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Proof step 20 from proof int2bits-len-neg (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
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, 2 ^ n→i(# bits) ≤ 2 ^ i * 2, 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

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ i + (j - i0) = i + j - i0;% ⊦ i < j + i0 ∧ j < j0 → i < j0 + i0;% ⊦ ~ i < 0 ↔ 0 < i;% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i + ~1 = i - 1;% ⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0);% ⊦ i + 0 = i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + ~ j = i - j;% ⊦ ~ i ≤ j ↔ 0 ≤ i + j;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ~ i + i = 0;