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Proof step 22 from proof getInt-theorem (previous step)

0 ≤ 24 ∧ 24 < 32, band(b2i(by1), 255) i<< 8 < 65536, 0 ≤ band(b2i(by1), 255) i<< 8 ⊦ bor(b2i(by3) * 2 ^ 24, (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255)) = bytes2int(by3 ' + by2 ' + by1 ') * 256, bor(65536 * band(b2i(by2), 255) + (band(b2i(by1), 255) i<< 8), bor(band(b2i(by0), 255), b2i(by3) * 2 ^ 24)) = bor(bytes2int(by3 ' + by2 ' + by1 ') * 256, band(b2i(by0), 255)), bor(b2i(by3) * 2 ^ 24, (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255)) = bytes2int(by3 ' + by2 ' + by1 ') * 256

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ j0 < 2 ^ i ∧ 0 ≤ i ∧ 0 ≤ j0 → bor(j * 2 ^ i, j0) = j * 2 ^ i + j0
Proof step 24:
¬ ( (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255) < 2 ^ 24 ∧ 0 ≤ 24 ∧ 0 ≤ (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255)), 0 ≤ 24 ∧ 24 < 32, band(b2i(by1), 255) i<< 8 < 65536, 0 ≤ band(b2i(by1), 255) i<< 8 ⊦ bor(b2i(by3) * 2 ^ 24, (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255)) = bytes2int(by3 ' + by2 ' + by1 ') * 256, bor(65536 * band(b2i(by2), 255) + (band(b2i(by1), 255) i<< 8), bor(band(b2i(by0), 255), b2i(by3) * 2 ^ 24)) = bor(bytes2int(by3 ' + by2 ' + by1 ') * 256, band(b2i(by0), 255)), bor(b2i(by3) * 2 ^ 24, (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255)) = bytes2int(by3 ' + by2 ' + by1 ') * 256
Proof step 31:
(band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255) < 2 ^ 24 ∧ 0 ≤ 24 ∧ 0 ≤ (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255), 0 ≤ 24 ∧ 24 < 32, band(b2i(by1), 255) i<< 8 < 65536, 0 ≤ band(b2i(by1), 255) i<< 8 ⊦ b2i(by3) * 2 ^ 24 + (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255) = bytes2int(by3 ' + by2 ' + by1 ') * 256, bor(65536 * band(b2i(by2), 255) + (band(b2i(by1), 255) i<< 8), bor(band(b2i(by0), 255), b2i(by3) * 2 ^ 24)) = bor(bytes2int(by3 ' + by2 ' + by1 ') * 256, band(b2i(by0), 255)), b2i(by3) * 2 ^ 24 + (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255) = bytes2int(by3 ' + by2 ' + by1 ') * 256

Rule argument

seq: ⊦ j0 < 2 ^ i ∧ 0 ≤ i ∧ 0 ≤ j0 → bor(j * 2 ^ i, j0) = j * 2 ^ i + j0 subst: [j0, i, j] → [ (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255), 24, b2i(by3)] rotate: F spec: intops inst: name: bor-2pot-plus-02

Additional information

Used simplifier rules: ⊦ i * j = j * i; Used terms: (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255), 24, b2i(by3)