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Proof step 84 from proof uint2bits-restsgn (previous step)

i = i1 + 1, 1 < 2 ^ i1, 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 ^ i1, 0 < 2 ^ i1, 0 ≤ 2 ^ i1, ∀ 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 = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 85:
1 < 2 ^ (i1 + 1), 2 < 2 ^ (i1 + 1), 1 < 2 ^ i1, 0 < i0, 0 ≤ i0, i0 < 2 ^ (i1 + 1), 2 ^ (i1 + 1) < i0 * 2, i0 * 2 < 4 * 2 ^ (i1 + 1), (¬ uint2bits(2 * 2 ^ (i1 + 1) - i0).rest.first) ∈ uint2bits(2 * 2 ^ (i1 + 1) - i0).rest.rest, i0 ≤ 2 ^ i1, ~1 ≤ i1, 0 < i1, 0 < 2 ^ i1, 0 < 2 ^ (i1 + 1), 0 ≤ 2 ^ i1, 0 ≤ 2 ^ (i1 + 1), 1 < i0, ∀ 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 ⊦ sgn(uint2bits(2 * 2 ^ (i1 + 1) - i0).rest) = uint2bits(2 * 2 ^ (i1 + 1) - i0).rest, 4 * 2 ^ (i1 + 1) = 1 + i0 * 2, uint2bits(2 * 2 ^ (i1 + 1) - i0).rest = [], 0 = i1

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < i → (i < i * j ↔ 1 < j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i + j < (* i1) * i ↔ j < (i1 - 1) * i;% ⊦ i +1 = i + 1;% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0;