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Proof step 15 from proof int2bits-len-range (previous step)

j < 0, 0 < 2 ^ i0, i→n(i + 1) < # int2bits(j), 0 ≤ 2 ^ i + j, j < 2 ^ i, 0 ≤ i0, 2 ^ i0 ≤ 0 - j, 0 - j < 2 ^ i0 * 2 ⊦ j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
0 ≤ 2 ^ i + j, 0 ≤ i0, 2 ^ i0 + j ≤ 0, 0 < 2 ^ i0 * 2 + j, j < 2 ^ i0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, 2 ^ i0 + j ≤ i, 0 ≤ i, 2 ^ i0 + j < 2 ^ i, j + 1 < 2 ^ i0, 0 < 2 ^ i, j < i0, i0 ≤ i, 2 ^ i0 + j ≤ i0, j < 0, 0 < 2 ^ i0, i→n(i) + 1 < # int2bits(j) ⊦ j = 0

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ i → i→n(i + 1) = i→n(i) + 1;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i +1 = i + 1;% ⊦ i < i + j ↔ 0 < j;% ⊦ i + j < j ↔ i < 0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (i < i * j ↔ 1 < j);% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ i → (2 ^ i ≤ 2 ^ j ↔ i ≤ j);% ⊦ i + j ≤ i + j0 ↔ j ≤ j0;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ 0 - i = ~ i;