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Proof step 62 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 % 256, i % 256 < i, 128 < i, 127 < i, 1 < i, 0 < i, 127 < i % 256, i % 256 ≤ 255 ⊦ # uint2bits(i % 256) = 9, i % 256 = bits2int(fillsgn(9, uint2bits(i % 256)).rest) + 256, fillsgn(9, uint2bits(i % 256)) = [], i2b(bits2int(fillsgn(9, uint2bits(i % 256)).rest)) = i2b(i % 256 - 256)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 63:
0 ≤ i, 2 ^ i ≤ j, j < 2 ^ (i + 1) ⊦ # uint2bits(j) = i→n(i + 2)

Rule argument

seq: 0 ≤ i, 2 ^ i ≤ j, j < 2 ^ (i + 1) ⊦ # uint2bits(j) = i→n(i + 2) subst: [i, j] → [7, i % 256] rotate: F spec: int2bits inst: name: uint2bits-len

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ 0 < j → i % j < j;% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) Used terms: 7, i % 256