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Proof step 41 from proof no-leading-zero (previous step)

int2bytes(band(b2i(by0), 255)) = i2b(0) ' + by ' + bytes1 + by0 ', band(b2i(by0), 255) < 256, 0 ≤ b2i(by), 0 ≤ band(b2i(by0), 255), i2b(0) ∈ int2bytes(band(b2i(by0), 255)), by0 ∈ int2bytes(band(b2i(by0), 255)), ∀ by, bytes0. # bytes0 < #(bytes1 + by0 ') → int2bytes(bytes2int(i2b(0) ' + by ' + bytes0)) = i2b(0) ' + by ' + bytes0 ∧ 0 ≤ b2i(by) → false ⊦ byte2bits(i2b(0)) + byte2bits(by) + bytes2bits(bytes1) = [], true ∈ bytes2bits(bytes1), true ∈ byte2bits(by), true ∈ byte2bits(i2b(0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ byte2bits(by) = fillsgn(8, int2bits(b2i(by)))
Proof step 43:
int2bytes(band(b2i(by0), 255)) = i2b(0) ' + by ' + bytes1 + by0 ', band(b2i(by0), 255) < 256, 0 ≤ b2i(by), 0 ≤ band(b2i(by0), 255), i2b(0) ∈ int2bytes(band(b2i(by0), 255)), by0 ∈ int2bytes(band(b2i(by0), 255)), ∀ by, bytes0. # bytes0 < #(bytes1 + by0 ') → int2bytes(bytes2int(i2b(0) ' + by ' + bytes0)) = i2b(0) ' + by ' + bytes0 ∧ 0 ≤ b2i(by) → false ⊦ byte2bits(i2b(0)) + fillsgn(8, int2bits(b2i(by))) + bytes2bits(bytes1) = [], true ∈ bytes2bits(bytes1), true ∈ fillsgn(8, int2bits(b2i(by))), true ∈ byte2bits(i2b(0))

Rule argument

name: byte2bits rotate: F subst: [by] → [by]

Additional information

Used terms: by