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Proof step 4 from proof cast-mod-neg-low (previous step)

¬(0 < i ∧ ~(2 ^ (i - 1)) ≤ j ∧ j < 2 ^ (i - 1)), 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0 ⊦ cast(i, j) = j % 2 ^ i + 2 ^ i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ cast(i, j) = j % 2 ^ i + 2 ^ i

Rule argument

Additional information

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