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Proof step 53 from proof bor-2pot-plus-01 (previous step)

int2bits(j0) ≠ [], # int2bits(j0) = i→n(i) + 1, 0 ≤ 2 ^ i, 0 < 2 ^ i, j + 1 < j0, 0 < j0, j < 0, j + 1 < i, 0 ≤ j0, j < i, j < j0, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < 2 ^ i, j0 < 2 ^ i, 0 < i, i→n(i) ≤ (# int2bits(j0))-1, 1 < 2 ^ i, 2 + j < 2 ^ i ⊦ int2bits(j0).rest = [], i = 0, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], (j + 1) * 2 ^ n→i((# int2bits(j0))-1) = 2 ^ (n→i(# uint2bits(j0)) - 1) + j * 2 ^ i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 54:
# int2bits(j0) = i→n(i) + 1, 2 + j < 2 ^ i, 1 < 2 ^ i, 0 < i, j0 < 2 ^ i, j < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < j0, j < i, 0 ≤ j0, j + 1 < i, j < 0, 0 < j0, j + 1 < j0, 0 < 2 ^ i, 0 ≤ 2 ^ i ⊦ bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, int2bits(j0).rest = [], 2 ^ i = 2 ^ (n→i(# uint2bits(j0)) - 1)

Rule argument

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 1 * i = i;% ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = j0;% ⊦ 0 < i → n→i(i→n(i)) = i;% ⊦ 0 + m = m;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;%⊦ 0 ≤ j ∧ j < 2 ^ i → (i→n(i) < # int2bits(j) ↔ # int2bits(j) = i→n(i) + 1);% ⊦ m + 1 ≤ n ↔ m < n;%n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n;% ⊦ # bits = 0 ↔ bits = [];% ⊦ m < 1 ↔ m = 0;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ int2bits(j) ≠ [];% ⊦ n -1 = n - 1;