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Proof step 13 from proof cast-mult-add (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
0 ≤ i0 + j, i0 < 0, 0 < i, i0 * 2 ^ i + j1 < 0, j1 < 0, 0 < 2 ^ i + j1, i0 < 2 ^ i + j1, 0 ≤ i ⊦ cast(i, 2 ^ i * (i0 + j) + j1) = cast(i, j1)

Rule argument

Additional information

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