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Proof step 39 from proof int2int-neg (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 40:
~2 * 2 ^ i + 1 < i, ~2 * 2 ^ i < i, ~2 * 2 ^ i < i + 2, j < ~1, j + 2 ^ i < i, j ≤ 4 * 2 ^ i, j < 4 * 2 ^ i, j < 3 * 2 ^ i, j ≤ 3 * 2 ^ i, 2 ^ i + j ≤ 0, 0 < 2 ^ i * 2 + j, 2 ^ i + j ≤ i + 2, 2 ^ i + j < i + 2, j < i + 2, j < 2 ^ i, j + 1 < 2 ^ i, j ≤ i, 0 ≤ i, j + 1 < 4 * 2 ^ i, 0 ≤ 4 * 2 ^ i + j, 0 < 4 * 2 ^ i + j, 2 ^ i + j ≤ i, j < i, j ≤ i + 2, j ≤ 0, j ≤ 2 ^ i, ~2 * 2 ^ i ≤ j, ~2 * 2 ^ i < j, j + 2 ^ i < 0, ~2 * 2 ^ i ≤ i, ~2 * 2 ^ i ≤ i + 2, 0 ≤ 2 ^ i, 0 < 2 ^ i, j < 0, ~2 * 2 ^ i < ~1, 1 < 2 ^ i ⊦ 3 * 2 ^ i + j = 0, 4 * 2 ^ i = 2 ^ (2 + i)

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 → (i * j < i ↔ 1 < j);% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i < (* j) ∧ j < 0 → i < 0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0);% ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0);% ⊦ 0 < i → (i * j ≤ i ↔ j ≤ 1);% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ i + (* j) < i0 + (* j0) ↔ i < i0 + (j0 - j);% ⊦ i < 0 ∧ j0 < j ∧ 0 ≤ j → i * j + j0 < 0;% ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0;% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0);% ⊦ 0 < i → (i * j < 0 ↔ j < 0);% ⊦ 0 < i → (i * j < i ↔ j < 1);% ⊦ 1 * i = i;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i + j < (* i1) * i ↔ j < (i1 - 1) * i;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ i + j < j ↔ i < 0;% ⊦ i +1 = i + 1;% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < ~ j ↔ i + j < 0;% ⊦ i * ~1 = ~ i;% ⊦ (* i2) * i + j < (* i1) * i ↔ j < (i1 - i2) * i;% ⊦ (* i2) * i < (* i1) * i + j0 ↔ (i2 - i1) * i < j0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ (* i2) * i ≤ (* i1) * i + j0 ↔ (i2 - i1) * i ≤ j0;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ i → abs(i) = i;% ⊦ 0 ≤ i ∧ 0 ≤ j → abs(i + j) = abs(i) + abs(j);% ⊦ n→i(i→n(i)) = abs(i);