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Proof step 69 from proof band-011x-pos (previous step)

bits2uint(bits) = (j / 2) % 2 ^ i0, bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2)) = false ' + bits, j % 2 = 1, 0 ≤ i0, 0 ≤ j, 0 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ j0 ∧ 0 ≤ j0 ∧ 0 ≤ i → bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j0))) = j0 % 2 ^ i ⊦ ~1 = i0, j = 1, bits2uint(bits) * 2 + 1 = j % (2 ^ i0 * 2)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 70:
j = i1 * 2 + j1 ∧ 0 ≤ j1 ∧ j1 < 2 ∧ 0 < 2, bits2uint(bits) = i1 % 2 ^ i0, bitsand(false ' + mklist(true, i→n(i0)), uint2bits(i1)) = false ' + bits, j1 = 1, 0 ≤ i0, 0 ≤ j, 0 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ j0 ∧ 0 ≤ j0 ∧ 0 ≤ i → bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j0))) = j0 % 2 ^ i ⊦ ~1 = i0, j = 1, bits2uint(bits) * 2 + 1 = j % (2 ^ i0 * 2)

Rule argument

seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [j, 2, i1, j1] rotate: T spec: int-div inst: name: div-elim-pos

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;%⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j); (elim) Used terms: j, 2, i1, j1