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Proof step 51 from proof int2bytes-len-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 52:
⊦ 0 ≤ i ∧ 0 ≤ j → i→n(i * j) = i→n(i) * i→n(j)
Proof step 53:
0 ≤ 8 ∧ 0 ≤ i + 1, m * 8 + n ≤ 1 + (i→n(8) * i→n(i + 1) - 1), 0 ≤ i, 2 ^ (8 * i) ≤ 2 ^ (8 * (i + 1) - 1), j < ~1, n < 8, 0 ≤ 2 ^ (n→i(m) * 8 + n→i(n) - 1) + j, j + 2 ^ (n→i(m) * 8 + n→i(n) - 2) < 0, 2 ^ (n→i(m) * 8 + n→i(n) - 2) < 2 ^ (n→i(m) * 8 + n→i(n) - 1), 1 + i→n(i) * 8 < m * 8 + n, j < 0, i→n(i) * 8 + 1 < i→n(8) * i→n(i + 1) ⊦ 8 * (i + 1) = 1, n = 0, m = i→n(i), j + 2 ^ (8 * i) = 0, ~1 = i, m = 0 ∧ n = 1

Rule argument

seq: ⊦ 0 ≤ i ∧ 0 ≤ j → i→n(i * j) = i→n(i) * i→n(j) subst: [j, i] → [i + 1, 8] rotate: F spec: intnat inst: name: i2n-mult

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;%i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j;% ⊦ i + j + i0 = (i + j) + i0; Used terms: i + 1, 8