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Proof step 5 from proof bertrand-estimate (previous step)

0 ≤ 4 ^ i, 4 ^ i / 4 ^ ((2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 0 ≤ j0 ∧ j0 ≤ j ∧ 0 < i → i ^ j / i ^ j0 = i ^ (j - j0)
Proof step 7:
0 ≤ (2 * i) / 3 ∧ (2 * i) / 3 ≤ i ∧ 0 < 4, 0 ≤ 4 ^ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i) ⊦

Rule argument

spec: int-pot inst: name: pot-div seq: ⊦ 0 ≤ j0 ∧ j0 ≤ j ∧ 0 < i → i ^ j / i ^ j0 = i ^ (j - j0) subst: [j0, j, i] → [(2 * i) / 3, i, 4] flags: rewrite

Additional information

Used simplifier rules: grz: ⊦ 0 < i → (0 < i ^ j ↔ i = 1 ∨ 0 ≤ j); from specification int-pot%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%nls-mult-div: ⊦ 0 ≤ i ∧ 0 < j ∧ i1 < j → ¬ i < (i * i1) / j; from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: (2 * i) / 3, i, 4