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Proof step 166 from proof uint2bits-restsgn (previous step)

i0 ≤ 2 ^ i, 2 ^ i = i0 + 1, false ∈ uint2bits(i0 + 1).rest, i0 < 2 * (i0 + 1), 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * (i0 + 1), 0 < i, 2 ^ (i - 1) < 2 * (i0 + 1), 0 < i0, 2 ^ (i - 1) ≤ i0, ~1 ≤ i0, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(uint2bits(i0 + 1).rest) = uint2bits(i0 + 1).rest, i = 1, i0 = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 167:
2 ^ i = i0 + 1, 2 ^ (i - 1) ≤ i0 + 1, ~2 ≤ i0, false ∈ uint2bits(i0 + 1).rest, i0 < 2 * (i0 + 1), 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * (i0 + 1), 0 < i, 2 ^ (i - 1) < 2 * (i0 + 1), 0 < i0, 2 ^ (i - 1) ≤ i0, ~1 ≤ i0, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(uint2bits(i0 + 1).rest) = uint2bits(i0 + 1).rest, i = 1, i0 = 1

Rule argument

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ 0 < i ∧ j < 0 → ((* j) < (* i) * i0 ↔ (j + 1 - i) / i < i0);% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i ≤ i + j ↔ 0 ≤ j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)