Queries collected from Github, each query is separated by "\\"

In Lean 4, if I have a variable (f : X) and a hypothesis (h : f ∈ Y), is there some way to convert f from type X to type Y? \\

Does Lean 4 have dual normed spaces?\\

In Mathlib 4, is there a theorem that shows `∀ (T : ℝ), ∫ t in (0 : ℝ)..T, f t = 0` implies `f = 0` almost everywhere?\\

In Lean 4, is there any way to specify a single implicit parameter for a theorem, but leave the rest to be inferred?\\

In Lean 4, if I have an AddCommGroup, is there any way to quickly create an instance of an AddCommMonoid out of it?\\

In mathlib? Exists.casesOn'\\

Does Lean 4 have continuous linear functionals defined anywhere in mathlib?\\

How do I find the supremum of a set in Lean 4? \\

In Lean 4, given a hypothesis h : A ∧ B ∧ C, how would I access B by itself?\\

Does Lean 4 have continuous linear functionals defined anywhere in mathlib?\\

Does Lean 4 have any version of the Weak Compactness Theorem (that's the one that says if X is a reflexive banach space with a bounded sequence u_k, then there is a weakly convergent subsequence of u_k)?\\

How do you prove a "for all" statement in Lean 4? Like, suppose I had the statement ∀ (T : ℝ), T^2 > 0. How would I start the proof?\\

In Lean 4, is there any way to convert a real number into an extended real number?\\

Given a sequence x_m : ℕ → ℝ in Lean 4, how would you state that the sequence converges to some x : ℝ?\\

When I try to type 𝓝, I get the error "unknown identifier '𝓝'". How do I fix that?\\

Does Lean 4 have a concise way to get a subsequence of a sequence ℕ → E?\\

Is there a shorthand way to state that a sequence is bounded in Lean 4?\\

How do you state that a function is surjective in Lean 4?\\

Does it have a formalization of weak convergence then?\\


Does Lean 4 have any definition for reflexive banach spaces?\\

How do you say a function is linear in Lean 4\\

In Lean 4, is there some way to keep external variables in scope when defining a function?\\

I meant more like if I'm proving a theorem with (g : \R \to \R) in the statement of the theorem and I want to define a function inside of the proof of the theorem. \\

If I wanted to define a function in Lean 4 that took in a variable (x : \R) and returned the type (P x) for some proposition P, how would I do that?\\

In Lean 4, if I have a lemma that ∀ (n : ℕ), ∃ (x : ℝ), (P x n) for some proposition P, how would I convert the "x"s into a function (f : ℕ → ℝ) so that for each (n : ℕ), (P (f n) n) is satisfied?\\

In Lean 4, if you're proving an epsilon-delta proof, how would you choose some arbitrary epsilon greater than zero\\


Is there some way in Lean 4 to assign a variable name like y to an expression that is used a lot in the statement of a theorem?\\

Given a theorem in Lean 4 that for each m : \N, there exists a unique u_m : X, how would I construct a sequence u : \N \to X where u m maps to u_m? \\

How do you write a scalar product of a vector space in Lean 4?\\

unfold g.property\\

Suppose in Lean 4 I had a type T I defined with a carrier set {f | h f}. How would I later access h f on some g : T?\\

Assuming I have a hypothesis that a function u is MemLp in Lean 4, how can I convert that function u to an element of Lp\\

Does Lean 4 have a formalization of distributions or functionals on normed spaces?\\

In Lean 4: instance instCoeWkp_to_Lp {d : ℕ+} {k : ℕ} {p : ℝ≥0∞}\\

Does Lean 4 have a way to define subspaces of measure spaces?\\

How do you define a restriction of a function f : A \to B to a subset C of A in Lean 4?\\

In mathlib? (f : Lp ℝ ((1 : ℕ+) : ℝ≥0∞) (volume : Measure (Fin 1 → ℝ)))\\

How would I define a coercion from a type (A k) depending on a parameter k to a type B in Lean 4 when k has no impact on the coercion?\\

How do you define subtypes in Lean 4?\\

So is {x | p x} equivalent to Set {x // p x}?\\

In lean 4, how do you write the closed interval of \R, [0, T]?\\

Are Hilbert spaces defined in Lean 4?\\

Is there a way to assign multiple variables in a \forall at the same time in Lean 4?

For instance, if I wanted to say "for all x, y of type \N", could you write something like "\forall (x, y : \N)"?\\

Can you extend two classes at the same time in Lean 4?\\

Is there a way to explicitly pass in implicit arguments in Lean 4?\\

How do you write scalar multiplication of a vector and a scalar in Lean 4?\\

Are infinite sums implemented in Lean 4?\\

How would I write an infinite sum of 1/2^n in Lean 4?\\

How would you say "there exists a unique number n so that n = m/2" in Lean 4?\\

Has Lean 4 implemented Banach spaces?\\

How do you state that a sequence x_n has a limit x in Lean 4?\\

Does Lean 4 have Schauder bases implemented yet?\\

In Lean 4, given a normed space X with a subspace S, how would you access the restriction of the norm of X to S?\\

In Lean 4, given a normed space X, how would I define a set S \subseteq X which is a normed subspace of X?\\

Assume in Lean 4 I had some normed space X and a subspace S \subseteq X. How would you define the closure of S in X?\\

In Lean 4, is there some way to introduce an arbitrary variable (x : ℝ) in the proof of a theorem?\\

In mathlib? x : ℝ := Classical.arbitrary ℝ \\

How can I show that the subtype {x // x > 0} is nonempty in Lean 4?\\

How do you get the set of infinitely differentiable functions in Lean 4?\\

Is there a simple proof build into the Lean 4 Mathlib library showing that fun x => 0 is continuous?\\

In Lean 4, given a function m : ℕ → ℕ and some N : ℕ, how would I show that {j : ℕ // m j > N} is nonempty?\\

In Lean 4, how would I start a proof of hm : ∀ (l : ℕ), m l ≥ l by induction, where m : ℕ → ℕ is StrictMono?\\

In Lean 4, if I have a function f : A → B → C and another function g : A → B, is there any way to pass g into f to obtain a variable of type C?\\

Given the type Fin m in Lean 4, where m > 0, how would you prove that Nonempty (Fin m)?\\

What does the Classical.choose at the end of this type mean in Lean 4? `coefs_inner_product d g w (Classical.choose ⋯) x`\\

In Lean 4, given a proposition Q : ∀ (n : ℕ), ∃! (x : A), P x, how would you construct variables xn : ℕ → A and hn : (n : ℕ) → P (xn n) mapping each n to its corresponding x and proof of P (xn n)?\\

In Lean 4, is there some way to discard a variable/hypothesis after it's been used to remove it from the current Tactic State?\\

In Lean 4, if I have an arbitrary m : ℕ generated with `let m : ℕ := Classical.arbitrary ℕ` and a variable d : Fin m which depends on m, is there some way to combine those to obtain a function dm : (m : ℕ) → Fin m returning the corresponding d for each m?\\

In Lean 4, I used Classical.choose to obtain a dm, then used Classical.choose_spec to get a hypothesis of type `(fun d_m => P d_m
      (Classical.choose ⋯)`
Is there any way to make it so that the hypothesis recognizes dm as the variable to apply to Classical.choose ...?\\

In Lean 4, given a proposition P : ℝ → Prop and a hypothesis h that ∀ (n : ℕ), ∃ (x : ℝ), P x, how would you define a function xn that returns the x corresponding to each n, and a second function hn that returns a proof that P (xn n) for each n?\\

How would you denote the space of infinitely differentiable functions in Lean 4?\\

In Lean 4, how would you get the laplacian of a function?\\

Suppose I have a variable (x : A) in Lean 4, and I want to check what x.property is. It returns something like "x ∈ A". I would like to get the actual explicit property. How do I do that?\\

When defining a function using def in Lean 4, can you define intermediate variables in the definition of the function using let?\\

In Lean 4, how would I say that a proposition holds for almost every t in ℝ?\\

Is there any way to get Lean 4 to infer that Modules over ℝ are AddCommMonoids?\\

How do I unfold the definition of a function inside of a hypothesis in Lean 4?\\

Given a theorem in Lean 4 that for some proposition P, ∃ x, P x, once you've constructed an x that satisfies P x, how would you conclude the proof?\\

In Lean 4, given nonegative terms A, B, C with A + B + C ≤ D, how would you prove that B ≤ D?\\

In Lean 4, given a hypothesis "let x := y; P x", how would I split that into two statements: a definition of x and a new hypothesis P x?\\

In Lean 4, is it possible to use the conv tactic to zoom in on an expression inside of a hypothesis rather than the current goal?\\

In Lean 4, is it possible to unfold a goal `f ∈ X` into the exact proposition you need to prove? For example, if `X := {f | P f}`, unfold `f ∈ X` into `P f`?\\

In Lean 4, given a function f : ℝ → ℝ where we know the output is always nonnegative, how would we show that Set.range f is nonnegative\\

In Lean 4, how would I convert the statement `T > 0` into the statement `0 ≤ T`?\\

In Lean 4, how would you prove `Bornology.isBounded S` for some set S?\\

What would I need to put after the := in this Lean 4 statement? "instance instL20THSeminormedAddCommGroup {d : ℕ+} {T : ℝ} : SeminormedAddCommGroup (L20THSpace d T) := "\\

In Lean 4, is there any easy way to instantiate a SeminormedAddCommGroup on a type without needing to fill in all the properties manually?\\

In Lean 4, if I have the type of functions A → M, and M is a Module over ℝ, is there some way to easily create an instance showing that A → M is also a Module over ℝ?\\

In Lean 4, both AddSubMonoids and AddCommGroups have a toAddCommMonoid method. How would I specify which one I want to use?\\

In Lean 4, find the code 
```
have (h : 0 = 0) := by
        rfl
```\\

I'm getting the error "failed to infer 'have' declaration type"\\

We're back to the error "tactic 'rfl' failed, expected goal to be a binary relation"\\

When I write the code 
```
have (hL : ∀ (T : ℝ), ∀ (m : ℕ), (h : T > 0) → (um T h m) ∈ L20THSpace d T) := by
        intro T
```
in Lean 4, I get the error "tactic 'introN' failed, insufficient number of binders" on the `intro T` part. Can you explain how to fix this?\\

In Lean 4, I have to prove `norm_nonneg` as part of defining a `SeminormedAddCommGroup.Core` on a custom space. Unfold the `‖x‖` to access the definition of the norm in that space in `∀ (x : ↥(L20THSpace d T)), 0 ≤ ‖x‖`\\

In Lean 4, how would I use a (T : ℝ) and a (h : T > 0) to convert T into an element (T : ℝ≥0)?\\

In Lean 4, how would you convert a hypothesis (h : T > 0) into a proof that (0 ≤ T), ideally inline?\\

In Lean 4, how would you expand `{y | Dist.dist y 0 < r}` so that it uses the definition of distances defined with the norm of the space of y?\\

In Lean 4, how would I change a subset statement like `S ⊆ {x | P x}` into a quantifier statement like `∀ (s : S), P s`?\\

In Lean 4, I have a goal of the form `∫ x in (0 : ℝ)..T, f x = 0`, and I would like to use the theorem "intervalIntegral.integral_zero", stating that the integral of zero is zero to reduce my goal to simply proving that `f x = 0`. How would I do that?\\

In Lean 4, how would I convert a statement of the form `∀ (x : Set.uIcc (0 : ℝ) T), f x = g x` into one of the form `Set.EqOn f g (Set.uIcc a b)`?\\

In Lean 4, if I have N : ℕ, m : ℕ, and h : m > N, how would I convert an element k : Fin N to type Fin m?\\

If my current goal in Lean 4 has a variable k : Fin N in it, and I want to convert it to a variable of type Fin m with m > N in the goal, how would I do that?\\

If I have t : Set.Icc 0 T in Lean 4, with T > 0, how would I prove that t ≥ 0?\\

In Lean 4, if you have 3 goals in cases, how do you swap to working on the second case?\\

In Lean 4, if I have an x : X, and I know that there is a way to interpret x as an element of type Y, how would I convert x to y in a proof?\\

In Lean 4, given a function f : X → ℝ with a normed space X, how would I show that it's an element of the NormedSpace.Dual of X and convert f to that type in a proof?\\

In Lean 4, if I have a theorem `Filter.Tendsto xj ⊤ (𝓝 x)`, how can I use that to apply the limit `j → ∞` on both sides of the equation `xj = 0` to obtain `x = 0`?\\

In Lean 4, if I have a theorem `Filter.Tendsto xj ⊤ (𝓝 x)`, and a statement `x = 0`, how can I convert that statement into one saying that the limit as `j → ∞` of xj is equal to 0?\\

In Lean 4, how would you get the limit of a sequence xj?\\

In Lean 4, if I have a goal `x = 0` and hypothesis `Filter.Tendsto xj ⊤ 0`, is there an easy way to change the goal to `Filter.Tendsto xj ⊤ x`?\\

In Lean 4, how would you rewrite expressions inside of a Filter.Tendsto statement? conv can't access that\\

In Lean 4, how would I unfold a Filter.Tendsto statement?\\

Can you apply at a hypothesis in Lean 4?\\

In Lean 4, how would I show that the top filter applied to the natural numbers is NeBot?\\

In Lean 4, if I have the goal `x = 0`, the hypothesis `xj = 0` and the hypothesis `Filter.Tendsto xj ⊤ x`, how can I prove the goal?\\

In Lean 4, if I have a function f : X → ℝ and a hypothesis h : IsBoundedLinearMap ℝ f. How can I use those to convert f into a NormedSpace.Dual of X?\\

In Lean 4, if I have a goal `Filter.Tendsto xj ⊤ (𝓝 0)`, and a hypothesis `h : ∀ (j : ℕ), xj j = 0`, how would I replace the xj in the `Filter.Tendsto xj` with just 0?\\

In Lean 4, is there any way to specify a single implicit parameter for a theorem, but leave the rest to be inferred?\\

In Lean 4, if you have a sequence xj which is zero whenever j > N, how would you prove that `xj =ᶠ[⊤] 0`?\\

Does Lean 4 have any version of the Weak Compactness Theorem (that's the one that says if X is a reflexive banach space with a bounded sequence u_k, then there is a weakly convergent subsequence of u_k)?\\

How do you prove a statement for all natural numbers with induction in Lean 4?\\

I want to say that for almost all x in (0, T), P x, using the sub measure space lebesgue measure on (0, T). When I try I get that Set.Icc 0 T does not have a measure on it. \\

In Lean 4, how would you convert an integral over Icc 0 T into an integral over 0..T?\\

In Lean 4, if I have a custom type X with a carrier set {x // P x} and a goal f ∈ X, how would I change the goal to simply proving P x?\\

In Lean 4, if f is a function with a root at 0, how would I show that the indicator function that takes the value of that function is equal to the function applied to the indicator?\\

In Lean 4, I'm trying to use `rw [←Set.indicator_comp_of_zero (g := fun x ↦ x ^ 2) (by simp)]` to rewrite `(s.indicator f t) ^ 2` as `s.indicator (f ^ 2) t`. 

However, I'm running into the issue that Set.indicator_comp_of_zero would only apply to `(s.indicator f) ^ 2`, without the `t` based on how the theorem is defined. How can I fix it to work in my situation?\\

In Lean 4, how would I show that an integral over a bounded set of a bounded function f is less than or equal to the measure of the set times the maximum of that function?\\

In Lean 4, how would I give a bound on the intervalIntegral of a bounded function?\\

