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Proof step 144 from proof uint2bits-add-high (previous step)

1 = i1 * 2 + j, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 ^ (i - 1) ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 0, 1 < 2 ^ i, j ≤ 1, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i, j < 0, 2 ^ (i - 1) + j < 0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ j = 0, i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), j + 2 ^ i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

User applied rule cut formula

And obtained the following new goals

Proof step 145:
j = 1 - i1 * 2, 1 = i1 * 2 + j, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 ^ (i - 1) ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 0, 1 < 2 ^ i, j ≤ 1, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i, j < 0, 2 ^ (i - 1) + j < 0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ j = 0, i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), j + 2 ^ i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Rule argument

fma: j = 1 - i1 * 2

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + j + i0 = (i + j) + i0; Used formulas: j = 1 - i1 * 2