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Proof step 16 from proof uint2bits-add-high (previous step)

0 ≤ i, 0 ≤ 2 ^ i + i, 0 ≤ 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 2 < 2 ^ i + i, 2 ^ (i - 1) + 2 < 2 ^ i + i, 2 ^ (i - 1) < 2 ^ i, 1 < 2 ^ i, 0 < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 < 2 ^ i + 2 ^ i, 2 ^ (i - 1) + 2 < 2 ^ i + 2 ^ i, ∀ 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) ⊦ false ' + true ' = sgn(sgn(int2bits(2 ^ i + 1).rest).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ 0 ≤ j → int2bits(j + 1) = sgn(false ' + int2bits(j) +1)
Proof step 18:
0 ≤ 2 ^ i, 0 ≤ i, 0 ≤ 2 ^ i + i, 0 ≤ 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 2 < 2 ^ i + i, 2 ^ (i - 1) + 2 < 2 ^ i + i, 2 ^ (i - 1) < 2 ^ i, 1 < 2 ^ i, 0 < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 < 2 ^ i + 2 ^ i, 2 ^ (i - 1) + 2 < 2 ^ i + 2 ^ i, ∀ 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) ⊦ false ' + true ' = sgn(sgn(sgn(false ' + int2bits(2 ^ i) +1).rest).rest)

Rule argument

name: int2bits-succ rotate: F subst: [j] → [2 ^ i]

Additional information

Used terms: 2 ^ i