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Proof step 332 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) = m * 8 + n, i→n(i * 8) % 8 = 0, 0 < i, 1 < i, 8 < m * 8 + n, n < 8 + m * 8, n < m * 8 + 7, n < 8, m < i→n(i), 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, m * 8 + n < i→n(i * 8), 0 ≤ bytes2int(i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) ' + bits2bytes(restn(n, int2bits(j % 2 ^ (i * 8))))) ⊦ n = 0, m = 0, i = 0, m * 8 + n = i→n(i * 8), i→n(i * 8) + 1 = m * 8 + n, mklist(false, 8 * (i→n(i) - m + 1)) + mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i) - m + 1 = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], m * 8 = 1, i2b(bits2int(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))))) = i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))), bits2int(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) = bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ m ≤ # bits + # bits0 ∧ # bits ≤ m → firstn(m, bits + bits0) = bits + firstn(m - # bits, bits0) subst: [bits0, bits, m] → [int2bits(j % 2 ^ (i * 8)), mklist(false, 8 - n), 8] rotate: F spec: list-del inst: bitlista name: firstn-app

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ # mklist(boolvar, n) = n;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ m < n → ¬ n < m;%n0 ≤ n ⊦ n - n0 ≤ m ↔ n ≤ m + n0;% ⊦ ¬ m + n < m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ n < n0 ∧ n0 < n1 → n +1 < n1; (forward)% ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward)% ⊦ n < m → (m - n) + n = m;% ⊦ m +1 = m + 1;% ⊦ m + n < m + n0 ↔ n < n0;% ⊦ m < n → n ≠ 0; Used terms: int2bits(j % 2 ^ (i * 8)), mklist(false, 8 - n), 8