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Proof step 21 from proof cast-add-2pot-h (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 22:
j % 2 ^ i + 2 ^ (i - 1) < 0, 0 ≤ i, 0 ≤ 2 ^ i, 0 ≤ 2 ^ i + j, 0 < 2 ^ i + j, 0 < i, j < 0, 0 ≤ j + 2 ^ (i - 1), j % 2 ^ i < j, j % 2 ^ i < 2 ^ (i - 1) ⊦ j % 2 ^ i = 0, cast(i, 2 ^ i + j) = j

Rule argument

Additional information

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