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Proof step 25 from proof int2bytes-rec-pos (previous step)

uint2bits(i) = uint2bits(i / 256) + fillsgn(9, uint2bits(i % 256)).rest, 8 < (# fillsgn(9, uint2bits(i % 256)) - 1) + # uint2bits(i / 256), 255 < i, 0 < i, 0 ≤ i ⊦ bits2bytes(firstn((# fillsgn(9, uint2bits(i % 256)) - 1) + # uint2bits(i / 256) - 8, uint2bits(i))) + i2b(bits2int(lastn(8, uint2bits(i)))) ' = bits2bytes(uint2bits(i / 256)) + bits2bytes(int2bits(i % 256)).last ', fillsgn(9, uint2bits(i % 256)) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 26:
⊦ # bits ≤ n → # fillsgn(n, bits) = n
Proof step 27:
¬ # uint2bits(i % 256) ≤ 9, uint2bits(i) = uint2bits(i / 256) + fillsgn(9, uint2bits(i % 256)).rest, 8 < (# fillsgn(9, uint2bits(i % 256)) - 1) + # uint2bits(i / 256), 255 < i, 0 < i, 0 ≤ i ⊦ bits2bytes(firstn((# fillsgn(9, uint2bits(i % 256)) - 1) + # uint2bits(i / 256) - 8, uint2bits(i))) + i2b(bits2int(lastn(8, uint2bits(i)))) ' = bits2bytes(uint2bits(i / 256)) + bits2bytes(int2bits(i % 256)).last ', fillsgn(9, uint2bits(i % 256)) = []
Proof step 30:
# uint2bits(i % 256) ≤ 9, uint2bits(i) = uint2bits(i / 256) + fillsgn(9, uint2bits(i % 256)).rest, 8 < (9 - 1) + # uint2bits(i / 256), 255 < i, 0 < i, 0 ≤ i ⊦ bits2bytes(firstn((9 - 1) + # uint2bits(i / 256) - 8, uint2bits(i))) + i2b(bits2int(lastn(8, uint2bits(i)))) ' = bits2bytes(uint2bits(i / 256)) + bits2bytes(int2bits(i % 256)).last ', fillsgn(9, uint2bits(i % 256)) = []

Rule argument

seq: ⊦ # bits ≤ n → # fillsgn(n, bits) = n subst: [bits, n] → [uint2bits(i % 256), 9] rotate: F spec: bitopsaux inst: name: fill-length

Additional information

Used terms: uint2bits(i % 256), 9