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Proof step 27 from proof bytes2bytes-zero (previous step)

b2i(by) = 0, b2i(by0) < 0, ∀ by0, by, bytes0. # bytes0 < # [] → b2i(by) = 0 ∧ b2i(by0) < 0 → bits2bytes(sgn(byte2bits(by) + byte2bits(by0) + bytes2bits(bytes0))) = by ' + by0 ' + bytes0 ⊦ bits2bytes(false ' + fillsgn(8, int2bits(b2i(by0)))) = by ' + by0 ', fillsgn(8, int2bits(b2i(by0))) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ 8 < # bits → bits2bytes(bits) = bits2bytes(firstn(# bits - 8, bits)) + i2b(bits2int(lastn(8, bits)))
Proof step 29:
¬ 8 < #(false ' + fillsgn(8, int2bits(b2i(by0)))), b2i(by) = 0, b2i(by0) < 0, ∀ by0, by, bytes0. # bytes0 < # [] → b2i(by) = 0 ∧ b2i(by0) < 0 → bits2bytes(sgn(byte2bits(by) + byte2bits(by0) + bytes2bits(bytes0))) = by ' + by0 ' + bytes0 ⊦ bits2bytes(false ' + fillsgn(8, int2bits(b2i(by0)))) = by ' + by0 ', fillsgn(8, int2bits(b2i(by0))) = []
Proof step 32:
8 < #(false ' + fillsgn(8, int2bits(b2i(by0)))), b2i(by) = 0, b2i(by0) < 0, ∀ by0, by, bytes0. # bytes0 < # [] → b2i(by) = 0 ∧ b2i(by0) < 0 → bits2bytes(sgn(byte2bits(by) + byte2bits(by0) + bytes2bits(bytes0))) = by ' + by0 ' + bytes0 ⊦ bits2bytes(firstn(#(false ' + fillsgn(8, int2bits(b2i(by0)))) - 8, false ' + fillsgn(8, int2bits(b2i(by0))))) + i2b(bits2int(lastn(8, false ' + fillsgn(8, int2bits(b2i(by0)))))) = by ' + by0 ', fillsgn(8, int2bits(b2i(by0))) = []

Rule argument

name: bits2bytes-rec rotate: F subst: [bits] → [false ' + fillsgn(8, int2bits(b2i(by0)))]

Additional information

Used terms: false ' + fillsgn(8, int2bits(b2i(by0)))