[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 16 from proof twopot-mone-div (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ 0 < i ∧ 0 < j0 → (2 * i - 1) / (2 * j0) = (i - 1) / j0
Proof step 18:
(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 0 < j, 0 < i, 0 < 2 ^ (i - 1), 0 < 2 ^ (j - 1), 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → j ≤ i0 ∧ 0 ≤ j → (2 ^ i0 - 1) / 2 ^ j = 2 ^ i0 / 2 ^ j - 1 ⊦ 2 ^ (i - 1) / 2 ^ (j - 1) = (2 ^ (i - 1) - 1) / 2 ^ (j - 1) + 1

Rule argument

spec: int-div name: two-mult seq: ⊦ 0 < i ∧ 0 < j0 → (2 * i - 1) / (2 * j0) = (i - 1) / j0 subst: [j0, i] → [2 ^ (j - 1), 2 ^ (i - 1)] flags: <all paths>

Additional information

Used simplifier rules: pot2-pos: ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s)% ⊦ 1 ≤ i ↔ 0 < i; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 2 ^ (j - 1), 2 ^ (i - 1)