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Proof step 8 from proof cast-mod-neg-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 10:
¬ i→n(i) ≤ # int2bits(j), 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i ⊦ j = ~1, bits2int(lastn(i→n(i), fillsgn(i→n(i), int2bits(j)))) = j % 2 ^ i
Proof step 13:
i→n(i) ≤ # int2bits(j), 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i ⊦ j = ~1, bits2int(lastn(i→n(i), int2bits(j))) = j % 2 ^ i

Rule argument

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

Additional information

Used terms: int2bits(j), i→n(i)