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Proof step 362 from proof int2bytes-rec-pos-any (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 363:
# int2bits(j % 2 ^ (i * 8)) = m * 8 + n ∧ n < 8 ∧ 8 ≠ 0, n = 0, i→n(i * 8) % 8 = 0, # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 0 < i, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), m ≤ i→n(i), 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) ⊦ mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i * 8) + 1 = # int2bits(j % 2 ^ (i * 8)), # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i = 0, bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8)))) = mklist(i2b(0), i→n(i) - m), (i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) % 8 = 0

Rule argument

seq: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n subst: [m, n, m0, n0] → [# int2bits(j % 2 ^ (i * 8)), 8, m, n] rotate: T spec: nat-div inst: name: div-elim

Additional information

Used simplifier rules: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n; (elim) Used terms: # int2bits(j % 2 ^ (i * 8)), 8, m, n