I am writing a 40-page (technical) paper about mathematics. Most grammar advice is easy to find. But, the hardest part I have been struggling the past days is when to use the definitive article.
The reason is that I would like to have my paper written as close as possible to what a native speaker would write (not caring whether it is British or American). My native language is German and a particular difficult problem is the use, or the omission, of the definitive article "the". I Did quite some research that is got me to my guess formulated below.
Sentence 1: This sounds like native English:
Using the variance 'v' of inductive type 'I' in its argument 'a', we can finish the proof.
In the above sentence I highlighted the mathematical variables with ' '.
My interpretation/guess: Since "the variance" and "the inductive type" is a repetition, we are allowed to omit the second "the". Thus, my guess, if we do not omit the second "the", the sentence sounds redundant and less like native English:
Using the variance 'v' of the inductive type 'I' in its argument 'a', we can finish the proof.
Sentence 2: Which version sound more native:
To obtain the right hand side of required type A, we map the function F over the list L.
To obtain the right hand side of the required type A, we map the function F over the list L.
Sentence 3: Here, I have no clue, which version sounds more like native English:
Furthermore, monotonicity of G gives us the isomorphism I.
Furthermore, the monotonicity of G gives us the isomorphism I.
Finally, what grammar rules am I missing? Something I can google in the future?