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Proof step 231 from proof int2bytes-rec-pos-any (previous step)

i→n(i * 8) % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))), (i→n(i) - 1) * 8 + n < i→n(i * 8), 0 < i, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8 ⊦ n = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i * 8) + 1 = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = i→n(i * 8), i = 0, bits2bytes(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) = mklist(i2b(0), i→n(i) - (i→n(i) - 1) + 1) + bits2bytes(int2bits(j % 2 ^ (i * 8)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 232:
0 ≤ i, 0 ≤ j ⊦ i→n(i) * i→n(j) = i→n(i * j)
Proof step 233:
0 ≤ i ∧ 0 ≤ 8, (i→n(i) * i→n(8)) % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))), (i→n(i) - 1) * 8 + n < i→n(i) * i→n(8), 0 < i, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8 ⊦ n = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i) * i→n(8) + 1 = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = i→n(i) * i→n(8), i = 0, bits2bytes(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) = mklist(i2b(0), i→n(i) - (i→n(i) - 1) + 1) + bits2bytes(int2bits(j % 2 ^ (i * 8)))

Rule argument

seq: 0 ≤ i, 0 ≤ j ⊦ i→n(i) * i→n(j) = i→n(i * j) subst: [j, i] → [8, i] rotate: T spec: intnat inst: name: i2np

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: 8, i