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Proof step 58 from proof uint2bits-restsgn (previous step)

0 ≤ i0 - 1 ∧ 0 ≤ i ∧ 0 ≤ n→i(# bits0) + 1, 0 ≤ n→i(# bits0), i0 - 1 = 2 ^ (n→i(# bits0) + 1) * (2 ^ (i + 1 - n→i(# bits0) + 1) - 1) ∧ n→i(# bits0) + 1 ≤ i + 1, i0 + 2 ^ (n→i(# bits0) + 1) = 2 * 2 ^ i + 1, i0 ≤ 2 * 2 ^ i, i0 < 2 ^ i * 2, 2 ^ i < 2 * i0, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, 0 ≤ 2 ^ i, ∀ j0, i. i→n(j0) < i→n(2 * i0 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ bits0 = [], 4 * 2 ^ i = 1 + 2 * i0, false ∈ bits0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 59:
2 ^ n→i(# bits0) * 2 ^ (i - n→i(# bits0)) = 2 ^ i, (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) < 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) + 1 < 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), n→i(# bits0) ≤ i, 0 ≤ 2 ^ i, 0 ≤ i, 0 < i, 0 < (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0), 1 < i, 0 ≤ (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * (2 ^ (i - n→i(# bits0)) - 1), 0 < 2 ^ i, 1 < 2 ^ i, (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1 < 2 ^ i, 2 ^ i < 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * (2 ^ (i - n→i(# bits0)) - 1) < 2 ^ i, (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) < 2 ^ i, ∀ j0, i1. i→n(j0) < i→n(2 * ((2 ^ (i - n→i(# bits0)) - 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 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ 1 * i = i;% ⊦ j < (* i1) * j ↔ 0 < (i1 - 1) * j;% ⊦ 0 < i → (i < i * j ↔ 1 < j);% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0);% ⊦ i + 1 ≤ j ↔ i < j;% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0);% ⊦ j0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 ≤ j0 → (i * j + j0 < i0 * j ↔ i < i0);% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ 0 < i → (i * j = i * j0 ↔ j = j0);% ⊦ i + j = i + i0 ↔ j = i0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i + (j - i) = j;% ⊦ i * j + i = i * (j + 1);% ⊦ i * j + i * j0 = i * (j + j0);% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i + j ≤ i + j0 ↔ j ≤ j0;% ⊦ 0 ≤ n→i(n);%i < 0 ⊦ (* i) ≤ n→i(n);% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ i * j = j * i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j - i0 + j = i - i0;