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Proof step 55 from proof cast-mod-pos-high (previous step)

i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, 2 ^ (i - 1) ≤ j % 2 ^ i ⊦ j % 2 ^ i = 2 ^ i + bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 57:
¬ i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, 2 ^ (i - 1) ≤ j % 2 ^ i ⊦ j % 2 ^ i = 2 ^ i + bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)
Proof step 60:
i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), i→n(i) = # int2bits(j % 2 ^ i).rest, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, 2 ^ (i - 1) ≤ j % 2 ^ i ⊦ j % 2 ^ i = 2 ^ i + bits2int(int2bits(j % 2 ^ i).rest)

Rule argument

seq: ⊦ n ≤ # bits → fillsgn(n, bits) = bits subst: [bits, n] → [int2bits(j % 2 ^ i), i→n(i) + 1] rotate: F spec: bitopsaux inst: name: fillsgn-longer

Additional information

Used terms: int2bits(j % 2 ^ i), i→n(i) + 1