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Proof step 120 from proof uint2bits-restsgn (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 121:
2 ^ i0 = i2 + 1, i + 1 = i0 + n→i(# bits), ~1 ≤ i2, 0 ≤ i, (2 ^ n→i(# bits) * i2) * 2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ (2 ^ n→i(# bits) * i2) * 2, 2 ^ n→i(# bits) * i2 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, 2 ^ n→i(# bits) * i2 < 2 * 2 ^ i, 0 < i2, ∀ j0, i1. i→n(j0) < i→n((2 ^ n→i(# bits) * i2) * 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) * i2 + 1), false ∈ bits

Rule argument

Additional information

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