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Proof step 24 from proof int2bits-len-range (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
0 ≤ i, 2 ^ i ≤ j, j < 2 ^ (i + 1) ⊦ # int2bits(j) = i→n(i + 2)
Proof step 26:
0 ≤ i0 ∧ 2 ^ i0 ≤ ~(j + 1) ∧ ~(j + 1) < 2 ^ (i0 + 1), 0 < i, j < ~1, (2 ^ i0 + j) + 1 < 2 ^ i, 1 < 2 ^ i0, 2 ^ i0 + j < i0, i0 < i, 2 ^ i0 + j < i, 0 ≤ i0, 0 ≤ 2 ^ i + j, i→n(i) + 1 < i→n(i0 + 2), 2 ^ i0 + j ≤ i0, i0 ≤ i, j < i0, 0 < 2 ^ i, j + 1 < 2 ^ i0, 2 ^ i0 + j < 2 ^ i, 0 ≤ i, 2 ^ i0 + j ≤ i, j < 2 ^ i, j < i, j < 2 ^ i0, 0 < 2 ^ i0 * 2 + j, 2 ^ i0 + j < 0, j < 0, 0 ≤ 2 ^ i0, 0 < 2 ^ i0, (2 ^ i0 + j) + 1 < i ⊦ j = 0, 2 ^ i0 + j = 0

Rule argument

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

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j;% ⊦ i ≤ ~1 ↔ i < 0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i < j → ¬ j < i;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i * j = j * i;% ⊦ 0 < i ∧ 0 < j → 0 < i + j; Used terms: i0, ~(j + 1)