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Proof step 64 from proof bytes2int-int2bytes-len-eq (previous step)

bytes1 = int2bytes(bytes2int(bytes1)), bytes2int(bytes1) * 256 + band(b2i(by), 255) < ~256, bytes2int(bytes1) < 0, 0 ≤ band(b2i(by), 255), band(b2i(by), 255) < 128, ∀ bytes0. # bytes0 < #(bytes1 + by ') → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ by = int2bytes(band(b2i(by), 255) - 256).last, bytes1 = [], bytes2int(bytes1) * 256 + band(b2i(by), 255) = ~256, band(b2i(by), 255) = 256, b2i(by) = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 65:
⊦ ~256 < i ∧ i < ~128 → int2bytes(i) = i2b(~1) ' + i2b(i) '
Proof step 66:
~256 < band(b2i(by), 255) - 256 ∧ band(b2i(by), 255) - 256 < ~128, bytes1 = int2bytes(bytes2int(bytes1)), bytes2int(bytes1) * 256 + band(b2i(by), 255) < ~256, bytes2int(bytes1) < 0, 0 ≤ band(b2i(by), 255), band(b2i(by), 255) < 128, ∀ bytes0. # bytes0 < #(bytes1 + by ') → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ by = (i2b(~1) ' + i2b(band(b2i(by), 255) - 256) ').last, bytes1 = [], bytes2int(bytes1) * 256 + band(b2i(by), 255) = ~256, band(b2i(by), 255) = 256, b2i(by) = 0

Rule argument

name: int2bytes-byte-neg rotate: F subst: [i] → [band(b2i(by), 255) - 256]

Additional information

Used simplifier rules: ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ band(255, b2i(by)) ≤ 0 ↔ by = i2b(0);% ⊦ band(i, band(i0, j)) = band(band(i, i0), j);% ⊦ band(i, j) = band(j, i);% ⊦ i - j < i0 ↔ i < j + i0;%j0 ≠ 0, j ≤ j0 ⊦ i < (* j) → i < (* j0);% ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ b2i(i2b(0)) = 0; Used terms: band(b2i(by), 255) - 256