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Proof step 120 from proof uint2bits-add-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 121:
⊦ j < 0 → abs(j) = 0 - j
Proof step 122:
j - 1 < 0, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 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) ⊦ j + 2 ^ i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), i = 1, j = 0, ¬ 2 ^ (i - 1) < (0 - (j - 1)) / 2 ∧ ¬ (0 - (j - 1)) / 2 ≤ 2 ^ (i - 2)

Rule argument

seq: ⊦ j < 0 → abs(j) = 0 - j subst: [j] → [j - 1] rotate: F spec: int-abs inst: name: Abs-neg

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: j - 1