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Proof step 12 from proof uint2bits-add-low (previous step)

1 < j + 2 ^ i, 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), ∀ 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((j + 2 ^ i) / 2) + ((j + 2 ^ i) % 2 = 1) = false ' + fillsgn(i→n(i), int2bits(j))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ j % 2 = 0 ∧ i ≤ 0 ∧ abs(i) ≤ j → (i + j) / 2 = (i - 1) / 2 + j / 2
Proof step 14:
¬(2 ^ i % 2 = 0 ∧ j ≤ 0 ∧ abs(j) ≤ 2 ^ i), 1 < j + 2 ^ i, 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), ∀ 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((j + 2 ^ i) / 2) + ((j + 2 ^ i) % 2 = 1) = false ' + fillsgn(i→n(i), int2bits(j))
Proof step 15:
2 ^ i % 2 = 0 ∧ j ≤ 0 ∧ abs(j) ≤ 2 ^ i, 1 < j + 2 ^ i, 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), ∀ 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((j - 1) / 2 + 2 ^ i / 2) + ((j + 2 ^ i) % 2 = 1) = false ' + fillsgn(i→n(i), int2bits(j))

Rule argument

seq: ⊦ j % 2 = 0 ∧ i ≤ 0 ∧ abs(i) ≤ j → (i + j) / 2 = (i - 1) / 2 + j / 2 subst: [j, i] → [2 ^ i, j] rotate: F spec: int-div inst: name: div2-add-neg

Additional information

Used terms: 2 ^ i, j