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Proof step 17 from proof uint2bits-mod2pot (previous step)

0 ≤ i0, j / 2 ≤ 2 ^ i0, 0 < j, 2 ^ i0 * 2 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ ~1 = i0, j = 1, uint2bits(j % (2 ^ i0 * 2)) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest) + (j % 2 = 1)')

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ 0 ≤ i ∧ 0 ≤ j → (j / 2 ≤ 2 ^ i ↔ j ≤ 2 ^ (i + 1) + 1)
Proof step 19:
0 ≤ i0 ∧ 0 ≤ j, 0 ≤ i0, j ≤ 2 ^ (i0 + 1) + 1, 0 < j, 2 ^ i0 * 2 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ ~1 = i0, j = 1, uint2bits(j % (2 ^ i0 * 2)) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest) + (j % 2 = 1)')

Rule argument

seq: ⊦ 0 ≤ i ∧ 0 ≤ j → (j / 2 ≤ 2 ^ i ↔ j ≤ 2 ^ (i + 1) + 1) subst: [i, j] → [i0, j] rotate: F spec: int-pot inst: name: le-2pot-01

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i < j → ¬ j < i; Used terms: i0, j