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Proof step 138 from proof uint2bits-restsgn (previous step)

i2 = 2 ^ i0 - 1, i - n→i(# bits) = i0 - 1, 2 ^ i0 = (2 ^ i0 - 1) + 1, i + 1 = i0 + n→i(# bits), 0 ≤ 2 ^ (i0 - 1), 0 < 2 ^ (i0 - 1), 1 < 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1), 0 < 2 ^ (n→i(# bits) + 1), ~1 ≤ 2 ^ i0 - 1, 0 ≤ i, 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1) + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1), 2 ^ i0 - 1 < 2 ^ (i0 - 1), 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, (2 ^ i0 - 1) * 2 ^ n→i(# bits) < 2 ^ i * 2, 1 < 2 ^ (i0 - 1), 0 ≤ 2 ^ i0 - 1, 0 < 2 ^ i0 - 1, ∀ j0, i1. i→n(j0) < i→n(2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1) + 1) → 0 < i1 ∧ 0 ≤ j0 ∧ 2 ^ i1 < j0 ∧ j0 < 2 * 2 ^ i1 → sgn(uint2bits(2 ^ (i1 + 2) - j0).rest) = uint2bits(2 ^ (i1 + 2) - j0).rest ⊦ sgn(bits + true ') = bits + true ', 4 * 2 ^ i = 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits) + 1), false ∈ bits

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 139:
n→i(# bits) + i0 = i + 1, (2 ^ i0 - 1) * 2 ^ n→i(# bits) < 2 ^ i * 2, n→i(# bits) ≤ i + 1, 0 < 2 ^ n→i(# bits), 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ i0 ≤ 2 ^ (i0 - 1), 2 ^ i ≤ 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1), 0 < i, 0 ≤ 2 ^ i, 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1) + 1 < 4 * 2 ^ i, 0 ≤ i, 0 < 2 ^ (n→i(# bits) + 1), 0 ≤ 2 ^ i0, 1 < 2 ^ (i0 - 1), 0 < 2 ^ (i0 - 1), 1 < 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1), 0 ≤ 2 ^ (i0 - 1), 0 < 2 ^ i0, 1 < 2 ^ i0, ∀ j0, i1. i→n(j0) < i→n(2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1) + 1) → 0 < i1 ∧ 0 ≤ j0 ∧ 2 ^ i1 < j0 ∧ j0 < 2 * 2 ^ i1 → sgn(uint2bits(2 ^ (i1 + 2) - j0).rest) = uint2bits(2 ^ (i1 + 2) - j0).rest ⊦ 4 * 2 ^ i = 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits) + 1), sgn(bits + true ') = bits + true ', false ∈ bits

Rule argument

Additional information

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