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Proof step 64 from proof uint2bits-add-high (previous step)

uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < 2 ^ (i - 1), ∀ 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 + 2 ^ i = 1, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) + (~1 = j % 2)' = sgn(int2bits((j - 1) / 2).rest + (~1 = j % 2)'), i = 1, j = 0, j = ~1, (~1 ≠ j % 2) ∈ int2bits((j - 1) / 2).rest

User applied rule cut formula

And obtained the following new goals

Proof step 65:
~1 ≠ j % 2, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < 2 ^ (i - 1), ∀ 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 + 2 ^ i = 1, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) + (~1 = j % 2)' = sgn(int2bits((j - 1) / 2).rest + (~1 = j % 2)'), i = 1, j = 0, j = ~1, (~1 ≠ j % 2) ∈ int2bits((j - 1) / 2).rest
Proof step 114:
uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < 2 ^ (i - 1), ∀ 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) ⊦ ~1 ≠ j % 2, j + 2 ^ i = 1, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) + (~1 = j % 2)' = sgn(int2bits((j - 1) / 2).rest + (~1 = j % 2)'), i = 1, j = 0, j = ~1, (~1 ≠ j % 2) ∈ int2bits((j - 1) / 2).rest

Rule argument

fma: ~1 ≠ j % 2

Additional information

Used formulas: ~1 ≠ j % 2