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Proof step 44 from proof int2bytes-len-neg-01 (previous step)

0 ≤ i, 0 < j + 2 ^ (8 * (i + 1)), j + 2 ^ (8 * (i + 1) - 1) < 0 ⊦ # int2bits(j) % 8 = 0, # int2bits(j) / 8 + 1 = i→n(i + 2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ i < 0 ∧ 0 ≤ j ∧ ~ i ≠ 2 ^ j → (0 < 2 ^ j + i ↔ # int2bits(i) ≤ i→n(j) + 1)
Proof step 46:
j < 0 ∧ 0 ≤ 8 * (i + 1) ∧ ~ j ≠ 2 ^ (8 * (i + 1)), 0 ≤ i, # int2bits(j) ≤ i→n(8 * (i + 1)) + 1, j + 2 ^ (8 * (i + 1) - 1) < 0 ⊦ # int2bits(j) % 8 = 0, # int2bits(j) / 8 + 1 = i→n(i + 2)

Rule argument

seq: ⊦ i < 0 ∧ 0 ≤ j ∧ ~ i ≠ 2 ^ j → (0 < 2 ^ j + i ↔ # int2bits(i) ≤ i→n(j) + 1) subst: [i, j] → [j, 8 * (i + 1)] rotate: F spec: int2bits inst: name: int2bits-lslen-04

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;%i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ~ i = j ↔ i + j = 0;% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ i + j < j ↔ i < 0;% ⊦ ¬ 2 ^ i < 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ i + j = j + i;% ⊦ i * j = j * i; Used terms: j, 8 * (i + 1)