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Proof step 82 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 ⊦ ¬ i0 ≤ 2 ^ (i - 1), 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 83:
1 < 2 ^ (i - 1), 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, i0 ≤ 2 ^ (i - 1), 0 < 2 ^ (i - 1), 0 ≤ 2 ^ (i - 1), ∀ 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 ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)