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Proof step 122 from proof int2bytes-rec-neg-05 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 123:
0 ≤ i, 0 ≤ j ⊦ i→n(i) * i→n(j) = i→n(i * j)
Proof step 124:
0 ≤ i ∧ 0 ≤ 8, i→n(i) * i→n(8) + 1 = # int2bits(j % 2 ^ (i * 8)), 0 < i, j + 2 ^ (i * 8 - 1) < 0, j + 2 ^ (i * 8) ≤ 0, j + 2 ^ (i * 8) < i, j < 0, j % 2 ^ (i * 8) < 0 ⊦ bits2bytes(int2bits(j / 2 ^ (i * 8) - 1) + int2bits(j % 2 ^ (i * 8)).rest) = bits2bytes(int2bits(j / 2 ^ (i * 8) - 1)) + fix(i→n(i) + 1, int2bytes(j % 2 ^ (i * 8))).rest, 2 ^ (i * 8) = 0, j % 2 ^ (i * 8) = 0, (i→n(i) * i→n(8)) % 8 = 0 ∧ int2bits(j % 2 ^ (i * 8)).rest ≠ []

Rule argument

seq: 0 ≤ i, 0 ≤ j ⊦ i→n(i) * i→n(j) = i→n(i * j) subst: [j, i] → [8, i] rotate: T spec: intnat inst: name: i2np

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: 8, i