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Proof step 35 from proof int2bits-add-low (previous step)

0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0, 2 ^ n→i(# int2bits(~ j)) + j ≤ 0 ⊦ j * 2 ^ i + j0 = 0, j0 = 0, sgn(uint2bits(2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
⊦ i < 0 ∧ 0 ≤ j ∧ ~ i ≠ 2 ^ j → (2 ^ j + i ≤ 0 ↔ ¬ # int2bits(i) ≤ i→n(j) + 1)
Proof step 37:
j < 0 ∧ 0 ≤ n→i(# int2bits(~ j)) ∧ ~ j ≠ 2 ^ n→i(# int2bits(~ j)), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0, ¬ # int2bits(j) ≤ i→n(n→i(# int2bits(~ j))) + 1 ⊦ j * 2 ^ i + j0 = 0, j0 = 0, sgn(uint2bits(2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

name: int2bits-lelen-04 rotate: F subst: [i, j] → [j, n→i(# int2bits(~ 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);% ⊦ 0 ≤ n→i(n);% ⊦ ~ i = j ↔ i + j = 0;% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ i * 0 = 0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + 0 = i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i < j → ¬ j < i; Used terms: j, n→i(# int2bits(~ j))