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Proof step 22 from proof cast-mod-neg-low (previous step)

i→n(i) ≤ # int2bits(j), 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j))) = j % 2 ^ i + 2 ^ i, ¬ 0 < j + 2 ^ i ∧ j % 2 ^ i ≠ 0

User applied rule cut formula

And obtained the following new goals

Proof step 23:
0 < j + 2 ^ i, i→n(i) ≤ # int2bits(j), 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j))) = j % 2 ^ i + 2 ^ i, ¬ 0 < j + 2 ^ i ∧ j % 2 ^ i ≠ 0

Rule argument

fma: 0 < j + 2 ^ i

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ i + 0 = i;% ⊦ ¬ 2 ^ i < 0; Used formulas: 0 < j + 2 ^ i