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Proof step 74 from proof cast-mod-neg-low (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 75:
i→n(i) = # int2bits(j % 2 ^ i).rest, i→n(i) < # int2bits(j % 2 ^ i), 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0, j + 2 ^ i ≤ 0, j + 2 ^ i < i ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(int2bits(j % 2 ^ i).rest) = j % 2 ^ i + 2 ^ i

Rule argument

Additional information

Used simplifier rules: ⊦ # bits.rest < # bits ↔ bits ≠ [];% ⊦ int2bits(j) ≠ [];% ⊦ # bits = 0 ↔ bits = [];% ⊦ m < n → n ≠ 0; (forward)% ⊦ m < n ∧ m = n0 → n0 < n; (forward)% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + 1 ≤ n ↔ m < n;