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Proof step 7 from proof cast-mod-pos-low (previous step)

j % 2 ^ i < 2 ^ (i - 1), 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, j % 2 ^ i < j, 0 ≤ 2 ^ (i - 1) + j ⊦ 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 8:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 9:
¬ i→n(i) ≤ # int2bits(j), j % 2 ^ i < 2 ^ (i - 1), 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, j % 2 ^ i < j, 0 ≤ 2 ^ (i - 1) + j ⊦ bits2int(lastn(i→n(i), fillsgn(i→n(i), int2bits(j)))) = j % 2 ^ i
Proof step 12:
i→n(i) ≤ # int2bits(j), j % 2 ^ i < 2 ^ (i - 1), 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, j % 2 ^ i < j, 0 ≤ 2 ^ (i - 1) + j ⊦ 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)