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Proof step 33 from proof cast-mod-pos-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ 0 < i → (m < # int2bits(i) ↔ ¬ i < 2 ^ (n→i(m) - 1))
Proof step 35:
¬ 0 < j % 2 ^ i, j % 2 ^ i < i, j % 2 ^ i ≤ 0, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, j % 2 ^ i < j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, j % 2 ^ i < 2 ^ (i - 1), i→n(i) ≤ (# int2bits(j % 2 ^ i) - 1) + # int2bits(j / 2 ^ i), i→n(i) + 1 < # int2bits(j % 2 ^ i), i→n(i) < # int2bits(j % 2 ^ i) ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + int2bits(j % 2 ^ i).rest)) = j % 2 ^ i, i→n(i) = # int2bits(j % 2 ^ i) - 1

Rule argument

seq: ⊦ 0 < i → (m < # int2bits(i) ↔ ¬ i < 2 ^ (n→i(m) - 1)) subst: [i, m] → [j % 2 ^ i, i→n(i)] rotate: F spec: int2bits inst: name: int2bits-lslen-06

Additional information

Used simplifier rules: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: j % 2 ^ i, i→n(i)