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Proof step 19 from proof uint2bits-div2pot (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ 0 < j ∧ 0 ≤ i → (i / j < j0 ↔ i < j * j0)
Proof step 21:
0 < 2 ∧ 0 ≤ j, j < 2 * 2 ^ i0, 0 < j, 2 ^ (i0 + 1) ≤ j, 0 ≤ i0, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i ≤ j0 → uint2bits(j0 / 2 ^ i) = butlastn(i→n(i), uint2bits(j0)) ⊦ j = 1, uint2bits(j / 2 ^ (i0 + 1)) = butlastn(i→n(i0), uint2bits(j / 2)), ~1 = 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; Used terms: 2 ^ i0, 2, j