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Proof step 4 from proof cast-mult-add (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

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