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Proof step 21 from proof cast-mod-pos-low (previous step)

i→n(i) ≤ # int2bits(j / 2 ^ i) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 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) = # 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 22:
⊦ bits ≠ [] → # bits.rest = (# bits)-1
Proof step 23:
fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) ≠ [], i→n(i) ≤ # int2bits(j / 2 ^ i) + (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)))-1, 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) = (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)))-1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i

Rule argument

seq: ⊦ bits ≠ [] → # bits.rest = (# bits)-1 subst: [bits] → [fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i))] rotate: F spec: list-data inst: bitlista name: length-rest

Additional information

Used simplifier rules: ⊦ fillsgn(n, bits) = [] ↔ bits = [] ∧ n = 0;% ⊦ int2bits(j) ≠ []; Used terms: fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i))