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Proof step 49 from proof int2int-neg (previous step)

j = ~(2 ^ i), j + 2 ^ i = 0, j < 0, j < i + 2, j ≤ 0, j ≤ i + 2, j ≤ 4 * 2 ^ i, j < 4 * 2 ^ i, j < 3 * 2 ^ i, j ≤ 3 * 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j ≤ i, j + 1 < 4 * 2 ^ i, j < i, j ≤ 2 ^ i, 0 < 2 ^ i * 2 + j, 0 < 4 * 2 ^ i + j, 0 < 2 ^ i, ~2 ≤ i, 0 ≤ 4 * 2 ^ i + j, 0 ≤ i, 0 ≤ 2 ^ i, ~2 < i ⊦ 4 * 2 ^ i = 2 ^ (n→i(# uint2bits(4 * 2 ^ i + j)) - 1), 3 * 2 ^ i + j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 50:
0 ≤ 2 ^ i + 2 ^ i, 0 < i + 2 ^ i, 0 ≤ 2 ^ i + i, 1 < 2 ^ i + 2 ^ i, ~2 ≤ 2 ^ i + i, ~2 < i + 2 ^ i, ~2 < i, 0 ≤ 2 ^ i, 0 ≤ i, ~2 ≤ i, 0 < 2 ^ i ⊦ 4 * 2 ^ i = 2 ^ (n→i(# uint2bits(2 ^ i * 3)) - 1)

Rule argument

Additional information

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