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Proof step 24 from proof int2int-neg (previous step)

0 < ~ j → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ ~ j ∧ ~ j < 2 ^ (i + 1)), j < 2 ^ n→i(# int2bits(~ j)), 1 < 2 ^ n→i(# int2bits(~ j)), j < 0, 0 < 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ 2 ^ n→i(# int2bits(~ j)), 0 < 2 ^ n→i(# int2bits(~ j)), j + 1 < 2 ^ n→i(# int2bits(~ j)) ⊦ 2 ^ n→i(# int2bits(~ j)) = 2 ^ (n→i(# uint2bits(2 ^ n→i(# int2bits(~ j)) + j)) - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
j < i, i < n→i(# int2bits(~ j)), 2 ^ i + j < 2 ^ n→i(# int2bits(~ j)), 0 < 2 ^ i, i ≤ n→i(# int2bits(~ j)), 2 ^ i + j ≤ 2 ^ n→i(# int2bits(~ j)), 2 ^ i + j ≤ i, j < 2 ^ n→i(# int2bits(~ j)), 1 < 2 ^ n→i(# int2bits(~ j)), 0 < 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ 2 ^ n→i(# int2bits(~ j)), 0 < 2 ^ n→i(# int2bits(~ j)), j + 1 < 2 ^ n→i(# int2bits(~ j)), 2 ^ i + j ≤ 0, 0 ≤ i, 0 < 2 ^ i * 2 + j, j < n→i(# int2bits(~ j)), j + 1 < n→i(# int2bits(~ j)), 2 ^ i + j ≤ n→i(# int2bits(~ j)), 2 ^ i + j < n→i(# int2bits(~ j)), j + 1 < 2 ^ i, j < 2 ^ i, j < 0 ⊦ 2 ^ n→i(# int2bits(~ j)) = 2 ^ (n→i(# uint2bits(2 ^ n→i(# int2bits(~ j)) + j)) - 1)

Rule argument

Additional information

Used simplifier rules: int2bits-not-empty: ⊦ int2bits(j) ≠ [];% ⊦ # bits = 0 ↔ bits = [];% ⊦ 0 < n ↔ n ≠ 0;%0 ≤ i ⊦ (* i) < n→i(n) ↔ i→n(i) < n;% ⊦ 0 ≤ n→i(n);% ⊦ i + j < j ↔ i < 0;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ i < i + j ↔ 0 < j;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i +1 = i + 1;% ⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0);% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 1 * i = i;% ⊦ j < (* i1) * j ↔ 0 < (i1 - 1) * j;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ 0 ≤ i → (2 ^ i ≤ 2 ^ j ↔ i ≤ j);% ⊦ i + j ≤ i + j0 ↔ j ≤ j0;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < ~ i ↔ i < 0;