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Proof step 10 from proof twopot-mone-div (previous step)

0 < j, j ≤ i, 0 ≤ i, 0 ≤ j, 0 < i, ∀ i0, j. i→n(i0) < i→n(i) → true → j ≤ i0 ∧ 0 ≤ j → (2 ^ i0 - 1) / 2 ^ j = 2 ^ i0 / 2 ^ j - 1 ⊦ (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) = (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 11:
j ≠ 0 ⊦ (2 * i) / (2 * j) = i / j
Proof step 12:
0 < j, j ≤ i, 0 ≤ i, 0 ≤ j, 0 < i, ∀ i0, j. i→n(i0) < i→n(i) → true → j ≤ i0 ∧ 0 ≤ j → (2 ^ i0 - 1) / 2 ^ j = 2 ^ i0 / 2 ^ j - 1 ⊦ 2 ^ j -1 ≠ 0, (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) = (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1
Proof step 14:
(2 * 2 ^ (i - 1)) / (2 * 2 ^ j -1) = 2 ^ (i - 1) / 2 ^ j -1, 0 < j, j ≤ i, 0 ≤ i, 0 ≤ j, 0 < i, ∀ i0, j. i→n(i0) < i→n(i) → true → j ≤ i0 ∧ 0 ≤ j → (2 ^ i0 - 1) / 2 ^ j = 2 ^ i0 / 2 ^ j - 1 ⊦ (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) = (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1

Rule argument

spec: int-div inst: name: div-two-01 seq: j ≠ 0 ⊦ (2 * i) / (2 * j) = i / j subst: [j, i] → [2 ^ j -1, 2 ^ (i - 1)] flags: []

Additional information

Used terms: 2 ^ j -1, 2 ^ (i - 1)