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Proof step 57 from proof cast-mod-neg-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 58:
⊦ i < ~1 → (m < # int2bits(i) ↔ ¬ 0 ≤ i + 2 ^ (n→i(m) - 1))
Proof step 59:
j % 2 ^ i < ~1, j % 2 ^ i < ~1, i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i - 1), j + 2 ^ i ≤ 0, j + 2 ^ i < i, 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0, j % 2 ^ i < 0, 2 ^ i + j % 2 ^ i < 0, ¬ 0 ≤ j % 2 ^ i + 2 ^ (n→i(i→n(i) + 1) - 1) ⊦ i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1, fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) = [], j % 2 ^ i = 0, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i + 2 ^ i

Rule argument

seq: ⊦ i < ~1 → (m < # int2bits(i) ↔ ¬ 0 ≤ i + 2 ^ (n→i(m) - 1)) subst: [i, m] → [j % 2 ^ i, i→n(i) + 1] rotate: F spec: int2bits inst: name: int2bits-lslen-05

Additional information

Used terms: j % 2 ^ i, i→n(i) + 1