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Proof step 12 from proof cast-law-add-pos-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
0 ≤ j1, j1 < 2 ^ i, j1 < 2 ^ (i - 1), j0 < 0, 0 ≤ i, 0 ≤ i0, 0 < i ⊦ cast(i, (i0 * 2 ^ i + j1) + j0) = cast(i, j1 + j0)

Rule argument

Additional information

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