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Proof step 12 from proof uint2bits-add-high (previous step)

1 - 2 ^ i < 0, (1 - 2 ^ i) + 2 ^ i = 1, (2 ^ (i - 1) + (1 - 2 ^ i)) + 1 < 2 ^ i, 1 - 2 ^ i < 2 ^ i, (1 - 2 ^ i) + 1 < 2 ^ i, 1 - 2 ^ i < i, 2 ^ (i - 1) + (1 - 2 ^ i) < 0, 0 < i, 1 - 2 ^ i < 0, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + (1 - 2 ^ i)) + 1 < i, 2 ^ (i - 1) + (1 - 2 ^ i) < i, 0 < 2 ^ i, (1 - 2 ^ i) + 1 < i, 2 ^ (i - 1) + (1 - 2 ^ i) < 2 ^ i, 2 ^ (i - 1) + (1 - 2 ^ i) ≤ 0, 1 - 2 ^ i ≤ 0, ∀ 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) ⊦ false ' + true ' = sgn(sgn(int2bits(2 ^ n→i(# int2bits(~(1 - 2 ^ i))) + (1 - 2 ^ i)).rest).rest)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
0 ≤ 2 ^ i + i, 0 ≤ 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 2 < 2 ^ i + i, 2 ^ (i - 1) + 2 < 2 ^ i + i, 2 ^ (i - 1) < 2 ^ i, 1 < 2 ^ i, 0 < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 < 2 ^ i + 2 ^ i, 2 ^ (i - 1) + 2 < 2 ^ i + 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) ⊦ false ' + true ' = sgn(sgn(int2bits(2 ^ (1 + i) + 1 - 2 ^ i).rest).rest)

Rule argument

Additional information

Used simplifier rules: int2bits-2pot-02: 0 ≤ i ⊦ int2bits(2 ^ i - 1) = false ' + mklist(true, i→n(i));% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)%i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i + 1 ≤ j ↔ i < j;% ⊦ i - j ≤ i0 ↔ i ≤ j + i0;% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ i + 0 = i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + (j - i0) = i + j - i0;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ 0 < i → n→i(i→n(i)) = i;% ⊦ n→i(m + n) = n→i(m) + n→i(n);% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ # mklist(boolvar, n) = n;% ⊦ # boolvar ' = 0 +1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ ~(i - j) = j - i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + (j - i) = j;