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Proof step 25 from proof bits2bytes-rec-02 (previous step)

n0 % 8 = 0, # bits = 16 + n0, ∀ bits0. # bits0 < (n0 + 8) + 8 → true → # bits0 % 8 = 0 ∧ 8 < # bits0 → bits2bytes(bits0) = bits2bytes(firstn(8, bits0)) + bits2bytes(restn(8, bits0)) ⊦ bits2bytes(firstn(n0 + 8, bits)) + i2b(bits2int(lastn(8, bits))) ' = bits2bytes(firstn(8, bits)) + bits2bytes(firstn(n0, restn(8, bits))) + i2b(bits2int(lastn(8, restn(8, bits)))) ', bits = [], n0 = 0

User applied rule apply induction

And obtained the following new goals

Proof step 26:
# firstn(n0 + 8, bits) % 8 = 0 ∧ 8 < # firstn(n0 + 8, bits) → bits2bytes(firstn(n0 + 8, bits)) = bits2bytes(firstn(8, firstn(n0 + 8, bits))) + bits2bytes(restn(8, firstn(n0 + 8, bits))), n0 % 8 = 0, # bits = 16 + n0, ∀ bits0. # bits0 < (n0 + 8) + 8 → true → # bits0 % 8 = 0 ∧ 8 < # bits0 → bits2bytes(bits0) = bits2bytes(firstn(8, bits0)) + bits2bytes(restn(8, bits0)) ⊦ bits2bytes(firstn(n0 + 8, bits)) + i2b(bits2int(lastn(8, bits))) ' = bits2bytes(firstn(8, bits)) + bits2bytes(firstn(n0, restn(8, bits))) + i2b(bits2int(lastn(8, restn(8, bits)))) ', bits = [], n0 = 0

Rule argument

subst: [bits0] → [firstn(n0 + 8, bits)]

Additional information

Used simplifier rules: ⊦ # firstn(n, bits) = n;% ⊦ m + n < m + n0 ↔ n < n0;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m; Used terms: firstn(n0 + 8, bits)