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Proof step 44 from proof uint2bits-add-high (previous step)

(2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 0 ≤ j + 2 ^ i, 2 ^ (i - 1) + j < 0, 0 < i, j < 0, 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, 0 ≤ 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) ⊦ uint2bits((j + 2 ^ i) / 2) + ((j + 2 ^ i) % 2 = 1)' = sgn(int2bits(j).rest), j + 2 ^ i = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ j % 2 = 0 ∧ i ≤ 0 ∧ abs(i) ≤ j → (i + j) / 2 = (i - 1) / 2 + j / 2
Proof step 46:
2 ^ i % 2 = 0 ∧ j ≤ 0 ∧ abs(j) ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 0 ≤ j + 2 ^ i, 2 ^ (i - 1) + j < 0, 0 < i, j < 0, 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, 0 ≤ 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) ⊦ uint2bits((j - 1) / 2 + 2 ^ i / 2) + ((j + 2 ^ i) % 2 = 1)' = sgn(int2bits(j).rest), j + 2 ^ i = 1

Rule argument

seq: ⊦ j % 2 = 0 ∧ i ≤ 0 ∧ abs(i) ≤ j → (i + j) / 2 = (i - 1) / 2 + j / 2 subst: [j, i] → [2 ^ i, j] rotate: F spec: int-div inst: name: div2-add-neg

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < i → 2 ^ i % 2 = 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ i < 0 → abs(i) = ~ i;% ⊦ ~ i ≤ j ↔ 0 ≤ i + j; Used terms: 2 ^ i, j