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Proof step 43 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) < i→n(i * 8 + 1), 0 ≤ j, 0 ≤ i, # int2bits(j % 2 ^ (i * 8)) ≤ 1 + 8 * i→n(i), 0 < i, 2 ^ (i * 8) ≤ j, 0 < j ⊦ # (mklist(¬ 0 ≤ j % 2 ^ (i * 8), i→n(i * 8 + 1) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))).rest % 8 = 0, i→n(i * 8 + 1) = # int2bits(j % 2 ^ (i * 8)), bits2bytes (int2bits(j / 2 ^ (i * 8)) + (mklist(¬ 0 ≤ j % 2 ^ (i * 8), 1 + 8 * i→n(i) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))).rest) = int2bytes(j / 2 ^ (i * 8)) + fix(i→n(i) + 1, int2bytes(j % 2 ^ (i * 8))).rest, ~1 = i * 8

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ bits ≠ [] → # bits.rest = (# bits)-1
Proof step 45:
mklist(¬ 0 ≤ j % 2 ^ (i * 8), i→n(i * 8 + 1) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8)) ≠ [], # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8 + 1), 0 ≤ j, 0 ≤ i, # int2bits(j % 2 ^ (i * 8)) ≤ 1 + 8 * i→n(i), 0 < i, 2 ^ (i * 8) ≤ j, 0 < j ⊦ (#(mklist(¬ 0 ≤ j % 2 ^ (i * 8), i→n(i * 8 + 1) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))))-1 % 8 = 0, i→n(i * 8 + 1) = # int2bits(j % 2 ^ (i * 8)), bits2bytes (int2bits(j / 2 ^ (i * 8)) + (mklist(¬ 0 ≤ j % 2 ^ (i * 8), 1 + 8 * i→n(i) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))).rest) = int2bytes(j / 2 ^ (i * 8)) + fix(i→n(i) + 1, int2bytes(j % 2 ^ (i * 8))).rest, ~1 = i * 8

Rule argument

seq: ⊦ bits ≠ [] → # bits.rest = (# bits)-1 subst: [bits] → [ mklist (¬ 0 ≤ j % 2 ^ (i * 8), i→n(i * 8 + 1) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))] rotate: F spec: list-data inst: bitlista name: length-rest

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ n < m → m - n ≠ 0;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: mklist (¬ 0 ≤ j % 2 ^ (i * 8), i→n(i * 8 + 1) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))