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Proof step 10 from proof uint2bits-len (previous step)

2 ^ (i0 + 1) ≤ j, j < 2 ^ (2 + i0), 0 < j, 0 ≤ i0, 2 ^ (i0 + 1) < 2 ^ (2 + i0), 0 < 2 ^ (2 + i0), 1 < 2 ^ (2 + i0), 0 ≤ 2 ^ (2 + i0), ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1) → # uint2bits(j0) = i→n(i + 2) ⊦ # uint2bits(j / 2) + 1 = i→n(3 + i0), j = 1, ~1 = i0, ¬ 2 ^ (i0 + 1) ≤ j / 2 ∧ ¬ j / 2 < 2 ^ i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ 0 < j ∧ 0 ≤ i → (i / j < j0 ↔ i < j * j0)
Proof step 12:
0 < 2 ∧ 0 ≤ j, 2 ^ (i0 + 1) ≤ j, j < 2 ^ (2 + i0), 0 < j, 0 ≤ i0, 2 ^ (i0 + 1) < 2 ^ (2 + i0), 0 < 2 ^ (2 + i0), 1 < 2 ^ (2 + i0), 0 ≤ 2 ^ (2 + i0), ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1) → # uint2bits(j0) = i→n(i + 2) ⊦ # uint2bits(j / 2) + 1 = i→n(3 + i0), j = 1, ~1 = i0, ¬ 2 ^ (i0 + 1) ≤ j / 2 ∧ ¬ j < 2 * 2 ^ i0

Rule argument

seq: ⊦ 0 < j ∧ 0 ≤ i → (i / (* j) < j0 ↔ i < j * j0) subst: [j0, j, i] → [2 ^ i0, 2, j] rotate: F spec: int-div inst: name: num-ls

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: 2 ^ i0, 2, j