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Proof step 3 from proof uint2bits-add-high (previous step)

2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 0 < 2 ^ i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, j < 0, 0 < i, 2 ^ (i - 1) + j < 0, 0 ≤ j + 2 ^ i, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 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) ⊦ uint2bits(j + 2 ^ i) = sgn(int2bits(j).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < i ∧ i ≠ 1 → uint2bits(i) = uint2bits(i / 2) + (i % 2 = 1)
Proof step 5:
¬(0 < j + 2 ^ i ∧ j + 2 ^ i ≠ 1), 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 0 < 2 ^ i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, j < 0, 0 < i, 2 ^ (i - 1) + j < 0, 0 ≤ j + 2 ^ i, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 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) ⊦ uint2bits(j + 2 ^ i) = sgn(int2bits(j).rest)
Proof step 43:
0 < j + 2 ^ i ∧ j + 2 ^ i ≠ 1, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 0 < 2 ^ i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, j < 0, 0 < i, 2 ^ (i - 1) + j < 0, 0 ≤ j + 2 ^ i, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 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) ⊦ uint2bits((j + 2 ^ i) / 2) + ((j + 2 ^ i) % 2 = 1) = sgn(int2bits(j).rest)

Rule argument

name: uint2bits-rec-01 rotate: F subst: [i] → [j + 2 ^ i]

Additional information

Used terms: j + 2 ^ i