Extended catching scale (exCS) is an extension of catching scale. In catching scale, {c 1`2} fails because {c #} usually reduces to separators with higher level than the grave accent. In exCS, what {c #} reduces to has limited level, and {c #} itself also has limited level, less than the grave accent. exCS puts separators with c into the process, gives them levels, so that higher separators, such as the grave accent and multiple comma can act on them.
Levels
Generally, if lv({#}) < lv(`) and lv({#′}) < lv(`), then lv({#}) < lv({c #′}) < lv(`), where # and #′ don’t begin with “c”. Comparisons between {c #} can be defined as:
- If lv({#1}) > lv({#2}) then lv({c #1}) > lv({c #2}); if lv({#1}) < lv({#2}) then lv({c #1}) < lv({c #2}); if lv({#1}) = lv({#2}) then lv({c #1}) = lv({c #2}).
Note that {c #} is valid only if lv({#}) < lv(`).
We say a separator A is a c-free if the symbol “c” doesn’t appear anywhere in the expression of A.
Case of c in the process
Let S = {A is separator|A is c-free∧ λx.s(x,x A 2) ≈ λx.ss(x,x A 2)} be the set of catching separators.
Adding this case to the main process forms exCS.
- Case: If the entry n is not 1, and the {c 1} is immediately before you, then
- Let separator M1 = max{A∈S|¬∃B∈S(λx.s(x,x A 2) >* λx.s(x,x B 2))} where the max is by level comparison, and Mi+1 = {1 Mi 2}.
- Find minimum of t such that lv(Mt) < lv(`).
- Change the “{c 1} n” into “Mt 2 {c 1} n-1″.
- The process ends.
- Case: If the entry n is not 1, and the “{c” is immediately before you, then
- Let separator M1 = max{A∈S|λx.s(x,x A 2) >* λx.s(x,x {c n-1 #} 2)∧ ¬∃B∈S(λx.s(x,x A 2) >* λx.s(x,x B 2) >* λx.s(x,x {c n-1 #} 2))} where the max is by level comparison, and Mi+1 = {1 Mi 2}.
- Find minimum of t such that lv(Mt) < lv(`).
- Change the “{c n #}” into “Mt“.
- The process ends.
Explanation
The M1 is identical to the separators in catching scale, but with a high level. Step 2 and 3 of the case of “c” makes a separator in the form {1{1…{1 M1 2}…2}2} to get lower level than the grave accent.
Now {c 1`2} reduces to {c 1…{c 1 {c 1,2} 2}…2}, which reduces to {c 1…{c 1 {1{1,,1,2}2} 2}…2}. Also, {c #} has higher level than what it reduces to. So in exCS the loop problem in catching scale is solved.
Comparison
Here’re some “separator reduction” in exCS.
{c 1,2} = {1{1,,1,,2}2}
{c 1{c 1,2}2} = {1{1,,1,,3}2}
{c 1{c 1{c 1,2}2}2} = {1{1,,1,,4}2}
{c 1`2} = {1{1,,1,,1,2}2}
{c 2`2} = {1{1,,1,,1{1,,1,2}2}2}
{c 1,2`2} = {1{1,,1,,1{1,,1{1,,1,,2}2}2}2}
{c 1{c 1,2}2`2} = {1{1,,1,,1{1,,1{1,,1,,3}2}2}2}
{c 1{c 1{c 1,2}2}2`2} = {1{1,,1,,1{1,,1{1,,1,,4}2}2}2}
{c 1{c 1`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,1,2}2}2}2}
{c 1{c 1`2}3`2} = {1{1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,1,2}2}2}2}2}2}
{c 1{c 1`2}1,2`2} = {1{1,,1,,1{1,,1,,2}2}2}
{c 2{c 1`2}1,2`2} = {1{1,,1{1,,1,,1{1,,1,,2}2}1{1,,1,2}2}2}
{c 1{c 1`2}1,1,2`2} = {1{1,,1{1,,1,,1{1,,1,,2}2}1{1,,1,,2}2}2}
{c 1{2{1,,1,,1,2}2}2`2} = {1{1,,1{2,,1,,1{1,,1,,2}2}2}2}
{c 1{1`2{1,,1,,1,2}2}2`2} = {1{1{1,,2,,2}2{1,,1,,1{1,,1,,2}2}2,,1,,2}2}
{c 1{1{1,,1,,1,2}1,2}2`2} = {1{1{1,,1,,1{1,,1,,2}2}1,2,,1,,2}2}
{c 1{1{1{1,,1,2}2,,1,,1,2}2}2`2} = {1{1{1,,1,2,,2}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,2}2,,1,,1,2}2}2`2} = {1{1{1,,1,,3}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,1,2}2,,1,,1,2}2}2`2} = {1{1{1,,1,,1,2}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,1,2}2,,1,,1,2}2}1,2`2} = {1{1{1,,1,,1{1,,1,,2}2}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1,,2,,1,2}2}2`2} = {1{1,,2,,1{1,,1,,2}2}2}
{c 1{1{1,,1,2,,1,2}2}2`2} = {1{1,,1,2,,1{1,,1,,2}2}2}
{c 1{1{1,,1,,2,2}2}2`2} = {1{1,,1,,2{1,,1,,2}2}2}
{c 1{1{1,,1,,1,3}2}2`2} = {1{1,,1,,1,2{1,,1,,2}2}2}
{c 1{1{1,,1,,1,3}2}1,2`2} = {1{1,,1,,1{1,,1,,2}3}2}
{c 1{1{1,,1,,1{2}2}2}2`2} = {1{1,,1,,1{2,,1,,2}2}2}
{c 1{1{1,,1,,1{1`2}2}2}2`2} = {1{1,,1,,1{1{1,,2,,2}2,,1,,2}2}2}
{c 1{1{1,,1,,1{1{1,,1,,1,2}2}2}2}2`2} = {1{1,,1,,1{1{1,,1,,1,2}2,,1,,2}2}2}
{c 1{1{1,,1,,1`2}2}2`2} = {1{1,,1,,1{1,,2,,2}2}2}
{c 1{1{1,,1,,1{1,,1,2}2}2}2`2} = {1{1,,1,,1{1,,1,2,,2}2}2}
{c 1{c 1,2`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,3}2,,2}2}2}
{c 1{1{1,,1,,1{1,,1,,2}2}2}2`2} = {1{1,,1,,1{1,,1,,3}2}2}
{c 1{c 1{c 1,2`2}2`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,4}2,,3}2}2}
{c 1{c 1{c 1{c 1,2`2}2`2}2`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,5}2,,4}2}2}
{c 1`3} = {1{1,,1,,1{1,,1,,1,2}2}2}
{c 1,2`3} = {1{1,,1,,1{1,,1,,1{1,,1{1,,1,,2}2}2}2}2}
{c 1{c 1`3}1,2`3} = {1{1,,1,,1{1,,1,,1{1,,1,,2}2}2}2}
{c 1{c 1,2`3}2`3} = {1{1,,1,,1{1,,1,,1{1,,1{1,,1,,3}2,,2}2}2}2}
{c 1{c 1{c 1,2`3}2`3}2`3} = {1{1,,1,,1{1,,1,,1{1,,1{1,,1,,4}2,,3}2}2}2}
{c 1`4} = {1{1,,1,,1{1,,1,,1{1,,1,,1,2}2}2}2}
{c 1`5} = {1{1,,1,,1{1,,1,,1{1,,1,,1{1,,1,,1,2}2}2}2}2}
{c 1`1,2} = {1{1,,1,,1,,2}2}
{c 2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,2}2}2}
{c 1{c 1`1,2}1,2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1`2{1,,1,,1,,2}2}2`1,2} = {1{1{1,,2,,2}2{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2,,1,,2}2}
{c 1{1{1,,1,,1,,2}1`2}2`1,2} = {1{1{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}1{1,,2,,2}2,,1,,2}2}
{c 1{1{1{1,,1,2}2,,1,,1{1,,1,,1,,2}2}2}2`1,2} = {1{1{1,,1,2,,2}2,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}2}2}2`1,2} = {1{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1,,2,,1{1,,1,,1,,2}2}2}2`1,2} = {1{1,,2,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1,,1,,2{1,,1,,1,,2}2}2}2`1,2} = {1{1,,1,,2{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1,,1,,1{1,,1,,1,,2}3}2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}2{1,,1,,2}2}2}
{c 1{1{1,,1,,1{1,,1,,1,,2}1,2}2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}1,2}2}
{c 1{1{1,,1,,1{1,,1,,1,,2}1`2}2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,2,,2}2}2}
{c 1{c 1{c 1`1,2}1,2`1,2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,3}2}2}
{c 1{c 1{c 1{c 1`1,2}1,2`1,2}2`1,2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,4}2}2}
{c 1`2,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,2}2}2}
{c 1`3,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,2}2}2}2}
{c 1`1,3} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2}2}
{c 1`1,4} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2}2}2}
{c 1`1,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2}
{c 2`1,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,2}2}2}
{c 1`2,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,2}2}2}
{c 1`1,2,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2}2}
{c 1`1,1,3} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2}2}
{c 1`1,1,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,,2}2}2}
{c 1`1{2}2} = {1{1,,1,,1{2,,1,,1,,2}2}2}
{c 1{c 1`1{2}2}1,2`1{2}2} = {1{1,,1,,1{1{1,,1,,2}2,,1,,1,,2}2}2}
{c 1{c 1{c 1`1{2}2}1,2`1{2}2}2`1{2}2} = {1{1,,1,,1{1{1,,1,,3}2,,1,,1,,2}2}2}
{c 1`2{2}2} = {1{1,,1,,1{1{1,,1,,1,2}2,,1,,1,,2}2}2}
{c 1`1{2}3} = {1{1,,1,,1{1{1,,1,,1{2,,1,,1,,2}2}2,,1,,1,,2}2}2}
{c 1`1{2}1,2} = {1{1{1,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{3}1,2} = {1{1{1,,1,,1,,2}3,,1,,1,,2}2}
{c 1`1{1,2}2} = {1{1{1,,1,,1,,2}1,2,,1,,1,,2}2}
{c 1`1{1{2}2}2} = {1{1{2,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1{1,2}2}2} = {1{1{1{1,,1,,1,,2}1,2,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`2}2} = {1{1,,2,,1,,2}2}
{c 2`1{1`2}2} = {1{1{1,,2,,1,,2}1{1,,1,2}2,,1,,1,,2}2}
{c 1`2{1`2}2} = {1{1{1,,2,,1,,2}1{1,,1,,1,2}2,,1,,1,,2}2}
{c 1`1{1`2}3} = {1{1{1,,2,,1,,2}1{1,,1,,1{1,,2,,1,,2}2}2,,1,,1,,2}2}
{c 1`1{1`2}1,2} = {1{1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`3}2} = {1{1{1,,2,,1,,2}2{1,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`1,2}2} = {1{1{1,,2,,1,,2}1{1,,1,,1,,2}1,2,,1,,1,,2}2}
{c 1`1{1`1{1`2}2}2} = {1{1{1,,2,,1,,2}1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`1`2}2} = {1{1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`1`1`2}2} = {1{1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,1,,2}2}
{c 1`1{1{2`}2}2} = {1{1{2,,2,,1,,2}2,,1,,1,,2}2}
{c 1`1{1{1`2`}2}2} = {1{1{1,,2,,1,,2}2,,2,,1,,2}2}
{c 1`1{1{1{1`2`}2`}2}2} = {1{1{1{1,,2,,1,,2}2,,2,,1,,2}2,,2,,1,,2}2}
{c 1`1{1` `2}2} = {1{1,,3,,1,,2}2}
{c 1`1{1{1,,1,2}2}2} = {1{1,,1,2,,1,,2}2}
{c 2`1{1{1,,1,2}2}2} = {1{1,,1{1,,1,2}2,,1,,2}2}
{c 1`2{1{1,,1,2}2}2} = {1{1,,1{1,,1,,1,2}2,,1,,2}2}
{c 1`1{1{1,,1,2}2}1,2} = {1{1,,1{1,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,2,2}2}2} = {1{1,,2{1,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1{2}2}2}2} = {1{1,,1{2,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1{1`2}2}2}2} = {1{1,,1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1`2}2}2} = {1{1,,1{1,,2,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1{1,,1,2}2}2}2} = {1{1,,1{1,,1,2,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1,,2}2}2} = {1{1,,1,,2,,2}2}
{c 1`1{1{1,,1,,3}2}2} = {1{1,,1,,3,,2}2}
{c 1`1{1{1,,1,,1,2}2}2} = {1{1,,1,,1,2,,2}2}
{c 1`1{c 1`1,2}2} = {1{1,,1,,1,,3}2}
{c 1`1{c 1`1{c 1`1,2}2}2} = {1{1,,1,,1,,4}2}
{c 1`1`2} = {1{1,,1,,1,,1,2}2}
{c 2`1`2} = {1{1,,1,,1,,1{1,,1,2}2}2}
{c 1{c 1`1`2}1,2`1`2} = {1{1,,1,,1,,1{1,,1,,2}2}2}
{c 1`2`2} = {1{1,,1,,1,,1{1,,1,,1,2}2}2}
{c 1`1{c 1`1`2}1,2`2} = {1{1,,1,,1,,1{1,,1,,1,,2}2}2}
{c 1`1`3} = {1{1,,1,,1,,1{1,,1,,1,,1,2}2}2}
{c 1`1`1,2} = {1{1,,1,,1,,1,,2}2}
{c 1`1`2,2} = {1{1,,1,,1,,1{1,,1,,1,,1,,2}1{1,,1,,1,,1,2}2}2}
{c 1`1`1,1,2} = {1{1,,1,,1,,1{1,,1,,1,,1,,2}1{1,,1,,1,,1,,2}2}2}
{c 1`1`1{2}2} = {1{1,,1,,1,,1{2,,1,,1,,1,,2}2}2}
{c 1`1`1{1`2}2} = {1{1,,2,,1,,1,,2}2}
{c 1`1`1{1{1,,1,2}2}2} = {1{1,,1,2,,1,,1,,2}2}
{c 1`1`1{c 1`1`1,2}2} = {1{1,,1,,1,,1,,3}2}
{c 1`1`1`2} = {1{1,,1,,1,,1,,1,2}2}
{c 1`1`1`1`2} = {1{1,,1,,1,,1,,1,,1,2}2}
{c 1{2`}2} = {1{1{2,,}2}2}
{c 2{2`}2} = {1{1{1{1,,1,2}2,,}2}2}
{c 1{c 1{2`}2}1,2{2`}2} = {1{1{1{1,,1,,2}2,,}2}2}
{c 1`2{2`}2} = {1{1{1{1,,1,,1,2}2,,}2}2}
{c 1`1`2{2`}2} = {1{1{1{1,,1,,1,,1,2}2,,}2}2}
{c 1{2`}3} = {1{1{1{1{2,,}2}2,,}2}2}
{c 1{2`}1,2} = {1{1{1,,2,,}2}2}
Let separator ◊ = {1{1{1,,2,,}2}2,,}, then
{c 2{2`}1,2} = {1{1◊1{1,,1,2}2}2}
{c 1{c 1{2`}1,2}1,2{2`}1,2} = {1{1◊1{1,,1,,2}2}2}
{c 1`2{2`}1,2} = {1{1◊1{1,,1,,1,2}2}2}
{c 1`1`2{2`}1,2} = {1{1◊1{1,,1,,1,,1,2}2}2}
{c 1{2`}2,2} = {1{1◊1{1{2,,}2}2}2}
{c 1{2`}1,3} = {1{1◊1{1◊2}2}2}
{c 1{2`}1,1,2} = {1{1◊1,,2}2}
{c 1{2`}2,1,2} = {1{1◊1{1◊1,,2}1{1{2,,}2}2}2}
{c 1{2`}1,1,1,2} = {1{1◊1{1◊1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{2}2} = {1{1◊1{2◊1,,2}2}2}
{c 1{2`}1{1`2}2} = {1{1,,2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}2}2} = {1{1{2,,}2◊1,,2}2}
{c 2{2`}1{1{1{2,,}2}2}2} = {1{1{1{1,,1,2}2,,}2◊1,,2}2}
{c 1`2{2`}1{1{1{2,,}2}2}2} = {1{1{1{1,,1,,1,2}2,,}2◊1,,2}2}
{c 1{2`}2{1{1{2,,}2}2}2} = {1{1{1{1{2,,}2}2,,}2◊1,,2}2}
{c 1{2`}1,2{1{1{2,,}2}2}2} = {1{1{1{1◊2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}2}3} = {1{1{1{1◊1{1{2,,}2◊1,,2}2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}2}1,2} = {1{1{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}2{1{1{2,,}2}2}1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1{2,,}2}2}2}
{c 1{2`}1,2{1{1{2,,}2}2}1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1◊2}2}2}
{c 1{2`}1{1{1{2,,}2}2}1,1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1{2,,}2}2}1{1{1{2,,}2}2}2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1{2,,}2◊1,,2}2}2}
{c 1{2`}1{1{1{2,,}2}2}1{1{1{2,,}2}2}1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1{1{1◊1,,2}2,,}2◊1,,2}2}2}
{c 1{2`}1{2{1{2,,}2}2}2} = {1{1◊1{2{1{1◊1,,2}2,,}2◊1,,2}2}2}
{c 1{2`}1{2{1{2,,}2}2}1,2} = {1{1{1◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1`2{1{2,,}2}2}2} = {1{1{1,,2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}3}2} = {1{1{1{1{2,,}2◊1,,2}2,,2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1,,1,2}2{2,,}2}2}2} = {1{1{1,,1,2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1{2,,}2}2{2,,}2}2}2} = {1{1{1{2,,}2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1,,2{2,,}2}2}2} = {1{1,,2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}3}2}2} = {1{1{2,,}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}3}2}1,2} = {1{1{1{1◊1,,2}2,,}3◊1,,2}2}
{c 1{2`}1{1{1{2,,}1,2}2}2} = {1{1{1{1◊1,,2}2,,}1,2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1,2}2}1,2} = {1{1{1{1◊1,,2}2,,}1{1◊1,,2}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1,,2}2}2} = {1{1{1{1◊1,,2}2,,}1,,2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1{2,,}2}2}2} = {1{1{1{1◊1,,2}2,,}1{2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1{2,,}2}2}1,2} = {1{1{1{1◊1,,2}2,,}1{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{3,,}2}2}2} = {1{1{2{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{3,,}2}2}1,2} = {1{1{1{1◊1,,2}3,,}2◊1,,2}2}
{c 1{2`}1{1{1{1,2,,}2}2}2} = {1{1{1{1◊1,,2}1,2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1`2,,}2}2}2} = {1{1{1{1,,2◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1{1{2,,}2}2,,}2}2}2} = {1{1{1{1{2,,}2◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{c 1{2`}1,2}2} = {1{1◊2,,2}2}
{c 2{2`}1{c 1{2`}1,2}2} = {1{1◊1{1,,1,2}2,,2}2}
{c 1{2`}2{c 1{2`}1,2}2} = {1{1◊1{1{2,,}2}2,,2}2}
{c 1{2`}1,2{c 1{2`}1,2}2} = {1{1◊1{1◊2}2,,2}2}
{c 1{2`}1{c 1{2`}1,2}1,2} = {1{1◊1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1,,2◊2}2}2} = {1{1,,2◊1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1{2,,}2◊2}2}2} = {1{1{2,,}2◊1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊3}2}2} = {1{1◊2{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊3}2}1,2} = {1{1{1◊1,,2}3,,2}2}
{c 1{2`}1{1{1◊1,2}2}2} = {1{1◊1{1◊1,,2}1,2,,2}2}
{c 1{2`}1{1{1◊1,2}2}1,2} = {1{1◊1{1◊1,,2}1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊1`2}2}2} = {1{1◊1{1,,2◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊1{1{2,,}2}2}2}2} = {1{1◊1{1{2,,}2◊1,,2}2,,2}2}
{c 1{2`}1{c 1{2`}1,3}2} = {1{1◊1{1◊2,,2}2,,2}2}
{c 1{2`}1{c 1{2`}1,1,2}2} = {1{1◊1,,3}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1,2}2}2} = {1{1◊2,,3}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1{c 1{2`}1,2}2}2}2} = {1{1◊2,,4}2}
{c 1{2`}1`2} = {1{1◊1,,1,2}2}
{c 2{2`}1`2} = {1{1◊1,,1{1,,1,2}2}2}
{c 1`2{2`}1`2} = {1{1◊1,,1{1,,1,,1,2}2}2}
{c 1{2`}2`2} = {1{1◊1,,1{1{2,,}2}2}2}
{c 1{2`}1,2`2} = {1{1◊1,,1{1◊2}2}2}
{c 1{2`}1{c 1{2`}1`2}2`2} = {1{1◊1,,1{1◊1{1◊1,,1,2}2}2}2}
{c 1{2`}1{c 1{2`}1`2}1,2`2} = {1{1◊1,,1{1◊1,,2}2}2}
{c 1{2`}1{1{1,,2◊1,,1,2}2}2`2} = {1{1,,2◊1,,1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊2,,1,2}2}2`2} = {1{1◊2,,1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,2,2}2}2`2} = {1{1◊1,,2{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1,3}2}2`2} = {1{1◊1,,1,2{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1,3}2}1,2`2} = {1{1◊1,,1{1◊1,,2}3}2}
{c 1{2`}1{1{1◊1,,1`2}2}2`2} = {1{1◊1,,1{1,,2◊1,,2}2}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1`2}1,2`2}2`2} = {1{1◊1,,1{1◊1,,3}2}2}
{c 1{2`}1`3} = {1{1◊1,,1{1◊1,,1,2}2}2}
{c 1{2`}1`4} = {1{1◊1,,1{1◊1,,1{1◊1,,1,2}2}2}2}
{c 1{2`}1`1,2} = {1{1◊1,,1,,2}2}
{c 1{2`}1{c 1{2`}1`1,2}1,2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊2,,1{1◊1,,1,,2}2}2}2`1,2} = {1{1◊2,,1{1◊1,,1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,2{1◊1,,1,,2}2}2}2`1,2} = {1{1◊1,,2{1◊1,,1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1{1◊1,,1,,2}3}2}2`1,2} = {1{1◊1,,1{1◊1,,1,,2}2{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1{1◊1,,1,,2}3}2}1,2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,2}3}2}
{c 1{2`}1{1{1◊1,,1{1◊1,,1,,2}1`2}2}2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1,,2◊1,,2}2}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1`1,2}1,2`1,2}2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,3}2}2}
{c 1{2`}1`2,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,1,2}2}2}
{c 1{2`}1`1,3} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,1{1◊1,,1,,2}2}2}2}
{c 1{2`}1`1,1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,1,,2}2}2}
{c 1{2`}1`1{2}2} = {1{1◊1,,1{2◊1,,1,,2}2}2}
{c 1{2`}1`1{1`2}2} = {1{1,,2◊1,,1,,2}2}
{c 1{2`}1`1{1{1{2,,}2}2}2} = {1{1{2,,}2◊1,,1,,2}2}
{c 1{2`}1`1{1{1{2,,}2}2}1,2} = {1{1{1{1◊1,,1,,2}2,,}2◊1,,1,,2}2}
{c 1{2`}1`1{1{1{1,,2,,}2}2}2} = {1{1◊2,,1,,2}2}
{c 1{2`}1`1{1{1◊1,,2}2}2} = {1{1◊1,,2,,2}2}
{c 1{2`}1`1{c 1{2`}1`1,2}2} = {1{1◊1,,1,,3}2}
{c 1{2`}1`1{c 1{2`}1`1{c 1{2`}1`1,2}2}2} = {1{1◊1,,1,,4}2}
{c 1{2`}1`1`2} = {1{1◊1,,1,,1,2}2}
{c 1{2`}1`1`1`2} = {1{1◊1,,1,,1,,1,2}2}
{c 1{2`}1{2`}2} = {1{1◊1{2,,}2}2}
{c 1{2`}1{2`}1,2} = {1{1◊1◊2}2}
{c 1{2`}1{2`}1`2} = {1{1◊1◊1,,1,2}2}
{c 1{2`}1{2`}1{2`}2} = {1{1◊1◊1{2,,}2}2}
{c 1{3`}2} = {1{1{2{1{1,,2,,}2}2,,}2}2}
{c 2{3`}2} = {1{1{1{1,,1,2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}2{3`}2} = {1{1{1{1{2,,}2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}1,2{3`}2} = {1{1{1{1◊2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}1`2{3`}2} = {1{1{1{1◊1,,1,2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}1{2`}2{3`}2} = {1{1{1{1◊1{2,,}2}2{1{1,,2,,}2}2,,}2}2}
{c 1{3`}3} = {1{1{1{1{2{1{1,,2,,}2}2,,}2}2{1{1,,2,,}2}2,,}2}2}
{c 1{3`}1,2} = {1{1{1{1{1,,2,,}2}3,,}2}2}
{c 1{3`}1,1,2} = {1{1{1{1{1,,2,,}2}3,,}1,,2}2}
{c 1{3`}1`2} = {1{1{1{1{1,,2,,}2}3,,}1,,1,2}2}
{c 1{3`}1{2`}2} = {1{1{1{1{1,,2,,}2}3,,}1{2,,}2}2}
{c 1{3`}1{2`}1,2} = {1{1{1{1{1,,2,,}2}3,,}1{1{1{1,,2,,}2}2,,}2}2}
{c 1{3`}1{3`}2} = {1{1{1{1{1,,2,,}2}3,,}1{2{1{1,,2,,}2}2,,}2}2}
{c 1{3`}1{3`}1,2} = {1{1{1{1{1,,2,,}2}3,,}1{1{1{1,,2,,}2}3,,}2}2}
{c 1{4`}2} = {1{1{2{1{1,,2,,}2}3,,}2}2}
{c 1{5`}2} = {1{1{2{1{1,,2,,}2}4,,}2}2}
{c 1{1,2`}2} = {1{1{1{1{1,,2,,}2}1,2,,}2}2}
{c 1{1,2`}1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}2}
{c 1{1,2`}1,1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2}2}
{c 1{1,2`}1`2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,1,2}2}
{c 1{1,2`}1{2`}2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1{2,,}2}2}
{c 1{1,2`}1{1,2`}2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1{1{1{1,,2,,}2}1,2,,}2}2}
{c 1{2,2`}2} = {1{1{2{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}2}
{c 1{1,3`}2} = {1{1{1{1{1,,2,,}2}1,2{1{1,,2,,}2}2,,}2}2}
{c 1{1,3`}1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}3,,}2}2}
{c 1{1,1,2`}1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}1,2,,}2}2}
{c 1{1,1,1,2`}2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}1{1{1,,2,,}2}1,2,,}2}2}
{c 1{1{2}2`}2} = {1{1{1{2{1,,2,,}2}2,,}2}2}
{c 1{1{2}2`}1,2} = {1{1{1{1,,2,,}2}2{1,,2,,}2}2}
{c 1{1{1`2}2`}2} = {1{1,,2{1,,2,,}2}2}
{c 1{1{c 1}2`}2} = {1{1,,1,2{1,,2,,}2}2}
{c 1{1{c 1{2`}2}2`}2} = {1{1{2,,}2{1,,2,,}2}2}
{c 1{1{c 1{2`}2}2`}1,2} = {1{1{1{1{1,,2,,}2}2,,}2{1,,2,,}2}2}
{c 1{1{c 1{2`}1,2}2`}2} = {1{1{1,,2,,}3}2}
{c 1{1{c 1{1{c 1{2`}2}2`}2}2`}2} = {1{1{2,,}2{1,,2,,}3}2}
{c 1{1{c 1{1{c 1{1{c 1{2`}2}2`}2}2`}2}2`}2} = {1{1{2,,}2{1,,2,,}4}2}
{c 1{1`2`}2} = {1{1{1,,2,,}1,2}2}
{c 2{1`2`}2} = {1{1{1,,2,,}1{1,,1,2}2}2}
{c 1`2{1`2`}2} = {1{1{1,,2,,}1{1,,1,,1,2}2}2}
{c 1{2`}2{1`2`}2} = {1{1{1,,2,,}1{1{2,,}2}2}2}
{c 1{2`}1,2{1`2`}2} = {1{1{1,,2,,}1{1{1{1{1,,2,,}2}2,,}2}2}2}
{c 1{3`}2{1`2`}2} = {1{1{1,,2,,}1{1{2{1{1,,2,,}2}2,,}2}2}2}
{c 1{1{1`2}2`}2{1`2`}2} = {1{1{1,,2,,}1{1{1{1,,2{1,,2,,}2}2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}2{1`2`}2} = {1{1{1,,2,,}1{1{1{1{1,,2,,}1,2}2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}3{1`2`}2} = {1{1{1,,2,,}1{1{1{1{1,,2,,}1{1{1{1{1,,2,,}1,2}2,,}2}2}2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}1,2{1`2`}2} = {1{1{1,,2,,}1{1{1,,2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}1,1,2{1`2`}2} = {1{1{1{1{1,,2,,}1{1{1,,2,,}2}2}2,,}1,,2}2}
{c 1{1{c 1{1`2`}2}2`}1{1{1{2,,}2}2}2{1`2`}2} = {1{1{2,,}2{1{1{1,,2,,}1{1{1,,2,,}2}2}2,,}1,,2}2}
{c 1{1{c 1{1`2`}2}2`}1{1{1{1{1{1,,2,,}1,,1,2}2,,}2}2}2{1`2`}2} = {1{1{1{1{1,,2,,}1,,1,2}2,,}2{1{1{1,,2,,}1{1{1,,2,,}2}2}2,,}1,,2}2}