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Proof step 77 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 ⊦ 0 < i - 1 ∧ 0 ≤ i0 ∧ 2 ^ (i - 1) < i0 ∧ i0 < 2 * 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 78:
(¬ 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 ⊦ 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, ¬ 2 * 2 ^ (i - 1) ≤ i0 ∧ ¬ i0 ≤ 2 ^ (i - 1)

Rule argument

Additional information

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