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Proof step 60 from proof uint2bits-restsgn (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 61:
2 ^ n→i(# bits0) * 2 ^ i0 = 2 ^ (i0 + n→i(# bits0)), 0 ≤ 2 ^ (i0 + n→i(# bits0)), (2 ^ i0 - 1) * 2 ^ n→i(# bits0) < 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), 0 < (2 ^ i0 - 1) * 2 ^ n→i(# bits0), 1 < i0 + n→i(# bits0), 0 ≤ (2 ^ i0 - 1) * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * (2 ^ i0 - 1), (2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1 < 2 ^ (i0 + n→i(# bits0)), 2 ^ (i0 + n→i(# bits0)) < 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * (2 ^ i0 - 1) < 2 ^ (i0 + n→i(# bits0)), (2 ^ i0 - 1) * 2 ^ n→i(# bits0) < 2 ^ (i0 + n→i(# bits0)), (2 ^ i0 - 1) * 2 ^ n→i(# bits0) + 1 < 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ i0, 1 < 2 ^ (i0 + n→i(# bits0)), 0 < i0 + n→i(# bits0), 0 ≤ i0 + n→i(# bits0), 0 < 2 ^ (i0 + n→i(# bits0)), ∀ j0, i1. i→n(j0) < i→n(2 * ((2 ^ i0 - 1) * 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 * 2 ^ (i0 + n→i(# bits0)) = 1 + 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

Rule argument

Additional information

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