How can we prove a noun phrase sentence?

It helps to remember that noun phrases are simply words that connect with nouns and that can be replaced by pronouns to avoid repetition. 

Some great noun phrase sentences and fun can be had by searching online , using good grammar books and following a great formula such as:

determiner + adverb/intensifier + adjective + premodifying noun + head noun + post modification.

Enjoy

THERE IS NOTHING TO PROVE ITS GRAMMAR ITS A SERIE OF RULES TO FOLLOW  TO HAVE A PHRASE SEMANTICALY CORRECT.

I have a french background and i think its the same rules in english  we call it " phrase nominale'

Example 1

The quick, brown fox jumped over the lazy dog

This noun phrase is the subject of the sentence. In other words, this sentence is about ‘the quick, brown fox.’ But, instead of just saying ‘fox,’ the rest of the noun phrase works to describe it.

Example 2

I think there’s a good pop song in pretty much anything -Kesha, pop singer

Here, the four words ‘a good pop song’ work together as a noun phrase. Instead of just saying “song,” Kesha sees “a good pop song” in anything. The phrase acts as a complement to the subject pronoun ‘there’. It is a complement because it is giving more information about the subject.

Example 3

The end of the season is hard for some athletes

The noun phrase is acting as the subject in this example.  The five words in the noun phrase work together to name a period of time that is hard for athletes.

So these are some examples of  noun phrase. hope it answered your question.

Good luck

I am not a specialist in English, but the proof of sentences, and their focus needs to be a multiplicity of techniques and technical and literary terms

I do not seem ti understand what you mean by prove - we don't prove those, we either use them or not