Bandits.UCB.ofReal_constSum
Lemma
No docstring.
Given
-
c : ℝ -
n : ℕ
Then
ENNReal.ofReal (constSum c n) = ∑ s ∈ Finset.range n, 1 / (↑s + 1) ^ (c - 1)Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
Nat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
Eq : {α : Sort u_1} → α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
Conventions for notations in identifiers:
* The recommended spelling of `=` in identifiers is `eq`.ENNReal.ofReal : ℝ → ENNReal`ofReal x` returns `x` if it is nonnegative, `0` otherwise.
Bandits.UCB.constSum : ℝ → ℕ → ℝA sum that appears in the UCB regret upper bound. For `c > 2` it is bounded uniformly in `n`, see `constSum_le`.Go to its page
Finset.range : ℕ → Finset ℕ`range n` is the set of natural numbers less than `n`.
HDiv.hDiv : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HDiv α β γ] → α → β → γ`a / b` computes the result of dividing `a` by `b`. The meaning of this notation is type-dependent. * For most types like `Nat`, `Int`, `Rat`, `Real`, `a / 0` is defined to be `0`. * For `Nat`, `a / b` rounds downwards. * For `Int`, `a / b` rounds downwards if `b` is positive or upwards if `b` is negative. It is implemented as `Int.ediv`, the unique function satisfying `a % b + b * (a / b) = a` and `0 ≤ a % b < natAbs b` for `b ≠ 0`. Other rounding conventions are available using the functions `Int.fdiv` (floor rounding) and `Int.tdiv` (truncation rounding). * For `Float`, `a / 0` follows the IEEE 754 semantics for division, usually resulting in `inf` or `nan`. Conventions for notations in identifiers: * The recommended spelling of `/` in identifiers is `div`.
HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ`a + b` computes the sum of `a` and `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `+` in identifiers is `add`.
HPow.hPow : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HPow α β γ] → α → β → γ`a ^ b` computes `a` to the power of `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `^` in identifiers is `pow`.
HSub.hSub : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HSub α β γ] → α → β → γ`a - b` computes the difference of `a` and `b`. The meaning of this notation is type-dependent. * For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. Conventions for notations in identifiers: * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).
Code
lemma ofReal_constSum (c : ℝ) (n : ℕ) :
ENNReal.ofReal (constSum c n) = ∑ s ∈ range n, 1 / ((s : ℝ≥0∞) + 1) ^ (c - 1)Proof
by
rw [constSum, ENNReal.ofReal_sum_of_nonneg (fun s _ ↦ by positivity)]
refine Finset.sum_congr rfl fun s _ ↦ ?_
rw [one_div, one_div, ENNReal.ofReal_inv_of_pos (by positivity),
← ENNReal.ofReal_rpow_of_pos (by positivity), ENNReal.ofReal_add (by positivity) zero_le_one,
ENNReal.ofReal_natCast, ENNReal.ofReal_one]New in v4.34.0-rc2-90-gdde3322 (2026-09-16), and its meaning has not changed since.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Audit surface: 1 project declarations, 34 external constants
✓ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.