## How is Grönwall’s inequality applied here?

Let $$h\in C^1(\mathbb R)$$ such that $$h’$$ is Lipschitz continuous and $$L\varphi:=-h’\varphi’+\varphi”\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ Now, let $$(X_t)_{t\ge0}$$ be the unique strong solution of $${\rm d}X_t=-h'(X_t){\rm d}t+\sqrt2{\rm d}W_t\tag1,$$ where $$(W_t)_{t\ge0}$$ is a Brownian motion.

I’ve read (in this paper, below Assumption 2.4) that if $$V:\mathbb R\to[0,\infty)$$ with $$V(x)\xrightarrow{x\to\infty}\infty\tag2$$ and $$a,d>0$$ with $$LV\le-aV+d\tag3,$$ then $$\operatorname E\left[V(X_t)\mid\mathcal F_s^X\right]\le e^{-a(t-s)}V(X_s)+\frac da(1-e^{-a(t-s)})\tag4.$$ Why does $$(4)$$ hold?

Obviously, $$(4)$$ is an application of (the Itō formula and) a Gronwall-type lemma. Actually, it’s precisely Theorem 6 here. However, in order to apply that theorem, we should need that the process $$\left(\operatorname E\left[V(X_t)\mid\mathcal F_s^X\right]\right)_{t\ge0}$$ is continuous. This shouldn’t hold, unless $$V$$ is (continuous and) bounded (which would allow an application of Lebesgue’s dominated convergence theorem). But $$V$$ is clearly assumed to be unbounded by $$(2)$$. So, what am I missing?

(Clearly, the authors of the paper are missing assumptions on $$V$$ anyway. In order for $$(3)$$ to make sense, $$V$$ needs to be twice differentiable (at least in some weak sense).

## Computational type theorists: how do you compare terms for equality here?

I am attempting to implement Simple Type Theory in the language D. How do you compare subterms to a term $$R$$ for the sake of computing the covering abstractors of $$R$$ in $$M$$?

By reference (class objects), syntactically by string value, or by some defined equality function?

Definition. (Body, scope, covering abstractors) Let $$\lambda x.P$$ be a component of a term $$M$$. The displayed component $$P$$ is called the body of or the scope of the abstractor $$\lambda x$$.

The covering abstractors of a component $$R$$ of $$M$$ are the abstractors in $$M$$ whose scopes contain $$R$$.

This is from the book “Basic Simple Type Theory”.

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## Why does the base class need to have a virtual destructor here if the derived class allocates no raw dynamic memory?

The following code causes a memory leak:

#include <iostream> #include <memory> #include <vector>  using namespace std;  class base {     void virtual initialize_vector() = 0; };  class derived : public base { private:     vector<int> vec;  public:     derived()     {         initialize_vector();     }      void initialize_vector()     {         for (int i = 0; i < 1000000; i++)         {             vec.push_back(i);         }     } };  int main() {     for (int i = 0; i < 100000; i++)     {         unique_ptr<base> pt = make_unique<derived>();     } }

It didn’t make much sense to me, since class derived allocates no raw dynamic memory, and unique_ptr deallocates itself. I get that class base’s implicit destructor is being called instead of derived’s, but I don’t get why that’s a problem here. If I were to write an explicit destructor for derived, I wouldn’t write anything for vec.

## Let’s put an end to the questions “what” here, they’re pointless

Let’s put an end to the questions “what” here, they’re pointless