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Proof step 117 from proof uint2bits-restsgn (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 118:
2 ^ (i + 1 - n→i(# bits)) ≤ (2 ^ (i + 1 - n→i(# bits)) - 1) * 2, 0 ≤ 2 ^ (i + 1 - n→i(# bits)), 0 ≤ i, (2 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1)) * 2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ (2 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1)) * 2, 2 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1) < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, 2 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1) < 2 * 2 ^ i, 1 < 2 ^ (i + 1 - n→i(# bits)), 0 < 2 ^ (i + 1 - n→i(# bits)), ∀ j0, i1. i→n(j0) < i→n((2 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1)) * 2 + 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 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1) + 1), false ∈ bits

Rule argument

Additional information

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