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Proof step 27 from proof int2bits-len-neg (previous step)

0 ≤ n→i(# bits), 0 ≤ i, 0 ≤ 2 ^ n→i(# bits) + 2 ^ i, 2 ^ i + 1 < 2 ^ n→i(# bits) + n→i(# bits), 2 ^ i < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 2 ^ i + 1 < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits), 2 ^ i < 2 ^ n→i(# bits) + i, n→i(# bits) ≤ i + 1, 0 ≤ i, 0 < i + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits) + 2 ^ i, i < n→i(# bits), 0 ≤ 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 0 ≤ 2 ^ n→i(# bits) + n→i(# bits), 1 < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits) + i, 0 ≤ 2 ^ n→i(# bits), 0 < 2 ^ n→i(# bits), 0 < 2 ^ i, 0 < 2 ^ n→i(# bits) + i, 0 ≤ 2 ^ n→i(# bits) + i ⊦ # bits = 1 + i→n(i), bits = [], true ∈ bits

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
n→i(# bits) = i + 1, 0 < 2 ^ i * 2 + i, ~1 ≤ 2 ^ i * 2 + i, 2 ^ i < 2 * (2 ^ i + 2 ^ i), 2 ^ i + 1 < 2 * (2 ^ i + 2 ^ i), 0 < 2 ^ i + i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 1 < 2 ^ i * 2 + i, 0 ≤ i ⊦ # bits = 1 + i→n(i), bits = [], true ∈ bits

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ j ∧ 0 ≤ j0 ∧ 0 ≤ i0 → 0 ≤ i0 * j + j0;% ⊦ i ≤ i;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ 1 * i = i;% ⊦ j < (* i1) * j + i ↔ 0 < (i1 - 1) * j + i;% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ (* i0) * i + i = (i0 + 1) * i;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i * j + i * j0 = i * (j + j0);% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1);% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ n→i(n);