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Proof step 18 from proof cast-law-mult-pos (previous step)

j1 = i1 * 2 ^ i + j ∧ 0 ≤ j ∧ j < 2 ^ i ∧ 0 ≤ i1 ∧ 0 < 2 ^ i, 0 ≤ j1, 0 < j1, j1 < 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i, 2 ^ (i - 1) ≤ j ⊦ cast(i, j1 * j0) = cast(i, (j - 2 ^ i) * j0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
0 ≤ j, j < 2 ^ i, 2 ^ (i - 1) ≤ j, 0 < i, 0 ≤ i0, 0 ≤ i, 0 < j ⊦ cast(i, j * j0) = cast(i, (j - 2 ^ i) * j0)

Rule argument

Additional information

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