Dropping array notation

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Dropping array notation (DAN) is the seventh part of my array notation.

Let’s have an overview of EAN, pDAN and sDAN. In EAN, the grave accent works as the least 1-separator, and it drops down once until the expansion. In pDAN, the double comma works as the least 2-separator, and it drops down twice until the expansion. In sDAN, the triple comma works as the least 3-separator, and it drops down 3 times until the expansion.

Continue upward, we have the least 4-separator – the quadruple comma (,,,,), the least 5-separator – the quintuple comma (,,,,,), and so on. Generally, an m-separator drops down m times until the expansion. Here comes the definition of dropping array notation.

Definition

Syntax:

• s(a,b) is an array where a and b are positive integers.
• s(#Am) is an array, where m is a positive integer, A is a separator, and # is a string such that s(#) is an array.

Definition of separators:

• The m-ple comma (,,…,) with m ≥ 2 are separators. (basic separators)
• {m} is a separator, where m is a positive integer.
• {#Am} is a separator, where m is a positive integer, A is a separator, and # is a string such that {#} is a separator.

Note:

1. The comma (,) is a shorthand for {1}.
2. The dot . = {1{1,,2}2}, the grave accent ` = {1,,2}, the double comma ,, = {1,,,2}, the triple comma ,,, = {1,,,,2}, and so on, the m-ple comma ,,…, = {1,,…,,2} with an (m+1)-ple comma inside.

Rules and process

• Rule 1: (base rule – only 2 entries) s(a,b) = a^b
• Rule 2:(recursion rule – neither the 2nd nor 3rd entry is 1) s(a,b,c #) = s(a, s(a,b-1,c #) ,c-1 #)
• Rule 3: (tailing rule – the last entry is 1) s(# A 1) = s(#) and {# A 1} = {#}
• Rule 4: (if lv(A) < lv(B)) {# A 1 B #′} = {# B #′}

where # is a string of entries and separators, it can also be empty. A and B are separators.

If none of the 4 rules above applies, start the process shown below. Note that case B1, B2 and B4 are terminal but case A and B3 are not. And note that red texts imply changes on the array. Before we start, let A₀ be the whole array. First start from the 3rd entry.

• Case A: If the entry is 1, then you jump to the next entry.
• Case B: If the entry n is not 1, look to your left:
  • Case B1: If the comma is immediately before you, then
    1. Change the “1,n” into “b,n-1” where n is this non-1 entry and the b is the iterator.
    2. Change all the entries at base layer before them into the base.
    3. The process ends.
  • Case B2: If the m-ple comma (m ≥ 2) M is immediately before you, then
    1. Change “M n” into “M 2 M n-1”, and let B_u(m+1) be the former M.
    2. Let t be such that the M is at layer t. And let B₀ be the whole array now.
    3. Repeat this:
       1. Subtract t by 1.
       2. Let separator B_t be such that it’s at layer t, and the M is inside it.
       3. If t = 1, then break the repeating, or else continue repeating.
    4. Find the maximal u(m) such that lv(A_u(m)) < lv(M).
    5. If lv(A_u(m)) < lv(,), then
       1. Let string P and Q be such that B_u(m) = “P B_u(m)+1 Q”.
       2. Change B_u(m) into “P {1 A_u(m)+1 2} Q”.
       3. The process ends.
    6. Set j = m. Set A_u(m+1) = M.
    7. Repeat this:
       1. Subtract j by 1.
       2. Find the maximal u(j) such that lv(A_u(j)) < lv(A_u(j+1)).
       3. Let string X and Y be such that B_u(j+1) = “X B_u(j+2) 2 Y”.
       4. If lv(A_u(j)) < lv(“X A_u(j+1) 2 Y”), then
          1. Find the minimal v(j) such that v(j) > u(j) and lv(A_v(j)) < lv(A_u(j+2)).
          2. Let string P and Q be such that B_u(j) = “P B_v(j) Q”.
          3. Change B_u(j) into “P X A_v(j) 2 Y Q”.
          4. The process ends.
       5. If j = 1, then break the repeating, or else continue repeating.
    8. Let string P and Q be such that B_u(1) = “P B_u(2) Q”.
    9. Change B_u(1) into S_b, where b is the iterator, S₁ is comma, and S_i+1 = “P S_i Q”.
    10. The process ends.
  • Case B3: If a separator K is immediately before you, and it doesn’t fit case B1 or B2, then
    1. Change the “K n” into “K 2 K n-1”.
    2. Set separator A_t = K, here K is at layer t.
    3. Jump to the first entry of the former K.
  • Case B4: If an lbrace is immediately before you, then
    1. Change separator {n #} into string S_b, where b is the iterator, S₁ = “{n-1 #}” and S_i+1 = “S_i 1 {n-1 #}”.
    2. The process ends.

Level comparison

All arrays have the lowest and the same level. And note that the same separators have the same level. To compare levels of other separators A and B, we follow these steps.

First some preparation.

1. Find the maximum of m that there’re some m-ple commas in the expressions of A or B.
2. Convert all the multiple comma of A and B into expressions of m-ple comma, using ,,…, (i commas) = {1,,…,,2} (i+1 commas)

Then the m-ple comma has the highest level, and other separators have level lower than it.

1. Apply rule 3 to A and B until rule 3 cannot apply any more.
2. Let A = {a₁A₁a₂A₂…a_k-1A_k-1a_k} and B = {b₁B₁b₂B₂…b_l-1B_l-1b_l}
3. If k = 1 and l > 1, then lv(A) < lv(B); if k > 1 and l = 1, then lv(A) > lv(B); if k = l = 1, follow step 4; if k > 1 and l > 1, follow step 5 ~ 10
4. If a₁ < b₁, then lv(A) < lv(B); if a₁ > b₁, then lv(A) > lv(B); if a₁ = b₁, then lv(A) = lv(B)
5. Let , and .
6. If lv(A_maxM(A)) < lv(B_maxM(B)), then lv(A) < lv(B); if lv(A_maxM(A)) > lv(B_maxM(B)), then lv(A) > lv(B); or else –
7. If |M(A)| < |M(B)|, then lv(A) < lv(B); if |M(A)| > |M(B)|, then lv(A) > lv(B); or else –
8. Let A = {#₁ A_maxM(A) #₂} and B = {#₃ B_maxM(B) #₄}
9. If lv({#₂}) < lv({#₄}), then lv(A) < lv(B); if lv({#₂}) > lv({#₄}), then lv(A) > lv(B); or else –
10. If lv({#₁}) < lv({#₃}), then lv(A) < lv(B); if lv({#₁}) > lv({#₃}), then lv(A) > lv(B); if lv({#₁}) = lv({#₃}), then lv(A) = lv(B)

Explanation

The A_t in case B3 and the B_t in step 1 ~ 3 of case B2 are preparation. Step 4 and 5 are where the m-ple comma (the least m-separator) drops down to an (m-1)-separator. Step 7 is where the (m-1)-separator drops down m-1 times, to a 0-separator, and step 6 is a preparation for it. And step 8 ~ 10 are the expansion. Here, the A_u(m+1) = M is an m-separator, and the A_u(j) is a (j-1)-separator.

The step 5 and step 7d are where the adding rules apply. Once an adding rule apply, the expansion doesn’t apply; only when all the adding rules don’t apply, the expansion applies.

The level comparison rules also change. The preparation step change A and B into such that the m-ple comma is the only “basic separator”.