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Proof step 79 from proof uint2bits-restsgn (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 80:
2 ^ (i - 1) < 4 * 2 ^ i, 2 + 2 ^ (i - 1) < i0 * 2, 2 ^ (i - 1) < i0 * 2, 2 * 2 ^ (i - 1) + 1 < 4 * 2 ^ i, 2 ^ (i - 1) < i0, 2 * 2 ^ (i - 1) + 1 < i0 * 2, 2 * 2 ^ (i - 1) < 2 ^ i, 0 ≤ i, 0 < i, 1 < i0, 1 < i, 0 < i0, 0 ≤ i0, 0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i < i0 * 2, 0 ≤ 2 ^ i, i0 * 2 < 4 * 2 ^ i, (¬ uint2bits(2 * 2 ^ i - i0).rest.first) ∈ uint2bits(2 * 2 ^ i - i0).rest.rest, 2 * 2 ^ (i - 1) ≤ i0, 2 ^ (i - 1) + 1 < 2 ^ i, 2 ^ (i - 1) < 2 ^ i, 2 * (2 ^ (i - 1) + 1) < 4 * 2 ^ i, 3 + 2 ^ (i - 1) < 4 * 2 ^ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ i = 1, sgn(uint2bits(2 * 2 ^ i - i0).rest) = uint2bits(2 * 2 ^ i - i0).rest, 4 * 2 ^ i = 1 + i0 * 2, uint2bits(2 * 2 ^ i - i0).rest = []

Rule argument

Additional information

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