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Proof step 215 from proof int2bytes-rec-neg-05 (previous step)

# int2bits(j % 2 ^ (i * 8)) = 8 * i→n(i), 0 ≤ bits2int(bytes2bits(bits2bytes(int2bits(j % 2 ^ (i * 8))))), 0 < i, j + 2 ^ (i * 8 - 1) < 0, j + 2 ^ (i * 8) ≤ 0, j + 2 ^ (i * 8) < i, j % 2 ^ (i * 8) < 0, 0 ≤ i ⊦ j % 2 ^ (i * 8) = 0, bits2bytes(int2bits(j % 2 ^ (i * 8))) = fix(i→n(i) + 1, bits2bytes(int2bits(j % 2 ^ (i * 8)))).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 216:
⊦ # bits % 8 = 0 ∧ bits ≠ [] → bytes2bits(bits2bytes(bits)) = bits

Rule argument

name: bits2bits-inj rotate: F subst: [bits] → [int2bits(j % 2 ^ (i * 8))]

Additional information

Used simplifier rules: ⊦ i < j → ¬ j < i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ bits2int(int2bits(i)) = i;% ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ int2bits(j) ≠ [];% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)%n ≠ 0 ⊦ (m * n) % n = 0;% ⊦ m0 * m * n = (m0 * m) * n;% ⊦ m * n = n * m; Used terms: int2bits(j % 2 ^ (i * 8))