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Proof step 148 from proof uint2bits-restsgn (previous step)

i0 = 1, ~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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 149:
1 < 2 ^ i, 0 ≤ 2 ^ i, false ∈ uint2bits(2 * (2 ^ i - 1)).rest, 0 < 2 ^ i, 0 < i, 2 ^ i ≤ 2, 1 < 2 ^ (i - 1), 0 < 2 ^ (i - 1), 0 ≤ 2 ^ (i - 1), ∀ j0, i. i→n(j0) < i→n(1 * 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 - 1)).rest) = uint2bits(2 * (2 ^ i - 1)).rest, i = 0, i = 1

Rule argument

Additional information

Used simplifier rules: uint2bits-pos-num: 1 < i ⊦ uint2bits((* i)) = uint2bits(i / 2) + (i % 2 = 1);%uint2bits-one: ⊦ uint2bits(1) = false + true;% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ bits + bits0 = bits + bits1 ↔ bits0 = bits1;% ⊦ sgn(true ' + false ') = true ' + false ';% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ (boolvar ' + bits).rest = bits;% ⊦ bits + boolvar = bits + boolvar ';% ⊦ boolvar + boolvar0 = boolvar ' + boolvar0 ';% ⊦ 0 < j → (j < 2 ↔ j = 1);% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ 2 ^ i ≤ 1 ↔ i < 1;% ⊦ 0 < i ∧ 0 ≤ j → ((* i) * i0 ≤ (* j) ↔ i0 ≤ j / i);% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ 2 ^ i = 1 ↔ i = 0;% ⊦ i * j = i ↔ i = 0 ∨ j = 1;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j - i = i * (j - 1);