minicomplexity
membership
definition

Given a number and a set of numbers from $[h]$ (in this order), check that the number is in the set.

details

For every $h\geq 1$, membership$h$ (or just mem$h$) is defined over the alphabet of all elements and subsets of $[h]=\{0,1,\dots,h{-}1\}$. Its instances are all strings of length $2$ where the first symbol is a number and the second symbol is a set:

$i\,\,\alpha$       with $i\in[h]$ and $\alpha\subseteq[h]$.
An instance is positive if $i\in\alpha$. Otherwise, $i\not\in\alpha$ and the instance is negative.

E.g., $\mathtt{3\{0{,}3{,}4\}}$ is a positive instance of mem$5$, whereas $\mathtt{2\{0{,}3{,}4\}}$ is a negative instance. (Note that here $\mathtt{\{0{,}3{,}4\}}$ is an individual symbol. Also note that strings like $\mathtt{3}$, $\mathtt{\{0{,}3{,}4\}}$, $\mathtt{\{0{,}3{,}4\}\{2\}1}$, $\mathtt{\{0{,}3{,}4\}3}$, $\mathtt{\{0{,}3{,}4\}\{3\}}$, and $\mathtt{3\;\,4}$ are not instances of the problem.)

notes

Introduced by Kapoutsis Kralovic Moemke 2008 (through its reverse), as a core problem from which one can construct other problems that can witness several separations between classes. Its importance stems from its simplicity and from the fact that it is in 1D\re-1D.

The name “membership” is suggested by this site. huh?

See list membership, ordered list membership, and projection for three related problems.

re-1D f1D/uny
re-1D f1D/uny rc-1D 1D/uny 21D/uny e1D/uny r1D/uny 1N/uny 21N/uny e1N/uny r1N/uny f1N/uny RD/uny 2RD/uny eRD/uny rRD/uny fRD/uny RN/uny 2RN/uny eRN/uny rRN/uny fRN/uny SD/uny 2SD/uny eSD/uny rSD/uny fSD/uny SN/uny 2SN/uny eSN/uny rSN/uny fSN/uny 2D/uny 22D/uny e2D/uny r2D/uny f2D/uny 2N/uny 22N/uny e2N/uny r2N/uny f2N/uny re-1D/uny re-21D/uny re-e1D/uny re-r1D/uny re-f1D/uny re-1N/uny re-21N/uny re-e1N/uny re-r1N/uny re-f1N/uny re-RD/uny re-2RD/uny re-eRD/uny re-rRD/uny re-fRD/uny re-RN/uny re-2RN/uny re-eRN/uny re-rRN/uny re-fRN/uny re-SD/uny re-2SD/uny re-eSD/uny re-rSD/uny re-fSD/uny re-SN/uny re-2SN/uny re-eSN/uny re-rSN/uny re-fSN/uny re-2D/uny re-22D/uny re-e2D/uny re-r2D/uny re-f2D/uny re-2N/uny re-22N/uny re-e2N/uny re-r2N/uny re-f2N/uny co-1D/uny co-21D/uny co-e1D/uny co-r1D/uny co-f1D/uny co-1N/uny co-21N/uny co-e1N/uny co-r1N/uny co-f1N/uny co-RD/uny co-2RD/uny co-eRD/uny co-rRD/uny co-fRD/uny co-RN/uny co-2RN/uny co-eRN/uny co-rRN/uny co-fRN/uny co-SD/uny co-2SD/uny co-eSD/uny co-rSD/uny co-fSD/uny co-SN/uny co-2SN/uny co-eSN/uny co-rSN/uny co-fSN/uny co-2D/uny co-22D/uny co-e2D/uny co-r2D/uny co-f2D/uny co-2N/uny co-22N/uny co-e2N/uny co-r2N/uny co-f2N/uny rc-1D/uny rc-21D/uny rc-e1D/uny rc-r1D/uny rc-f1D/uny rc-1N/uny rc-21N/uny rc-e1N/uny rc-r1N/uny rc-f1N/uny rc-RD/uny rc-2RD/uny rc-eRD/uny rc-rRD/uny rc-fRD/uny rc-RN/uny rc-2RN/uny rc-eRN/uny rc-rRN/uny rc-fRN/uny rc-SD/uny rc-2SD/uny rc-eSD/uny rc-rSD/uny rc-fSD/uny rc-SN/uny rc-2SN/uny rc-eSN/uny rc-rSN/uny rc-fSN/uny rc-2D/uny rc-22D/uny rc-e2D/uny rc-r2D/uny rc-f2D/uny rc-2N/uny rc-22N/uny rc-e2N/uny rc-r2N/uny rc-f2N/uny
1D re-RD rc-RD
1D re-RD rc-RD 21D e1D r1D f1D 1N 21N e1N r1N f1N RD 2RD eRD rRD fRD RN 2RN eRN rRN fRN SD 2SD eSD rSD fSD SN 2SN eSN rSN fSN 2D 22D e2D r2D f2D 2N 22N e2N r2N f2N re-21D re-e1D re-r1D re-f1D re-1N re-21N re-e1N re-r1N re-f1N re-2RD re-eRD re-rRD re-fRD re-RN re-2RN re-eRN re-rRN re-fRN re-SD re-2SD re-eSD re-rSD re-fSD re-SN re-2SN re-eSN re-rSN re-fSN re-2D re-22D re-e2D re-r2D re-f2D re-2N re-22N re-e2N re-r2N re-f2N co-1D co-21D co-e1D co-r1D co-f1D co-1N co-21N co-e1N co-r1N co-f1N co-RD co-2RD co-eRD co-rRD co-fRD co-RN co-2RN co-eRN co-rRN co-fRN co-SD co-2SD co-eSD co-rSD co-fSD co-SN co-2SN co-eSN co-rSN co-fSN co-2D co-22D co-e2D co-r2D co-f2D co-2N co-22N co-e2N co-r2N co-f2N rc-21D rc-e1D rc-r1D rc-f1D rc-1N rc-21N rc-e1N rc-r1N rc-f1N rc-2RD rc-eRD rc-rRD rc-fRD rc-RN rc-2RN rc-eRN rc-rRN rc-fRN rc-SD rc-2SD rc-eSD rc-rSD rc-fSD rc-SN rc-2SN rc-eSN rc-rSN rc-fSN rc-2D rc-22D rc-e2D rc-r2D rc-f2D rc-2N rc-22N rc-e2N rc-r2N rc-f2N
complete for:
– no record –
also hard for:
– no record –
at least as hard as:
– no record –
fcomp$^r$:$\leq^\text{lac}_\text{RD}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{RD}$ mem$^c$:$\leq^\text{t}_\text{h}$ mem$^c$:$\leq^\text{lac}_\text{1D}$ proj:$\leq^\text{lac}_\text{RD}$ proj$^c$:$\leq^\text{lac}_\text{RD}$
fcomp$^r$:$\leq^\text{lac}_\text{RD}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{RD}$ mem$^c$:$\leq^\text{t}_\text{h}$ mem$^c$:$\leq^\text{lac}_\text{1D}$ proj:$\leq^\text{lac}_\text{RD}$ proj$^c$:$\leq^\text{lac}_\text{RD}$ fcomp$^r$:$\leq_\text{RD}$ fcomp$^r$:$\leq^\text{lac}_\text{2D}$ fcomp$^r$:$\leq_\text{2D}$ fcomp$^{rc}$:$\leq_\text{RD}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{2D}$ fcomp$^{rc}$:$\leq_\text{2D}$ mem$^c$:$\leq_\text{h}$ mem$^c$:$\leq_\text{1D}$ mem$^c$:$\leq^\text{lac}_\text{RD}$ mem$^c$:$\leq_\text{RD}$ mem$^c$:$\leq^\text{lac}_\text{2D}$ mem$^c$:$\leq_\text{2D}$ mem$^c$:$\leq^\text{lac}_\text{1N}$ mem$^c$:$\leq_\text{1N}$ proj:$\leq_\text{RD}$ proj:$\leq^\text{lac}_\text{2D}$ proj:$\leq_\text{2D}$ proj$^c$:$\leq_\text{RD}$ proj$^c$:$\leq^\text{lac}_\text{2D}$ proj$^c$:$\leq_\text{2D}$
cowl:$\leq^\text{t}_\text{h}$ cowl:$\leq^\text{lac}_\text{1D}$ cowl$^r$:$\leq^\text{t}_\text{h}$ cowl$^r$:$\leq^\text{lac}_\text{1D}$ cowl$^c$:$\leq^\text{t}_\text{h}$ cowl$^c$:$\leq^\text{lac}_\text{1D}$ cowl$^{rc}$:$\leq^\text{t}_\text{h}$ cowl$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj:$\leq^\text{t}_\text{h}$ disj:$\leq^\text{lac}_\text{1D}$ disj$^c$:$\leq^\text{t}_\text{h}$ disj$^c$:$\leq^\text{lac}_\text{1D}$ fcomp$^r$:$\leq^\text{t}_\text{h}$ fcomp$^r$:$\leq^\text{lac}_\text{1D}$ fcomp$^{rc}$:$\leq^\text{t}_\text{h}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{1D}$ fmatch:$\leq^\text{lac}_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{1D}$ fmatch$^c$:$\leq^\text{lac}_\text{1D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ fpath:$\leq^\text{lac}_\text{1D}$ fpath$^r$:$\leq^\text{lac}_\text{1D}$ fpath$^c$:$\leq^\text{lac}_\text{1D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1D}$ fzmatch$^c$:$\leq^\text{lac}_\text{1D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ incl:$\leq^\text{t}_\text{h}$ incl:$\leq^\text{lac}_\text{1D}$ incl$^r$:$\leq^\text{t}_\text{h}$ incl$^r$:$\leq^\text{lac}_\text{1D}$ incl$^c$:$\leq^\text{t}_\text{h}$ incl$^c$:$\leq^\text{lac}_\text{1D}$ incl$^{rc}$:$\leq^\text{t}_\text{h}$ incl$^{rc}$:$\leq^\text{lac}_\text{1D}$ msp:$\leq^\text{lac}_\text{1D}$ msp$^r$:$\leq^\text{lac}_\text{1D}$ msp$^c$:$\leq^\text{lac}_\text{1D}$ msp$^{rc}$:$\leq^\text{lac}_\text{1D}$ olmem:$\leq^\text{t}_\text{h}$ olmem:$\leq^\text{lac}_\text{1D}$ olmem$^c$:$\leq^\text{t}_\text{h}$ olmem$^c$:$\leq^\text{lac}_\text{1D}$ proj:$\leq^\text{t}_\text{h}$ proj:$\leq^\text{lac}_\text{1D}$ proj$^c$:$\leq^\text{t}_\text{h}$ proj$^c$:$\leq^\text{lac}_\text{1D}$ sep:$\leq^\text{t}_\text{h}$ sep:$\leq^\text{lac}_\text{1D}$ sep$^r$:$\leq^\text{t}_\text{h}$ sep$^r$:$\leq^\text{lac}_\text{1D}$ sep$^c$:$\leq^\text{t}_\text{h}$ sep$^c$:$\leq^\text{lac}_\text{1D}$ sep$^{rc}$:$\leq^\text{t}_\text{h}$ sep$^{rc}$:$\leq^\text{lac}_\text{1D}$ sp:$\leq^\text{t}_\text{h}$ sp:$\leq^\text{lac}_\text{1D}$ sp$^r$:$\leq^\text{t}_\text{h}$ sp$^r$:$\leq^\text{lac}_\text{1D}$ sp$^c$:$\leq^\text{t}_\text{h}$ sp$^c$:$\leq^\text{lac}_\text{1D}$ sp$^{rc}$:$\leq^\text{t}_\text{h}$ sp$^{rc}$:$\leq^\text{lac}_\text{1D}$ srtall:$\leq^\text{t}_\text{h}$ srtall:$\leq^\text{lac}_\text{1D}$ srtall$^c$:$\leq^\text{t}_\text{h}$ srtall$^c$:$\leq^\text{lac}_\text{1D}$ srtrc$^r$:$\leq^\text{t}_\text{h}$ srtrc$^r$:$\leq^\text{lac}_\text{1D}$ srtrc$^{rc}$:$\leq^\text{t}_\text{h}$ srtrc$^{rc}$:$\leq^\text{lac}_\text{1D}$
cowl:$\leq^\text{t}_\text{h}$ cowl:$\leq^\text{lac}_\text{1D}$ cowl$^r$:$\leq^\text{t}_\text{h}$ cowl$^r$:$\leq^\text{lac}_\text{1D}$ cowl$^c$:$\leq^\text{t}_\text{h}$ cowl$^c$:$\leq^\text{lac}_\text{1D}$ cowl$^{rc}$:$\leq^\text{t}_\text{h}$ cowl$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj:$\leq^\text{t}_\text{h}$ disj:$\leq^\text{lac}_\text{1D}$ disj$^c$:$\leq^\text{t}_\text{h}$ disj$^c$:$\leq^\text{lac}_\text{1D}$ fcomp$^r$:$\leq^\text{t}_\text{h}$ fcomp$^r$:$\leq^\text{lac}_\text{1D}$ fcomp$^{rc}$:$\leq^\text{t}_\text{h}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{1D}$ fmatch:$\leq^\text{lac}_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{1D}$ fmatch$^c$:$\leq^\text{lac}_\text{1D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ fpath:$\leq^\text{lac}_\text{1D}$ fpath$^r$:$\leq^\text{lac}_\text{1D}$ fpath$^c$:$\leq^\text{lac}_\text{1D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1D}$ fzmatch$^c$:$\leq^\text{lac}_\text{1D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ incl:$\leq^\text{t}_\text{h}$ incl:$\leq^\text{lac}_\text{1D}$ incl$^r$:$\leq^\text{t}_\text{h}$ incl$^r$:$\leq^\text{lac}_\text{1D}$ incl$^c$:$\leq^\text{t}_\text{h}$ incl$^c$:$\leq^\text{lac}_\text{1D}$ incl$^{rc}$:$\leq^\text{t}_\text{h}$ incl$^{rc}$:$\leq^\text{lac}_\text{1D}$ msp:$\leq^\text{lac}_\text{1D}$ msp$^r$:$\leq^\text{lac}_\text{1D}$ msp$^c$:$\leq^\text{lac}_\text{1D}$ msp$^{rc}$:$\leq^\text{lac}_\text{1D}$ olmem:$\leq^\text{t}_\text{h}$ olmem:$\leq^\text{lac}_\text{1D}$ olmem$^c$:$\leq^\text{t}_\text{h}$ olmem$^c$:$\leq^\text{lac}_\text{1D}$ proj:$\leq^\text{t}_\text{h}$ proj:$\leq^\text{lac}_\text{1D}$ proj$^c$:$\leq^\text{t}_\text{h}$ proj$^c$:$\leq^\text{lac}_\text{1D}$ sep:$\leq^\text{t}_\text{h}$ sep:$\leq^\text{lac}_\text{1D}$ sep$^r$:$\leq^\text{t}_\text{h}$ sep$^r$:$\leq^\text{lac}_\text{1D}$ sep$^c$:$\leq^\text{t}_\text{h}$ sep$^c$:$\leq^\text{lac}_\text{1D}$ sep$^{rc}$:$\leq^\text{t}_\text{h}$ sep$^{rc}$:$\leq^\text{lac}_\text{1D}$ sp:$\leq^\text{t}_\text{h}$ sp:$\leq^\text{lac}_\text{1D}$ sp$^r$:$\leq^\text{t}_\text{h}$ sp$^r$:$\leq^\text{lac}_\text{1D}$ sp$^c$:$\leq^\text{t}_\text{h}$ sp$^c$:$\leq^\text{lac}_\text{1D}$ sp$^{rc}$:$\leq^\text{t}_\text{h}$ sp$^{rc}$:$\leq^\text{lac}_\text{1D}$ srtall:$\leq^\text{t}_\text{h}$ srtall:$\leq^\text{lac}_\text{1D}$ srtall$^c$:$\leq^\text{t}_\text{h}$ srtall$^c$:$\leq^\text{lac}_\text{1D}$ srtrc$^r$:$\leq^\text{t}_\text{h}$ srtrc$^r$:$\leq^\text{lac}_\text{1D}$ srtrc$^{rc}$:$\leq^\text{t}_\text{h}$ srtrc$^{rc}$:$\leq^\text{lac}_\text{1D}$ all:$\leq^\text{t}_\text{h}$ all:$\leq_\text{h}$ all:$\leq^\text{lac}_\text{1D}$ all:$\leq_\text{1D}$ all:$\leq^\text{lac}_\text{RD}$ all:$\leq_\text{RD}$ all:$\leq^\text{lac}_\text{2D}$ all:$\leq_\text{2D}$ all:$\leq^\text{lac}_\text{1N}$ all:$\leq_\text{1N}$ all$^r$:$\leq^\text{t}_\text{h}$ all$^r$:$\leq_\text{h}$ all$^r$:$\leq^\text{lac}_\text{1D}$ all$^r$:$\leq_\text{1D}$ all$^r$:$\leq^\text{lac}_\text{RD}$ all$^r$:$\leq_\text{RD}$ all$^r$:$\leq^\text{lac}_\text{2D}$ all$^r$:$\leq_\text{2D}$ all$^r$:$\leq^\text{lac}_\text{1N}$ all$^r$:$\leq_\text{1N}$ all$^c$:$\leq^\text{t}_\text{h}$ all$^c$:$\leq_\text{h}$ all$^c$:$\leq^\text{lac}_\text{1D}$ all$^c$:$\leq_\text{1D}$ all$^c$:$\leq^\text{lac}_\text{RD}$ all$^c$:$\leq_\text{RD}$ all$^c$:$\leq^\text{lac}_\text{2D}$ all$^c$:$\leq_\text{2D}$ all$^c$:$\leq^\text{lac}_\text{1N}$ all$^c$:$\leq_\text{1N}$ all$^{rc}$:$\leq^\text{t}_\text{h}$ all$^{rc}$:$\leq_\text{h}$ all$^{rc}$:$\leq^\text{lac}_\text{1D}$ all$^{rc}$:$\leq_\text{1D}$ all$^{rc}$:$\leq^\text{lac}_\text{RD}$ all$^{rc}$:$\leq_\text{RD}$ all$^{rc}$:$\leq^\text{lac}_\text{2D}$ all$^{rc}$:$\leq_\text{2D}$ all$^{rc}$:$\leq^\text{lac}_\text{1N}$ all$^{rc}$:$\leq_\text{1N}$ cowl:$\leq_\text{h}$ cowl:$\leq_\text{1D}$ cowl:$\leq^\text{lac}_\text{RD}$ cowl:$\leq_\text{RD}$ cowl:$\leq^\text{lac}_\text{2D}$ cowl:$\leq_\text{2D}$ cowl:$\leq^\text{lac}_\text{1N}$ cowl:$\leq_\text{1N}$ cowl$^r$:$\leq_\text{h}$ cowl$^r$:$\leq_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{RD}$ cowl$^r$:$\leq_\text{RD}$ cowl$^r$:$\leq^\text{lac}_\text{2D}$ cowl$^r$:$\leq_\text{2D}$ cowl$^r$:$\leq^\text{lac}_\text{1N}$ cowl$^r$:$\leq_\text{1N}$ cowl$^c$:$\leq_\text{h}$ cowl$^c$:$\leq_\text{1D}$ cowl$^c$:$\leq^\text{lac}_\text{RD}$ cowl$^c$:$\leq_\text{RD}$ cowl$^c$:$\leq^\text{lac}_\text{2D}$ cowl$^c$:$\leq_\text{2D}$ cowl$^c$:$\leq^\text{lac}_\text{1N}$ cowl$^c$:$\leq_\text{1N}$ cowl$^{rc}$:$\leq_\text{h}$ cowl$^{rc}$:$\leq_\text{1D}$ cowl$^{rc}$:$\leq^\text{lac}_\text{RD}$ cowl$^{rc}$:$\leq_\text{RD}$ cowl$^{rc}$:$\leq^\text{lac}_\text{2D}$ cowl$^{rc}$:$\leq_\text{2D}$ cowl$^{rc}$:$\leq^\text{lac}_\text{1N}$ cowl$^{rc}$:$\leq_\text{1N}$ ctwl:$\leq^\text{t}_\text{h}$ ctwl:$\leq_\text{h}$ ctwl:$\leq^\text{lac}_\text{1D}$ ctwl:$\leq_\text{1D}$ ctwl:$\leq^\text{lac}_\text{RD}$ ctwl:$\leq_\text{RD}$ ctwl:$\leq^\text{lac}_\text{2D}$ ctwl:$\leq_\text{2D}$ ctwl:$\leq^\text{lac}_\text{1N}$ ctwl:$\leq_\text{1N}$ ctwl$^r$:$\leq^\text{t}_\text{h}$ ctwl$^r$:$\leq_\text{h}$ ctwl$^r$:$\leq^\text{lac}_\text{1D}$ ctwl$^r$:$\leq_\text{1D}$ ctwl$^r$:$\leq^\text{lac}_\text{RD}$ ctwl$^r$:$\leq_\text{RD}$ ctwl$^r$:$\leq^\text{lac}_\text{2D}$ ctwl$^r$:$\leq_\text{2D}$ ctwl$^r$:$\leq^\text{lac}_\text{1N}$ ctwl$^r$:$\leq_\text{1N}$ ctwl$^c$:$\leq^\text{t}_\text{h}$ ctwl$^c$:$\leq_\text{h}$ ctwl$^c$:$\leq^\text{lac}_\text{1D}$ ctwl$^c$:$\leq_\text{1D}$ ctwl$^c$:$\leq^\text{lac}_\text{RD}$ ctwl$^c$:$\leq_\text{RD}$ ctwl$^c$:$\leq^\text{lac}_\text{2D}$ ctwl$^c$:$\leq_\text{2D}$ ctwl$^c$:$\leq^\text{lac}_\text{1N}$ ctwl$^c$:$\leq_\text{1N}$ ctwl$^{rc}$:$\leq^\text{t}_\text{h}$ ctwl$^{rc}$:$\leq_\text{h}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{1D}$ ctwl$^{rc}$:$\leq_\text{1D}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{RD}$ ctwl$^{rc}$:$\leq_\text{RD}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{2D}$ ctwl$^{rc}$:$\leq_\text{2D}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{1N}$ ctwl$^{rc}$:$\leq_\text{1N}$ disj:$\leq_\text{h}$ disj:$\leq_\text{1D}$ disj:$\leq^\text{lac}_\text{RD}$ disj:$\leq_\text{RD}$ disj:$\leq^\text{lac}_\text{2D}$ disj:$\leq_\text{2D}$ disj:$\leq^\text{lac}_\text{1N}$ disj:$\leq_\text{1N}$ disj$^r$:$\leq^\text{t}_\text{h}$ disj$^r$:$\leq_\text{h}$ disj$^r$:$\leq^\text{lac}_\text{1D}$ disj$^r$:$\leq_\text{1D}$ disj$^r$:$\leq^\text{lac}_\text{RD}$ disj$^r$:$\leq_\text{RD}$ disj$^r$:$\leq^\text{lac}_\text{2D}$ disj$^r$:$\leq_\text{2D}$ disj$^r$:$\leq^\text{lac}_\text{1N}$ disj$^r$:$\leq_\text{1N}$ disj$^c$:$\leq_\text{h}$ disj$^c$:$\leq_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{RD}$ disj$^c$:$\leq_\text{RD}$ disj$^c$:$\leq^\text{lac}_\text{2D}$ disj$^c$:$\leq_\text{2D}$ disj$^c$:$\leq^\text{lac}_\text{1N}$ disj$^c$:$\leq_\text{1N}$ disj$^{rc}$:$\leq^\text{t}_\text{h}$ disj$^{rc}$:$\leq_\text{h}$ disj$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj$^{rc}$:$\leq_\text{1D}$ disj$^{rc}$:$\leq^\text{lac}_\text{RD}$ disj$^{rc}$:$\leq_\text{RD}$ disj$^{rc}$:$\leq^\text{lac}_\text{2D}$ disj$^{rc}$:$\leq_\text{2D}$ disj$^{rc}$:$\leq^\text{lac}_\text{1N}$ disj$^{rc}$:$\leq_\text{1N}$ fcomp$^r$:$\leq_\text{h}$ fcomp$^r$:$\leq_\text{1D}$ fcomp$^r$:$\leq^\text{lac}_\text{1N}$ fcomp$^r$:$\leq_\text{1N}$ fcomp$^{rc}$:$\leq_\text{h}$ fcomp$^{rc}$:$\leq_\text{1D}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{1N}$ fcomp$^{rc}$:$\leq_\text{1N}$ fmatch:$\leq_\text{1D}$ fmatch:$\leq^\text{lac}_\text{RD}$ fmatch:$\leq_\text{RD}$ fmatch:$\leq^\text{lac}_\text{2D}$ fmatch:$\leq_\text{2D}$ fmatch:$\leq^\text{lac}_\text{1N}$ fmatch:$\leq_\text{1N}$ fmatch$^r$:$\leq_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{RD}$ fmatch$^r$:$\leq_\text{RD}$ fmatch$^r$:$\leq^\text{lac}_\text{2D}$ fmatch$^r$:$\leq_\text{2D}$ fmatch$^r$:$\leq^\text{lac}_\text{1N}$ fmatch$^r$:$\leq_\text{1N}$ fmatch$^c$:$\leq_\text{1D}$ fmatch$^c$:$\leq^\text{lac}_\text{RD}$ fmatch$^c$:$\leq_\text{RD}$ fmatch$^c$:$\leq^\text{lac}_\text{2D}$ fmatch$^c$:$\leq_\text{2D}$ fmatch$^c$:$\leq^\text{lac}_\text{1N}$ fmatch$^c$:$\leq_\text{1N}$ fmatch$^{rc}$:$\leq_\text{1D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{RD}$ fmatch$^{rc}$:$\leq_\text{RD}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{2D}$ fmatch$^{rc}$:$\leq_\text{2D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{1N}$ fmatch$^{rc}$:$\leq_\text{1N}$ fpath:$\leq_\text{1D}$ fpath:$\leq^\text{lac}_\text{RD}$ fpath:$\leq_\text{RD}$ fpath:$\leq^\text{lac}_\text{2D}$ fpath:$\leq_\text{2D}$ fpath:$\leq^\text{lac}_\text{1N}$ fpath:$\leq_\text{1N}$ fpath$^r$:$\leq_\text{1D}$ fpath$^r$:$\leq^\text{lac}_\text{RD}$ fpath$^r$:$\leq_\text{RD}$ fpath$^r$:$\leq^\text{lac}_\text{2D}$ fpath$^r$:$\leq_\text{2D}$ fpath$^r$:$\leq^\text{lac}_\text{1N}$ fpath$^r$:$\leq_\text{1N}$ fpath$^c$:$\leq_\text{1D}$ fpath$^c$:$\leq^\text{lac}_\text{RD}$ fpath$^c$:$\leq_\text{RD}$ fpath$^c$:$\leq^\text{lac}_\text{2D}$ fpath$^c$:$\leq_\text{2D}$ fpath$^c$:$\leq^\text{lac}_\text{1N}$ fpath$^c$:$\leq_\text{1N}$ fpath$^{rc}$:$\leq_\text{1D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{RD}$ fpath$^{rc}$:$\leq_\text{RD}$ fpath$^{rc}$:$\leq^\text{lac}_\text{2D}$ fpath$^{rc}$:$\leq_\text{2D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{1N}$ fpath$^{rc}$:$\leq_\text{1N}$ fzmatch:$\leq_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{RD}$ fzmatch:$\leq_\text{RD}$ fzmatch:$\leq^\text{lac}_\text{2D}$ fzmatch:$\leq_\text{2D}$ fzmatch:$\leq^\text{lac}_\text{1N}$ fzmatch:$\leq_\text{1N}$ fzmatch$^r$:$\leq_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{RD}$ fzmatch$^r$:$\leq_\text{RD}$ fzmatch$^r$:$\leq^\text{lac}_\text{2D}$ fzmatch$^r$:$\leq_\text{2D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1N}$ fzmatch$^r$:$\leq_\text{1N}$ fzmatch$^c$:$\leq_\text{1D}$ fzmatch$^c$:$\leq^\text{lac}_\text{RD}$ fzmatch$^c$:$\leq_\text{RD}$ fzmatch$^c$:$\leq^\text{lac}_\text{2D}$ fzmatch$^c$:$\leq_\text{2D}$ fzmatch$^c$:$\leq^\text{lac}_\text{1N}$ fzmatch$^c$:$\leq_\text{1N}$ fzmatch$^{rc}$:$\leq_\text{1D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{RD}$ fzmatch$^{rc}$:$\leq_\text{RD}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{2D}$ fzmatch$^{rc}$:$\leq_\text{2D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{1N}$ fzmatch$^{rc}$:$\leq_\text{1N}$ incl:$\leq_\text{h}$ incl:$\leq_\text{1D}$ incl:$\leq^\text{lac}_\text{RD}$ incl:$\leq_\text{RD}$ incl:$\leq^\text{lac}_\text{2D}$ incl:$\leq_\text{2D}$ incl:$\leq^\text{lac}_\text{1N}$ incl:$\leq_\text{1N}$ incl$^r$:$\leq_\text{h}$ incl$^r$:$\leq_\text{1D}$ incl$^r$:$\leq^\text{lac}_\text{RD}$ incl$^r$:$\leq_\text{RD}$ incl$^r$:$\leq^\text{lac}_\text{2D}$ incl$^r$:$\leq_\text{2D}$ incl$^r$:$\leq^\text{lac}_\text{1N}$ incl$^r$:$\leq_\text{1N}$ incl$^c$:$\leq_\text{h}$ incl$^c$:$\leq_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{RD}$ incl$^c$:$\leq_\text{RD}$ incl$^c$:$\leq^\text{lac}_\text{2D}$ incl$^c$:$\leq_\text{2D}$ incl$^c$:$\leq^\text{lac}_\text{1N}$ incl$^c$:$\leq_\text{1N}$ incl$^{rc}$:$\leq_\text{h}$ incl$^{rc}$:$\leq_\text{1D}$ incl$^{rc}$:$\leq^\text{lac}_\text{RD}$ incl$^{rc}$:$\leq_\text{RD}$ incl$^{rc}$:$\leq^\text{lac}_\text{2D}$ incl$^{rc}$:$\leq_\text{2D}$ incl$^{rc}$:$\leq^\text{lac}_\text{1N}$ incl$^{rc}$:$\leq_\text{1N}$ irc$^r$:$\leq^\text{t}_\text{h}$ irc$^r$:$\leq_\text{h}$ irc$^r$:$\leq^\text{lac}_\text{1D}$ irc$^r$:$\leq_\text{1D}$ irc$^r$:$\leq^\text{lac}_\text{RD}$ irc$^r$:$\leq_\text{RD}$ irc$^r$:$\leq^\text{lac}_\text{2D}$ irc$^r$:$\leq_\text{2D}$ irc$^r$:$\leq^\text{lac}_\text{1N}$ irc$^r$:$\leq_\text{1N}$ irc$^{rc}$:$\leq^\text{t}_\text{h}$ irc$^{rc}$:$\leq_\text{h}$ irc$^{rc}$:$\leq^\text{lac}_\text{1D}$ irc$^{rc}$:$\leq_\text{1D}$ irc$^{rc}$:$\leq^\text{lac}_\text{RD}$ irc$^{rc}$:$\leq_\text{RD}$ irc$^{rc}$:$\leq^\text{lac}_\text{2D}$ irc$^{rc}$:$\leq_\text{2D}$ irc$^{rc}$:$\leq^\text{lac}_\text{1N}$ irc$^{rc}$:$\leq_\text{1N}$ lmem:$\leq^\text{t}_\text{h}$ lmem:$\leq_\text{h}$ lmem:$\leq^\text{lac}_\text{1D}$ lmem:$\leq_\text{1D}$ lmem:$\leq^\text{lac}_\text{RD}$ lmem:$\leq_\text{RD}$ lmem:$\leq^\text{lac}_\text{2D}$ lmem:$\leq_\text{2D}$ lmem:$\leq^\text{lac}_\text{1N}$ lmem:$\leq_\text{1N}$ lmem$^c$:$\leq^\text{t}_\text{h}$ lmem$^c$:$\leq_\text{h}$ lmem$^c$:$\leq^\text{lac}_\text{1D}$ lmem$^c$:$\leq_\text{1D}$ lmem$^c$:$\leq^\text{lac}_\text{RD}$ lmem$^c$:$\leq_\text{RD}$ lmem$^c$:$\leq^\text{lac}_\text{2D}$ lmem$^c$:$\leq_\text{2D}$ lmem$^c$:$\leq^\text{lac}_\text{1N}$ lmem$^c$:$\leq_\text{1N}$ msp:$\leq_\text{1D}$ msp:$\leq^\text{lac}_\text{RD}$ msp:$\leq_\text{RD}$ msp:$\leq^\text{lac}_\text{2D}$ msp:$\leq_\text{2D}$ msp:$\leq^\text{lac}_\text{1N}$ msp:$\leq_\text{1N}$ msp$^r$:$\leq_\text{1D}$ msp$^r$:$\leq^\text{lac}_\text{RD}$ msp$^r$:$\leq_\text{RD}$ msp$^r$:$\leq^\text{lac}_\text{2D}$ msp$^r$:$\leq_\text{2D}$ msp$^r$:$\leq^\text{lac}_\text{1N}$ msp$^r$:$\leq_\text{1N}$ msp$^c$:$\leq_\text{1D}$ msp$^c$:$\leq^\text{lac}_\text{RD}$ msp$^c$:$\leq_\text{RD}$ msp$^c$:$\leq^\text{lac}_\text{2D}$ msp$^c$:$\leq_\text{2D}$ msp$^c$:$\leq^\text{lac}_\text{1N}$ msp$^c$:$\leq_\text{1N}$ msp$^{rc}$:$\leq_\text{1D}$ msp$^{rc}$:$\leq^\text{lac}_\text{RD}$ msp$^{rc}$:$\leq_\text{RD}$ msp$^{rc}$:$\leq^\text{lac}_\text{2D}$ msp$^{rc}$:$\leq_\text{2D}$ msp$^{rc}$:$\leq^\text{lac}_\text{1N}$ msp$^{rc}$:$\leq_\text{1N}$ owl:$\leq^\text{t}_\text{h}$ owl:$\leq_\text{h}$ owl:$\leq^\text{lac}_\text{1D}$ owl:$\leq_\text{1D}$ owl:$\leq^\text{lac}_\text{RD}$ owl:$\leq_\text{RD}$ owl:$\leq^\text{lac}_\text{2D}$ owl:$\leq_\text{2D}$ owl:$\leq^\text{lac}_\text{1N}$ owl:$\leq_\text{1N}$ owl$^r$:$\leq^\text{t}_\text{h}$ owl$^r$:$\leq_\text{h}$ owl$^r$:$\leq^\text{lac}_\text{1D}$ owl$^r$:$\leq_\text{1D}$ owl$^r$:$\leq^\text{lac}_\text{RD}$ owl$^r$:$\leq_\text{RD}$ owl$^r$:$\leq^\text{lac}_\text{2D}$ owl$^r$:$\leq_\text{2D}$ owl$^r$:$\leq^\text{lac}_\text{1N}$ owl$^r$:$\leq_\text{1N}$ owl$^c$:$\leq^\text{t}_\text{h}$ owl$^c$:$\leq_\text{h}$ owl$^c$:$\leq^\text{lac}_\text{1D}$ owl$^c$:$\leq_\text{1D}$ owl$^c$:$\leq^\text{lac}_\text{RD}$ owl$^c$:$\leq_\text{RD}$ owl$^c$:$\leq^\text{lac}_\text{2D}$ owl$^c$:$\leq_\text{2D}$ owl$^c$:$\leq^\text{lac}_\text{1N}$ owl$^c$:$\leq_\text{1N}$ owl$^{rc}$:$\leq^\text{t}_\text{h}$ owl$^{rc}$:$\leq_\text{h}$ owl$^{rc}$:$\leq^\text{lac}_\text{1D}$ owl$^{rc}$:$\leq_\text{1D}$ owl$^{rc}$:$\leq^\text{lac}_\text{RD}$ owl$^{rc}$:$\leq_\text{RD}$ owl$^{rc}$:$\leq^\text{lac}_\text{2D}$ owl$^{rc}$:$\leq_\text{2D}$ owl$^{rc}$:$\leq^\text{lac}_\text{1N}$ owl$^{rc}$:$\leq_\text{1N}$ olmem:$\leq_\text{h}$ olmem:$\leq_\text{1D}$ olmem:$\leq^\text{lac}_\text{RD}$ olmem:$\leq_\text{RD}$ olmem:$\leq^\text{lac}_\text{2D}$ olmem:$\leq_\text{2D}$ olmem:$\leq^\text{lac}_\text{1N}$ olmem:$\leq_\text{1N}$ olmem$^c$:$\leq_\text{h}$ olmem$^c$:$\leq_\text{1D}$ olmem$^c$:$\leq^\text{lac}_\text{RD}$ olmem$^c$:$\leq_\text{RD}$ olmem$^c$:$\leq^\text{lac}_\text{2D}$ olmem$^c$:$\leq_\text{2D}$ olmem$^c$:$\leq^\text{lac}_\text{1N}$ olmem$^c$:$\leq_\text{1N}$ proj:$\leq_\text{h}$ proj:$\leq_\text{1D}$ proj:$\leq^\text{lac}_\text{1N}$ proj:$\leq_\text{1N}$ proj$^c$:$\leq_\text{h}$ proj$^c$:$\leq_\text{1D}$ proj$^c$:$\leq^\text{lac}_\text{1N}$ proj$^c$:$\leq_\text{1N}$ rmatch:$\leq^\text{lac}_\text{1D}$ rmatch:$\leq_\text{1D}$ rmatch:$\leq^\text{lac}_\text{RD}$ rmatch:$\leq_\text{RD}$ rmatch:$\leq^\text{lac}_\text{2D}$ rmatch:$\leq_\text{2D}$ rmatch:$\leq^\text{lac}_\text{1N}$ rmatch:$\leq_\text{1N}$ rmatch$^r$:$\leq^\text{lac}_\text{1D}$ rmatch$^r$:$\leq_\text{1D}$ rmatch$^r$:$\leq^\text{lac}_\text{RD}$ rmatch$^r$:$\leq_\text{RD}$ rmatch$^r$:$\leq^\text{lac}_\text{2D}$ rmatch$^r$:$\leq_\text{2D}$ rmatch$^r$:$\leq^\text{lac}_\text{1N}$ rmatch$^r$:$\leq_\text{1N}$ rmatch$^c$:$\leq^\text{lac}_\text{1D}$ rmatch$^c$:$\leq_\text{1D}$ rmatch$^c$:$\leq^\text{lac}_\text{RD}$ rmatch$^c$:$\leq_\text{RD}$ rmatch$^c$:$\leq^\text{lac}_\text{2D}$ rmatch$^c$:$\leq_\text{2D}$ rmatch$^c$:$\leq^\text{lac}_\text{1N}$ rmatch$^c$:$\leq_\text{1N}$ rmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ rmatch$^{rc}$:$\leq_\text{1D}$ rmatch$^{rc}$:$\leq^\text{lac}_\text{RD}$ rmatch$^{rc}$:$\leq_\text{RD}$ rmatch$^{rc}$:$\leq^\text{lac}_\text{2D}$ rmatch$^{rc}$:$\leq_\text{2D}$ rmatch$^{rc}$:$\leq^\text{lac}_\text{1N}$ rmatch$^{rc}$:$\leq_\text{1N}$ rpath:$\leq^\text{lac}_\text{1D}$ rpath:$\leq_\text{1D}$ rpath:$\leq^\text{lac}_\text{RD}$ rpath:$\leq_\text{RD}$ rpath:$\leq^\text{lac}_\text{2D}$ rpath:$\leq_\text{2D}$ rpath:$\leq^\text{lac}_\text{1N}$ rpath:$\leq_\text{1N}$ rpath$^r$:$\leq^\text{lac}_\text{1D}$ rpath$^r$:$\leq_\text{1D}$ rpath$^r$:$\leq^\text{lac}_\text{RD}$ rpath$^r$:$\leq_\text{RD}$ rpath$^r$:$\leq^\text{lac}_\text{2D}$ rpath$^r$:$\leq_\text{2D}$ rpath$^r$:$\leq^\text{lac}_\text{1N}$ rpath$^r$:$\leq_\text{1N}$ rpath$^c$:$\leq^\text{lac}_\text{1D}$ rpath$^c$:$\leq_\text{1D}$ rpath$^c$:$\leq^\text{lac}_\text{RD}$ rpath$^c$:$\leq_\text{RD}$ rpath$^c$:$\leq^\text{lac}_\text{2D}$ rpath$^c$:$\leq_\text{2D}$ rpath$^c$:$\leq^\text{lac}_\text{1N}$ rpath$^c$:$\leq_\text{1N}$ rpath$^{rc}$:$\leq^\text{lac}_\text{1D}$ rpath$^{rc}$:$\leq_\text{1D}$ rpath$^{rc}$:$\leq^\text{lac}_\text{RD}$ rpath$^{rc}$:$\leq_\text{RD}$ rpath$^{rc}$:$\leq^\text{lac}_\text{2D}$ rpath$^{rc}$:$\leq_\text{2D}$ rpath$^{rc}$:$\leq^\text{lac}_\text{1N}$ rpath$^{rc}$:$\leq_\text{1N}$ rzmatch:$\leq^\text{lac}_\text{1D}$ rzmatch:$\leq_\text{1D}$ rzmatch:$\leq^\text{lac}_\text{RD}$ rzmatch:$\leq_\text{RD}$ rzmatch:$\leq^\text{lac}_\text{2D}$ rzmatch:$\leq_\text{2D}$ rzmatch:$\leq^\text{lac}_\text{1N}$ rzmatch:$\leq_\text{1N}$ rzmatch$^r$:$\leq^\text{lac}_\text{1D}$ rzmatch$^r$:$\leq_\text{1D}$ rzmatch$^r$:$\leq^\text{lac}_\text{RD}$ rzmatch$^r$:$\leq_\text{RD}$ rzmatch$^r$:$\leq^\text{lac}_\text{2D}$ rzmatch$^r$:$\leq_\text{2D}$ rzmatch$^r$:$\leq^\text{lac}_\text{1N}$ rzmatch$^r$:$\leq_\text{1N}$ rzmatch$^c$:$\leq^\text{lac}_\text{1D}$ rzmatch$^c$:$\leq_\text{1D}$ rzmatch$^c$:$\leq^\text{lac}_\text{RD}$ rzmatch$^c$:$\leq_\text{RD}$ rzmatch$^c$:$\leq^\text{lac}_\text{2D}$ rzmatch$^c$:$\leq_\text{2D}$ rzmatch$^c$:$\leq^\text{lac}_\text{1N}$ rzmatch$^c$:$\leq_\text{1N}$ rzmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ rzmatch$^{rc}$:$\leq_\text{1D}$ rzmatch$^{rc}$:$\leq^\text{lac}_\text{RD}$ rzmatch$^{rc}$:$\leq_\text{RD}$ rzmatch$^{rc}$:$\leq^\text{lac}_\text{2D}$ rzmatch$^{rc}$:$\leq_\text{2D}$ rzmatch$^{rc}$:$\leq^\text{lac}_\text{1N}$ rzmatch$^{rc}$:$\leq_\text{1N}$ rc$^r$:$\leq^\text{t}_\text{h}$ rc$^r$:$\leq_\text{h}$ rc$^r$:$\leq^\text{lac}_\text{1D}$ rc$^r$:$\leq_\text{1D}$ rc$^r$:$\leq^\text{lac}_\text{RD}$ rc$^r$:$\leq_\text{RD}$ rc$^r$:$\leq^\text{lac}_\text{2D}$ rc$^r$:$\leq_\text{2D}$ rc$^r$:$\leq^\text{lac}_\text{1N}$ rc$^r$:$\leq_\text{1N}$ rc$^{rc}$:$\leq^\text{t}_\text{h}$ rc$^{rc}$:$\leq_\text{h}$ rc$^{rc}$:$\leq^\text{lac}_\text{1D}$ rc$^{rc}$:$\leq_\text{1D}$ rc$^{rc}$:$\leq^\text{lac}_\text{RD}$ rc$^{rc}$:$\leq_\text{RD}$ rc$^{rc}$:$\leq^\text{lac}_\text{2D}$ rc$^{rc}$:$\leq_\text{2D}$ rc$^{rc}$:$\leq^\text{lac}_\text{1N}$ rc$^{rc}$:$\leq_\text{1N}$ sep:$\leq_\text{h}$ sep:$\leq_\text{1D}$ sep:$\leq^\text{lac}_\text{RD}$ sep:$\leq_\text{RD}$ sep:$\leq^\text{lac}_\text{2D}$ sep:$\leq_\text{2D}$ sep:$\leq^\text{lac}_\text{1N}$ sep:$\leq_\text{1N}$ sep$^r$:$\leq_\text{h}$ sep$^r$:$\leq_\text{1D}$ sep$^r$:$\leq^\text{lac}_\text{RD}$ sep$^r$:$\leq_\text{RD}$ sep$^r$:$\leq^\text{lac}_\text{2D}$ sep$^r$:$\leq_\text{2D}$ sep$^r$:$\leq^\text{lac}_\text{1N}$ sep$^r$:$\leq_\text{1N}$ sep$^c$:$\leq_\text{h}$ sep$^c$:$\leq_\text{1D}$ sep$^c$:$\leq^\text{lac}_\text{RD}$ sep$^c$:$\leq_\text{RD}$ sep$^c$:$\leq^\text{lac}_\text{2D}$ sep$^c$:$\leq_\text{2D}$ sep$^c$:$\leq^\text{lac}_\text{1N}$ sep$^c$:$\leq_\text{1N}$ sep$^{rc}$:$\leq_\text{h}$ sep$^{rc}$:$\leq_\text{1D}$ sep$^{rc}$:$\leq^\text{lac}_\text{RD}$ sep$^{rc}$:$\leq_\text{RD}$ sep$^{rc}$:$\leq^\text{lac}_\text{2D}$ sep$^{rc}$:$\leq_\text{2D}$ sep$^{rc}$:$\leq^\text{lac}_\text{1N}$ sep$^{rc}$:$\leq_\text{1N}$ sp:$\leq_\text{h}$ sp:$\leq_\text{1D}$ sp:$\leq^\text{lac}_\text{RD}$ sp:$\leq_\text{RD}$ sp:$\leq^\text{lac}_\text{2D}$ sp:$\leq_\text{2D}$ sp:$\leq^\text{lac}_\text{1N}$ sp:$\leq_\text{1N}$ sp$^r$:$\leq_\text{h}$ sp$^r$:$\leq_\text{1D}$ sp$^r$:$\leq^\text{lac}_\text{RD}$ sp$^r$:$\leq_\text{RD}$ sp$^r$:$\leq^\text{lac}_\text{2D}$ sp$^r$:$\leq_\text{2D}$ sp$^r$:$\leq^\text{lac}_\text{1N}$ sp$^r$:$\leq_\text{1N}$ sp$^c$:$\leq_\text{h}$ sp$^c$:$\leq_\text{1D}$ sp$^c$:$\leq^\text{lac}_\text{RD}$ sp$^c$:$\leq_\text{RD}$ sp$^c$:$\leq^\text{lac}_\text{2D}$ sp$^c$:$\leq_\text{2D}$ sp$^c$:$\leq^\text{lac}_\text{1N}$ sp$^c$:$\leq_\text{1N}$ sp$^{rc}$:$\leq_\text{h}$ sp$^{rc}$:$\leq_\text{1D}$ sp$^{rc}$:$\leq^\text{lac}_\text{RD}$ sp$^{rc}$:$\leq_\text{RD}$ sp$^{rc}$:$\leq^\text{lac}_\text{2D}$ sp$^{rc}$:$\leq_\text{2D}$ sp$^{rc}$:$\leq^\text{lac}_\text{1N}$ sp$^{rc}$:$\leq_\text{1N}$ srtall:$\leq_\text{h}$ srtall:$\leq_\text{1D}$ srtall:$\leq^\text{lac}_\text{RD}$ srtall:$\leq_\text{RD}$ srtall:$\leq^\text{lac}_\text{2D}$ srtall:$\leq_\text{2D}$ srtall:$\leq^\text{lac}_\text{1N}$ srtall:$\leq_\text{1N}$ srtall$^r$:$\leq^\text{t}_\text{h}$ srtall$^r$:$\leq_\text{h}$ srtall$^r$:$\leq^\text{lac}_\text{1D}$ srtall$^r$:$\leq_\text{1D}$ srtall$^r$:$\leq^\text{lac}_\text{RD}$ srtall$^r$:$\leq_\text{RD}$ srtall$^r$:$\leq^\text{lac}_\text{2D}$ srtall$^r$:$\leq_\text{2D}$ srtall$^r$:$\leq^\text{lac}_\text{1N}$ srtall$^r$:$\leq_\text{1N}$ srtall$^c$:$\leq_\text{h}$ srtall$^c$:$\leq_\text{1D}$ srtall$^c$:$\leq^\text{lac}_\text{RD}$ srtall$^c$:$\leq_\text{RD}$ srtall$^c$:$\leq^\text{lac}_\text{2D}$ srtall$^c$:$\leq_\text{2D}$ srtall$^c$:$\leq^\text{lac}_\text{1N}$ srtall$^c$:$\leq_\text{1N}$ srtall$^{rc}$:$\leq^\text{t}_\text{h}$ srtall$^{rc}$:$\leq_\text{h}$ srtall$^{rc}$:$\leq^\text{lac}_\text{1D}$ srtall$^{rc}$:$\leq_\text{1D}$ srtall$^{rc}$:$\leq^\text{lac}_\text{RD}$ srtall$^{rc}$:$\leq_\text{RD}$ srtall$^{rc}$:$\leq^\text{lac}_\text{2D}$ srtall$^{rc}$:$\leq_\text{2D}$ srtall$^{rc}$:$\leq^\text{lac}_\text{1N}$ srtall$^{rc}$:$\leq_\text{1N}$ srtrc$^r$:$\leq_\text{h}$ srtrc$^r$:$\leq_\text{1D}$ srtrc$^r$:$\leq^\text{lac}_\text{RD}$ srtrc$^r$:$\leq_\text{RD}$ srtrc$^r$:$\leq^\text{lac}_\text{2D}$ srtrc$^r$:$\leq_\text{2D}$ srtrc$^r$:$\leq^\text{lac}_\text{1N}$ srtrc$^r$:$\leq_\text{1N}$ srtrc$^{rc}$:$\leq_\text{h}$ srtrc$^{rc}$:$\leq_\text{1D}$ srtrc$^{rc}$:$\leq^\text{lac}_\text{RD}$ srtrc$^{rc}$:$\leq_\text{RD}$ srtrc$^{rc}$:$\leq^\text{lac}_\text{2D}$ srtrc$^{rc}$:$\leq_\text{2D}$ srtrc$^{rc}$:$\leq^\text{lac}_\text{1N}$ srtrc$^{rc}$:$\leq_\text{1N}$ twl:$\leq^\text{t}_\text{h}$ twl:$\leq_\text{h}$ twl:$\leq^\text{lac}_\text{1D}$ twl:$\leq_\text{1D}$ twl:$\leq^\text{lac}_\text{RD}$ twl:$\leq_\text{RD}$ twl:$\leq^\text{lac}_\text{2D}$ twl:$\leq_\text{2D}$ twl:$\leq^\text{lac}_\text{1N}$ twl:$\leq_\text{1N}$ twl$^r$:$\leq^\text{t}_\text{h}$ twl$^r$:$\leq_\text{h}$ twl$^r$:$\leq^\text{lac}_\text{1D}$ twl$^r$:$\leq_\text{1D}$ twl$^r$:$\leq^\text{lac}_\text{RD}$ twl$^r$:$\leq_\text{RD}$ twl$^r$:$\leq^\text{lac}_\text{2D}$ twl$^r$:$\leq_\text{2D}$ twl$^r$:$\leq^\text{lac}_\text{1N}$ twl$^r$:$\leq_\text{1N}$ twl$^c$:$\leq^\text{t}_\text{h}$ twl$^c$:$\leq_\text{h}$ twl$^c$:$\leq^\text{lac}_\text{1D}$ twl$^c$:$\leq_\text{1D}$ twl$^c$:$\leq^\text{lac}_\text{RD}$ twl$^c$:$\leq_\text{RD}$ twl$^c$:$\leq^\text{lac}_\text{2D}$ twl$^c$:$\leq_\text{2D}$ twl$^c$:$\leq^\text{lac}_\text{1N}$ twl$^c$:$\leq_\text{1N}$ twl$^{rc}$:$\leq^\text{t}_\text{h}$ twl$^{rc}$:$\leq_\text{h}$ twl$^{rc}$:$\leq^\text{lac}_\text{1D}$ twl$^{rc}$:$\leq_\text{1D}$ twl$^{rc}$:$\leq^\text{lac}_\text{RD}$ twl$^{rc}$:$\leq_\text{RD}$ twl$^{rc}$:$\leq^\text{lac}_\text{2D}$ twl$^{rc}$:$\leq_\text{2D}$ twl$^{rc}$:$\leq^\text{lac}_\text{1N}$ twl$^{rc}$:$\leq_\text{1N}$