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Proof step 146 from proof uint2bits-restsgn (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 147:
~1 ≤ i0, false ∈ uint2bits(2 * 2 ^ i - i0 + 1).rest, i0 < 2 * 2 ^ i, 0 < i0, 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 < i, 2 ^ i ≤ i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(uint2bits(2 * 2 ^ i - i0 + 1).rest) = uint2bits(2 * 2 ^ i - i0 + 1).rest, 4 * 2 ^ i = 2 * (i0 + 1), ¬ 2 * 2 ^ (i - 1) ≤ i0 + 1 ∧ ¬ i0 < 2 ^ (i - 1) ∧ i ≠ 1

Rule argument

Additional information

Used simplifier rules: ⊦ i = 1 → i * j = j;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ ¬ i0 < j → ¬ i0 < i;% ⊦ 0 < j → (j < 2 ↔ j = 1);% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) ≤ (* i) * i0 ↔ (i + j - 1) / i ≤ i0);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i ∧ 0 ≤ j → ((* i) * i0 < (* j) ↔ i0 < (i + j - 1) / i);% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ i * j = j * i;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ 0 < i → (i ≤ 1 ↔ i = 1);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j - i0 ↔ i + i0 < j;