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Proof step 4 from proof cast-law-mult-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
0 ≤ j1, 0 < j1, j1 < 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i ⊦ cast(i, (i0 * 2 ^ i + j1) * j0) = cast(i, cast(i, j1) * j0)

Rule argument

Additional information

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