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Proof step 45 from proof bytes2int-append (previous step)

0 ≤ 2 ∧ 0 ≤ 8 ∧ 0 ≤ n→i(# bytes0), bytes2bits(bytes).first, bytes2bits(bytes0).first ⊦ bytes = [], bytes0 = [], bytes2bits(bytes) + bytes2bits(bytes0) = [], bitsinvert(bytes2bits(bytes)) + bitsinvert(bytes2bits(bytes0)) = [], band(bits2int(bytes2bits(bytes0)), 2 ^ (8 * n→i(# bytes0)) - 1) = 2 ^ (8 * n→i(# bytes0)) + bits2int(bytes2bits(bytes0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ j < 0 ∧ 0 ≤ i → band(j, 2 ^ i - 1) = (j + 1) % 2 ^ i + 2 ^ i - 1
Proof step 47:
bits2int(bytes2bits(bytes0)) < 0 ∧ 0 ≤ 8 * n→i(# bytes0), 0 ≤ 2 ∧ 0 ≤ 8 ∧ 0 ≤ n→i(# bytes0), bytes2bits(bytes).first, bytes2bits(bytes0).first ⊦ bytes = [], bytes0 = [], bytes2bits(bytes) + bytes2bits(bytes0) = [], bitsinvert(bytes2bits(bytes)) + bitsinvert(bytes2bits(bytes0)) = [], (bits2int(bytes2bits(bytes0)) + 1) % 2 ^ (8 * n→i(# bytes0)) + 2 ^ (8 * n→i(# bytes0)) - 1 = 2 ^ (8 * n→i(# bytes0)) + bits2int(bytes2bits(bytes0))

Rule argument

seq: ⊦ j < 0 ∧ 0 ≤ i → band(j, 2 ^ i - 1) = (j + 1) % 2 ^ i + 2 ^ i - 1 subst: [i, j] → [8 * n→i(# bytes0), bits2int(bytes2bits(bytes0))] rotate: F spec: intops inst: name: band-011x-neg

Additional information

Used simplifier rules: bytes2bits-empty-01: ⊦ bytes2bits(bytes) = [] ↔ bytes = [];% ⊦ 0 ≤ n→i(n);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;%bits ≠ [] ⊦ 0 ≤ bits2int(bits) → ¬ bits.first; Used terms: 8 * n→i(# bytes0), bits2int(bytes2bits(bytes0))