inc.), \nemploying the LGR r.f. coil (resonating at 350 MHz) desribed above.\n\n\nThe MRI and CSI data were acquired using spin-echo imaging pulse sequence\nwithout slice selection \\cite{callaghan,haacke} (yielding sum total of signal\nconributions from the non imaged dimensions).\nFrequency encoding gradient was\nemployed for the directly detected dimension and phase encoding gradients for\nthe indirect dimensions \\cite{callaghan,haacke}.\nThe CSI experiments were carried out with the NMR chemical shift as the directly\ndetected dimension, with phase encoding gradients along the indirectly detected \nimaging dimensions.\nThe r.f. pulses were applied at a carrier frequency of 261 ppm (to excite the\nmetallic \\mbox{$^7$Li} nuclear spins in the Knight shift region\n\\cite{rangeet,abragam}),\ntypically with a strength of 12.5 $kHz$, with a recycle (relaxation) delay of\n0.5 $s$.\nThe gradient dephasing delay and phase encoding gradient duration were 0.5 $ms$.\n\nThroughout this manuscript,\nthe first axis label ($x,y,z$) describing an MRI experiment stands for frequency\nencoding dimension and the remaining ones correspond to phase econded\ndimensions.   For e.g., MRI($xyz$) implies frequency encoding along $x$ axis,\nand phase encoding along the remaining directions.\n\n$G_x,G_y,G_z$ and $N_x,N_y,N_z$ denote respectively the gradient strengths in\nunits of $T\/m$ and number of data points in $k$-space ($^*$ denoting complex\nnumber of points acquired in quadrature) \\cite{callaghan,haacke}, along\n$x,y,z$ axes.\n$L_x,L_y,L_z$ and $\\Delta x,\\Delta y,\\Delta z$ are respectively the resultant\nnominal field of view (FOV) and resolution, in units of $mm$, along $x,y,z$ axes\n\\cite{callaghan,haacke}.\n\nAlso, $n$ is the number of transients accumulated for signal averaging and\n$SW$ is the spectral width (in units of $kHz$) for the directly detected\ndimension in MRI and CSI.\n\\\\ \\\\\n{\\bf 1d MRI({\\em y}):} \\\\\n$n=64,\\ SW=50$ \\\\\n$G_y=0.42,\\ N_y^*=200,\\ L_y=7.143,\\ \\Delta y= 0.0357$\n\\\\\n{\\bf 2d MRI({\\em xy}):} \\\\\n$n=32,\\ SW=100$ \\\\\n$G_x=0.24,\\ N_x^*=200,\\ L_x=25,\\ \\Delta x= 0.125$ \\\\\n$L_y=10$ \\\\\n(1) $G_y=0.12,\\ N_y=20,\\  \\Delta y= 0.500$\n     (Figs.\\ref{fig0twoStrps}a,\\ref{figSI0td20td40td80}a) \\\\\n(2) $G_y=0.24,\\ N_y=40,\\  \\Delta y= 0.250$\n     (Figs.\\ref{fig0twoStrps}b,\\ref{fig0xy}b,\\ref{fig0xy0yz}a,\n     \\ref{figSI0td20td40td80}b) \\\\\n(3) $G_y=0.48,\\ N_y=80,\\  \\Delta y= 0.125$ (Fig.\\ref{figSI0td20td40td80}c) \\\\\n{\\bf 2d MRI({\\em yz}):} \\\\\n$n=32$ \n$SW=50$ \\\\\n$G_y=0.42,\\ N_y^*=200,\\ L_y=7.143,\\ \\Delta y= 0.0357$ \\\\\n$G_z=0.06,\\  N_z=16,\\    L_z=16,\\ \\Delta z= 1.000$\n\\\\\n{\\bf 2d MRI({\\em zy}):} \\\\\n$n=32,\\ SW=50$ \\\\\n$G_z=0.42,\\ N_z^*=200,\\ L_z=7.143,\\ \\Delta z= 0.0357$ \\\\\n$G_y=0.06,\\  N_y=16,\\    L_y=16,\\ \\Delta y= 1.000$\n\\\\\n{\\bf MRI({\\em xyz}):} \\\\\n$n=16,\\ SW=100$ \\\\ \n$G_x=0.24,\\ N_x^*=200,\\ L_x=25,\\ \\Delta x= 0.125$ \\\\\n$G_y=0.24,\\ N_y=40,\\    L_y=10,\\ \\Delta y= 0.250$ \\\\\n$G_z=0.06,\\  N_z=16,\\    L_z=16,\\ \\Delta z= 1.000$\n\\\\\n{\\bf 2d CSI({\\em y}):} \\\\\n$n=8,\\  SW=100$, number of data points(complex)=$1024$ \\\\ \n$G_y=0.24,\\ N_y=40,\\    L_y=10,\\ \\Delta y= 0.250$\n\\\\\n{\\bf 3d CSI({\\em yz}):} \\\\\n$n=24,\\ SW=100$, number of data points(complex)=$1024$ \\\\ \n$G_y=0.12,\\ N_y=20,\\    L_y=10,\\ \\Delta y= 0.500$ \\\\\n$G_z=0.06,\\  N_z=16,\\    L_z=16,\\ \\Delta z= 1.000$ \\\\\n\\\\\nAll data were processed in Bruker's Topspin, with one zero fill prior to complex\nfast Fourier Transform (FFT) along each dimension either without any window\nfunction or with sine-bell window function.\nAll data were 'normalized' (to $\\approx 10$, for plotting convenience) to aid\ncomparing {\\em relative} intensities from different regions {\\em within} a given\nimage.\nFor the purpose of determining the ratios of signal intensities associated\nwith different regions of the bulk metal, the intensity values were measured\ndirectly from the processed images either in Topspin or Matlab (for e.g.,\n'datatip' utility in Matlab, yields the coordinates and the 'value' (intensity) \nof a data point by clicking on it, in 1d, 2d and 3d plots). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nApplication of external pressure is a powerful method to tune the intricate interplay of competing energy scales in correlated materials and the emergence of novel unconventional phases in a clean fashion. It offers significant advantages as a control parameter compared with chemical substitution and application of magnetic fields, because it does not introduce additional disorder as in the case of substitution of one element by an other or polarize the electrons as a magnetic field does.\n\nA large variety of experimental setups has been developed to probe physical properties under hydrostatic pressure.\\cite{Nicklas2015} In contrast, experiments under uniaxial pressure appeared to be limited to low pressure and only few experimental probes. Recently, however, the development of piezoelectric-driven pressure devices opened a new perspective.\\cite{Hicks2014a,Barber2019} These devices allow the application of large positive and negative pressures and the amplitude of the applied pressure can be easily changed at low temperatures. In a short period of time experimental stages to access a large number of physical properties of materials have been developed. These include electrical transport,\\cite{Stern2017,Steppke2017} magnetic susceptibility,\\cite{Hicks2014b} nuclear-magnetic resonance,\\cite{Kissikov2017,Luo2019,Pustogow2019} muon-spin resonance,\\cite{Grinenko2020} and angle-resolved photo emission, for which mechanically or thermally activated cells have also been introduced.\\cite{Flototto2018,Ricco2018,Pfau2019a,Pfau2019b,Sunko2019}\n\nAn important quantity to characterize a material is the specific heat, which is the fundamental thermodynamic property giving information on the internal degrees of freedom of a material and the entropy related with them.\nTo address the experimental challenge of studying the heat capacity under large uniaxial pressures, we employ a variation of known AC heat-capacity measurement techniques.\\cite{Sullivan1968} Heat capacity measurement has been combined with uniaxial pressure previously,\\cite{Jin1992,Reinders1994,Miclea2002,Zieve2004,Dix2009} but with traditional, anvil-based uniaxial pressure cells. Samples have been thermally isolated by using low thermal conductivity materials, such as stainless steel or superconducting NbTi, as piston or additional spacer.  However in previous anvil-based uniaxial-pressure measurements, e.g.\\ on the unconventional superconductor Sr$_2$RuO$_4$, it did not prove practical to maintain high stress homogeneity,\\cite{Kittaka2010,Taniguchi2015} which is one of the main challenges in carrying out this kind of experiments. Furthermore, under applied uniaxial pressure the samples even may deform plastically. To reduce these effects we apply force to the sample through a layer of epoxy,\\cite{Hicks2014a} which acts as a conformal layer that dramatically improves stress homogeneity. However, it also makes heat-capacity measurement more challenging, because the epoxy layer provides an unavoidable strong thermal link to the pressure cell.\n\nFor our study we have used Sr$_2$RuO$_4$ that provides a demanding test of our new apparatus. Sr$_2$RuO$_4$ is an unconventional superconductor with a superconducting transition temperature up to $T_c=1.5$~K in the best crystals.\\cite{Mackenzie1998,Mackenzie2003,Mackenzie2017} From resistivity and magnetic susceptibility experiments it is known that $T_c$ shows a pronounced dependence on the applied uniaxial pressure.\\cite{Hicks2014b,Steppke2017,Barber2018} This and the sharp superconducting transition anomaly make it an ideal material to demonstrate the potential of our technique for the study of correlated materials. A successful experiment on Sr$_2$RuO$_4$ can only be done using a technique that introduces no disorder or plastic deformations, and that probes a region in which the strain induced in the sample is highly homogeneous.\n\n\n\n\\section{METHOD}\n\nFor a setup in which the sample is strongly coupled to the environment as it is in a pressure cell, whether hydrostatic or uniaxial, standard quasi-adiabatic or relaxation techniques are limited to cases where the heat capacity of the whole pressure cell including the sample is measured and the heat capacity of the sample can then be separated from the (large) addenda. That implies restrictions on the materials which can be investigated and limits experiments to low temperatures. The advantage of such a technique is that one obtains absolute values of the heat capacity, but the resolution and the pressure regime are limited. In heat-capacity measurements at higher pressure, where anvil-type cells are used, or for uniaxial pressure experiments, the application of this technique is not possible anymore. The mass of sample is negligible with respect to that of the pressure apparatus. In these cases the heat capacity can only be determined using an AC heat capacity measurement technique.\\cite{Sullivan1968} With the AC technique it is possible to record heat capacity data in a wide range of parameter space on a sample which is not well thermally isolated from its environment by adjusting the measurement frequency. The drawback is that it is generally challenging to obtain absolute values of the heat capacity and one usually has to be content with data having arbitrary units. As we demonstrate, however, it can still yield a wealth of useful information.\n\n\\subsection{AC Heat Capacity}\\label{AC Heat Capacity}\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.9\\linewidth]{Scheme_HC}\n\\centering\n\\caption{\n(a) Schematic drawing of the thermal couplings of a sample in an AC heat-capacity setup.\n(b) A schematic diagram of the frequency response curve $F$ against $\\omega$. $\\omega$ is the angular frequency of the temperature oscillation. The curve can be divided into three regions separated by $\\omega_1$ and $\\omega_2$.\n }\n\\label{Scheme_HC}\n\\end{figure}\n\nIn the AC heat capacity measurement technique an alternating current is applied at frequency $\\omega\/2$ to the heater, leading to an AC heat power at frequency $\\omega$ to determine the heat capacity $C_{AC}$. Here $\\omega=2\\pi f$ is the angular frequency.\nThe governing relationship for measurements of the AC heat capacity is\n\\begin{equation}\n  C_{AC}=  \\frac{P}{\\omega T_{AC} } F(\\omega).\n\\label{Cac}\n\\end{equation}\n$P$ is the average power and $F(\\omega)$ is a frequency response curve that characterizes the thermalization of the sample, and differs from sample to sample, because it depends on time constants determined by thermal conductances and heat capacities of the system.\n$F(\\omega)$ depends on the time constants $\\tau_1$ and $\\tau_2$:\n\\begin{equation}\\label{F_omega}\n  F(\\omega)=\\left[ 1+\\frac{1}{\\omega^2\\tau_1^2}+\\omega^2\\tau_2^2\\right] ^{-1\/2}.\n\\end{equation}\n$\\tau_1=C_{AC}\/k_b$ describes the time scale of the applied heat power decaying to the environment, whereas  $\\tau_2=\\sqrt{\\tau_h^2+\\tau_{\\theta}^2+\\tau_{\\rm int}^2}$ describes the internal thermal time scale within the system itself.\nHere $\\tau_h=C_h\/k_h$,$\\tau_{\\theta}=C_{\\theta}\/k_{\\theta}$  and $\\tau_{\\rm int}=C_s\/k_s$ (see also Fig.\\ \\ref{Scheme_HC}a). The time constants $\\tau_h$ $\\tau_{\\theta}$ and $\\tau_{\\rm int}$ describe the time scales for the heater and, thermometer and sample to be thermalized, respectively. $C_h$, $C_{\\theta}$, and $C_s$ are the heat capacity of the heater, thermometer, and sample, respectively. For a good design, the responses of heater and thermometer need to be fast so one should aim at $\\tau_{\\rm int}\\gg \\tau_h$ and $\\tau_{\\theta}$.\n\nA schematic diagram  of $F(\\omega)$ is shown in Fig.\\ \\ref{Scheme_HC}b. At low frequencies, indicated as regime I, $\\omega\\ll\\omega_1= 1\/\\tau_1=k_b\/C_{AC}$, $F(\\omega)$ is reduced due to dissipation of temperature oscillations into the environment and at high frequencies, $\\omega\\gg\\omega_2= 1\/\\tau_2\\cong k_s\/C_s$, because the heater-sample-temperature sensor system does not thermalize, marked as regime III.\nIn the plateau region between these limits $F(\\omega)\\approx 1$ and\n\\begin{equation}\\label{C_ac_prox}\nC_{AC}\\approx  \\frac{P}{\\omega T_{AC} }.\n\\end{equation}\n\nIn addition to the temperature oscillations the application of the oscillatory heating power leads to an temperature offset $T_{DC}$ in the sample, which can be determined in the low frequency limit $\\omega\\ll\\omega_1$. Here $F(\\omega)=\\omega\\tau_1$ and the temperature offset can be estimated as\n\\begin{equation}\\label{T_DC}\nT_{DC}\\approx  \\frac{P}{k_b}.\n\\end{equation}\n\n\n\n\n\\subsection{Experimental Setup}\n\n\nThe general considerations in the section above show that it should be possible to measure the heat capacity in an uniaxial pressure apparatus as shown in Fig.\\ \\ref{Scheme_thermalConductivity}a by choosing the correct set of experimental parameters. In the following we will explain the experimental setup and describe the details of the preparation process using the example of a Sr$_2$RuO$_4$ single crystal.\n\n\nThe sample is marked by a red circle in Fig.\\ \\ref{Scheme_thermalConductivity}a and shown in detail in Fig.\\ \\ref{Scheme_thermalConductivity}b. In this setup the applied force results in a normal strain\n\\begin{equation}\\label{strain}\n\\varepsilon_{xx}=\\frac{l-l_0}{l_0}\n\\end{equation}\nin the sample. Here $l_0$ is the length of the unstrained sample and $l$ the length of the strained sample. The length change is measured capacitively and can be controlled. \\MN{}{The applied strain can go beyond $1\\%$. In Sr$_2$RuO$_4$ the Young's modulus is about 180~GPa and correspondingly the applied uniaxial pressure can reliably reach up to about 2~GPa. However, the maximum uniaxial pressure depends strongly on the mechanical properties of the investigated material.} Further details can be found in Ref.\\ \\onlinecite{Hicks2014a}.\nThe present AC heat-capacity technique can be adapted to different types of uniaxial pressure devices, e.g.\\ to a stress-controlled apparatus \\cite{Barber2019} \\MN{}{and  is fully compatible with experiments in magnetic fields}\n\n\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.9\\linewidth]{Scheme_thermalConductivity}\n\\centering\n\\caption{\n(a) Photograph of the uniaxial pressure apparatus used in the present study. The red circle marks the sample region.\n(b) Photograph of the setup of the heat capacity measurements under strain including heater and thermometer. The sample is glued between the jaws of the uniaxial pressure device.  \\MN{}{The exposed length, width and thickness of the shown sample are 2~mm, 200~$\\mu$m and 150~$\\mu$m, respectively. The device} allows the application of compressive and tensile strains. The red, yellow, and white rectangles represent the (quasi)homogeneous, inhomogeneous, and unstrained regions, respectively, see text for details.\n(c) Schematic diagram of the setup illustrating the photograph in (b).}\n\\label{Scheme_thermalConductivity}\n\\end{figure}\n\n\nIn Fig.\\ \\ref{Scheme_thermalConductivity}b we show a photograph of the bar-shaped sample that has been carefully cut, polished, and then mounted within the jaws of the uniaxial pressure rig.\nThe nature of the apparatus means that only the central part of the sample is homogeneously strained. Force is transferred to the sample through the epoxy layer around the sample. The sample ends which are protruding beyond it are unstrained, and there are intermediate regions, marked in yellow in Fig.\\ \\ref{Scheme_thermalConductivity}b, where the strain is built up. Therefore, we have to choose the measurement conditions in a way that we only probe the homogeneous part of the sample. On the example of a Sr$_2$RuO$_4$ single crystal we will demonstrate that this is in principle possible by varying the excitation frequency $f_{\\rm exc}=f\/2$ of the heater, if the characteristic parameters of the setup, such as the different thermal conductances, have been chosen in the appropriate range.\n\nFor the experiments single crystalline Sr$_2$RuO$_4$ was aligned using a bespoke Laue x-ray camera, and cut using a wire saw into thin bars with whose long axis aligned with the [100] direction of the crystal. For the best results these bars were polished using home-made apparatus based on diamond impregnated paper with a minimum grit size of 1~$\\mu$m. The bar was then mounted within the jaws of the uniaxial pressure rig using Stycast 2850FT epoxy with Catalyst 23LV (Henkel Loctide). A resistive thin film resistor chip (State of the Art, Inc., Series No.:\\ S0202DS1001FKW) as heater and a Au-AuFe(0.07\\%) thermocouple are fixed to opposite sides of the sample using Dupont 6838 single component silver-filled epoxy. \\MN{}{The resistance of heater is about 640~$\\Omega$ and the applied power is in the range of $\\mu$W. The heater is connected electrically using manganin wires providing a low thermal conductance to the bath. At 1~K the thermal conductance of the Stycast layers and the manganin wires is about $10^{-4}$ and $10^{-7}$~ W\/K, respectively. Thus, the heat loss is largely dominated by the Stycast layers.} The thermocouple was spot-welded in-house and its calibration fixed by reference to that of a calibrated RuO$_2$ thermometer.\\footnote{In the temperature range between 0.15 and 4.5~K the thermopower $S$ of the Au-AuFe(0.07\\%) thermocouple is described by $S(T)=[10.1483\\cdot T\/{\\rm K}-8.75772\\cdot (T\/{\\rm K})^2+4.00231\\cdot (T\/{\\rm K})^3-0.838741\\cdot (T\/{\\rm K})^4+0.0667604\\cdot (T\/{\\rm K})^5]{\\rm ~\\mu V\/K}$ .} Special care was taken when epoxying to the pressure cell to minimize tilt and ensure an as homogeneous strain field as possible.\n\n\nThe uniaxial pressure apparatus was mounted on a dilution refrigerator (Oxford Instruments), with thermal coupling to the mixing chamber via a high purity silver wire. The data were acquired between 500~mK and 4.2~K, with operation above 1.5~K achieved by circulating a small fraction of the mixture.  The extremely low noise level of 20~${\\rm pV\/\\sqrt{Hz}}$ on the thermocouple readout was achieved by the combination of an EG\\&G 7265 lock-in amplifier and a \\MN{}{high frequency} low temperature transformer (\\MN{}{LTT-h from} CMR direct) mounted on the 1~K pot of the dilution refrigerator, operating at a gain of 300. \\MN{}{The input impedance of the transformer is about $0.1~\\Omega$, which ensured a flat frequency response from several hundred Hz to several tens of kHz.} A Keithley 6221 low-noise current source was used to drive the heater. The piezo-electric actuators were driven at up to $\\pm400$ V using a bespoke high-voltage amplifier.\n\n\n\n\\subsection{Strain inhomogeneity}\n\nThe nature of our setup is that the strain profile along the direction of the application of the uniaxial pressure is not homogeneous.\nAs we will describe below, by adjusting the excitation frequency $f_{exc}$ to an appropriate value the actual heat-capacity measurement can be confined to the quasi-homogeneously strained region of the sample. Besides this source of strain inhomogeneity there are other sources which can be reduced in the preparation and mounting process of the sample in the apparatus.\n\n\n\\subsubsection*{Imperfections of sample surface and geometry}\n\nThe bar-shaped needles cut from crystals have typically terraces and irregular shapes on their surface which can induce inhomogeneous strain fields when they are under uniaxial pressure. Imperfections may also lead to an early failure of pressurized samples reducing the maximum achievable pressure.\nA perfect sample is a cuboid, i.e.\\ each surface is parallel to the opposite one, and has a smooth surface roughness. Therefore, we carefully polish our samples and inspect the shape and the surface quality under a microscope before mounting in the uniaxial pressure apparatus.\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.95\\linewidth]{heater_mounting}\n\\centering\n\\caption{(a) Heater fixed with a silver foil to the sample. The contact to the sample is on the whole plane.\n(b) The simulation of the strain $\\varepsilon_{xx}$ pattern corresponding to the setup in (a). One of the silver-filled epoxy blocks was set to be invisible, as indicated by the\ndash-dotted lines, such that the strain profile on the edge of the sample is visible.\n(c) Heater fixed to the sample by four thin silver wires on the edges of the sample.\n(d) Corresponding simulation to (c). The strain inhomogeneity is reduced in the center compared with the setup shown in (a).\n }\n\\label{heater_mounting}\n\\end{figure}\n\n\\subsubsection*{Bending}\n\nAsymmetric mounting of a sample leads to bending.\\cite{Barber2017} An ideal sample mounting is a sample mounted between two plates with symmetrical epoxy layers on top and at the bottom. However, the sample might end up with a small offset in height. To reduce inhomogeneity in preparing the sample we aim for an aspect ratio $l_s\/t >10$,  where $l_s$ is the exposed length and $t$ the thickness of the sample\n\n\\subsubsection*{Mounting of the heater}\n\nOne of the main sources of inhomogeneous strain fields originates from the sample configuration in the AC heat-capacity setup. In order to transmit the heating power from the heater resistance to the sample, we use thermal contacts made by silver wires glued to the sample using silver-filled epoxy. Since the Young's modulus of silver and the sample are generally very different, as in our example of Sr$_2$RuO$_4$, the contacts create inhomogeneous strain fields. We tried to minimize this effect. We realized two different types of silver contacts to the sample. Figures \\ref{heater_mounting}a and \\ref{heater_mounting}c show photographs of the setups. In the first one a silver strip was glued on a contact length of about 300~$\\mu$m on both edges to the sample using silver-filled epoxy. In the second, the thermal contact is divided into 4 smaller areas instead of a large one, by gluing 8 silver wires with diameter of 50~$\\mu$m on both edges. The total contact area in both cases is almost the same.\nThe experiments on our test sample Sr$_2$RuO$_4$ showed indeed a significant sharpening of the superconducting transition anomaly in the latter case.\n\n\nIn addition to the experiments we simulated the strain fields in the sample by a finite element method using a commercial software package.\\footnote{Autodesk Inventor 2015, Autodesk Inc..}\nFor the simulation we set the Young's modulus of the sample to 180 GPa and the Poisson's ratio to 0.33. For the dimensions of the sample we used the values from the experiment, a thickness 100~$\\mu$m, width 300~$\\mu$m, and length 2~mm. One of the sample ends was set to be fixed and the other end was subjected to a pressure of 0.18~GPa, leading to $\\varepsilon_{xx}=0.1$\\%. The silver and silver-filled epoxy were set to have Poisson's ratio of 0.35. The Young's modulus for the silver epoxy was set to be $1\/3$ of that of the silver, which is 110~GPa. The results for both configurations are shown in Figs.\\ \\ref{heater_mounting}b and \\ref{heater_mounting}d. The color bar shows the strain scale, ranging from 0.07 to 0.13\\%. In the first configuration, with silver strip glued with silver-filled epoxy on both edges to the sample, the strain inhomogeneity on the sample is greater than 60\\% in the center (see Fig.\\ \\ref{heater_mounting}b).\nIn the second design using 8 silver wires with diameter of 50~$\\mu$m on both edges for the thermal contact, the strain inhomogeneity is strongly reduced in the bulk, except in the regions very close to the contact surfaces. Since heat capacity is a bulk-sensitive measurement, the inhomogeneity near the surface is negligible. The strain inhomogeneity in this configuration is only about $10$\\% in the center region of the sample. This shows that it is highly desirable to have separated smaller contact areas to transmit the heat to the sample in order to reduce strain inhomogeneities in accordance with the experimental results. In the following we continued with the second configuration.\n\n\n\n\n\\section{RESULTS}\n\nWe demonstrate the capabilities of our setup and discuss its advantages and limitations by showing representative data from experiments on Sr$_{2}$RuO$_{4}$. The first step in an AC heat-capacity experiment is to find a suitable measurement frequency in the plateau region of the frequency response curve $F(\\omega)$. We note that the existence of this plateau depends on the respective characteristics of the setup as discussed in Sec.\\ \\ref{AC Heat Capacity}. If a suitable frequency has been found, temperature \\MN{}{sweeps, also in applied magnetic field, or pressure\/magnetic field ramps} can be conducted and the heat capacity recorded. According to Eq.\\ \\ref{C_ac_prox} we will plot our results on Sr$_{2}$RuO$_{4}$ as $P\/[\\omega T_{AC}(T)]$. As we will discuss in Sec.\\ \\ref{Cv} $C_{AC}\\approx P\/(\\omega T_{AC})$ is not strictly valid in our setup and has to be treated with caution.\n\n\n\\subsection{Measuring frequency}\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.95\\linewidth]{frequency_effect}\n\\centering\n\\caption{(a) Frequency sweeps at 1 and 4.23~K.\n(b) Data  recorded at 313~Hz for zero and small strains up to $\\varepsilon_{xx}=-0.19$\\%.\n(c) Data on  Sr$_2$RuO$_4$ in the region around its superconducting transition at $\\varepsilon_{xx}=-0.19$\\% for different frequencies.\n }\n\\label{frequency_effect}\n\\end{figure}\n\n\n\nFigure \\ref{frequency_effect}a shows the frequency response at 1 and 4.23~K in case of our example Sr$_2$RuO$_4$ crystal. It shows a broad plateau between a few hundred hertz and several kilohertz at both temperatures, attesting that in principle heat-capacity measurements should be possible in the desired temperature range. By raising the temperature from 1 to 4.23~K the plateau narrows slightly but remains well-defined.\n\nIn the lower frequency part of the plateau in Fig.\\ \\ref{frequency_effect}a, temperature oscillations extend throughout the sample and all three regions the homogenously strained in the center, the unstrained portions at the ends and the regions where strain builds up are probed in a measurement (see Fig.\\ \\ref{Scheme_thermalConductivity}b). Figure \\ref{frequency_effect}b shows $P\/[\\omega T_{AC}(T)]$ recorded at $f_{exc}=313$~Hz for different $\\varepsilon_{xx}$. At zero strain we see a single sharp transition anomaly at $T_c\\approx1.45$~K. Upon increasing $|\\varepsilon_{xx}|$ the step-like feature moves to higher temperatures, consistent with the increase in $T_c$ with strain,\\cite{Hicks2014b} but a second feature remains at the original zero-strain transition. This latter feature stems from the unstrained part of the sample.\n\nTo reduce the size of the probed part of the sample and restrict it to the homogenously strained region in the center, we increased the measurement frequency. We note that we still stay in the plateau region of frequency response curve. To demonstrate the importance of this increase in measurement frequency, we applied modest strain $\\varepsilon_{xx}=-0.19$~\\% and increased $f_{exc}$ from 313~Hz in steps to 2503~Hz. The data are displayed in Fig.\\ \\ref{frequency_effect}c.\nAt 313 and 613~Hz, in addition to the peak at $\\approx 1.65$~K corresponding to the transition in the central, strained, portion of the sample, a smaller peak is visible at $\\approx 1.45$~K, corresponding to the transition in the end portions.  This feature shows that temperature oscillations extend into the sample ends at these frequencies.\nTo avoid this, one has to work at the high end of the feasible range of frequencies. For this particular sample, a measurement frequency above $\\sim1.5$~kHz was required.\nWorking at high frequencies with low enough power to avoid heating gives a very small signal, an r.m.s.\\ thermocouple voltage of only $1 - 2$~nV. Therefore, the described low temperature passive amplification was employed to achieve an r.m.s.\\ noise level of 20~pVHz$^{-1\/2}$, ensuring a signal-to-noise ratio in excess of 50.\n\n\n\n\n\\subsection{Heat-capacity results on Sr$_2$RuO$_4$}\n\nBased on considerations outlined in the previous section we selected an excitation frequency of $f_{exc}=1503$~Hz to measure the heat capacity of Sr$_2$RuO$_4$. The results for three different strains $\\varepsilon_{xx}=0$\\%, $-0.25$\\%, and $-0.37$\\% are presented in Fig.\\ \\ref{HC_Sr2RuO4} as $P\/[\\omega T_{AC}(T)]$. Additionally the inset shows the results from a standard relaxation-type heat-capacity measurement from a piece of sample cut from the same crystal. It is qualitatively similar to the results in the uniaxial pressure cell at zero strain.\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.95\\linewidth]{HC_Sr2RuO4}\n\\centering\n\\caption{Recorded signal $P\/[\\omega T_{AC}(T)]$ of Sr$_2$RuO$_4$ as function of temperature for three different strains $\\varepsilon_{xx}=0$\\%, $-0.25$\\%, and $-0.37$\\%. The inset shows a specific-heat experiment on a piece from the same crystal using a standard relaxation time method.\n}\n\\label{HC_Sr2RuO4}\n\\end{figure}\n\nAccording to Eq.\\ \\ref{C_ac_prox} we find $C_{AC}(T)\\approx P\/[\\omega T_{AC}(T)]$. However, this relation has to be taken with caution since the probed sample volume is not constant as function of temperature. We have selected $f_{exc}$ in order to probe the homogenously strained portion of the sample, but we have to notice that the thermal conductivity $\\kappa$ of any studied material varies as function of temperature and strain, and as a consequence the probed sample volume also changes. To obtain absolute values of the volume specific heat $c_v(T)$ at a certain strain the temperature dependence of the thermal conductivity $\\kappa(T)$ has to be known at that strain too.\n\n\n\n\\subsection{Determination of the volume specific heat}\\label{Cv}\n\nThe conversion between the measured signal and the volume or molar specific heat is trivial in a conventional setup, because the volume (or mass) of the sample is constant. In our measurements, the probed sample volume varies since the thermal diffusion length $l_d$, which depends on thermal conductivity, specific heat, and frequency, changes as a function of temperature. Therefore, it is nontrivial to convert our data $P\/[\\omega T_{AC}(T)]$ to volume specific heat $c_v$. We start with an ideal case to demonstrate the relation between $P\/[\\omega T_{AC}(T)]$ and $c_v$ in case of our experimental setup.\nSuppose that the heater contact is point-like in the center of a very narrow sample such that the heat flow is one-dimensional propagating in the left and right direction. The probed volume $V$ is equal to the cross-sectional area $A$ times twice the diffusion length $l_d$, which is a function of the angular frequency $\\omega$, the volume specific heat $c_v$ and the thermal conductivity $\\kappa$.\n\\begin{equation}\n  l_d=\\sqrt{\\frac{2\\kappa(T)}{\\omega c_v(T)}}\t\n  \\label{S1}\n\\end{equation}\n$C_{AC}$ can be expressed as follows:\n\\begin{equation}\\label{S2}\n  C_{AC}=c_v \\times V = c_v \\times A\\times l_d = \\frac{2A}{\\sqrt{\\omega}} \\sqrt{2\\kappa(T) c_v(T)}.\n\\end{equation}\nBy using Eq.\\ \\ref{Cac} and \\ref{S2} we finally obtain the volume specific heat $c_v$:\n\\begin{equation}\\label{S3}\n  c_{v}(T)=\\left (\\frac{P \\times F(\\omega)}{2A}\\right )^2 \\times \\frac{1}{\\omega\\times 2\\kappa(T)} \\times \\frac{1}{[T_{AC}(T)]^2}.\n\\end{equation}\t\nThis exemplifies the reciprocal dependence of $c_v(T)$ on the thermal conductivity and the square of the temperature-oscillation amplitude in case of a simplified one dimensional model.\n\nWe further note that the excitation frequency in our current measurement is not too far away from the upper cut-off frequency, which describes the time scale for the heat propagating from the heater to the thermocouple. At this excitation frequency $F(\\omega)<1$ and depends on temperature adding a further uncertainty on the determination of $c_v(T)$.\n\n\nThe validity of the Eqs.\\ \\ref{S2} and \\ref{S3} is based on the above-mentioned assumptions that the heater contact is point-like and the heat flow is one-dimensional. In reality, both the sample width and the heater contact size are finite. This implies for the experimental setup to satisfy the assumptions of the examined model system, the exposed sample length ($l_s$) must be far longer than the heater length ($l_h$) and the sample width ($w_s$), $l_s \\gg l_h,w_s$. Our present setup is already a good approximation to an ideal configuration but could in principle be further optimized.\n\nIn spite of the above caveats, we note that in some cases quantitative statements on the evolution of the specific heat on varying uniaxial pressure are possible based on the presently accessible data. For example, in superconductors, as in the case of Sr$_2$RuO$_4$, it is possible to obtain information on the evolution of the size of the superconducting transition anomaly with pressure, which is an important quantity characterizing superconductivity. In that case the thermal conductivity does not show any abrupt change across the transition and close to $T_c$\n\\begin{equation}\\label{S4}\n  \\frac{c_{v}^s}{c_{v}^n}=\\frac{\\kappa_n}{\\kappa_s} \\times \\left(\\frac{T_{AC}^n}{T_{AC}^s}\\right)^2 \\approx \\left(\\frac{T_{AC}^n}{T_{AC}^s}\\right)^2\n\\end{equation}\t\nwith $\\kappa_n\\approx\\kappa_s$.\nThe indices $s$ and $n$ indicate the corresponding values in the superconducting and in the normal state, respectively.\n\n\n\\section{CONCLUSION}\n\nWe have developed a new experimental setup using piezoelectric-driven uniaxial pressure cells for probing heat capacity at low temperatures. By optimizing our preparation and measuring processes we achieve an extremely high resolution and a high strain homogeneity in the probed sample volume.\nThe technique can be easily extended to different temperature regions. In addition to temperature sweeps, heat capacity can be recorded as function of applied pressure, and our apparatus is also fully compatible with work in magnetic fields.\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank A.\\ S.\\ Gibbs, F.\\ Jerzembeck, N.\\ Kikugawa, Y.\\ Maeno, D.\\ A.\\ Sokolov for providing and characterizing the samples and M.\\ Brando and U.\\ Stockert for experimental support.\n\\end{acknowledgments}\n\n\\section*{DATA AVAILABILITY}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\\section*{REFERENCES}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe ubiquitous appearance of translation symmetry in physical systems signals the importance of having a complete picture of the complex role it may play. In particular, although the ground state energy (associated with time-translation symmetry) of a many-body quantum system or a quantum field theory is frequently studied, the ground state \\textit{momentum} (associated with space-translation symmetry) is rarely discussed. Rather, in most cases one focuses on the momentum difference between excited states and the ground state. In this work we reveal a connection between the momentum and the entanglement structure of a quantum state, in the context of lattice spin (boson) systems:\n\\begin{theorem}\nIf a quantum state $|\\Psi\\rangle$ in a lattice spin (boson) system is an eigenstate of the lattice translation operator $T:|\\Psi\\rangle\\to e^{iP}|\\Psi\\rangle$ with a non-trivial momentum $e^{iP}\\neq1$, then $|\\Psi\\rangle$ must be long-range entangled, namely $|\\Psi\\rangle$ cannot be transformed to an un-entangled product state $|000...\\rangle$ through an adiabatic evolution or a finite-depth quantum circuit (local unitary). \n\n\n\\label{Thm}\n\\end{theorem}\n\nThe intuition behind this statement follows from the sharp difference between translation $T$ and an ordinary onsite symmetry $G$ that is defined as a tensor product of operators acting on each lattice-site (such as the electromagnetic $U(1)$). A product state may recreate any total symmetry charge $Q$ under $G$ by simply assigning individual local Hilbert space states to carry some charge $Q_\\alpha$ such that $Q=\\sum_{\\alpha}Q_\\alpha$. However in the case of non-onsite translation symmetry, all translation-symmetric product states, which take the form $|\\alpha\\rangle^{\\otimes L}$, can only carry trivial charge (lattice momentum). This suggests that non-trivial momentum is an inherently non-local quantity that cannot be reproduced without faraway regions still retaining some entanglement knowledge of each other, i.e. the state must be long-range entangled.\n\nIn condensed matter physics, we are often interested in ground states of translational-invariant local Hamiltonians. If the ground state is short-range entangled~\\cite{PhysRevB.82.155138} (SRE) in the sense that it is connected to a product state through a finite-depth (FD) quantum circuit, then we expect the ground state to be unique, with a finite gap separating it from the excited states. In contrast for long-range entangled~\\cite{PhysRevB.82.155138} (LRE) ground states, we expect certain ``exotic'' features: possible options include spontaneous symmetry-breaking cat states (e.g. GHZ-like states), topological orders (e.g. fractional quantum Hall states), and gapless states (e.g. metallic or quantum critical states). Theorem~\\ref{Thm} provides us an opportunity to explore the interplay between translation symmetry and the above modern notions. An immediate corollary is\n\\begin{corollary}\nIf a non-zero momentum state $|\\Psi\\rangle$ is realized as a ground state of a local spin Hamiltonian, then the ground state cannot be simultaneously unique and gapped. Possible options include (1) gapless spectrum, (2) intrinsic topological order and (3)  spontaneous translation symmetry breaking.\n\\end{corollary}\nIn fact, we show in Sec.~\\ref{sec:wCDW} that option (2) is a special subset of option (3) through the mechanism of ``weak symmetry-breaking\"~\\cite{Kitaev06}.\n\nOur result is reminiscent of the celebrated Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) theorems \\cite{LSM,Oshikawa1999,Hastings04}, which state that in systems with charge $U(1)$ and translation symmetries, a ground state with fractional $U(1)$ charge filling (per unit cell) cannot be SRE. In our case the non-trivial lattice momentum $e^{iP}\\neq1$ plays a very similar role as the fractional charge density in LSMOH. In fact, as we discuss in Sec.~\\ref{sec: LSMOH}, our theorem can be viewed as a more basic version of LSMOH that only involves translation symmetry, from which the standard LSMOH can be easily derived. As a by-product, we also discover a previously unknown version of LSMOH constraint that involves an onsite $\\mathbb{Z}_n$ symmetry and lattice translations. \n\nThe rest of this paper will be structured as follows: in Sec.~\\ref{sec:proof} we provide a proof of Theorem~\\ref{Thm} via a quantum circuit approach, and generalize it to fermion systems. Three consequences of Theorem~\\ref{Thm} are discussed in Sec.~\\ref{sec:consequences}: in Sec.~\\ref{sec: LSMOH} we discuss several LSMOH-type theorems; in Sec.~\\ref{sec:wCDW} we show that a gapped topological order must \\textit{weakly} break translation symmetry if one of its ground states on torus has nonzero momentum -- this is a generalization of the Tao-Thouless physics in fractional quantum Hall effect~\\cite{PhysRevB.28.1142,PhysRevB.77.155308}; in Sec.~\\ref{sec:cSPT} we discuss the implication of Theorem~\\ref{Thm} for the classification of crystalline symmetry-protected topological (SPT) phases. We end with some discussions in Sec.~\\ref{sec:discussions}.\n\n\n\\section{Proof}\n\\label{sec:proof}\n\nIn this section we prove that SRE states necessarily possess trivial momentum, conversely implying that all non-trivial momentum ground states must be LRE. The approach that we take utilizes the quantum circuit formalism, which is equivalent to the usual adiabatic Hamiltonian evolution formulation~\\cite{doi:10.1126\/science.273.5278.1073,PhysRevB.82.155138} but conceptually cleaner. In particular we will harness the causal structure of quantum circuits, which will allow us to `cut and paste' existing circuits to create useful new ones.\n\nWe shall first prove Theorem~\\ref{Thm} in one space dimension, from which the higher-dimensional version follows immediately.\n\n\\subsection{Proof in $1$d}\n\\label{sec:1dproof}\n\nFirst let us specify our setup more carefully. We consider a spin (boson) system with a local tensor product Hilbert space $\\mathcal{H}=\\otimes_i\\mathcal{H}_i$ where $\\mathcal{H}_i$ is the local Hilbert space at unit cell $i$. The system is put on a periodic ring with $L$ unit cells so $i\\in\\{1,2...L\\}$. In each unit cell the Hilbert space $\\mathcal{H}_i$ is $q$-dimensional ($q$ does not depend on $i$), with a basis labeled by $\\{|a_i\\rangle_i\\}$ ($a_i\\in\\{0,1...q-1\\}$). The translation symmetry is implemented by a unitary operator that is uniquely defined through its action on the tensor product basis \n\\begin{eqnarray}\n\\label{eq:Tboson}\n    T:\\hspace{5pt} & & |a_1\\rangle_1\\otimes|a_2\\rangle_2\\otimes...|a_{L-1}\\rangle_{L-1}\\otimes|a_L\\rangle_L \\nonumber \\\\ & &\\longrightarrow |a_L\\rangle_1\\otimes|a_1\\rangle_2\\otimes...|a_{L-2}\\rangle_{L-1}\\otimes|a_{L-1}\\rangle_L.\n\\end{eqnarray}\nUnder this definition of translation symmetry (which is the usual definition), we have\\footnote{Importantly, we are not dealing with translation under twisted boundary condition, in which case $T^L=g$ for some global symmetry $g$. Many of our conclusions in this work need to be rephrased or reexamined for such twisted translations.} $T^L=1$ and any translational-symmetric product state $|\\varphi\\rangle^{\\otimes L}$ has trivial lattice momentum $e^{iP}=1$.\n\nNow consider a SRE state $|\\Psi_{P(L)}\\rangle$ with momentum $P(L)$. By SRE we mean that there is a quantum circuit $U$ with depth $\\xi\\ll L$ that sends $|\\Psi_{P(L)}\\rangle$ to the product state $|\\mathbf{0}\\rangle\\equiv |0\\rangle^{\\otimes L}$ (we do not assume $U$ to commute with translation). The depth $\\xi$ will be roughly the correlation length of $|\\Psi_{P(L)}\\rangle$. Our task is to prove that $P(L)=0$ mod $2\\pi$ as long as $\\xi\\ll L$. Notice that this statement is in fact stronger than that for FD circuit which requires $\\xi\\sim O(1)$ as $L\\to \\infty$. For example, our result holds even if $\\xi\\sim {\\rm PolyLog}(L)$, which is relevant if we want the quantum circuit to simulate an adiabatic evolution more accurately~\\cite{Haah2018}. Our result is also applicable if the existence of $U$ requires extra ancilla degrees of freedom (DOF) that enlarges the onsite Hilbert space to $\\tilde{\\mathcal{H}}_i$ with dimension $\\tilde{q}>q$ (for example see Ref.~\\cite{ElsePoWatanabe2019}), since ancilla DOFs by definition come in product states and therefore cannot change the momentum.\n\nThe proof will be split into two steps where in \\textit{Step 1} we first prove that the momentum is trivial for all $L=mn$ where $m,n\\in\\mathbb{Z}^+$ are mutually coprime satisfying $m,n\\gg\\xi$. In \\textit{Step 2} we use the results of \\textit{Step 1} to show that this may be extended to all other lengths.\n\n\n\n\\underline{\\textit{Step 1:}} A key ingredient of the proof is to recognize that the entanglement structure of the SRE state $|\\Psi_{P(L)}\\rangle$ on system size $L=mn$, where $m,n\\in\\mathbb{Z}^+$ and $n\\gg\\xi$, is adiabatically connected to that of $m$ identical unentangled length $n$ SRE systems. The existence of such an adiabatic deformation, which is of a similar flavour to those presented in Refs.~\\cite{PhysRevX.7.011020} and \\cite{PhysRevB.96.205106}, is due to the finite correlation length of SRE systems, and will be explained in the following paragraph.\n\n\n\nTake the SRE state $|\\Psi_{P(L)}\\rangle$ placed on a periodic chain of length $L=mn$ with $m,n\\in\\mathbb{Z}^+$ and $n\\gg\\xi$. Let us try to decouple this system at some point (say between site $i$ and $i+1$) via an adiabatic evolution, creating an `open' chain.\nTo show that such a decoupling cut exists, we use the fact that SRE states always have a FD quantum circuit $U$ that sends the ground state to the $|\\mathbf{0}\\rangle\\equiv |0\\rangle^{\\otimes L}$ product state (see Fig.~\\ref{fig:unitaries}(a)). The appropriate cut is then created by modifying this circuit to form a new lightcone-like FD quantum circuit $\\tilde{U}$ with all unitaries outside the `lightcone', i.e. those that do not affect the transformation that sends the two sites $i$ and $i+1$ to $|0\\rangle$, set to identity (see Fig.~\\ref{fig:unitaries}(b)). Such a modified circuit would span $\\sim\\xi$ qudits on either side of the cut and by construction takes the two sites on either side of the cut to $|0\\rangle$, thus completely removing any entanglements across the link\\footnote{This can be better understood in reverse: consider the state constructed by $\\tilde{U} U^\\dag|\\mathbf{0}\\rangle$ ($=\\tilde{U}|\\Psi_{P(L)}\\rangle$) which never directly couples qudits on either side of the cut. Thus $\\tilde{U}$ can be understood as completely removing the entanglement across the applied link.}.\nLet us concretely take $\\tilde{U}^{[0]}$ to denote the appropriate lightcone cut between the last and first qudit (recall that we are on a ring), and define the shifted adiabatic cut between the $x-1$ and $x$th qudits to be $\\tilde{U}^{[x]}\\equiv T^x \\tilde{U}^{[0]}T^{-x}$. If the ground state is translation-symmetric we have $\\tilde{U}^{[x]}|\\Psi_{P(L)}\\rangle=e^{-i x P(L)}T^x\\tilde{U}^{[0]}|\\Psi_{P(L)}\\rangle$\nso we see that $\\tilde{U}^{[x]}$ performs the same cut (up to a phase factor) at any link. By construction this means that the local density matrices of a region surrounding the cut obeys $\\rho_{lr}=\\rho_l\\otimes|00\\rangle\\langle00|\\otimes\\rho_r$, where the left (right) region to the cut is denoted $l$ ($r$), which in turn implies that the operations $\\tilde{U}^{[x]}$ disentangles the system along that cut.\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=0.98\\columnwidth]{unitaries.pdf}\n    \\caption{\\label{fig:unitaries} (Color online) Depiction of finite-depth quantum circuits applied on $|\\Psi_P\\rangle$. Here qudits are depicted as solid circles while unitaries are depicted as rectangles. (a) A SRE state $|\\Psi_P\\rangle$ is always connected to the $|\\mathbf{0}\\rangle$ trivial state via a FD quantum circuit $U$. From $U$ a lightcone-like `adiabatic cut' $\\tilde{U}$ can be created (framed in blue). (b) $\\tilde{U}$ connects $|\\Psi_P\\rangle$ to a state that is completely decoupled across the cut.}\n\\end{figure}\n\nThe cutting procedure may be simultaneously applied to two separate links, as long as they are separated by a distance much greater than the correlation length. With this in mind, let us identically apply the cut on an $L=mn$ length system with a cut after every $n$th qudit, as depicted in Fig.~\\ref{fig:adiabaticcutting}, via the FD quantum circuit $\\tilde{U}^{[0]}\\tilde{U}^{[n]}...\\tilde{U}^{[(m-1)n]}$. Since the adiabatic deformation fully disentangles the system across the cuts, the resulting state should take the form $|\\tilde{\\Psi}_1\\rangle\\otimes|\\tilde{\\Psi}_2\\rangle\\otimes...|\\tilde{\\Psi}_m\\rangle$ where each $|\\tilde{\\Psi}_i\\rangle$ is an $n$-block SRE state. \n\nNow let us examine the symmetries of this resultant system. The original $\\mathbb{Z}_{mn}$ translation symmetry, generated by operator $T$, of the original system is broken by the adiabatic deformation. However the $\\mathbb{Z}_m$ translation symmetry subgroup, generated by operator $T^n$, is preserved since by construction identical cuts occurs at every $n$th junction. This immediately implies that all the $n$-block states are identical $|\\tilde{\\Psi}_i\\rangle=|\\tilde{\\Psi}\\rangle$ and the total state after the cut is simply $|\\tilde{\\Psi}\\rangle^{\\otimes m}$.\nThus we know that the original $\\mathbb{Z}_m$ quantum number is the same as the final one which must be trivial since we are dealing with an $n$-block product state $|\\tilde{\\Psi}\\rangle^{\\otimes m}$. This implies\n\\begin{align}\nnP(L)=0\\mod 2\\pi\\quad,\n\\label{eq:LP}\n\\end{align}\n$\\forall L=mn$ with $m,n\\in\\mathbb{Z}^+$ and $n\\gg\\xi$.\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=0.95\\columnwidth]{cutting_chains.pdf}\n    \\caption{\\label{fig:adiabaticcutting} (Color online) Illustration of the adiabatic cutting procedure on a periodic length $L=mn$ chain. Here we take $m=4$ example to demonstrate how four identical cuts, applied by $\\tilde{U}$ (blue rectangle) at every $n$th link, on a length $L=4n$ state $|\\Psi_{P(L)}\\rangle$ (purple circle) produces four decoupled length $n$ SRE states.}\n\\end{figure}\n\nUsing this relation on a general system length $L=p_1^{q_1}p_2^{q_2}...p_d^{q_d}$ (here we are using prime factorisation notation) we arrive at the condition\n\\begin{align}\nP(L)=0\\mod \\frac{2\\pi}{p_1^{r_1}p_2^{r_2}...p_d^{r_d}}\\quad,\n\\end{align}\n$\\forall{r_i\\in\\{1,...,q_i\\}}$ such that $p_1^{r_1}p_2^{r_2}...p_d^{r_d}\\gg\\xi$. If $L$ factorises into at least two mutually coprime numbers $m,n$ with $m,n\\gg\\xi$ then these conditions can only be satisfied if \n\\begin{align}\nP(L)=0\\mod 2\\pi\\quad,\n\\label{eq:zeroP}\n\\end{align}\nwhich is satisfied for almost all large enough $L$.\n\n\\underline{\\textit{Step 2:}} There are a sparse set of cases for which \\textit{Step 1} does not enforce trivial momentum, the most notable case being when $\\tilde{L}=p^{q}$ with $p$ prime and $q\\in\\mathbb{Z}^+$. Factorisations such as $\\tilde{L}=p_1^{q_1}p_2$ are also not covered if $p_1^{q_1}\\not\\gg \\xi$. \n\nTo show that these cases also possess trivial momentum, once again take a SRE state $|\\Psi_{P(L)}\\rangle$ on a general length $L$ system with momentum $P(L)\\mod 2\\pi$. By the definition of a SRE state, there exists a FD quantum circuit $V_{L}$ such that $|\\Psi_{P(L)}\\rangle=V_L|\\mathbf{0}\\rangle$. This circuit obeys $TV_LT^\\dag|\\mathbf{0}\\rangle=e^{iP(L)}V_L|\\mathbf{0}\\rangle$, meaning that it boosts the trivial momentum of the $|\\mathbf{0}\\rangle$ state by $P(L)\\mod 2\\pi$. Consider the composition of a circuit\n\\begin{align}\n    \\left(TV_L^\\dag T^\\dag\\right) V_L|\\mathbf{0}\\rangle=e^{-iP(L)}|\\mathbf{0}\\rangle\\quad.\n    \\label{eq:boostL}\n\\end{align}\nAs may be understood via the causality structure the phase $e^{-iP(L)}$ will come piecewise from lightcone circuits. Let us understand this in detail: split $\\tilde{V}_L\\equiv TV_L^\\dag T^\\dag V_L$ into a light-cone circuit $\\tilde{V}_{L,1}$ and reverse lightcone circuit $\\tilde{V}_{L,2}$ such that $\\tilde{V}_L=\\tilde{V}_{L,1}\\tilde{V}_{L,2}$, as depicted in Fig.~\\ref{fig:UgeneralL}. The causal structure of the light cone guarantees that a gate $U_1$ in $\\tilde{V}_{L,1}$ and a gate $U_2$ in $\\tilde{V}_{L,1}$ must commute if $U_2$ appears in a layer after $U_1$, which then allows for the decomposition $\\tilde{V}_L=\\tilde{V}_{L,1}\\tilde{V}_{L,2}$.\nAlthough the exact form of this decomposition is quite malleable, for concreteness let us define $\\tilde{V}_{L,1}$ to be constructed causally such that the 1st (lowest) layer consists of a single 2-qudit gate (as seen in Fig.~\\ref{fig:UgeneralL}).\n$\\tilde{V}_{L,1}$ will have support over qudits in the range $[L-\\eta,L]$, where by the SRE nature $\\eta\\ll L$. Due to Eq.~\\ref{eq:boostL} we see that\n\\begin{align}\n    \\tilde{V}_{L,2} |\\mathbf{0}\\rangle=|0...0\\rangle^{[1,L-\\eta-1]}\\otimes|\\alpha\\rangle^{[L-\\eta,L]}\n\\end{align}\nfor some $|\\alpha\\rangle$. By construction, we have\n\\begin{align}\n    \\tilde{V}_{L,1} |\\alpha\\rangle=e^{-iP(L)}|0...0\\rangle^{[L-\\eta,L]}\\quad,\n    \\label{eq:circ1tildeU}\n\\end{align}\nsuch that we satisfy Eq.~\\ref{eq:boostL}.\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=\\columnwidth]{UgeneralLf.pdf}\n    \\caption{\\label{fig:UgeneralL} (Color online) Illustration of splitting $TV_L^\\dag T^\\dag V_L=\\tilde{V}_{L,1}\\tilde{V}_{L,2}$ with $\\tilde{V}_{L,1}\\tilde{V}_{L,2}|\\mathbf{0}\\rangle=e^{-iP(L)}|\\mathbf{0}\\rangle$. Here we have taken a snapshot of the circuit to focus on $\\tilde{V}_{L,1}$ (framed in blue), however the support of $\\tilde{V}_{L,1}$ (in the depicted example 16 qudits) is actually much smaller than the system length. Recall that the circuit is periodic such that the orange arrows, corresponding to components of $\\tilde{V}_{L,2}$ (framed in orange), eventually connect on the far side of the ring.\n    }\n\\end{figure}\n\nNow we will extend the circuit $V_L$ from length $L$ to $nL$ for some $n\\in\\mathbb{Z}^+$, where $n,L$ are coprime and $\\gg\\xi$, and denote this extended circuit $V_{nL}$. To do this we simply unstitch the circuit $V_L$ at some link and reconnect the ends of $n$ consecutive copies of this unstitched $V_L$ circuit to create a FD quantum circuit $V_{nL}$. Let us see what happens to $\\tilde{V}_{nL}\\equiv TV_{nL}^\\dag T^\\dag V_{nL}$ by once again splitting the circuit into two $\\tilde{V}_{nL}=\\tilde{V}_{nL,1}\\tilde{V}_{nL,2}$, where $\\tilde{V}_{nL,k}=\\prod_{j=0}^{n-1} T^{jL}\\tilde{V}_{L,k}(T^\\dag)^{jL}$ with $k\\in\\{1,2\\}$. By construction and due to the SRE nature of state construction\n\\begin{align}\n    \\tilde{V}_{nL,2} |\\mathbf{0}\\rangle^{\\otimes n}=\\left(|0...0\\rangle^{[1,L-\\eta-1]}\\otimes|\\alpha\\rangle^{[L-\\eta,L]}\\right)^{\\otimes n}\\quad.\n\\end{align}\nHowever, by Eq.~\\ref{eq:circ1tildeU}, we have\n\\begin{align}\n    \\tilde{V}_{nL,1} \\tilde{V}_{nL,2} |\\mathbf{0}\\rangle^{\\otimes n}=e^{-inP(L)}|\\mathbf{0}\\rangle^{\\otimes n}\\quad,\n    \\label{eq:circ1tildeUnL}\n\\end{align}\nso this implies\n\\begin{align}\n    TV_{nL} T^\\dag|\\mathbf{0}\\rangle^{\\otimes n} =e^{inP(L)}V_{nL}|\\mathbf{0}\\rangle^{\\otimes n}\\quad,\n    \\label{eq:boostnL}\n\\end{align}\nwhich means that $V_{nL}$ boosts the momentum of $|\\mathbf{0}\\rangle$ on a length $nL$ system to a state with momentum $P(nL)=nP(L)\\mod 2\\pi$. In \\textit{Step 1} we showed that $P(nL)=0\\mod 2\\pi$, so this implies $nP(L)=0\\mod2\\pi$. Since this holds for two mutually coprime values of $n$, one concludes that $1$d SRE translation-symmetric states have $P(L)=0\\mod2\\pi$ for all $L\\gg\\xi$.\n\n\n\n\n\n\\subsection{Higher-dimensional extension}\n\nOur result can be extended to higher dimensions. Consider a $d$-dimensional lattice system and a state $|\\Psi\\rangle$ that has nontrivial momentum $P$ along, say, $\\hat{x}$ direction. We can view the state as a 1d state along the $\\hat{x}$ axis, with an enlarged Hilbert space per unit cell (generally exponentially large in $\\prod_i L_i$ with $i$ denoting the transverse directions). A finite-depth quantum circuit of the $d$-dimensional system will also be a finite-depth quantum circuit when viewed as a $1$d circuit along the $\\hat{x}$-direction (a proof and a somewhat subtle example are presented in Appendix~\\ref{app:higherdSRE}; the converse is not true but that does not concern us here). This immediately implies that a SRE state on the $d$-dimensional system must also be SRE when viewed as a $1$d state along $\\hat{x}$. What we proved in Sec.~\\ref{sec:1dproof} thus implies that the non-trivial momentum state $|\\Psi\\rangle$ must be long-range entangled. In particular, imposing locality in the transverse directions will only further restrict possible FD circuit, and will certainly not lead to possibilities beyond the $1$d proof. This completes the proof of Theorem~\\ref{Thm}.\\hfill$\\blacksquare$\n\n\n\\subsection{Fermion systems}\n\\label{sec:Fermions}\n\n\n\nIt is not difficult to generalize our Theorem~\\ref{Thm} to fermionic system. The only subtlety is that the usual definition of translation symmetry in fermion systems has an extra $\\mathbb{Z}_2$ sign structure compared to the naive implementation in Eq.~\\ref{eq:Tboson}. Instead of specifying the sign structure in the tensor product basis as in Eq.~\\ref{eq:Tboson}, it is more convenient to define translation operator through $Tc_{i,\\alpha}T^{-1}=c_{i+1,\\alpha}$ where $c_{i,\\alpha}$ is a fermion operator in unit cell $i$ with some internal index $\\alpha$, and $c_{L+1,\\alpha}=c_{1,\\alpha}$. This operator relation, together with $T|\\mathbf{0}\\rangle=|\\mathbf{0}\\rangle$ for the fermion vacuum, uniquely determines the action of $T$ on any state. Now consider a product state $|\\varphi\\rangle^{\\otimes L}$, it is easy to verify that the momentum is $e^{iP}=1$ for odd $L$ and $e^{iP}=\\pm1$ for even $L$, where the sign is the fermion parity on each site $\\langle\\varphi|(-1)^{\\sum_\\alpha c^{\\dagger}_{\\alpha}c_{\\alpha}}|\\varphi\\rangle$. We can then go through the proof in Sec.~\\ref{sec:proof}, but now with fermion parity preserving FD quantum circuits, and conclude the following:\n\\begin{theorem}\nAny short-range entangled translation eigenstate $|\\Psi\\rangle$ in a lattice fermion system must have momentum (say in the $x$-direction) $e^{iP_x}=1$ if $L_x$ is odd, and $e^{iP_x}=\\pm1$ if $L_x$ is even. States violating this condition must in turn be long-range entangled.\n\\label{FermionThm}\n\\end{theorem}\nThe details of the proof are presented in Appendix~\\ref{app:fermionproof}.\n\nUsing the same proof technique, we can extend the above result further in various directions. We mention two such extensions without going into the details: (1) for $L_x$ even, the option of $e^{iP_x}=-1$ is possible for a SRE state only if $V\/L_x$ is odd ($V=L_xL_y...$ being the volume); (2) if the total fermion parity is odd in a system with even $V$, then any translation eigenstate must be LRE.\n\n\n\\section{Consequences}\n\\label{sec:consequences}\n\nOne of the beauties of Theorem~\\ref{Thm} lies in the non-trivial consequences that easily follow. For this section, it is useful to introduce an alternative, but equivalent, formulation of Theorem~\\ref{Thm}\n\\begin{manualtheorem}{1}[Equivalent]\nIf there exists a finite-depth local unitary that boosts a state's momentum to a different value (mod $2\\pi$), then the state is necessarily long-range entangled.\n\\end{manualtheorem}\nThe equivalence of this new formulation with the one introduced in Sec.~\\ref{sec:intro} can be understood as follows: if all translation-symmetric SRE states possess trivial momentum then non-trivial momentum states must be LRE. Thus if there exists a finite-depth local unitary that can boost a state's momentum to a different value then at least one of either the original or final state possesses non-trivial momentum and must be LRE. The other state is connected to the LRE state via a finite-depth local unitary and thus must also be LRE. The converse follows by contradiction: assume there exists a SRE state that has non-trivial momentum. Such a state (by definition of SRE) is connected via a FD local unitary to the translation-symmetric direct-product state $|\\alpha\\rangle^{\\otimes L}$ which in turn has trivial momentum. Since there now exists a FD local unitary that boosts the momentum to a different value, this implies that the original state was LRE which leads to the contradiction.\n\nThis equivalent formulation allows for a direct test for long-range entanglement that we will demonstrate on known and previously unknown LSM theories, and topological orders. In the following discussions we will mostly focus on spin (boson) systems for simplicity, but the results can be generalized quite readily to fermion systems as well.\n\n\n\\subsection{LSMOH constraints}\n\\label{sec: LSMOH}\n\n\nThe original Lieb-Schultz-Mattis (LSM) theory~\\cite{LSM} along with the extensions by Oshikawa~\\cite{Oshikawa1999} and Hastings~\\cite{Hastings04}, collectively referred to as LSMOH, and their descendants are powerful tools for understanding the low-energy nature of lattice systems. In one of its most potent forms the theorem states that systems with $U(1)$ and translational symmetry that have non-commensurate $U(1)$ charge filling must be `exotic', meaning that they cannot be SRE states. Since the conception of the original LSM theory the field has flourished rapidly with many extensions that impose similar simple constraints based on symmetry~\\cite{Nachtergaele2007,Watanabe14551,Po2017,LU2020168060,lu2017liebschultzmattis,PhysRevB.98.125120,PhysRevB.99.075143}, and connections to various fields of physics such as symmetry-protected topological (SPT) order and t'Hooft anomaly in quantum field theory~\\cite{Cheng2015,Jian2017,Cho2017,Metlitski2017,PhysRevB.101.224437,Ye2021}. These sort of constraints also have immediate experimental consequences, as they provide general constraints in determining candidate materials of exotic states such as quantum spin liquids~\\cite{Balents2010}. Thus, unsurprisingly, there has been a lot of interest in generating more LSMOH-like theorems that provide simple rules to find exotic states. In the following section we provide simple proofs of some known and previously unknown LSMOH theorems.\n\n\n\nThe first example we consider is the aforementioned non-commensurate 1d $U(1)\\times T$ LSM ($T$ denotes the translation symmetry). In this case there exists a local unitary momentum boost that is the large gauge transformation $U=e^{i\\frac{2\\pi }{L}\\sum x\\hat{n}_x}$, where $\\hat{n}_x$ is the local number operator at $x$. Notice that this transformation is an on-site phase transformation and thus a FD quantum circuit of depth 1. The commutation relation with translation is $TUT^\\dag=e^{i 2\\pi \\frac{\\hat{N}}{L}}U$ ($\\hat{N}$ being the total charge) which means that for non-commensurate filling $\\frac{\\hat{N}}{L}\\notin\\mathbb{Z}$, we may always boost the momentum by a non-trivial value $2\\pi \\frac{\\hat{N}}{L}$. Via the equivalent formulation of Theorem~\\ref{Thm}, this immediately implies that non-commensurate filling leads to a LRE state. This observation may be summarised as\n\\begin{corollary}\n{\\normalfont($U(1)\\times T$ LSM) A $1$d translation and $U(1)$ symmetric state that possesses non-commensurate $U(1)$ charge filling must be long-range entangled.}\n\\end{corollary}\nThe standard LSM theorem follows from this statement since we may now apply it to a \\textit{ground} state of a $1$d translation and $U(1)$ symmetric local spin Hamiltonian to show that the state must be either gapless or a spontaneously symmetry-broken cat state.\nNotice that, strictly speaking, the statement we proved differs slightly from the standard LSM theorem, in that we did not directly prove the vanishing of the energy gap. Rather we showed that any simultaneous eigenstate of translation and $\\hat{N}$ such that $\\langle\\hat{N}\\rangle\/L\\notin\\mathbb{Z}$ must be LRE. In principle we do not even need to assume the parent Hamiltonian to be translation or $U(1)$ symmetric, just that the state itself be translation and $U(1)$ symmetric. In fact the statement encompasses all states, not just the ground state, which is perhaps unsurprising since LRE is fundamentally a property of a state and not the Hamiltonian.\n\nThe higher-dimensional $U(1)\\times T$ LSMOH theorem may be proved following the same logic if $\\langle\\hat{N}\\rangle\/L_i\\notin\\mathbb{Z}$ for some direction $i$ (similar to what was done in Ref.~\\cite{Oshikawa1999}). For generic values of $L_i$ the above condition may not hold, and more elaborate arguments are needed (for example see Ref.~\\cite{YaoOshikawa2020}) which we will not discuss here. \n\nOur proof of the LSM theorem has an appealing feature compared to the classic proof~\\cite{LSM}: we did not need to show that the state $|\\Omega'\\rangle=U|\\Omega\\rangle$ had excitation energy $\\sim O(1\/L)$ (relative to the ground state $|\\Omega\\rangle$); rather it suffices for us to show that $|\\Omega'\\rangle$ has a different lattice momentum compared to $|\\Omega\\rangle$, which is enough to establish the LRE nature of $|\\Omega\\rangle$. Next we shall use this simplifying feature to generalize the $U(1)\\times T$ LSM theorem to a new constraint involving only discrete $\\mathbb{Z}_n\\times T$ symmetries.\n\n\n\nLet us consider a spin chain ($1$d) with translation symmetry and an onsite $\\mathbb{Z}_n$ symmetry generated by $Z\\equiv\\otimes_iZ_i$ ($Z_i^n=1$). We consider the case when the system size $L=nM$ for some $M\\in\\mathbb{N}$, and study simultaneous eigenstates of the translation and $\\mathbb{Z}_n$ symmetries. If such a state is an unentangled product state $\\otimes_i|\\varphi\\rangle_i$, then by definition $Z=1$ when acting on this state, namely the state carries trivial $\\mathbb{Z}_n$ charge. This turns out to be true for any symmetric SRE state, which we now prove. Suppose a translation eigenstate $|\\Psi\\rangle$ has $Z|\\Psi_P\\rangle=e^{i2\\pi Q\/n}|\\Psi\\rangle$ for some $Q\\neq0$ (mod $n$). We can construct a local unitary which is an $\\mathbb{Z}_n$-analogue of the large gauge transform\n\\begin{equation}\nU=\\otimes_iZ_i^i,\n\\end{equation}\nwhere $i$ is the unit cell index. For system size $L=nM$ one can verify that $TUT^{-1}U^{\\dagger}=Z^{\\dagger}$. This means that the momentum of the twisted state $U|\\Psi\\rangle$ will differ from that of the untwisted $|\\Psi\\rangle$ by $\\langle\\Psi|Z^{\\dagger}|\\Psi\\rangle=e^{-i2\\pi Q\/n}\\neq1$. By the equivalent form of Theorem~\\ref{Thm} $|\\Psi\\rangle$ must be LRE. We therefore have\n\\begin{corollary}\n\\label{ZnLSM}\n{\\normalfont($\\mathbb{Z}_n\\times T$ LSM)} A $1$d translation and $\\mathbb{Z}_n$ symmetric ground state that possesses non-trivial $\\mathbb{Z}_n$ charge on system lengths $L=nM$ for some $M\\in\\mathbb{N}$ cannot be short-range entangled, and thus is either gapless or spontaneously symmetry-broken cat state.\n\\end{corollary}\n\nThe above statement also generalizes to higher dimensions if $L_i=nM$ for some direction $i$. For systems with $U(1)$ symmetry, we can choose to consider a $Z_L$ subgroup of the $U(1)$, and the above $\\mathbb{Z}_n\\times T$ LSM theorem leads to the familiar $U(1)\\times T$ LSMOH theorem.\n\n\nThe two LSM-type theorems discussed so far, together with our Theorem~\\ref{Thm}, can all be viewed as ``filling-type\" LSM theorems, in the sense that these theorems constraint a symmetric many-body state $|\\Psi\\rangle$ to be LRE when $|\\Psi\\rangle$ carries certain non-trivial quantum numbers, such as lattice momentum $e^{iP}\\neq1$, total $U(1)$ charge $Q\\notin L\\mathbb{Z}$ or total $\\mathbb{Z}_n$ charge $Q\\notin L\\mathbb{Z}\/n\\mathbb{Z}$.\n\nThere is another type of LSM theorems that involve projective symmetry representations in the onsite Hilbert space, the most familiar example being the spin-$1\/2$ chain with $SO(3)$ symmetry. Our Theorem~\\ref{Thm} can also be used to understand some (but possibly not all) of the projective symmetry types of LSM. Here we discuss one illuminating example with onsite $\\mathbb{Z}_2\\times\\mathbb{Z}_2$ symmetry in one dimension~\\cite{PhysRevB.83.035107,Ogata2019,Ogata2021}, such that the generators of the two $\\mathbb{Z}_2$ group anti-commutes when acting on the local Hilbert space: $X_iZ_i=-Z_iX_i$ (this can simply be represented by the Pauli matrices $\\sigma_x$, $\\sigma_z$). Now set the length $L=2N$ with odd $N$, and consider the three local unitaries $U_x=(\\mathds{1}\\otimes\\sigma_x)^{\\otimes N}$, $U_z=(\\mathds{1}\\otimes\\sigma_z)^{\\otimes N}$, and $U_{xz}=(\\sigma_x\\otimes\\sigma_z)^{\\otimes N}$. One can verify the commutation relations $TU_{x}T^\\dag=(-1)^{Q_x}U_x$, $TU_{z}T^\\dag=(-1)^{Q_z}U_z$, and $TU_{xz}T^\\dag=(-1)^{N+Q_x+Q_z} U_{xz}$. These commutation relations imply that the momentum of any symmetric state $|\\Psi\\rangle$ will be boosted by $\\Delta P=\\pi$ by at least one of the three unitaries, therefore $|\\Psi\\rangle$ must be LRE by Theorem~\\ref{Thm}.\n\n\n\n\n\\subsection{Topological orders: weak CDW}\n\\label{sec:wCDW}\n\n\nWe now consider an intrinsic (bosonic) topological order on a $d$-dimensional torus. By definition there will be multiple degenerate ground states, separated from the excitation continuum by a finite energy gap. If one of the ground states $|\\Psi_a\\rangle$ has a non-trivial momentum, say along the $\\hat{x}$ direction, then according to Theorem 1 this state should be LRE even when viewed as a one-dimensional system in $\\hat{x}$ direction (with the other dimensions $y,z...$ viewed as internal indices). Since there is no intrinsic topological order in one dimension, the only mechanism for the LRE ground state is spontaneous symmetry breaking. The lattice translation symmetry is the only relevant symmetry here -- all the other symmetries can be explicitly broken without affecting the LRE nature of $|\\Psi_a\\rangle$, since the state will still have nontrivial momentum. Therefore $|\\Psi_a\\rangle$ must be a cat state that spontaneously breaks the $\\hat{x}$-translation symmetry~\\footnote{Another way to see this is to note that a cat state is composed of individual SRE states. Since we have proven that translation symmetric SRE states possess trivial momentum, it follows that the cat state may only achieve non-trivial momentum when the individual SRE states break translation symmetry, i.e. the cat state must correspond to translation symmetry breaking.}, also known as a charge density wave (CDW) state~\\cite{RevModPhys.60.1129}. Furthermore, any other ground state $|\\Psi_{b\\neq a}\\rangle$ can be obtained from $|\\Psi_a\\rangle$ by a unitary operator $U_{ba}$ that is non-local in the directions transverse to $\\hat{x}$, but crucially is local in $\\hat{x}$ -- for example in two dimensions $U_{ba}$ corresponds to moving an anyon around the transverse cycle. By Theorem~\\ref{Thm} we then conclude that $|\\Psi_b\\rangle$ is also a CDW in $\\hat{x}$. \n\nPerhaps the most familiar example of the above statement is the fractional quantum Hall effect. It is known that the $1\/k$ Laughlin state on the torus is adiabatically connected to a quasi-one-dimensional CDW state in the Landau gauge, also known as the Tao-Thouless state~\\cite{PhysRevB.28.1142,PhysRevB.77.155308}. For example for $k=2$ the Tao-Thouless state with momentum $P=\\pi n$, in the Landau orbit occupation number basis, reads\n\\begin{equation}\n    |101010...\\rangle+e^{i\\pi n}|010101...\\rangle.\n\\end{equation}\n\n\nThe CDW nature of the ground states is perfectly compatible with the topological order being a symmetric state, since there is no \\textit{local} CDW order parameters with nonzero expectation value. The CDW order parameter in this case is non-local in the directions transverse to $\\hat{x}$. For example, in two-dimensions the CDW order parameter is defined on a large loop that wraps around the cycle transverse to $\\hat{x}$. This phenomenon is dubbed \\textit{weak} symmetry breaking in Ref.~\\cite{Kitaev06}. The weak spontaneous symmetry breaking requires a certain degeneracy for the ground state. This degeneracy is naturally accommodated by the ground state manifold of the topological order. For example for the above Tao-Thouless state at $k=2$ the CDW order requires a two-fold ground state degeneracy, which is nothing but the two degenerate Laughlin states on torus.\n\n\nThe above results can be summarized as follows:\n\\begin{corollary}\n\\label{cor:toporderCDW}\nIf a ground state of a gapped topological order on a $d$-dimensional torus ($d>1$) has a non-trivial momentum in $\\hat{x}$, then any ground state of this topological order must \\textit{weakly} break translation symmetry in $\\hat{x}$.\n\\end{corollary}\nA further example of these results, alongside the effects of anyon condensation, applied upon the $\\mathbb{Z}_2$ topologically ordered Toric code is demonstrated in the Appendix~\\ref{app:Toriccode}. The above result also implies the following constraint on possible momentum carried by a topologically ordered ground state:\n\\begin{corollary}\nIf a gapped topological order has $q$ degenerate ground states on torus, then the momentum of any ground state in any direction is quantized: \n\\begin{equation}\nP^{(a)}_i=2\\pi N^{(a)}_i\/q,\n\\end{equation}\nwhere $N^{(a)}_i$ is an integer depending on the ground state (labeled by $a$) and direction $i$.\n\\end{corollary}\nThis is simply because for other values of the momentum, the ground state degeneracy required by the spontaneous translation-symmetry-breaking order will be larger than the ground state degeneracy from the topological order, which results in an inconsistency. An immediate consequence of the above corollary is that invertible topological orders (higher-dimensional states that are LRE by our definition but has only a unique gapped ground state on closed manifolds), such as the chiral $E_8$ state\\cite{Kitaev06}, cannot have nontrivial momentum on a closed manifolds since $q=1$.\n\n\n\nThe above statement immediately implies that the momenta of topological ordered ground states are robust under adiabatic deformations, as long as the gap remains open and translation symmetries remain unbroken. For the Tao-Thouless states this conclusion can also be drawn from the LSM theorem if the $U(1)$ symmetry is unbroken. Our result implies that the momenta of Laughlin-Tao-Thouless states are robustly quantized even if the $U(1)$ symmetry is explicitly broken.\n\n\n\n\n\\subsection{Crystalline symmetry-protected topological phases}\n\\label{sec:cSPT}\n\n\nThere has been growing interest and successes in understanding the symmetry-protected topological (SPT) phases associated with crystalline symmetries~\\cite{PhysRevX.7.011020,ThorngrenElse,PhysRevB.96.205106,Shiozaki2018,SongFangQi2018,Else2018}. When the protecting symmetry involves lattice translation, a crucial ``smoothness'' assumption~\\cite{ThorngrenElse,PhysRevB.96.205106} is used. Essentially one assumes that for such SPT phases the inter unit-cell entanglement can be adiabatically removed, possibly with the help of additional ancilla degrees of freedom. This allows one to formally ``gauge'' the translation symmetry~\\cite{ThorngrenElse} and build crystalline topological phases out of lower-dimensional states~\\cite{PhysRevB.96.205106,SongFangQi2018,Else2018}. \n\nOur result, namely Theorem~\\ref{Thm}, serves as a non-trivial check on the smoothness assumption in the following sense. If there were SRE states with non-trivial lattice momenta, such states would have irremovable inter-unit cell entanglement since unentangled states cannot have non-trivial momentum. Equivalently the correlation length $\\xi$ cannot be tuned to be smaller than the unit cell size $a$. In fact, if such states exist, they would by definition be non-trivial SPT states protected solely by translation symmetry -- such SPT states would be beyond all the recent classifications. \n\nWe note that our result is a necessary condition, but not a proof, for the smoothness assumption, as there may be other ways to violate the assumption without involving a ground state momentum. It will be interesting to see if the arguments used in this work can be extended to fully justify the smoothness assumption.\n\n\n\\section{Discussions}\n\\label{sec:discussions}\n\nIn this paper we have shown that a quantum many-body state with non-trivial lattice momentum is necessarily long-range entangled, hence establishing a simple yet intriguing connection between two extremely familiar concepts in physics: translation symmetry and quantum entanglement. Many directions can be further explored, which we briefly comment on in the remainder of this Section.\n\nOne important aspect that we have so far skipped over is that LSM theory is in fact intimately connected to quantum anomalies~\\cite{Cheng2015,Jian2017,Cho2017,Metlitski2017,PhysRevB.101.224437,Ye2021}. This is natural since they both provide UV conditions that constraint the low-energy behaviours. For the ``projective symmetry\" type of LSM theorems, this connection has been precisely established and it is known that such LSM constraints correspond to certain discrete (quantized) t'Hooft anomalies. For the ``partial filling'' type of LSM such as the familiar $U(1)\\times T$ constraint, however, the connection has been discussed~\\cite{Song2021Polarization,Else2021FL,Wang20,PhysRevResearch.3.043067} but has yet to be fully developed. As we discussed in Sec.~\\ref{sec: LSMOH}, our main result (Theorem~\\ref{Thm}) can be viewed as a ``partial filling'' type of LSM that only involves translation symmetry. It is therefore natural to ask whether Theorem~\\ref{Thm} can be understood from an anomaly perspective. To achieve this goal, it is clear that the standard quantized t'Hooft anomaly is insufficient (a point which was also emphasized in Ref.~\\cite{Else2021FL} for the $U(1)\\times T$ LSM) -- for example, the toric code discussed in Appendix~\\ref{app:Toriccode} has no t'Hooft anomaly since one can condense the $e$ particle to obtain a trivial symmetric state. One would therefore need to expand the notion of anomaly to accommodate the partial-filling type of LSM constraints including the one discussed in this work, possibly along the line of the ``unquantized anomaly\" discussed in Ref.~\\cite{PhysRevResearch.3.043067}. We leave this aspect to future work.\n\nAnother powerful consequence of the traditional $U(1)\\times T$ LSM theorem is on the stability of the LRE ground states (with partial charge filling) under symmetric perturbations: assuming the charge compressibility is finite (could be zero), then a small perturbation will not change the charge filling discontinuously, so the system remains LRE under small symmetric perturbations (unless the perturbation leads to spontaneous symmetry-breaking like the BCS attraction). It is natural to ask whether the other ``partial filling\" types of LSM theorems can serve similar purposes. In fact Ref.~\\cite{PhysRevResearch.3.043067} discussed precisely this point under the notion of ``unquantized anomaly''. The unquantized anomalies are very similar to Theorem~\\ref{Thm} and \\ref{FermionThm} and Corollary~\\ref{ZnLSM}, except that the key quantity is not the discrete charges (lattice momentum or $\\mathbb{Z}_n$ charges) on a specific systems size $L$, but the charge densities (momentum density or $\\mathbb{Z}_n$ charge density). Such discrete charge densities can not be defined for a fixed $L$, but may be defined for a sequence of systems with $L\\to \\infty$. Ref.~\\cite{PhysRevResearch.3.043067} argued that, in the context of Weyl and Dirac semimetals, as long as these discrete charge densities are well behaved in the $L\\to\\infty$ limit, the unquantized anomalies will protect the LRE nature of the states under symmetric perturbations. Our work here can be viewed as a rigorous justification of the unquantized anomalies in Ref.~\\cite{PhysRevResearch.3.043067} on fixed system sizes.\n\nAssuming a well-behaving momentum density in the thermodynamic limit, we can also apply our results to a Fermi liquid with a generic Fermi surface shape, such that the ground state from the filled Fermi sea has a non-vanishing momentum density (this requires breaking of time-reversal, inversion and reflection symmetries). This can be viewed as a non-perturbative explanation for the stability of such low-symmetry Fermi surface, even in the absence of the charge $U(1)$ symmetry. (Recall that perturbatively the stability comes from the fact that the Cooper pairing terms no longer connect opposite points on the Fermi surface).\n\nAnother question one may ask is whether a broader group of non-onsite symmetries obey similar charge and entanglement restrictions. It is easy to see that exactly the same constraint holds for glide reflections and screw rotations, since when the system is viewed as $1$d there is no difference between glide reflection, screw rotation and translation. It is also easy to see that the constraint does \\textit{not} hold for point group symmetries (rotations and reflections), because such symmetries will be onsite at some points in space (the fixed points of point groups). It is therefore important that translation symmetry is \\textit{everywhere} non-onsite. The question becomes even more intriguing if we consider more general unitary operators (such as quantum cellular automata~\\cite{GNVW}).\n\nThere are many more natural avenues for further exploration. The interplay between the non-local nature of translation symmetry with crystalline symmetry anomalies is not yet well understood and requires more concrete mathematical grounding such as a rigorous proof of when the smoothness condition is valid. Relatedly it remains to be determined whether translation symmetry may be truly treated as an onsite symmetry and gauged, or whether its non-locality and non-trivial momentum may hinder or require modifications to the usual gauging process. Implications of our results on the ``emergibility\" of phases may also provide fruitful insights to achievable and unachievable states on the lattice~\\cite{PhysRevX.11.031043,Ye2021}. Our work has shown without a doubt that translation symmetry is many-faceted and plays a crucial role in the entanglement properties of crystalline materials.\n\n\n\n\\begin{acknowledgments}\nWe acknowledge insightful discussions with Yin-Chen He, Timothy Hsieh, and especially Liujun Zou. We thank Anton Burkov for a previous collaboration that inspired this work. We thank the anonymous referees for their careful reading of our manuscript and their illuminating comments and questions. LG was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by a Vanier Canada Graduate Scholarship. \nResearch at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn many models of set theory, Souslin trees offer a variety of different\nhomogeneity or rigidity properties.\nProbably the most prominent homogeneity property for Souslin trees is\n\\emph{strong homogeneity} (cf. Section \\ref{sec:str_hom} for the definition)\nwhich implies that the tree is in a certain sense minimal with respect to its\nautomorphism group.\nOn the other hand, a great number of rigidity notions\n(i.e. absence of nontrivial automorphisms) for Souslin trees\nand an array of implications between most of them are known.\nIn this paper, which resulted out of a part of the authors PhD thesis\n\\cite{diss}, we present some interrelations between the class\nof strongly homogeneous Souslin trees and that of free trees,\nthe latter consisting of those Souslin trees which have the strongest known\nrigidity properties.\n\nThe key result which leads to these correspondences\nis a certain method for decomposing a strongly homogeneous Souslin tree\ninto $n$ free factors\n(Theorem \\ref{thm:str_hom=free_x_free}, which is a strengthening of a known\nthough unpublished result).\nThis decomposition uses an elementary,\nbut apparently new combinatorial tool,\nan $n$-optimal matrix of partitions,\nwhich we introduce in the first section.\nAs will be seen in Section 2, there are several ways to decompose a strongly\nhomogeneous Souslin tree into $n$ free trees.\nBut the construction we give using an $n$-optimal matrix of partitions\nenables us to prove strong consequences about the behaviour of the factors\nwhich finally are used in the third section to separate certain notions\nof parametrized rigidity for Souslin trees (which are all weakenings of\nfreeness) in Corollaries \\ref{cor:n-free} and \\ref{cor:n-free_not_UBP}.\n\nA few words on the structure of the paper and the assumed background\nwhich differs strongly from section to section.\nThe first section\nis about the very elementary notion of $n$-optimal matrices of\npartitions and does not assume any prerequisites.\nThe other two sections treat Souslin trees and their structural properties.\nIn Section 2 we review strong homogeneity and freeness for Souslin trees\nand prove two decomposition theorems for strongly homogeneous\nSouslin trees.\nThe final section collects several rigidity notions for Souslin trees\n(most of them taken from \\cite{degrigST}) and gives the aforementioned\nseparation results.\nSome definitions and proofs in Section 3 refer to the technique of forcing\nwhich we do not review here.\nAnd even though we give the necessary definitions concerning Souslin trees\nat the beginning of Section 2, some acquaintance with this subject will\ncertainly enhance the reader's understanding\nof the constructions in Section 2\n(very good references, also on forcing, are,\ne.g. \\cite{devlin-johnsbraten,kunen,jechneu}).\nAnyway, we have made an effort to write a paper that is accessible to an\naudience exceeding the circle of experts on Souslin trees.\n\n\\section{Optimal matrices of partitions}\n\nThe main idea is as follows:\nConsider an infinite matrix with $\\omega$  rows and $n$ columns\nwhere $n$ is a natural number larger than 1:\n$$\n\\begin{pmatrix}\nP_{0,0}&\\ldots&P_{0,m}&\\ldots&P_{0,n-1}\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\\nP_{k,0}&\\ldots&P_{k,m}&\\ldots&P_{k,n-1}\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\\n\\end{pmatrix}\n$$\nSuppose that the entries of this matrix are partitions of the set $\\omega$\nof natural numbers.\nWe want to choose these partitions in a way such that\n(i) we get an infinite set whenever we intersect a finite family of subsets\nof $\\omega$ coming from (distinct) partitions of a single column and\n(ii) we get a singleton whenever we intersect $n$ sets belonging\nto partitions each coming from different columns.\nIn the following definition the latter requirement is stated\nin a slightly stronger form: we want to obtain a singleton whenever we\nintersect $n$ sets not all coming from the same column.\nThe construction in the proof of Lemma \\ref{lm:opt_mtx_prt} actually yields\nmatrices that satisfy this stronger condition,\nand we will use it in the proof of Proposition \\ref{prp:decomp}\nto derive an additional result.\n\n\\begin{defi}\n\\label{defi:opt_mtx_prt}\nFor $n\\in\\omega$,\nan $n$\\emph{-optimal matrix of partitions}\nis a family $(P_{k,m}\\mid k\\in\\omega,\\,m<n)$\nof infinite partitions $P_{k,m}=(a_i^{k,m}\\mid i\\in\\omega)$ of $\\omega$\nwith the following properties.\n\\begin{enumerate}[(i)]\n\\item \\emph{Column-wise agreement:}\nFor all $m<n$ and all $i:k\\to\\omega$ where $k\\in\\omega$,\nthe intersection $\\bigcap_{\\ell<k}a_{i(\\ell)}^{\\ell,m}$ is infinite.\n\\item $n$\\emph{-optimality:}\nFor all maps $(i,k,m):n\\to\\omega\\times\\omega\\times n$\nwith $(k(j),m(j))\\neq(k(\\ell),m(\\ell))$ for all $j<\\ell<n$\nand $m(j)\\neq m(\\ell)$ for at least one pair $j,\\ell<n$\nthe intersection\n$$\\bigcap_{j<n} a^{k(j),m(j)}_{i(j)}\\quad\\text{ is a singleton.}$$\n\\end{enumerate}\n\\end{defi}\n\nNote that if in (ii) the domain of $(i,k,m)$ is restricted\nto a proper subset of $n$,\ni.e., if we intersect over less than $n$ sets,\nthen the corresponding intersection has to be infinite as well.\n\n\\begin{lm}\n\\label{lm:opt_mtx_prt}\nThere is an $n$-optimal matrix of partitions for every natural number $n>1$.\n\\end{lm}\n\\begin{proof}\nTo start we fix a bijective enumeration\n$h=(h_0,\\ldots,h_{n-1}):\\omega\\to\\omega^n$ and define $a_i^{0,m}$\nto be the pre-image of $i$ under $h_m$.\nLet $P_{0,m}:=\\{a_i^{0,m}\\mid i\\in\\omega\\}$.\n\nThe rest of the construction consists of a three-fold recursion.\nThe outer loop is indexed with $(k,m)\\in\\omega\\times n$,\nand goes row by row, from the left to the right.\nOne could also say that the progression of the indices\nfollows the lexicographic order of $\\omega\\times n$, i.e.,\n$m$ grows up to $n-1$ and then drops down to 0 while $k$ increases to $k+1$.\n(The first $n$ stages of the outer loop, where $k=0$,\nhave been included in the recursive anchor in the first line of the proof.)\n\nThe inner recursion loops are common $\\omega$-recursions.\nIn each stage of the middle one we define one element $a^{k,m}_i$\nof the partition $P_{k,m}$,\nand the innermost consists of a choice procedure for the elements\nof that set $a^{k,m}_i$.\n\nSo assume that the partitions $P_{\\ell,m}=\\{a_i^{\\ell,m}\\mid i\\in\\omega\\}$\nhave already been defined for $(\\ell,m)<_\\mathrm{lex}(k,n)$\nand also the $i$ first sets\n$a^{k,m}_0=a_0,\\ldots,a^{k,m}_{i-1}=a_{i-1}$ of $P_{k,m}$\nhave been fixed.\nAssume also,\nthat the family constructed so far\nhas the properties (i) and (ii) from Definition~\\ref{defi:opt_mtx_prt}.\nWe inductively choose three sequences\n$x_\\ell,\\,y_\\ell$ and $z_\\ell$ of members of $\\omega\\setminus\\bigcup_{h<i}a_h$\nand afterwards set $a_i:=\\{x_\\ell,y_\\ell\\mid\\ell\\in\\omega\\}$.\nThe members of the $x$ sequence will make the matrix satisfy\ncolumn-wise agreement (requirement (i))\nwhile the $y_\\ell$ guarantee that the intersections\nfor (ii) ($n$-optimality) are non-empty.\nThe elements $z_\\ell$ go back to the stack\nand build the pool for the construction of the further members of $P_{k,m}$.\n\nWe need the following objects $f,\\,b,\\,I,\\,c,\\,\\tau$ and $d$\nfor book-keeping and assume that they\nhave been fixed at the start of the definition\nof the members of the partition $P_{m,k}$, before the construction of $a_0$.\n\nLet $f:\\omega\\to{^k \\omega}$ be onto and $\\aleph_0$-to-1\nand set for $\\ell\\in\\omega$\n$$b(\\ell):=\\bigcap_{j<k} a^{j,m}_{f(\\ell)(j)}.$$\nThese sets $b(\\ell)$ have to be met by $a_i$ infinitely many times.\nSo we will choose $x_\\ell$ from $b(\\ell)$.\nLet $I$ be the set of subsets $\\sigma$ of\n$(\\omega\\times k\\times n)\\cup(\\omega\\times(k+1)\\times m)$ of\ncardinality $n-1$ such that $\\mathrm{pr}_{2,3}\\!\\!\\upharpoonright\\!\\!\\sigma$ is injective and \n$\\mathrm{pr}_3\\!\\!\\upharpoonright\\!\\!\\sigma$ is not equal to to the constant function whose only\nvalue is $m$,\nwhere $\\mathrm{pr}_3$ and $\\mathrm{pr}_{2,3}$ are the projections to the third\nand to the second and third component of a triple respectively.\nSo if $\\sigma\\in I$ then the elements of $\\sigma$\nare indices for $n-1$ members of pairwise distinct partitions and these \npartitions lie in at least two distinct columns of the $n$-optimal matrix.\nFor $\\sigma\\in I$ let\n$$c(\\sigma):=\\bigcap_{(j,p,q)\\in\\sigma} a^{p,q}_j$$\nand note that this set is infinite,\nbecause if $P_{r,s}$ is a partition which is not involved in $\\sigma$\nthen $c(\\sigma)$ meets every element of $P_{r,s}$ in exactly one natural\nnumber by $n$-optimality.\n\nAlso the sets $c(\\sigma)$ have to be met by $a_i$.\nSo we fix a bijective enumeration $\\tau$ of $I$ and\nchoose $y_\\ell$ from $c(\\tau(\\ell))$\nunless that set has already been hit by earlier members of $a_i$.\n\nCondition (ii) imposes that each set $c(\\sigma)$ be met in only one element.\nSo once the intersection of $a_i$ and $c(\\sigma)$ is non-empty,\nthat particular set $c(\\sigma)$ has to be avoided in later choices\nof members of $a_i$. \nWe thus define for every natural number $x$ the set\n$$d(x):=\\bigcup\\left\\{ c(\\sigma)\\mid \\, \\sigma\\in I\\text{ and }\nx\\in c(\\sigma)\n\\right\\}$$\nand choose $x_\\ell$ and $y_\\ell$ from outside\n$\\bigcup_{j<\\ell} d(x_j)\\cup d(y_j)$.\n\nWe now turn to the formal definition of our three sequences\n$x_\\ell,\\,y_\\ell$ and $z_\\ell$\nand argue afterwards, why these choices are always possible.\nSet $e=\\bigcup_{h<i}a_h$ and let inductively\n\\begin{eqnarray*}\nx_\\ell&:=&\n\\min\n \\left(\n  b(\\ell)\n  \\setminus\n  \\left(\n   e\\cup\\bigcup_{j<\\ell}(d(x_j)\\cup d(y_j)\\cup\\{z_j\\})\n  \\right)\n \\right),\\\\\ny_\\ell&:=&\n\\begin{cases}\nx_\\ell,&\nc(\\tau(\\ell))\n\\cap\n\\bigcup_{j<\\ell}\\{x_j,y_j,x_{j+1}\\}\n \\neq\n\\emptyset\\\\\n\\min \n c(\\tau(\\ell))\n \\setminus \n  (e\\cup\\{z_0,..,z_{\\ell-1}\\}),&\n\\text{otherwise,}\n\\end{cases}\\\\\nz_\\ell&:=&\n\\min\n \\left(\n  b(\\ell)\n  \\setminus\n  \\left(\n   e\\cup\\bigcup_{j<\\ell}\\{x_j,y_j,z_j,x_{j+1},y_{j+1}\\}\n  \\right)\n \\right).\n\\end{eqnarray*}\nWe finally show that this construction does not break down, i.e., that the \nsets, from which $x_\\ell$ or $z_\\ell$ are picked, are non-empty for all\n$\\ell$.  (A variation shows that the choice $y_\\ell$ is always possible, as well.)\nThe argument splits depending on whether $k$ is less than or at least $n-1$.\nSo let $k<n-1$ and $\\ell\\in\\omega$, set\n$\\rho=\\rho(\\ell)=\\{(f(\\ell)(j),j,m)\\,:\\,j<k\\}$ and fix $\\sigma\\in I$ such that $\\rho\\subset\\sigma$,\nsay $\\sigma=\\rho\\cup\\{(j_1,p_1,q_1),\\ldots,(j_r,p_r,q_r)\\}$ for $r=n-1-k$.\nNow let\n$$F:=F(\\ell,\\sigma):=\n\\left\\{\n\\{(g_1,p_1,q_1),\\ldots,(g_r,p_r,q_r)\\}\n\\,:\\,\n(g_1,\\ldots,g_r)\\in\\omega^r\n\\right\\}$$\nand note that $b(\\ell)$ is the disjoint union of the family\n$\\{c(\\rho\\cup\\tau)\\,:\\,\\tau\\in F\\}$:\n$$b(\\ell)=\\dot{\\bigcup_{\\tau\\in F}}c(\\rho\\cup\\tau).$$\nIn order to prove that we always can choose our new element of the $x$-, $y$-\nor $z$-sequence, we first state two consequences of $n$-optimality:\n\\begin{enumerate}\n  \\item For every $\\tau\\in F$ and $h<i$ the intersection $c(\\rho\\cup\\tau)\\cap a_h$\n    is a singleton.\n  \\item Given any $\\sigma'\\in I$, for every $\\tau\\in F$ with\n    $\\sigma'\\ne\\rho\\cup\\tau$ the intersection $c(\\rho\\cup\\tau)\\cap c(\\sigma') $ has at\n    most one element.\n  \\end{enumerate}\nNow note that in order to define $x_\\ell$ or $z_\\ell$,\nwe subtract only finitely many sets of the form\n$a_h$ or $c(\\sigma')$ from $b(\\ell)$ .\nSo in any case, infintely many natural numbers remain in\nthe pool.\n\nNext let $k\\ge n-1$. Then any set of the form $c(\\sigma)$ meets $b(\\ell)$ in\nat most one member by a direct application of $n$-optimality.\nFurthermore the $z$-part of the construction yields by induction on\n$i$ that $b(\\ell)\\setminus e=b(\\ell)\\setminus(\\bigcup_{h<i}a_h)$ is an\ninfinite set.\n\n\n\nWe leave the verification that this construction indeed yields an\n$n$-optimal matrix of partitions for the reader.\n\\end{proof}\n\nThere is also a proof of Lemma \\ref{lm:opt_mtx_prt}\nwhich uses a more sophisticated\nforcing style argument in the construction of the partition $P_{k,m}$,\nbut since it does not significantly decrease the length of the proof\nwe stuck to the elementary recursive construction given above.\n\n\n\\section{Strongly homogeneous and free Souslin trees}\n\\label{sec:str_hom_+_free}\n\n\\subsection{Preliminaries on Souslin trees}\n\\label{sec:prel_ST}\n\nA \\emph{tree} is a partial order $(T,<_T)$ where for all $t\\in T$\nthe set of predecessors $\\{s\\mid s<_T t\\}$\nis well-ordered by $<_T$.\nThe elements of a tree are called \\emph{nodes}.\nFor a node $t\\in T$ we let $\\NF(t)$ be the set of $t$'s immediate successors.\nThe \\emph{height} of the node $t$ in $T$ is the order\ntype of the set of its predecessors under the ordering of $T$,\n$\\hgt_T(t):=\\ot(\\{s\\mid s<_T t\\},<_T)$.\nFor an ordinal $\\alpha$ we let $T_\\alpha$ denote\nthe set of nodes of $T$ with height $\\alpha$.\nIf $\\hgt_T(s)>\\alpha$ we let $s\\!\\!\\upharpoonright\\!\\!\\alpha$ be the unique\npredecessor of $s$ in level $\\alpha$.\n\nThe \\emph{height of a tree} $T$, $\\hgt T$,\nis the minimal ordinal $\\alpha$ such that $T_\\alpha$ is empty.\nAn \\emph{antichain} is a set of pairwise incomparable nodes of $T$,\nso for $\\alpha<\\hgt T$,\nthe level $T_\\alpha$ is an antichain of $T$.\n\nNodes, that do not have $<_T$-successors, are called \\emph{leaves}, and\n$T$ is called \\emph{$\\kappa$-splitting} or \\emph{$\\kappa$-branching},\n$\\kappa$ a cardinal, if all nodes of $T$\nhave exactly $\\kappa$ immediate successors, except for the leaves.\n\nA \\emph{branch} is a subset $b$ of $T$ that is linearly ordered by $<_T$ and\nclosed downwards, i.e. if $s<_T t\\in b$ then $s\\in b$.\nUnder the notion of a \\emph{normal} tree we subsume the following\nfour conditions:\n\\begin{enumerate}[a)]\n\\item there is a single minimal node called the \\emph{root};\n\\item each node $s$ with $\\hgt(s)+1<\\hgt T$ has at least two immediate successors;\n\\item each node has successors in every higher non-empty level;\n\\item branches of limit length have unique limits (if they are extended in the tree),\ni.e., if $s,t$ are nodes of $T$ of limit height\nwhose sets of predecessors coincide, then $s=t$.\n\\end{enumerate}\nNote that by condition c) leaves can\nonly appear in the top level of a normal tree.\n\nFor a node $t\\in T$ we denote by $T(t)$ the set $\\{s\\in T:t\\leq_T s\\}$\nof nodes above (and including) $t$ which becomes a tree when equipped\nwith the ordering inherited from $T$.\nA tree $T$ is said to be \\emph{homogeneous},\nif for all pairs $s,t\\in T$ of the same height there is\na tree isomorphism (of partial orders) between $T(s)$ and $T(t)$,\nthe trees of nodes in $T$ above $s$ and $t$ respectively.\nFor many classes of trees, such as Souslin trees, this is equivalent to \nthe condition that for each pair $s,t\\in T$ of nodes of the same height there\nis an automorphism of $T$ mapping $s$ to $t$.\nA tree is \\emph{rigid} if it does not admit any non-trivial automorphism.\n\nWe will consider two operations on the class of trees: sum and product.\nGiven trees $(T^i,<_i)$ for $i\\in I$, the \\emph{tree sum} of this family,\ndenoted by $\\bigoplus_{i\\in I}T^i$ is the disjoint union of the sets $T^i$\nplus a common root $r\\notin \\bigcup T^i$.\nThe tree order $<$ on $\\bigoplus T^i$ is\ngiven by the (disjoint) union of the tree orders of summands as well as\nthe relation $r< t$ for all $t\\in \\bigcup T^i$.\nThe height of $\\bigoplus T^i$ is given by the ordinal\n$1 + \\sup\\{\\hgt T^i: i\\in I\\}$.\n\nLet now all trees $T^i$ be of height $\\mu$.\nThe \\emph{tree product} $\\bigotimes_{i\\in I}T^i$\nover the family $(T^i)_{i\\in I}$\nis given by the union over the cartesian products of the levels\n$T^i_\\alpha$:\n$$\\bigotimes_{i\\in I}T^i := \\bigcup_{\\alpha<\\mu}\\prod_{i\\in I} T^i_\\alpha.$$\nThe product tree order is simply the conjunction of the relations $<_i$.\n\nIn order to make a decomposition of a tree into a product feasible\nwe also introduce the notion of a nice tree equivalence relation.\nLet $T$ be a normal and $\\aleph_0$-splitting tree and $\\equiv$\nan equivalence relation on $T$.\nThen we say that $\\equiv$ is a \\emph{nice tree equivalence relation (nice\n  t.e.r.)} if $\\equiv$ respects levels (i.e., it refines $T\\otimes T$),\nis compatible with the tree order (i.e., $\\hgt(s)=\\hgt(r) $ and\n$s<t\\equiv u > r$ imply $s\\equiv r$), the quotient partial order\n$T\/\\!\\!\\equiv$ of $\\equiv$-classes ordered by the inherited partial order, i.e.\n$$[s]<_\\equiv [t]\\quad\\iff\\quad s< t\\,,$$\nis a normal and $\\aleph_0$-splitting tree and the relation is nice,\nby which we mean that for all triples of nodes $s,r,t$ such that\n$s\\equiv r$ and $t$ is above $s$ there is a node $u\\equiv t$, $u$ above\n$r$.\nAnother way to formulate this last property ``niceness'' associates to each\nbranch $b$ through $T$ a subtree $T^{b}_\\equiv:=\\bigcup_{s\\in\n  b}s\/\\!\\!\\equiv$ of $T$ and requires that it satisfies point c) in our\ndefinition of normal trees, i.e., every node $t\\in T^{b}_\\equiv$ has\nsuccessors in every higher level of $T^b_\\equiv$. \n\nNow consider the case that a tree $T$ carries nice tree equivalence relations\n$\\equiv_i$ for $i<n$ such that for every level $\\alpha$\nand every $n$-sequence of equivalence classes\n$c_i\\in (T\/\\!\\!\\equiv_i)_\\alpha$, $i<n$\nthe intersection $\\bigcap c_i$ is a singleton $\\{t\\}$ with $t\\in T_\\alpha$.\nThen we have a natural isomorphism\nbetween the tree $T$ and the product of its quotient trees,\n$\\bigotimes_{i<n} T\/\\!\\!\\equiv_i$ given by\n$t\\mapsto (t\/\\!\\!\\equiv_i : i< n)$.\n\nWe finally come to Souslin trees.\nIn general,\na \\emph{Souslin tree} is a tree $T$ of height $\\omega_1$\nsuch that\nevery family of pairwise incomparable nodes\nand also every branch of $T$ is at most countable.\n\\emph{Unless stated otherwise,\nwe will only consider normal and $\\aleph_0$-splitting Souslin trees.}\nIn this case the sole absence of uncountable\nantichains -- referred to as \\emph{the contable chain condition (c.c.c.)} --\nalready implies that the tree has no cofinal branch.\nA main elementary feature of Souslin trees is that their square is\nno longer Souslin:\n$T\\otimes T$ violates the c.c.c. for every Souslin tree $T$.\n\nThe naming \\emph{Souslin} of these trees stems from their tight connection\nto the famous question of\nMikhail Yakovlevich Souslin that was published as \\emph{Probl\\`{e}me 3)}\non page 223 of the first issue of Fundamenta Mathematicae in 1920\n(cf. also \\cite{devlin-johnsbraten} or \\cite[II.4]{kunen};\nthe latter transliterates the name as \\emph{Suslin}). \nIt is well known that the existence of Souslin trees is independent of ZFC,\nso whenever we assume that there is some Souslin tree, we make an extra\nassumption beyond the realm of standard set theory.\n\n\\subsection{Strongly homogeneous and free trees}\n\\label{sec:str_hom}\n\nWe take a closer look at two classes of Souslin trees,\nthat are widely known among set theorists, although\noften under different names, of which we try to state as many as possible.\n\nStrongly homogeneous Souslin trees occur quite often\nin set theoretic literature.\nIn \\cite{larson-todorcevic} they are called\ncoherent Souslin trees and play a central\nrole in the solution of Katetov's Problem\non the metrizability of certain compact spaces.\nShelah and Zapletal show in \\cite[Theorem 4.12]{shelah-zapletal} that\nTodorcevic's term for a Souslin tree\nin one Cohen real is strongly homogeneous,\nLarson gives a direct $\\diamondsuit$-construction (\\cite[Lemma 1.2]{larson}),\nand also Jensen's construc\\-tion (under the same hypothesis)\nof a 2-splitting, homogeneous tree,\nas carried out in \\cite[Chapter IV]{devlin-johnsbraten},\nis easily seen to yield a strongly homogeneous tree.\n\n\\begin{defi}\nA Souslin tree $T$ is called \\emph{strongly homogeneous} if there is a family\n$(\\psi_{st}\\mid s,t\\in T,\\,\\hgt s=\\hgt t)$ which has the following properties:\n\\begin{enumerate}[1)]\n\\item $\\psi_{st}$ is an isomorphism between\nthe tree $T(s)$ of nodes above $s$ and the tree $T(t)$ of nodes\nabove $t$ and $\\psi_{ss}$ is the identity.\n\\item (commutativity) For all nodes $r,s,t$ of the same level of $T$\nwe have $\\psi_{rt}=\\psi_{st}\\circ\\psi_{rs}$.\n\\item (coherence) For nodes $r,s$ from the same level,\n$t$ above $r$ and $u=\\psi_{rs}(t)$ we require that $\\psi_{tu}$ is\nthe restriction of $\\psi_{rs}$ to the tree $T(t)\\subset T(r)$.\n\\item (transitivity) If $t$ and $u$ are nodes\non the same limit level $T_\\alpha$, then there is a level $T_\\gamma$\nbelow such that for the corresponding predecessors $r$ of $t$ and $s$ of $u$\nwe have $\\psi_{rs}(t)=u$.\n\\end{enumerate}\n\\end{defi}\n\nSome authors call such a family of tree isomorphisms associated to a strongly\nhomogeneous tree a \\emph{coherent family}.\n\nGiven any homogeneous tree,\nit is easy to define a family on $T$ with the properties 1-3) above.\nThe crucial property of a coherent family is that of \\emph{transitivity},\nwhich means that every limit level is a minimal extension of the initial\nsegment below with respect to the coherent family on that initial segment. \nAlso the automorphism group of a strongly homogeneous Souslin tree $T$\nis in a sense minimal, as shown in the following proposition.\n\n\\begin{prp}\n\\label{prp:str_hom_aut}\nEvery automorphism $\\varphi$ of a strongly homogeneous Souslin tree $T$ is\neventually equal to the union of a subset of the coherent family\n$(\\psi_{st})$, i.e.,\nthere is a countable ordinal $\\alpha$ such that for all\nnodes $t$ of height greater than $\\alpha$ we have\n$$\\varphi(t) =\n\\psi_{t\\upharpoonright\\alpha,\\varphi(t\\upharpoonright\\alpha)}(t).$$\n\\end{prp}\nThis implies that the automorphism group of such a tree has exactly\n$2^{\\aleph_0}$ elements.\n\\begin{proof}\nIt suffices to show that above every node $r\\in T$ there is a node\n$s$, such that all $t>s$ are mapped by the automorphism\n$\\varphi$ according to the rule\nstated above with $\\alpha=\\hgt(s)$.\n\nTo reach a statement contradicting the transitivity\nof the family $(\\psi_{st})$,\nwe assume that there is a node $r\\in T$,\nsuch that for each successor $s$ of $r$ there is\na node $t\\geq s$, such that $\\varphi(t)\\neq\\psi_{s\\varphi(s)}(t)$.\nWe can inductively choose an increasing sequence of ordinals\n$\\alpha_n$ such that for all nodes $t\\in T_{\\alpha_{n+1}}$ we have\n$$\\varphi(t)\\neq\n\\psi_{t\\upharpoonright\\alpha_n \\varphi(t\\upharpoonright\\alpha_n)}(t).$$\nLet $\\alpha$ be the supremum of the $\\alpha_n$ and pick any node\n$t\\in T_\\alpha$.\nSince $\\alpha$ is a limit ordinal and by transitivity of the coherent family\nwe find an $n\\in\\omega$ such that\n$\\varphi(t)=\\psi_{t\\upharpoonright\\alpha_n \\varphi(t\\upharpoonright\\alpha_n)}(t)$\nwhich is of course impossible by the choice of the $\\alpha_n$.\n\\end{proof}\n\nNow we come to \\emph{free} trees. Also this property has several different\nnames, e.g. full\n(Jensen, Todor\\c{c}evi\\'{c}, \\cite{GKH,todorcevic_trees_and_orders}) or\n'Souslin and all derived trees Souslin'\n(Abraham and Shelah, \\cite{abraham-shelah_aronszajn_trees,abraham-shelah}).\nIn the context of \\cite{degrigST}\n(cf. Section \\ref{sec:free} of the present article)\nfree trees could also be called\n'$<\\!\\!\\omega$-fold Souslin off the generic branch'.\n\n\\begin{defi}\nA normal tree $T$ of height $\\omega_1$ is \\emph{free}\nif for every finite (and non-empty) set of nodes\n$s_0,\\ldots,s_n$ of $T$ of the same height,\nthe tree product $\\bigotimes_{i=0}^n T(s_i)$ satisfies the c.c.c.\n\\end{defi}\n\nFree trees are easily seen to be rigid Souslin trees\nas the product of two isomorphic relative trees $T(s)$ and $T(t)$\nwould clearly not be Souslin.\nIn Section \\ref{sec:separating} we will also consider weaker,\nparametrized forms of freeness.\n\n\\subsection{Decompositions of strongly homogeneous Souslin trees}\n\\label{sec:dec_free}\n\nWe now come to the key result of this paper.\nThe following theorem is stated in \\cite[p.246]{shelah-zapletal}\nin the case $n=2$ without proof.\nLarson gives the construction of a single free subalgebra\nof a strongly homogeneous Souslin algebra in terms of trees\nin the proof of Theorem 8.5 in his paper \\cite{larson}.\nSome ideas in the following proof are borrowed from that construction.\n\n\\begin{thm}\\label{thm:str_hom=free_x_free}\nFor every natural number $n>1$ and every\n$\\aleph_0$-branching, strongly homogeneous Souslin tree $T$\nthere are free Souslin trees $S_0,\\ldots,S_{n-1}$\nsuch that $T\\cong \\bigotimes_{m<n} S_m$.\n\\end{thm}\n\n\\begin{proof}\nLet $T$ be a strongly homogeneous Souslin tree and\ndenote by $\\psi_{s,t}$ the members of the coherent family of $T$.\nWe inductively (level by level) define $n$ nice t.e.r.s\n$\\equiv_0,\\ldots,\\equiv_{n-1}$ with the following properties:\n\\begin{itemize}\n\\item $T\/\\!\\!\\equiv_m$ is a free Souslin tree for $m<n$.\n\\item For any sequence $(s_0,\\ldots,s_{n-1})\\in T_\\alpha$\nthe intersection of the classes $s_m\/\\!\\!\\equiv_m$ for $m<n$ is\na singleton: $\\bigcap_{m<n} (s_m\/\\!\\!\\equiv_m)=\\{r\\}$ for some $r\\in T_\\alpha$.\n\\end{itemize}\nThe second claim entails the existence of the isomorphism between\n$T$ and $\\bigotimes_{m<n} T\/\\!\\!\\equiv_m$.\n\nLet $(P_{k,m}\\,:\\, m<n,k\\in\\omega)$ be an $n$-optimal matrix of partitions,\nwhere we view each $P_{k,m}$ as enumerated by $a_i^{k,m}$, $i\\in\\omega$.\nIn order to define t.e.r.s we transfer the whole\nmatrix of the $P_{k,m}$ to every set $\\NF(s)$ for $s\\in T$ in a coherent way:\nChoose for every $\\alpha<\\omega_1$\nan anchor node $r_\\alpha\\in T_\\alpha$\nand a bijection $\\sigma_\\alpha:\\omega\\to\\NF(r_\\alpha)$,\nand define for $s\\in T_\\alpha$ and all indices $i,k,m$ the sets\n$$a_i^{k,m}(s):=(\\psi_{r_\\alpha,s}\\circ\\sigma_\\alpha)''a_i^{k,m}.$$\nThen clearly for every $s\\in T$, $k\\in\\omega$ and $m<n$,\nthe set $P_{k,m}(s):=\\{a_i^{k,m}(s)\\mid i\\in\\omega\\}$ forms\na partition of $\\NF(s)$, and these partitions are linked to each other\nby the coherent family in a coherent way, i.e., \n$\\psi_{s,t}$ transfers $P_{k,m}(s)$ to $P_{k,m}(t)$.\n\nFix $m<n$ in order to define $\\equiv_m$ on $T$ by recursion on the height.\nWe will also enumerate the $\\equiv_m$-classes of each level in\norder type $\\omega$, i.e.,\nwe will fix an onto mapping $h:T\\to\\omega$,\nsuch that for $s,t\\in T_\\alpha$ we have\n$s\\equiv_m t$ if and only if $h(s)=h(t)$.\n\nChoose $P_{0,m}(\\text{root})$ as the partition\nof the set $T_1=\\NF(\\text{root})$ and let $\\equiv_m$ on level $T_1$\nbe the equivalence relation with classes\n$a_i^{0,m}(\\text{root})$ for $i\\in\\omega$.\nLet $h(\\text{root})=0$ and choose $h$ on $T_1$ in a way,\nsuch that nodes $s$ and $t$ are $\\equiv_m$-equivalent\njust in case that their $h$-values coincide.\n\nNext we consider the case where $\\alpha$ is a successor ordinal,\n$\\alpha=\\gamma+1$ for some $\\gamma<\\omega_1$.\nLet $s,t\\in T_\\alpha$ and let $s^-<_T s$ and $t^-<_T t$ be\ntheir direct predecessors on level $\\gamma$.\nWe let  $s\\equiv_m t$ if and only if their direct predecessors\nare $\\equiv_m$-equivalent, $s^-\\equiv_m t^-$\n(so in particular $h(s^-,m)=h(t^-,m)$),\nand if there is $i\\in\\omega$ such that\n$$s\\in a_i^{h(s^-),m}(s^-)\n\\text{ and }t\\in a_i^{h(t^-),m}(t^-).$$\nIn words, the $\\equiv_m$-equivalence of the direct predecessors\ngives us a natural number $h(s^-)$ and we apply\n$P_{h(s^-),m}$ on level $\\alpha$ to decide\nwhether or not $s$ and $t$ are $\\equiv_m$-equivalent. \nExtend $h$ to level $T_\\alpha$ as described above.\n\nOn limit stages $\\lambda$ the relation $\\equiv_m$ is already determined\nby its behaviour below,\nand we choose the $h\\!\\!\\upharpoonright\\!\\! T_\\lambda$ once more in any way such that\n$h(s)=h(t)$ is equivalent to\n$\\equiv_m$-equivalence for nodes $s,t\\in T_\\lambda$.\n\nHaving finished the construction of the relation $\\equiv_m$,\nwe show that it produces a nice t.e.r,\nwhere all properties but niceness follow rather easily from the construction.\nSo we only deduce niceness.\nLetting $s\\equiv_m r$ on level $\\alpha$ and $t$ above $s$ we claim \nthat $\\psi_{s,r}(t)\\equiv_m t$ and show this\nby induction on the height of $t$ above $s$.\nFor successor stages the claim follows directly\nfrom the construction and the inductive hypothesis,\nsince the relevant partition $P_{j,m}$\nis transferred via $\\psi_{s,r}$ by the coherence of the coherent family.\nThe limit case follows directly from the inductive assumption.\n(This property of $\\equiv_m$,\nthat $\\equiv_m$-equivalence lifts from $s$ and $r$\nto preimages and images under\n$\\psi_{s,r}$ will be used again in the proof of the Claim below.)\n\nIt remains to prove the two properties stated before the construction.\nWe start with the freeness of $T\/\\!\\!\\equiv_m$.\nLet $s_0,\\ldots,s_{k-1}$ be pairwise non-$m$-equivalent nodes\nof the same height $\\alpha$ for some natural number $k$.\nWe write $S_i$ for $(T\/\\!\\!\\equiv_m)(s_i\/\\!\\!\\equiv_m)$ and try to find\nfor every antichain $A$ of $\\bigotimes_{i<k}S_i$\nan antichain $B$ of $T$ of the same cardinality.\nThis would prove that $T\/\\!\\!\\equiv_m$ is a free tree.\nWe get a hint about where to look for the members of such an antichain\n$B$ from the following\\\\\n{\\bf Claim.} \nFix $m<n$ and pairwise non-$m$-equivalent nodes\n$s_0,\\ldots,s_{k-1}\\in T_\\alpha$.\nFor any sequence $(t_0,\\ldots,t_{k-1})$ of nodes in $T_\\beta$,\nwith $\\alpha<\\beta$ and $s_i<t_i$ for $i<k$,\nthe intersection of the classes\n$t_i\/\\!\\!\\equiv_m\\cap\\, T(s_i)$ above the nodes $s_i$,\nshifted above $s_0$ by $\\psi_{s_i,s_0}$, i.e., the set\n$$\\bigcap_{i<k} {\\psi_{s_i,s_0}}''(t_i\/\\!\\!\\equiv_m)$$\nis infinite and therefore non-empty.\n\\begin{proof}[Proof of the Claim.]\nBy induction on the height $\\beta$ of the nodes $t_i$,\nstarting with $\\beta=\\alpha+1$.\nIn this minimal case we have $t_i^-=s_i$.\nSo the sets ${\\psi_{s_i,s_0}}\"(t_i\/\\!\\!\\equiv_m)$\nbelong to distinct partitions $P_{h(s_i),m}(s_0)$, $i<k$\nand therefore have an infinite intersection\nby property (i) of the $n$-optimal matrix.\n\nFor the higher successor case $\\beta=\\gamma+1,\\,\\alpha<\\gamma$,\nwe simulate this initial situation.\nBy the inductive hypothesis pick a node\n$$r_0\\in\\bigcup_{i<k}{\\psi_{s_i,s_0}}''(t^-_i\/\\!\\!\\equiv_m)>s_0,$$\nand let $r_i:=\\psi_{s_0,s_i}(r_0)>s_i$ for $i<k$.\nWe then know that $r_i\\equiv_m t_i^-$,\nso $t_i\/\\!\\!\\equiv_m$ has elements above $r_i$.\nAs a consequence $\\bigcup_{i<k}{\\psi_{r_i,r_0}}''(t_i\/\\!\\!\\equiv_m)$\nis infinite by the same argument as above and furthermore\na subset of $\\bigcup_{i<k}{\\psi_{s_i,s_0}}''(t_i\/\\!\\!\\equiv_m)$.\n\nFor the case where $\\beta$ is a limit ordinal\nwe choose $\\gamma<\\beta$ large enough,\nsuch that letting $q_i=t_i\\!\\!\\upharpoonright\\!\\!\\gamma$ for all $i,j<k$ we have\n$\\psi_{q_i,q_j}(t_i)=t_j$.\nThis is possible due to the transitivity of the coherent family.\nWe also require $\\alpha<\\gamma$.\nThe inductive hypothesis gives us a node\n$$r_0\\in\\bigcup_{i<k}{\\psi_{s_i,s_0}}''(q_i\/\\!\\!\\equiv_m)\\subset T_\\gamma,$$\nwhich  we copy to $r_i:=\\psi_{s_0,s_i}(r_0)$.\nBy this choice, we also have $r_i\\equiv_m q_i$.\nWe consider $u=\\psi_{q_i,r_0}(t_i)$.\nBy the commutativity of the coherent family this definition of $u$\nis independent from the choice of $i<k$.\nBut then\n$$\\psi_{s_0,s_i}(u)=\\psi_{r_0,r_i}(u)=\n\\psi_{r_0,r_i}\\circ\\psi_{q_i,r_0}(t_i)=\\psi_{q_i,r_i}(t_i)$$\nwhere the first equation follows from coherence,\nthe second from the definition of $u$ and the third one from commutativity.\nSo the property stated above right after the construction of $\\equiv_m$\nimplies that $\\psi_{s_i,s_0}(t_i)\\equiv_m u$\nsince $r_i\\equiv_m t_i$ for all $i<k$.\nThis completes the proof of the Claim.\n\\end{proof}\nBy virtue of the Claim we can pick for every tuple\n$(t_0\/\\!\\!\\equiv_m,\\ldots,t_{k-1}\/\\!\\!\\equiv_m)$\nof our antichain $A\\subset \\bigotimes_{i<k} S_i$\na node $u\\in\\bigcap_{i<k}{\\psi_{s_i,s_0}}''(t_i\/\\!\\!\\equiv_m)$ and\ncollect all these nodes in a set $B$.\nThen $B$ is clearly an antichain of $T$ with the same cardinality as $A$.\nSo we have shown that $T\/\\!\\!\\equiv_m$ is indeed a free tree.\n\nNow for the second property.\nLet $(s_0,\\ldots,s_{n-1})$ be any sequence of nodes of some $T_\\alpha$.\nWe need to show that $\\bigcap_{m<n}s_m\/\\!\\!\\equiv_m$ has a unique element.\nThis is done by induction on $\\alpha>0$.\nStarting with $\\alpha=1$ we know that\n$(s_m\/\\!\\!\\equiv_m)=a_{i_m}^{k_m,m}(\\mathrm{root})$\nfor some $i_m$ and $k_m$.\nSo property (ii) of our $n$-optimal matrix is all we need here.\nFor $\\alpha=\\gamma+1$ we assume that the classes $s_m^-\/\\!\\!\\equiv_m$\nmeet in a single node, say $r\\in T_\\gamma$.\nThe set of elements of $s_m\/\\!\\!\\equiv_m$ which lie above $r$\nis then just $a_{i_m}^{h_m(r),m}(r)$ \nand again property (ii) of the matrix proves the claim.\nIn the limit case we once more use the transitivity of the coherent family.\nSo let $\\alpha$ be a limit and $\\gamma<\\alpha$ large enough such that\n$\\psi_{q_m,q_\\ell}(s_m)=s_\\ell$ where we abbreviate $s_m\\!\\!\\upharpoonright\\!\\!\\gamma=q_m$.\nFor a last time in this proof we use the commutativity of the\ncoherent family:\nLet $r$ be the unique element of the intersection\nof the classes $q_m\/\\!\\!\\equiv_m$.\nThen $t=\\psi_{q_m,r}(s_m)$ is well defined and\nindependent from the choice of $m<n$.\nBy the lifting property of the equivalence relations stated above,\nit follows from $q_m\\equiv_m r$, that $s_m\\equiv_m t$.\n\\end{proof}\n\nWe now state an algebraic feature which distinguishes our\nmethod of decomposition as just carried out from other decompositions,\nnamely that partial products of our decomposition are always \\emph{rigid},\ncf. Remark \\ref{rem:decomp}.\n\n\\begin{prp}\n\\label{prp:decomp}\nLet $T$ be a strongly homogeneous Souslin tree and assume that\nit has been decomposed into a product of $n$ free trees $S_0,\\ldots, S_{n-1}$\nby the procedure presented in the last proof.\nThen the product of less than $n$ pairwise distinct trees from the \nsequence $S_0,\\ldots,S_{n-1}$ is a rigid Souslin tree.\n\\end{prp}\n\\begin{proof}\nIt is clear that the product tree is Souslin and that it\nis sufficient to show rigidity only for the case of $n-1$ factors, where $n>2$.\nSo assume that $R:=\\bigotimes_{i<n-1}S_i$ admits the automorphism\n$\\varphi'\\ne \\id$ and derive a contradiction as follows.\n\nIdentifying $T$ and $R\\otimes S_{n-1}$ we can lift $\\varphi'$\nto an automorphism $\\varphi = \\varphi'\\otimes \\id$ of $T$.\nBy Proposition \\ref{prp:str_hom_aut} there is a countable ordinal $\\alpha$\nsuch that above level $T_\\alpha$ the mapping $\\varphi$ is given by\na subfamily of the coherent family of $T$.\nAs $\\varphi' \\ne \\id$ there must be a node\n$\\bar{s}=(s\/\\!\\!\\equiv_0,\\ldots,s\/\\!\\!\\equiv_{n-1})\\in R$ such that \n$\\varphi'(\\bar{s})\\ne \\bar{s}$.\nWe certainly can assume that $\\hgt_R(\\bar{s})=\\alpha$.\n\nPick an $\\ell<n-1$ such that the $\\ell$th component $\\bar{s}_\\ell$\nof $\\bar{s}$ is not mapped to itself under $\\varphi'$.\nWe fix a representative $s\\in \\bar{s}_\\ell=s\/\\!\\!\\equiv_\\ell\\subset T_\\alpha$\nand let $q:=\\varphi(s)$ in order to get\n$\\varphi(r)=\\psi_{sq}(r)$ for all (immediate) successors of $s$.\nThis is where we will find a contradiction.\n\nIn the construction above we have associated to class $s\/\\!\\!\\equiv_\\ell$\na natural number $h(s,\\ell)$ which defined the index of the partition \nthat was used to extend $\\equiv_\\ell$ to the successors of the members of\nthe class $s\/\\!\\!\\equiv_\\ell$.\nThe same holds for $q=\\varphi(s)$, but as $q\\not\\equiv_\\ell s$\nwe have $h(q,\\ell)\\ne h(s,\\ell)$.\nThis implies that $\\equiv_\\ell$ for immediate succesors of $s$ and of $q$\nis formed by virtue of different partitions of column $\\ell$ of our\npartition matrix.\nFix any immediate successor $r$ of $s$.\nSince $\\varphi'$ is assumed to be a well-defined mapping\n$\\varphi$ should map the intersection $\\bigcap_{i<n-1} r\/\\!\\!\\equiv_i$\nonto the set $\\bigcap_{i<n-1} \\varphi(r)\/\\!\\!\\equiv_i$. \nIn particular, the equation\n$${\\psi_{sq}}''\\left(T(s)\\cap\\bigcap_{i<n-1} r\/\\!\\!\\equiv_i\\right)\n= \\left( T(q) \\cap \\bigcap_{i<n-1} \\psi_{sq}(r)\/\\!\\!\\equiv_i\\right)$$\nshould be true.\nBut this is not the case, because, by $n$-optimality of the matrix\nof partitions, the left-hand-side of the above equation intersects\nevery $\\equiv_\\ell$-class of the immediate successors of $q$\nin exactly one element, while the right-hand-side is a subset\nof such an $\\equiv_\\ell$-class and at the same time an infinite set.\n\\end{proof}\n\nIn the next section we will find corollaries of\nTheorem \\ref{thm:str_hom=free_x_free} in a similar vein.\nI especially mention Lemma \\ref{lm:free_to_str_hom} which states that\nforcing with such a partial product (as above) turns the complementary\npartial product, which was rigid in the ground model,\ninto a strongly homogeneous Souslin tree in the generic extension.\n\nThe following complements Theorem \\ref{thm:str_hom=free_x_free}.\n\n\\begin{thm}\\label{thm:str_hom=str_hom_x_str_hom}\nEvery $\\aleph_0$-branching, strongly homogeneous Souslin tree $T$\nis (iso\\-morphic to) the tree product\nof $n$ strongly homogeneous Souslin trees for any given natural number $n>0$.\n\\end{thm}\n\\begin{proof}\nThis is just a simpler variant of the construction in the proof\nof Theorem \\ref{thm:str_hom=free_x_free} where\nwe use only the first row of the matrix of partitions\n(or just any bijection between $\\omega^n$ and $\\omega$).\nIt is then easy to verify that the coherent family of $T$ descends\nto the thus obtained factor trees and renders them strongly homogeneous.\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:decomp}\n\\begin{enumerate}[(i)]\n\\item Though, of course, not every tree product\nof two strongly homogeneous Souslin trees is Souslin again\n(e.g.\\ take $T\\otimes T$),\nthere is a converse to the last theorem:\nIf $S$ and $T$ are strongly homogeneous Souslin trees and\nthe tree product $S\\otimes T$ satisfies the c.c.c.,\nthen $S\\otimes T$ is a strongly homogeneous Souslin tree as well.\n\\item We see that there are two essentially distinct ways to decompose\na strongly homogeneous tree into (at least three) free factors.\nAn application of Theorem \\ref{thm:str_hom=str_hom_x_str_hom}\nto decompose a given strongly homogeneous Souslin tree $T$ into $\\ell$\nstrongly homogenous factors $S_0,\\ldots,S_{\\ell-1}$\nfollowed by an $\\ell$-fold application of the procedure used in the proof of\nTheorem \\ref{thm:str_hom=free_x_free} to decompose the tree $S_k$ into $m_k$\nfree trees $R^k_i$ for $0\\leq i < m_k$\nnever results in the same decomposition\nas directly using the proof of Theorem\n\\ref{thm:str_hom=free_x_free} to decompose $T$ into\n$\\sum_{k=0}^{\\ell-1} m_k$ free factors.\nThe partial products of the latter decomposition are all rigid by\nProposition \\ref{prp:decomp} while\nthe first also has partial products that are strongly homogeneous.\n\\end{enumerate}\n\\end{rem}\n\n\\section{Separating high degrees of rigidity}\n\\label{sec:separating}\n\nIn this chapter we review several families of rigidity notions\nfor Souslin trees, all of them weaker than freeness.\nThese definitions (except for that of an \\emph{$n$-free} Souslin tree)\nare all taken from \\cite{degrigST}.\nMost of these definitions refer to the technique of forcing applied\nwith a Souslin tree as the forcing partial order.\nWe do not review forcing here.\nBut recall,\nthat forcing with a Souslin tree always assumes the inverse order on the tree\n(i.e., trees grow downwards when considered as forcing partial orders,\nthe root is the maximal element, etc.) and adjoins a cofinal branch.\n\nThis section is divided in five short subsections.\nThe first two introduce the rigidity notions to be considered\nand the last three state many and prove some separations between them.\nWe only give proofs that either are elementary or use the\nproof of the Decomposition\nTheorem \\ref{thm:str_hom=free_x_free}.\n\n\\subsection{Parametrized freeness}\n\\label{sec:free}\n\nConsidering the definition of the property \\emph{free} for Souslin trees\nit is natural to ask whether or not it makes any difference\nif the number of the factors in the tree products,\nthat are required to be Souslin, is bounded.\nThis leads to the following definition which we rightaway connect\nto the definition of \\emph{being $n$-fold Souslin off the generic branch}\nmet in \\cite{degrigST}.\n\n\\begin{defi}\nLet $n$ be a positive natural number.\n\\begin{enumerate}[a)]\n\\item We say that a Souslin tree $T$ is \\emph{$n$-free} if for every subset\n$P$ of size $n$ of some level $T_\\alpha$, $\\alpha<\\omega_1$, the\ntree product $\\bigotimes_{s\\in P} T(s)$ satisfies the c.c.c.\n\\item A Souslin tree is said to be\n$n$\\emph{-fold Souslin off the generic branch},\nif for any sequence $\\vec{b}=(b_0,\\ldots,b_{n-1})$\ngeneric for the $n$-fold forcing product of (the inverse partial order\nof) $T$ and any node $s\\in T\\setminus\\bigcup_{i\\in n}b_i$,\nthe subtree $T(s)$ of all nodes of $T$ above $s$ is a Souslin tree\nin the generic extension $M[\\vec{b}]$ (which amounts to requiring that the\nadjunction of $\\vec{b}$ does not collapse $\\omega_1$ and preserves the\nc.c.c. of the $T(s)$, $s\\notin\\bigcup b_i$).\n\\end{enumerate}\n\\end{defi}\nIt is easy to see that a 2-free Souslin tree or a tree which is Souslin\noff the generic branch cannot be decomposed as the product of two Souslin\ntrees. And this common feature is no coincidence.\n\n\\begin{prp}\\label{prp:free_Sotgb}\nFor a positive natural number $n$ and a normal Souslin tree $T$\nthe following statements are equivalent.\n\\begin{enumerate}[a)]\n\\item $T$ is $n$-fold Souslin off the generic branch.\n\\item $T$ is $(n\\!+\\!1)$-free.\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nWe start with the implication (b$\\to$a).\nAssume that $T$ is $n+1$-free and let $\\vec{b}=(b_0,\\ldots,b_{n-1})$ be\ngeneric for $T^{\\otimes n}$, the $n$-fold tree product of $T$ with itself.\nChoose $\\alpha<\\omega_1$ large enough,\nsuch that the nodes $t_i:=b_i(\\alpha)$ are pairwise incompatible.\nFinally, pick a node $t_n\\in T_\\alpha$ distinct from all the $b_i(\\alpha)$.\nBy our freeness assumption on $T$,\nthe product tree $\\bigotimes_{i\\in{n+1}}T(t_i)$\nsatisfies the countable chain condition.\nBut then $M[\\vec{b}]\\vDash$''$T(t_n)$ is Souslin''\nby a standard argument concerning chain conditions in forcing iterations.\nNow it is easy to see that $T$ is $n$-fold Souslin off the generic branch.\n\nFor the other direction we inductively show that\n$T$ is $m$-free for $m\\leq n+1$,\nassuming that $T$ is $n$-fold Souslin off the generic branch.\nThe inductive claim is trivial for $m=1$. \nSo let $m\\geq1$ and let $s_0,\\ldots,s_m$ be pairwise distinct nodes\nof the same height.\nThen for any generic sequence\n$\\vec{b}=(b_0,\\ldots,b_{m-1})$ for $\\bigotimes_{i\\in m}T(s_i)$\nwe know that $T(s_m)$ is Souslin in the  generic extension $M[\\vec{b}]$.\nFinally the two-step iteration $\\bigotimes_{i\\in m}T(s_i)\\ast \\check{T}(s_m)$\nis isomorphic to $\\bigotimes_{i\\in m+1}T(s_i)$\nand satisfies the countable chain condition.\n\\end{proof}\n\nThis proposition implies that a free tree $T$ is also\n\\emph{free off the generic branch} in the sense that\nin the generic extension obtained by adjoining\na cofinal branch $b$ through $T$, for every node $t\\in T\\setminus b$,\nthe tree $T(t)$ is still free.\n\n\\subsection{Further types of rigidity}\n\nIn Sections 1-4 of \\cite{degrigST} different notions of rigidity\nfor Souslin trees are collected:\n(ordinary) rigidity, total rigidity and the unique branch property and their\nabsolute counterparts,\nwhere absoluteness refers to forcing extensions obtained by\nadjoining a generic branch to the Souslin tree under consideration.\nIn this context also the stronger notion of being\n($n$-fold) Souslin off the generic branch is introduced \nwhich we already considered in the last section.\n\n\\begin{defi}\n\\begin{enumerate}[a)]\n\\item A Souslin tree $T$ is called \\emph{$n$-absolutely rigid}, if $T$ is a rigid tree in the generic extension\nobtained by forcing with $T^n$ (or equivalently $T^{\\otimes n}$).\n\\item A Souslin tree is \\emph{totally rigid}, if the trees $T(s)$ and $T(t)$ are non-isomorphic for all pairs\nof distinct nodes $s$ and $t$ of $T$. It is \\emph{$n$-absolutely totally rigid} if it is totally rigid\nafter forcing with $T^n$.\n\\item A Souslin tree $T$ has the \\emph{unique branch property (UBP)}, if forcing with $T$ adjoins only a single\ncofinal branch to $T$. For $n>0$ we say, that $T$ has the $n$-\\emph{absolute UBP}, if forcing with\n$T^{n+1}$ adjoins exactly $n+1$ cofinal branches to $T$.\n\\end{enumerate}\n\\end{defi}\n\n\nFuchs and Hamkins prove implications as well as some independencies\nbetween these rigidity notions.\nThey also give in \\cite[Section 4]{degrigST}\na diagram of implications between the\ndegrees of rigidity that we have approximately\nreconstructed here for the convenience of the reader.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{ccccccc}\n2-free&$\\longleftarrow$&\\makebox[2.5cm][c]{3-free}&$\\longleftarrow$&\\makebox[2.5cm][c]{4-free}&$\\longleftarrow$&$\\ldots$\\\\\n$\\downarrow$&&$\\downarrow$&&$\\downarrow$&&\\\\\nUBP&$\\longleftarrow$&\\makebox[2.5cm][c]{absolutely UBP}&$\\longleftarrow$&\\makebox[2.5cm][c]{2-absolutely UBP}&$\\longleftarrow$&$\\ldots$\\\\\n$\\downarrow$&&$\\downarrow$&&$\\downarrow$&&\\\\\ntotally rigid&$\\longleftarrow$&\\makebox[3cm][c]{abs. totally rigid}&$\\longleftarrow$&\n\\makebox[3cm][c]{2-abs. totally rigid}&$\\longleftarrow$&$\\ldots$\\\\\n$\\downarrow$&&$\\downarrow$&&$\\downarrow$&&\\\\\nrigid&$\\longleftarrow$&\\makebox[2.5cm][c]{absolutely rigid}&$\\longleftarrow$&\\makebox[2.5cm][c]{2-absolutely rigid}&$\\longleftarrow$&$\\ldots$\\\\\n&&&&&&\n\\end{tabular} \\end{small}\n\\end{center}\n\\caption{Implications between degrees of rigidity for Souslin trees.}\n\\label{diagram}\n\\end{table}\n\nFuchs and Hamkins show that\nthe part of the diagram to the left and below ``absolutely UBP''\nis complete in the sense that there are no further\ngeneral implications between these rigidity properties.\nThey ask whether the rest of the diagram\nis complete as well, cf.~\\cite[Question 4.1]{degrigST}.\nWe will show (resp. state) below that there are\nneither implications from left to the right\n(including downwards diagonals,\ncf.~Corollaries \\ref{cor:n-free} and \\ref{cor:n-free_not_UBP}),\nnor from the second to the upper row (Theorem \\ref{thm:not_simple_UBP}).\n\n\\begin{rem}\n\\label{rem:diagram}\n\\item Using a standard $\\diamondsuit$-construction\nscheme for a Souslin tree (e.g., cf. \\cite[Section 2]{degrigST})\nit is not hard to construct a Souslin tree $T$ with the following two\nfeatures:\n\\begin{itemize}\n\\item On each level $T_\\alpha$ no two distinct nodes\nhave the same number of immediate successors.\nSo in particular $T$ is $n$-absolutely totally rigid\nfor every $n\\in \\omega$.\n\\item The substructure $R$ of $T$ obtained by\nrestricting the supporting set to the nodes on the limit levels of $T$\nplus the root, is a homogeneous Souslin tree.\nThen in a generic extension obtained by forcing with $T$ there are\nmany cofinal branches in $R$ and each of them gives rise to a cofinal branch\nof $T$, which is thus not a UBP tree.\n(In fact, every $\\aleph_0$-spiltting Souslin tree can be extended to an\n$n$-absolutely totally rigid Souslin tree by inserting new successor levels \nsuch that every two nodes of the same height have a different number of\nimmediate successors.)\n\\end{itemize}\nThis shows that in Diagram \\ref{diagram} there can be no arrows that\npoint upwards from the two lower rows.\nSo the only question left open is whether there should be any more arrows\nbetween the two lower rows, but a similar construction as the one alluded to\nabove should also eliminate those.\n\\end{rem}\n\n\\subsection{Distinct degrees of freeness}\n\nOur next corollary of (the proof of)\nTheorem~\\ref{thm:str_hom=free_x_free} gives the separation of\nthe finite degrees of freeness, i.e.,\nit shows that the family of parametrized freeness\nconditions is properly increasing in strength.\n\n\\begin{cor}\\label{cor:n-free}\nIf there is a strongly homogeneous Souslin tree,\nthen there is an $n$-free, but not $n+1$-free tree.\n\\end{cor}\n\\begin{proof}\nLet the strongly homogeneous Souslin tree $T$ be decomposed\nas the tree product of $n$ free trees $S_i$ for $i<n$\nas carried out in the proof of Theorem~\\ref{thm:str_hom=free_x_free}.\nWe show, that the tree sum of the factors,\n$$R:=\\bigoplus_{i<n}S_i$$\nis an $n$-free but not $n+1$-free Souslin tree.\nThe Claim used in the proof of Theorem~\\ref{thm:str_hom=free_x_free}\nremains true in the following variant:\\\\\n{\\bf Claim'.} For any pair of sequences $(s_0,\\ldots,s_{n-1})$ in $T_\\alpha$ and $t_i>s_i$ in $T_\\beta$ and\nany sequence $m:n\\to n$ the intersection\n$$\\bigcap_{i<n}{\\psi_{s_i,s_0}}''t_i\/\\!\\!\\equiv_{m(i)}$$\nis not empty.\n\nModulo the obvious changes in the notation,\nthe proof of the Claim' remains completely the same as before,\nexploiting the $n$-optimality of the matrix.\nAnd also with the same argument as above we can construct\nan antichain of $T$ from any given antichain of $R$\nmaintaining the cardinality. So $R$ is $n$-free.\n\nWe argue that $R$ is not $n$-fold Souslin off the generic branch.\n\nIf $b_i$ is a cofinal branch through $S_i$\nthen in the generic extension obtained by adjoining\n$\\vec{b}=(b_0,\\ldots,b_{n-1})$, the strongly homogeneous\ntree $T$ has a cofinal branch as well,\nthus destroying the Souslinity of all subtrees of $R$.\n\\end{proof}\n\n\\subsection{Freeness and absolute rigidity}\n\nIn this section we improve upon the result of the last one\nby showing that $n$-freeness of a tree does not even imply\n$(n\\!-\\!1)$-absolute rigidity.\n\n\\begin{lm}\\label{lm:free_to_str_hom}\nLet the strongly homogeneous Souslin tree $T$ be\ndecomposed as the tree product of $n$ free trees $S_i$ for $i<n$\nas carried out in the proof of Theorem~\\ref{thm:str_hom=free_x_free}.\nLet $\\{a,b\\}$ be a partition of the set $n$ with $a,b\\neq\\varnothing$\nand set $\\mathbb{P}:=\\bigotimes_{i\\in a}S_i$ and $R:=\\bigotimes_{i\\in b}S_i$.\nThen\n$$\\vDash_{\\mathbb{P}}\\text{``}\\check{R}\\text{ is strongly homogeneous.''}$$\n\\end{lm}\n\\begin{proof}\nWe adopt the notation from the statement of the lemma and\nargue inside the generic extension obtained by adjoining a generic branch\n$c$ to the Souslin tree $\\mathbb{P}$.\nThen we have the natural isomorphism $\\rho:R\\cong c\\otimes R\\subset T$.\nDenote the canonical projection $T\\to R$ by $\\pi$.\n\nWe define the tree isomorphisms $\\varphi_{{r}{s}}$\n(members of the coherent family of $R$ in the generic extension)\nfor nodes ${r}$ and ${s}$ of $R_\\alpha$, $\\alpha<\\omega_1$\nfrom the members $\\psi_{\\rho({r})\\rho({s})}$ of the coherent family of $T$.\nFor this, we refer to the maps $h(\\cdot,j):T\\to\\omega$\nused in the construction of the t.e.r.s $\\equiv_j$ in the\nproof of Theorem~\\ref{thm:str_hom=free_x_free}.\nWe collect them and define $h:T\\to\\omega^{|b|}$ by\nsimply concatenating the values $h(t,j)$ for $t\\in T$ and $j\\in b$.\n\nIf ${r},\\,{s}\\in R_\\alpha$ and $h(\\rho({r}))=h(\\rho({s}))$,\nthen we let\n$$\\varphi_{{r}{s}}:=\n\\pi\\circ\\psi_{\\rho({r})\\rho({s})}\\circ\\rho:R({r})\\to R({s}).$$\nIt follows from the fact that the $n$-optimal partition matrices\nare transported between the (sets of immediate successors of the)\nnodes by the members $\\psi_{tu}$ of the coherent family of $T$,\nthat this definition is sound and indeed yields an isomorphism.\n\nNow let $r,s\\in R_\\alpha$ with $h(\\rho({r}))\\neq h(\\rho({s}))$.\nIn order to define $\\varphi_{{r}{s}}$ we compose the tree isomorphisms,\nthat we have already defined for the immediate successors of ${r}$ and ${s}$.\nFor every direct successor ${u}\\in\\NF({r})$\nthere is exactly one ${v}\\in\\NF({s})$ with $h(\\rho({u}))=h(\\rho({v}))$.\nThis follows from the $n$-optimality of the partition matrix.\nLet $\\varphi_{{r}{s}}({u})$ be just this ${v}$.\nIf ${x}$ is a non-immediate successor of ${r}$,\nthen first find the immediate successor ${u}$ of ${r}$\nbelow $x$ and the image $v=\\varphi_{rs}(u)$,\nand set\n$$\\varphi_{{r}{s}}({x}):=\\varphi_{{u}{v}}({x}).$$\n\nIt remains to prove, that the family of tree isomorphisms just defined\nis coherent, commutative and transitive.\nCommutativity and coherence are inherited from the coherent family of $T$.\n(Note that $\\varphi_{rs}(x)=y$ implies that $h(\\rho(x))=h(\\rho(y))$,\nso the two cases do not interfere.)\nAs for transitivity,\nlet $x,y\\in R_\\lambda$ for some countable limit ordinal $\\lambda$.\nThen by the transitivity of the family of the $\\psi_{tu}$ for $T$\nthere are $t<\\rho(x)$ and $u<\\rho(y)$ with $\\psi_{tu}(\\rho(x))=\\rho(y)$.\nBut then $t$ and $u$ lie in $b\\otimes R$,\nso there are $r,s\\in R$ such that\n$\\rho(r)=t$ and $\\rho(s)=u$ and thus $\\varphi_{rs}(x)=y$.\n\\end{proof}\n\nSo, e.g.\\ in the case $n=2$,\nforcing with one free tree does not only destroy the freeness of another one,\nbut even turns the latter into a strongly homogeneous Souslin tree,\ni.e., it adjoins many generic automorphisms.\n\n\n\\begin{cor}\n\\label{cor:n-free_not_UBP}\nLet $n>1$.\nIf there is a strongly homogeneous Souslin tree,\nthen there is an $n$-free tree\nwhich is not $(n\\!-\\!1)$-absolutely rigid.\n\\end{cor}\n\\begin{proof}\nWe fix $n>1$ and\nuse the tree $R$ from the proof of Corollary~\\ref{cor:n-free}\nobtained from a strongly homogeneous tree $T$\nas the tree sum $R=\\bigoplus_{i<n}S_i$\nof the free factors $S_i$, $i<n$ of $T$.\nFrom Corollary~\\ref{cor:n-free} we know, that $R$ is $n$-free.\n\nTo show that $R$ is not $(n\\!-\\!1)$-absolutely rigid we refer\nto Lemma~\\ref{lm:free_to_str_hom}.\nIt follows directly from the case that $a=n\\setminus\\{i\\}$ for some $i<n$\nthat $R$ is not rigid in the generic extension obtained by adjoining\na cofinal branch through the trees $S_j$ for $j<n$ and $j\\neq i$.\nBut this generic extension can also be reached by forcing with\n$R^{\\otimes n-1}$.\n\\end{proof}\n\n\\subsection{Freeness and the unique branch property}\n\nWe start with an easy result deduced from the elementary properties\nof finitely free trees for the second column of the diagram.\n\n\\begin{prp}\n\\label{prp:3-free+UBP}\nIf there is a 3-free Souslin tree,\nthen there is also a Souslin tree which has the UBP and is not 2-free.\n\\end{prp}\n\\begin{proof}\nLet $T$ be $3$-free and pick distinct nodes $s,t\\in T$ of the same height.\nWe show, that the Souslin tree $S=T(s)\\otimes T(t)$ has the UBP\\@.\nLet $b\\otimes c$ be a generic,\ncofinal branch in $S$ (viewing $b$ and $c$ as trees).\nBy the 2-fold Souslinity off the generic branch of $T$,\nevery tree of the form $T(r)$ with $r\\in T\\setminus (b\\cup c)$\nis Souslin in the generic extension by $b\\otimes c$.\nOn the other hand,\nif there was a second cofinal branch through $S$ in the generic extension,\nthen one of its components would have to pass\nthrough such a node $r\\notin b\\cup c$, which yields a contradiction.\n\nTo prove that $S$ is not $2$-free,\nlet $u>s$ and $v,w>t$ be of the same height, where $v\\ne w$.\nThen\n$$S(u,v)\\otimes S(u,w) \\cong T(u)\\otimes T(v)\\otimes T(u)\\otimes T(w)$$\nhas an uncountable antichain,\nbecause it has the square of the Souslin tree $T(u)$ as a factor.\n\\end{proof}\n\nThis result cannot be improved by simply requiring $T$ to be free, because\nby iterating the forcing with a tree product of two factors\n$n\\!+\\!1$ times, we adjoin at least $2^n$ cofinal branches.\n\nWe do have the following non-implication result for the $n$-absolute UBP\nand $2$-freeness under the stronger assumption of $\\diamondsuit$.\n\n\\begin{thm}\n\\label{thm:not_simple_UBP}\nAssume $\\diamondsuit$.\nThen there is a Souslin tree\nwhich is not 2-free but has the $n$-absolute UBP for all $n\\in\\omega$.\n\\end{thm}\n\nThe methods of proof for this theorem lie beyond the scope of this paper.\nIt uses ideas from \\cite{degrigST} and \\cite{SAE}.\nA proof-sketch can be found in \\cite[Theorem 1.6.3]{diss}.\n\n\\subsection{Further directions}\n\nAs a closing remark we mention how Diagram \\ref{diagram},\nwhich captures the implications between four families of rigidity notions and\nimplications between them, could possibly be extended.\n\n\\begin{description}\n\\item[Real rigidity] \nIn \\cite{abraham-shelah_aronszajn_trees} two Aronszajn trees are called\n\\emph{really different} if there is no isomorphism between any of their\nrestrictions to some club set of levels. In this vein, we could call a Souslin\ntree \n\\emph{really rigid} if all of its restrictions to club sets of levels are\nrigid. This property is clearly stronger than ordinary rigidity yet\nindependent of total rigidity (cf. Remark \\ref{rem:diagram}) and is implied\nby the unique branching property. \nAlso the variant of \\emph{real, total\n  rigidity} and the $n$-absolute versions of real and of real, total rigidity\ncould be considered. \n\n\\item[Self-specializing trees] A normal tree $T$ of height $\\omega_1$ is called \\emph{special} if there is a\ncountable family $(A_n)_{n\\in\\omega}$ of antichains of $T$ that covers all of\n$T$. As $T$ is uncountable, one of the $A_n$ has to be uncountable as well,\nso a special tree $T$ is not Souslin.\nOn the other hand, every branch of $T$ meets each antichain $A_n$\nin at most one node and is therefore countable.\n\nA \\emph{self-specializing tree} is a Souslin tree $T$ that specializes itself\nby forcing a generic branch $b$ through it, i.e., in the generic extension\nobtained by adjoining $b$ to the universe, the tree $T\\setminus b$ is special.\nSelf-specializing trees can be found in models of $\\diamondsuit$.\nThey are UBP: a second cofinal branch in $T$ would prevent $T\\setminus b$\nfrom being special. But of course they are not Souslin off the generic branch, \nand they can neither be 2-absolutely really rigid nor absolutely UBP,\nbecause forcing with a special tree collapses $\\omega_1$,\nand in this second generic extension the limit levels of $T$ form an\n$\\aleph_0$-splitting tree of countable height which must be homogeneous\nby a result of Kurepa (cf.\\cite[p.102]{kurepa}). \n\nNow let us call a Souslin tree $T$ \\emph{$n$-self-specializing} if it is\n$n$-free (i.e. $(n\\!-\\!1)$-fold Souslin off the generic branch) and\nforcing a generic branch $\\vec{b}$ through $T^n$ makes $T\\setminus\\tilde{b}$\nspecial where $\\tilde{b}$ is the set of components of the elements of\n$\\vec{b}$. It is not yet verified but seems quite plausible\nthat one can construct an $n$-self-specializing tree under $\\diamondsuit$.\nIn the implication diagram its place could be between $n$-free\nand $(n\\!-\\!1)$-absolutely UBP, yet it is stronger than both of these\nproperties. And there would be no horizontal implications, for\nan $n$-self-specializing tree is neither $(n\\!-\\!1)$-self-specializing nor\n$(n\\!+\\!1)$-self-specializing.\n\n\n\\end{description}\nAs is clear from the outset, adding these families to Diagram\n\\ref{diagram} results in a far more complicated directed graph which is in\nparticular non-planar.\nWe leave such considerations for future work.\n\n\\subsection*{Acknowledgements}\nThanks are due to Piet Rodenburg for pointing out a flaw in the proof of Lemma\n\\ref{lm:opt_mtx_prt}.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}