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Proof step 1 from proof cast-mod-pos-low (conclusion)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: ⊦ 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 + 1 ≤ j ↔ i < j;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;