LEMMA | Local Epidemic Modeling for Management & Action | Topic Modeling library

So I'm trying to perform a simple proof using cardinalities. It looks like:

I have been working on an algorithm (Dafny cannot prove function-method equivalence, with High-Order-Polymorphic Recursive vs Linear Iterative) to count the number of subsequences of a sequence that hold a property P. For instance, how many subsequences hold that 'the number of positives on its left part are more that on its right part'.

But this was just to offer some context.

The important function is this CountAux function below. Given a proposition P (like, x is positive), a sequ sequence of sequences, an index i to move through the sequences, and an upper bound j:

I've been working with Isabelle/HOL for a few months now, but I've been unable to figure out the exact intention of the use of _tac.

Specifically, I'm talking about cases vs case_tac and induct vs indut_tac (although it would be nice to know the meaning of tac in general, since I'm also using other methods such as cut_tac).

I've noticed I can't use cases or induct using apply with ⋀-bound variables, but I can if it's an structured proof. Why?

An example of this:

The input word is standalone and not part of a sentence but I would like to get all of its possible lemmas as if the input word were in different sentences with all possible POS tags. I would also like to get the lookup version of the word's lemma.

Why am I doing this?

I have extracted lemmas from all the documents and I have also calculated the number of dependency links between lemmas. Both of which I have done using en_core_web_sm. Now, given an input word, I would like to return the lemmas that are linked most frequently to all the possible lemmas of the input word.

So in short, I would like to replicate the behaviour of token._lemma for the input word with all possible POS tags to maintain consistency with the lemma links I have counted.

In a large corpus of text, I am interested in extracting every sentence which has a specific list of (Verb-Noun) or (Adjective-Noun) somewhere in the sentence. I have a long list but here is a sample. In my MWE I am trying to extract sentences with "write/wrote/writing/writes" and "book/s". I have around 30 such pairs of words.

Here is what I have tried but it's not catching most of the sentences:

I have following simple Isabelle/HOL theory:

I have found the same problem when proving several lemmas: rules with an equality sometimes only work in one direction.

For instance, I'd like to use append_assoc to get from xs @ ys @ zs to (xs @ ys) @ zs, but since append_assoc is defined as (xs @ ys) @ zs = xs @ ys @ zs, I can't.

Is there any way to indicate I want to use some rule backwards?

Thanks in advance.

I'm having a bit of trouble understanding the difference between strong and weak specification in Coq. For instance, if I wanted to write the replicate function (given a number n and a value x, it creates a list of length n, with all elements equal to x) using the strong specification way, how would I be able to do that? Apparently I have to write an Inductive "version" of the function but how?

Definition in Haskell:

Here's a minimal example of my problem

Lemma arith: forall T (G: seq T), (size G + 1 + 1).+1 = (size G + 3).

I would like to be able to reduce this to forall T (G: seq T), (size G + 2).+1 = (size G + 3).

by the simplest possible means. Trying simpl or auto immediately does nothing.

If I rewrite with associativity first, that is,

intros. rewrite - addnA. simpl. auto.,

simpl and auto still do nothing. I am left with a goal of

(size G + (1 + 1)).+1 = size G + 3

I guess the .+1 is "in the way" of simpl and auto working on the (1+1) somehow. It seems like I must first remove the .+1 before I can simplify the 1+1.

However, in my actual proof, there is a lot more stuff than the .+1 "in the way" and I would really like to simplify my copious amount of +1s first. As a hack, I'm using 'replace' on individual occurrences but this feels very clumsy (and there are a lot of different arithmetic expressions to replace). Is there any better way to do this?

I am using the ssrnat library.

Thanks.