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Proof step 30 from proof cast-mod-neg-high (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 31:
0 < j + 2 ^ i, i→n(i) ≤ # int2bits(j), 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i ⊦ i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j))) = j % 2 ^ i, ¬ 0 < j + 2 ^ i ∧ j % 2 ^ i ≠ 0
Proof step 43:
i→n(i) ≤ # int2bits(j), 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i ⊦ 0 < j + 2 ^ i, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j))) = j % 2 ^ i, ¬ 0 < j + 2 ^ i ∧ j % 2 ^ i ≠ 0

Rule argument

fma: 0 < j + 2 ^ i

Additional information

Used formulas: 0 < j + 2 ^ i