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Proof step 63 from proof cast-mod-neg-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 64:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 65:
¬ i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), j % 2 ^ i + 2 ^ (i - 1) < 0, j < 0, 0 < i ⊦ bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i + 2 ^ i, j = ~1, i = 1, j % 2 ^ i = 0
Proof step 74:
i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), i→n(i) = # int2bits(j % 2 ^ i).rest, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), j % 2 ^ i + 2 ^ (i - 1) < 0, j < 0, 0 < i ⊦ bits2int(int2bits(j % 2 ^ i).rest) = j % 2 ^ i + 2 ^ i, j = ~1, i = 1, j % 2 ^ i = 0

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