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Proof step 7 from proof int2bits-add-high (previous step)

j * 2 ^ i + j0 < 0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0 ⊦ sgn(int2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ j < 0 ∧ j0 < 0 ∧ abs(j0) < 2 ^ i ∧ 0 ≤ i → # int2bits(~(j * 2 ^ i + j0)) = # int2bits(~ j) + i→n(i)
Proof step 9:
j < 0 ∧ j0 < 0 ∧ abs(j0) < 2 ^ i ∧ 0 ≤ i, j * 2 ^ i + j0 < 0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0 ⊦ sgn(int2bits(2 ^ n→i(# int2bits(~ j) + i→n(i)) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)

Rule argument

name: int2bits-len-add rotate: F subst: [j0, i, j] → [j0, i, j]

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i < 0 → abs(i) = ~ i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: j0, i, j