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Proof step 21 from proof int2bits-len-range (previous step)

0 ≤ 2 ^ i0 + 2 ^ i, 0 ≤ 2 ^ i0 + 2 ^ i0, 0 < i + 2 ^ i0, 1 < 2 ^ i0 + 2 ^ i, 0 ≤ i, 1 < 2 ^ i0 + 2 ^ i0, 0 < 2 ^ i, 0 < i0 + 2 ^ i0, i0 ≤ i, 0 ≤ i0, 0 < 2 ^ i0, i→n(i) + 1 < # int2bits(~(1 - 2 ^ i0)) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

name: int2bits-len-pos rotate: F subst: [i, j] → [i0 - 1, ~(1 - 2 ^ i0)]

Additional information

Used simplifier rules: int2bits-zero: ⊦ int2bits(0) = false ';% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ 0 < j → abs(j) = j;% ⊦ 0 ≤ i → abs(i) = i;% ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j);% ⊦ m + n < m + n0 ↔ n < n0;% ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i + (j - i0) = i + j - i0;% ⊦ i - j ≤ i0 ↔ i ≤ j + i0;% ⊦ i + 0 = i;% ⊦ i + 1 ≤ j ↔ i < j;% ⊦ 0 ≤ i → (2 ^ i ≤ 2 ^ j ↔ i ≤ j);% ⊦ i + j ≤ i ↔ j ≤ 0;% ⊦ i + (j - i) = j;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ # boolvar ' = 0 +1;% ⊦ ¬ m + n < m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m; Used terms: i0 - 1, ~(1 - 2 ^ i0)