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Proof step 47 from proof int2bytes-rec-pos (previous step)

uint2bits(i) = uint2bits(i / 256) + fillsgn(9, uint2bits(i % 256)).rest, # uint2bits(i % 256) ≤ 9, 255 < i, 0 < i, 0 ≤ i, i % 256 ≤ 127 ⊦ fillsgn(9, uint2bits(i % 256)) = [], i2b(bits2int(fillsgn(9, uint2bits(i % 256)).rest)) = i2b(i % 256), bits2int(fillsgn(9, uint2bits(i % 256)).rest) = i % 256

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ bits ≠ [] ∧ # bits < n → fillsgn(n, bits) = bits.first ' + fillsgn(n -1, bits)
Proof step 49:
¬(uint2bits(i % 256) ≠ [] ∧ # uint2bits(i % 256) < 9), uint2bits(i) = uint2bits(i / 256) + fillsgn(9, uint2bits(i % 256)).rest, # uint2bits(i % 256) ≤ 9, 255 < i, 0 < i, 0 ≤ i, i % 256 ≤ 127 ⊦ fillsgn(9, uint2bits(i % 256)) = [], i2b(bits2int(fillsgn(9, uint2bits(i % 256)).rest)) = i2b(i % 256), bits2int(fillsgn(9, uint2bits(i % 256)).rest) = i % 256
Proof step 55:
uint2bits(i % 256) ≠ [] ∧ # uint2bits(i % 256) < 9, uint2bits(i) = uint2bits(i / 256) + (uint2bits(i % 256).first ' + fillsgn(9 -1, uint2bits(i % 256))).rest, # uint2bits(i % 256) ≤ 9, 255 < i, 0 < i, 0 ≤ i, i % 256 ≤ 127 ⊦ uint2bits(i % 256).first ' + fillsgn(9 -1, uint2bits(i % 256)) = [], i2b(bits2int((uint2bits(i % 256).first ' + fillsgn(9 -1, uint2bits(i % 256))).rest)) = i2b(i % 256), bits2int((uint2bits(i % 256).first ' + fillsgn(9 -1, uint2bits(i % 256))).rest) = i % 256

Rule argument

seq: ⊦ bits ≠ [] ∧ # bits < n → fillsgn(n, bits) = bits.first ' + fillsgn(n -1, bits) subst: [bits, n] → [uint2bits(i % 256), 9] rotate: F spec: bitopsaux inst: name: fill-rec

Additional information

Used terms: uint2bits(i % 256), 9