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Proof step 7 from proof cast-mod-neg-low (previous step)

0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ bits2int(lastn(i→n(i), fillsgn(i→n(i), int2bits(j)))) = j % 2 ^ i + 2 ^ i

User applied rule cut formula

And obtained the following new goals

Proof step 8:
0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ j = ~1, bits2int(lastn(i→n(i), fillsgn(i→n(i), int2bits(j)))) = j % 2 ^ i + 2 ^ i

Rule argument

fma: j = ~1

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ~1 < i ↔ 0 ≤ i;%i < 0 ⊦ (* i) + j = j - ~ i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ 2 ^ i < 1 ↔ i < 0;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used formulas: j = ~1