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Proof step 72 from proof uint2bits-add-high (previous step)

(j - 1) / 2 = ~(2 ^ i0), j % 2 = 0, uint2bits(~(2 ^ i0) + 2 ^ (i - 1)) = sgn(int2bits(~(2 ^ i0)).rest), j < i0, 2 ^ (i - 1) + j < i0, 0 ≤ i0, 2 ^ (i - 2) < 2 ^ (i - 1), abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 ≤ 2 ^ (i - 1), j < 0, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 2 ^ (i - 1) + j < 0, 0 < i, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 2 ^ (i - 2) < 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ j = ~1, j = 0, i = 1, uint2bits(~(2 ^ i0) + 2 ^ (i - 1)) + false ' = sgn(int2bits(~(2 ^ i0)).rest + false '), j + 2 ^ i = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 73:
2 ^ i0 + (j - 1) / 2 = 0, j % 2 = 0, uint2bits(2 ^ (i - 1) - 2 ^ i0) = sgn(mklist(false, i→n(i0))), 2 ^ (i - 2) < 2 ^ (i - 1), abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 ≤ 2 ^ (i - 1), j < 0, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 2 ^ (i - 1) + j < 0, 0 < i, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 2 ^ (i - 2) < 2 ^ i, j < i0, 2 ^ (i - 1) + j < i0, 0 < i0, j + 1 < i0, (2 ^ (i - 1) + j) + 1 < i0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ i0 = 0, j + 2 ^ i = 1, j = ~1, j = 0, i = 1

Rule argument

Additional information

Used simplifier rules: int2bits-2pot-01: 0 ≤ i ⊦ int2bits(~(2 ^ i)) = true ' + mklist(false, i→n(i));% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ sgn(bits) = [] ↔ bits = [];% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ (boolvar ' + bits).rest = bits;% ⊦ mklist(boolvar, n) + boolvar ' = boolvar ' + mklist(boolvar, n);% ⊦ sgn(false ' + mklist(false, n)) = false ';% ⊦ bits = bits0 + bits ↔ bits0 = [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + ~ j = i - j;% ⊦ ~ i = j ↔ i + j = 0;