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Proof step 9 from proof uint2bits-add-low (previous step)

j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), j < 0, 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), 2 * 2 ^ (i - 1) + j ≤ 1, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 * 2 ^ (i - 1) + j) = false ' + fillsgn(i→n(i), int2bits(j))

User applied rule cut formula

And obtained the following new goals

Proof step 10:
0 ≤ 2 * 2 ^ (i - 1) + j, j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), j < 0, 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), 2 * 2 ^ (i - 1) + j ≤ 1, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 * 2 ^ (i - 1) + j) = false ' + fillsgn(i→n(i), int2bits(j))

Rule argument

fma: 0 ≤ 2 * 2 ^ (i - 1) + j

Additional information

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