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Proof step 32 from proof getInt-theorem (previous step)

band(b2i(by1), 255) i<< 8 < 65536, 0 ≤ band(b2i(by1), 255) i<< 8, (band(b2i(by1), 255) i<< 8) + 65536 * band(b2i(by2), 255) < 16777216, 0 ≤ band(b2i(by2), 255) ⊦ bor(16777216 * b2i(by3), bor(band(b2i(by0), 255), 65536 * band(b2i(by2), 255) + (band(b2i(by1), 255) i<< 8))) = bor(bytes2int(by3 ' + by2 ' + by1 ') * 256, band(b2i(by0), 255)), 65536 * band(b2i(by2), 255) + (band(b2i(by1), 255) i<< 8) + 16777216 * b2i(by3) = bytes2int(by3 ' + by2 ' + by1 ') * 256, bor(16777216 * b2i(by3), 65536 * band(b2i(by2), 255) + (band(b2i(by1), 255) i<< 8)) = bytes2int(by3 ' + by2 ' + by1 ') * 256

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ 0 ≤ j ∧ j < 32 → i i<< j = i * 2 ^ j
Proof step 34:
0 ≤ 8 ∧ 8 < 32, band(b2i(by1), 255) * 2 ^ 8 < 65536, 0 ≤ band(b2i(by1), 255) * 2 ^ 8, band(b2i(by1), 255) * 2 ^ 8 + 65536 * band(b2i(by2), 255) < 16777216, 0 ≤ band(b2i(by2), 255) ⊦ bor(16777216 * b2i(by3), bor(band(b2i(by0), 255), 65536 * band(b2i(by2), 255) + band(b2i(by1), 255) * 2 ^ 8)) = bor(bytes2int(by3 ' + by2 ' + by1 ') * 256, band(b2i(by0), 255)), 65536 * band(b2i(by2), 255) + band(b2i(by1), 255) * 2 ^ 8 + 16777216 * b2i(by3) = bytes2int(by3 ' + by2 ' + by1 ') * 256, bor(16777216 * b2i(by3), 65536 * band(b2i(by2), 255) + band(b2i(by1), 255) * 2 ^ 8) = bytes2int(by3 ' + by2 ' + by1 ') * 256

Rule argument

seq: ⊦ 0 ≤ j ∧ j < 32 → i i<< j = i * 2 ^ j subst: [j, i] → [8, band(b2i(by1), 255)] rotate: F spec: intops inst: name: intshiftleft-is-mult

Additional information

Used simplifier rules: ⊦ i + j = j + i; Used terms: 8, band(b2i(by1), 255)