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Proof step 133 from proof uint2bits-restsgn (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 134:
2 ^ i0 = i2 + 1, i + 1 = i0 + n→i(# bits), 0 ≤ 2 ^ (i - n→i(# bits)), 0 < 2 ^ (i - n→i(# bits)), 1 < 2 ^ (n→i(# bits) + 1) * i2, 0 < 2 ^ (n→i(# bits) + 1), ~1 ≤ i2, 0 ≤ i, 2 ^ (n→i(# bits) + 1) * i2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ 2 ^ (n→i(# bits) + 1) * i2, i2 < 2 ^ (i - n→i(# bits)), 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, i2 * 2 ^ n→i(# bits) < 2 ^ i * 2, 1 < 2 ^ (i - n→i(# bits)), 0 ≤ i2, 0 < i2, ∀ j0, i1. i→n(j0) < i→n(2 ^ (n→i(# bits) + 1) * i2 + 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 * (i2 * 2 ^ n→i(# bits) + 1), false ∈ bits

Rule argument

Additional information

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