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Proof step 105 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, i→n(i) + 1 = # bits2bytes(false ' + firstn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + # bits2bytes(restn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)), 2 ^ (i * 8) ≤ j, 0 < i, 0 ≤ j % 2 ^ (i * 8) ⊦ (i→n(i * 8) + 1) % 8 = 0, int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = (bits2bytes(false ' + firstn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + bits2bytes(restn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)) ).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 106:
0 ≤ i, 0 ≤ j ⊦ i→n(i) * i→n(j) = i→n(i * j)
Proof step 107:
0 ≤ i ∧ 0 ≤ 8, # int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i) * i→n(8) + 1, i→n(i) + 1 = # bits2bytes(false ' + firstn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + # bits2bytes(restn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)), 2 ^ (i * 8) ≤ j, 0 < i, 0 ≤ j % 2 ^ (i * 8) ⊦ (i→n(i) * i→n(8) + 1) % 8 = 0, int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = ( bits2bytes(false ' + firstn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + bits2bytes(restn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)) ).rest

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