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Proof step 74 from proof cast-mod-neg-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
⊦ i < ~1 → (# int2bits(i) ≤ m ↔ 0 ≤ i + 2 ^ (n→i(m) - 1))
Proof step 76:
¬ j % 2 ^ i < ~1, i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 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 + 2 ^ i ≤ 0, j + 2 ^ i < i, # int2bits(j % 2 ^ i) ≤ i→n(i) ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i
Proof step 88:
j % 2 ^ i < ~1, i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 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 + 2 ^ i ≤ 0, j + 2 ^ i < i, 0 ≤ j % 2 ^ i + 2 ^ (n→i(i→n(i)) - 1) ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i

Rule argument

seq: ⊦ i < ~1 → (# int2bits(i) ≤ m ↔ 0 ≤ i + 2 ^ (n→i(m) - 1)) subst: [m, i] → [i→n(i), j % 2 ^ i] rotate: F spec: int2bits inst: name: int2bits-lelen-03

Additional information

Used terms: i→n(i), j % 2 ^ i