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Proof step 187 from proof int2bytes-rec-neg-05 (previous step)

1 + 8 * i→n(i) = # int2bits(j % 2 ^ (i * 8)), i→n(i) + 1 = # bits2bytes(int2bits(j % 2 ^ (i * 8))), 8 * i→n(i) ≤ 7, j + 2 ^ (i * 8 - 1) < 0, j + 2 ^ (i * 8) ≤ 0, j + 2 ^ (i * 8) < i, j % 2 ^ (i * 8) < 0, 0 < i ⊦ i = 0, int2bits(j % 2 ^ (i * 8)).rest = [], j % 2 ^ (i * 8) = 0, bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = bits2bytes(int2bits(j % 2 ^ (i * 8))).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 188:
⊦ # bits % 8 ≠ 0 → # bits2bytes(bits) = # bits / 8 + 1
Proof step 189:
# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0, 1 + 8 * i→n(i) = # int2bits(j % 2 ^ (i * 8)), i→n(i) + 1 = # int2bits(j % 2 ^ (i * 8)) / 8 + 1, 8 * i→n(i) ≤ 7, j + 2 ^ (i * 8 - 1) < 0, j + 2 ^ (i * 8) ≤ 0, j + 2 ^ (i * 8) < i, j % 2 ^ (i * 8) < 0, 0 < i ⊦ i = 0, int2bits(j % 2 ^ (i * 8)).rest = [], j % 2 ^ (i * 8) = 0, bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = bits2bytes(int2bits(j % 2 ^ (i * 8))).rest

Rule argument

name: bits2bytes-len-02 rotate: F subst: [bits] → [int2bits(j % 2 ^ (i * 8))]

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;%n ≠ 0 ⊦ (m + n0 * n) % n = m % n;% ⊦ m0 * m * n = (m0 * m) * n;% ⊦ m * n = n * m; Used terms: int2bits(j % 2 ^ (i * 8))