Create my own 'Refine' function

I’d like to understand how the Refine function works internally.

I tried to create my own Refine function to support basic cases, my goal is not to reproduce the whole Refine function but to understand how it is built in the kernel.

For example:

In[0]:= Refine[Sqrt[x^2], x > 0]
Out[0]:= x

A naive attempt is to create the rule:

MyRefine[Sqrt[x_ ^ 2], x_ > 0] := x

Obvioulsy, it works with the following:

In[1]:= MyRefine[Sqrt[a ^ 2], a > 0]
Out[1]:= a

But not with:

In[2]:= MyRefine[Sqrt[a ^ 2], a > 1]
Out[2]:= MyRefine[Sqrt[a^2], a > 1]

How to write a rule which works for any positive value in assumptions (a > 1 also means a > 0)?
For example, I’d like that MyRefine function behave as follows:

In[3]:= $Assumptions:= z > 1
In[4]:= MyRefine[Sqrt[a ^ 2], a > z]
Out[4]:=a

My current output is of course Out[4]:= MyRefine[Sqrt[a ^ 2], a > z]

Actually, my question is more general. Another example:

In[5]:= Refine[Sqrt[x^2], Element[x, Reals]]
Out[5]:= Abs[x]

Naive attempt:

MyRefine[Sqrt[x_ ^ 2], Element[x_, Reals]]:= Abs[x]

Obvioulsy working with:

In[6]:= MyRefine[Sqrt[a^2], Element[a, Reals]]
Out[6]:= Abs[a]

But not with the following (whereas Integers is a subset of Reals):

In[7]:= MyRefine[Sqrt[a^2], Element[a, Integers]]
Out[7]:= MyRefine[Sqrt[a^2], Element[a, Integers]]

I have no idea about the approach to achieve this. Which kind of rules must I write? Is there a known algorithm behind?
Any help would be greatly appreciated.

ClearAll[MyRefine]
MyRefine[expr_, assumptions_] := Simplify[expr, assumptions]

MyRefine[Sqrt[a^2], 2 a - 1 > 1]                  (* Out: a *)
Assuming[z > 1, MyRefine[Sqrt[a^2], a > z]]       (* Out: a *)