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Proof step 20 from proof band-011x-is-unsigned-03 (previous step)

2 ^ i + j = 0, j < 0, # int2bits(j) < # int2bits(~ j) ⊦ j + 2 ^ n→i(# int2bits(j) - 1) = 0, j = 0, band(j, 2 ^ n→i(# int2bits(j)) - 1) = bits2uint(int2bits(j))

User applied rule cut formula

And closed the goal!

Rule argument

fma: j = ~(2 ^ i)

Additional information

Used simplifier rules: ⊦ ~ i = j ↔ i + j = 0;% ⊦ ~ i + i = 0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < 0 → 2 ^ i = 0;% ⊦ ¬ i ≤ j ↔ j < i;%0 ≤ i ⊦ int2bits(~(2 ^ i)) = true ' + mklist(false, i→n(i));% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ # mklist(boolvar, n) = n;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0;% ⊦ 0 + m = m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ n→i(i→n(i)) = abs(i);% ⊦ i + ~ j = i - j;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + 0 = i;% ⊦ ~ i = 0 ↔ i = 0;% ⊦ ~ i < 0 ↔ 0 < i;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ n→i(m + n) = n→i(m) + n→i(n);% ⊦ 0 ≤ i → n→i(i→n(i)) = i;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ bits2uint(true ' + mklist(false, n)) = 2 ^ n→i(n);% ⊦ 0 ≤ i → abs(i) = i;% ⊦ 0 ≤ i ∧ 0 ≤ j → (2 ^ i = 2 ^ j ↔ i = j); Used formulas: j = ~(2 ^ i)