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Proof step 18 from proof int2bits-len-neg (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
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

Rule argument

Additional information

Used simplifier rules: ⊦ # bits = 0 ↔ bits = [];% ⊦ 0 < n ↔ n ≠ 0;%0 ≤ i ⊦ (* i) < n→i(n) ↔ i→n(i) < n;% ⊦ i +1 = i + 1;% ⊦ i + j < j ↔ i < 0;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0);% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i < i + j ↔ 0 < j;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ i ≤ i + j ↔ 0 ≤ j;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ ~ i = j ↔ i + j = 0;