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Proof step 70 from proof uint2bits-restsgn (previous step)

(2 - 1) * i ≤ 1, 0 ≤ i0 ∧ 0 ≤ n→i(# bits0), 2 ^ i0 = i + 1, 2 ^ n→i(# bits0) * (i + 1) = (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ (i + 1) * 2 ^ n→i(# bits0), 0 < i * 2 ^ n→i(# bits0), 1 < i0 + n→i(# bits0), 0 ≤ i * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * i, i * 2 ^ n→i(# bits0) * 2 + 1 < (i + 1) * 2 ^ n→i(# bits0), (i + 1) * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * i < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) + 1 < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ i0, 1 < (i + 1) * 2 ^ n→i(# bits0), 0 < i0 + n→i(# bits0), 0 ≤ i0 + n→i(# bits0), 0 < (i + 1) * 2 ^ n→i(# bits0), ∀ j0, i1. i→n(j0) < i→n(2 * (i * 2 ^ n→i(# bits0) * 2 + 1) + 0) → 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 * (i + 1) * 2 ^ n→i(# bits0) = 1 + 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 71:
2 ^ i0 = i + 1, i < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), i * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 < i * 2 ^ n→i(# bits0), 1 < i0 + n→i(# bits0), 0 ≤ i * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * i, i * 2 ^ n→i(# bits0) * 2 + 1 < (i + 1) * 2 ^ n→i(# bits0), (i + 1) * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * i < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) + 1 < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ i0, 1 < (i + 1) * 2 ^ n→i(# bits0), 0 < i0 + n→i(# bits0), 0 ≤ i0 + n→i(# bits0), i ≤ 1, 0 < (i + 1) * 2 ^ n→i(# bits0), 0 ≤ (i + 1) * 2 ^ n→i(# bits0), i < i0 + n→i(# bits0), i < (i + 1) * 2 ^ n→i(# bits0), i + 1 < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), ∀ j0, i1. i→n(j0) < i→n(2 * (i * 2 ^ n→i(# bits0) * 2 + 1) + 0) → 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 * (i + 1) * 2 ^ n→i(# bits0) = 1 + 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

Rule argument

Additional information

Used simplifier rules: ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ 2 ^ n→i(m) ≠ 0;% ⊦ i * j < i * j0 → i ≠ 0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ n→i(n);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 1 * i = i;% ⊦ i * j = j * i;