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Proof step 18 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, 2 ^ i ≤ j, i→n(i) ≤ # int2bits(j / 2 ^ i) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + 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 19:
⊦ n = # bits → lastn(n, bits0 + bits) = bits
Proof step 20:
i→n(i) ≠ # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j % 2 ^ i < 2 ^ (i - 1), 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, j % 2 ^ i < j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ i ≤ j, i→n(i) ≤ # int2bits(j / 2 ^ i) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i
Proof step 38:
i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j % 2 ^ i < 2 ^ (i - 1), 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, j % 2 ^ i < j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ i ≤ j, i→n(i) ≤ # int2bits(j / 2 ^ i) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest ⊦ bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i

Rule argument

seq: ⊦ n = # bits → lastn(n, bits0 + bits) = bits subst: [bits, bits0, n] → [fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, int2bits(j / 2 ^ i), i→n(i)] rotate: F spec: list-del inst: bitlista name: lastN-append

Additional information

Used terms: fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, int2bits(j / 2 ^ i), i→n(i)