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Proof step 19 from proof int2bytes-rec-pos-any (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
i→n(i * 8) + 1 < # int2bits(j % 2 ^ (i * 8)), i→n(i * 8) < 8 * i→n(i), 0 < j, 2 ^ (i * 8) ≤ j, # int2bits(j % 2 ^ (i * 8)) ≤ 1 + 8 * i→n(i), 0 < i ⊦ bits2bytes(int2bits(j / 2 ^ (i * 8)) + fillsgn(1 + 8 * i→n(i), int2bits(j % 2 ^ (i * 8))).rest) = int2bytes(j / 2 ^ (i * 8)) + fix(i→n(i) + 1, int2bytes(j % 2 ^ (i * 8))).rest, i = 0, # fillsgn(i→n(i * 8) + 1, int2bits(j % 2 ^ (i * 8))).rest % 8 = 0 ∧ fillsgn(i→n(i * 8) + 1, int2bits(j % 2 ^ (i * 8))).rest ≠ []

Rule argument

Additional information

Used simplifier rules: ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ m + n < m + n0 ↔ n < n0;% ⊦ m0 * m * n = (m0 * m) * n;% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ m0 * m = 0 ↔ m0 = 0 ∨ m = 0;% ⊦ m < n ∧ n ≤ n0 → m < n0; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + 1 ≤ n ↔ m < n;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ m + n = n + m;% ⊦ m * n = n * m;% ⊦ i * j = j * i;