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Proof step 22 from proof int2bytes-len-pos-01 (previous step)

0 < n→i(m) * 8 + n→i(n), 8 * (i + 1) < n→i(m) * 8 + n→i(n), 0 ≤ j, n→i(m) * 8 + n→i(n) < 2 + 8 * (i + 1), 2 ^ (8 * (i + 1) - 1) ≤ j, j < 2 ^ (8 * (i + 1)), 0 ≤ i, n < 8, 2 ^ (n→i(m) * 8 + n→i(n) - 2) ≤ j, j < 2 ^ (n→i(m) * 8 + n→i(n) - 1) ⊦ m + 1 = i→n(i + 2), n = 0, m * 8 + n = 0, m = 0 ∧ n = 1

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 23:
j * i < j * i0 + j0, 0 ≤ j0, j0 < j ⊦ i ≤ i0
Proof step 24:
i + 1 ≤ n→i(m), 0 < n→i(m) * 8 + n→i(n), 8 * (i + 1) < n→i(m) * 8 + n→i(n), 0 ≤ j, n→i(m) * 8 + n→i(n) < 2 + 8 * (i + 1), 2 ^ (8 * (i + 1) - 1) ≤ j, j < 2 ^ (8 * (i + 1)), 0 ≤ i, n < 8, 2 ^ (n→i(m) * 8 + n→i(n) - 2) ≤ j, j < 2 ^ (n→i(m) * 8 + n→i(n) - 1) ⊦ m + 1 = i→n(i + 2), n = 0, m * 8 + n = 0, m = 0 ∧ n = 1

Rule argument

seq: j * i < j * i0 + j0, 0 ≤ j0, j0 < j ⊦ i ≤ i0 subst: [j, i, i0, j0] → [8, i + 1, n→i(m), n→i(n)] rotate: F spec: int-mult inst: name: modlsp

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ 0 ≤ n→i(n);%0 ≤ i ⊦ n→i(n) < (* i) ↔ n < i→n(i);% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ m < n → (n ≤ m ↔ false); Used terms: 8, i + 1, n→i(m), n→i(n)