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Proof step 4 from proof cast-mod-pos-high (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

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