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Proof step 44 from proof cast-mod-neg-low (previous step)

¬ # int2bits(j % 2 ^ i) ≤ i→n(i) + 1, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), j % 2 ^ i + 2 ^ (i - 1) < 0, j < 0, 0 < i, j + 2 ^ i < i, j + 2 ^ i ≤ 0, i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i - 1) ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i + 2 ^ i, j = ~1, i = 1, j % 2 ^ i = 0, fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) = [], i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 45:
i→n(i) + 1 < # int2bits(j % 2 ^ i), i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i - 1), j + 2 ^ i ≤ 0, j + 2 ^ i < i, 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1, fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) = [], j % 2 ^ i = 0, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i + 2 ^ i

Rule argument

Additional information

Used simplifier rules: ⊦ int2bits(j) ≠ [];% ⊦ # bits = 0 ↔ bits = [];% ⊦ m < n → n ≠ 0; (forward)% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;