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Proof step 7 from proof int2bytes-len-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < i ∧ 0 ≤ j → (2 ^ j ≤ i ↔ ¬ # int2bits(i) ≤ i→n(j) + 1)
Proof step 9:
0 < j ∧ 0 ≤ 8 * i, # int2bits(j) % 8 = 0, # int2bits(j) % 8 = 0, 8 * i + 1 < 8 * (i + 1), ¬ # int2bits(j) ≤ i→n(8 * i) + 1, j < 2 ^ (8 * (i + 1) - 1), 0 ≤ i ⊦ # int2bits(j) / 8 = i→n(i) + 1

Rule argument

seq: ⊦ 0 < i ∧ 0 ≤ j → (2 ^ j ≤ i ↔ ¬ # int2bits(i) ≤ i→n(j) + 1) subst: [i, j] → [j, 8 * i] rotate: F spec: int2bits inst: name: int2bits-lelen-01

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ 2 ^ i ≤ 0 ↔ i < 0;% ⊦ 0 < i → (i * j < 0 ↔ j < 0); Used terms: j, 8 * i