Congruence | A command line interface to Atlassian Confluence | Command Line Interface library

your if conditions need some modifications:

use

You can try this one (as I understood to your questions). First, add name to your input for example:

To prove that theorem, you need to be able to write a function cardinality : fset_expr -> nat that computes the cardinality of a finite set, and to do that, you will need to decide whether or not two elements (a1 a2 : A) are equal (so that you don't double-count).

If you include an assumption forall a1 a2 : A, {a1 = a2} + {a1 <> a2}, then your theorem should be provable.

If you have Python 3.8 or later, you can do everything you need to with a very small number of lines of code.

First some mathematics: I'm assuming that you want to solve ax = b (mod m) for an integer x, given integers a, b and m. I'm also assuming that m is positive.

The first thing you need to compute is the greatest common divisor g of a and m. There are two cases:

• if b is not a multiple of g, then the congruence has no solutions (if ax + my = b for some integers x and y, then any common divisor of a and m must also be a divisor of b)

• if b is a multiple of g, then the congruence is exactly equivalent to (a/g)x = (b/g) (mod (m/g)). Now a/g and m/g are relatively prime, so we can compute an inverse to a/g modulo m/g. Multiplying that inverse by b/g gives a solution, and the general solution can be obtained by adding an arbitrary multiple of m/g to that solution.

Python's math module has had a gcd function since Python 3.5, and the built-in pow function can be used to compute modular inverses since Python 3.8.

Putting it all together, here's some code. First a function that finds the general solution, or raises an exception if no solution exists. If it succeeds, it returns two integers. The first gives a particular solution; the second gives the modulus that provides the general solution.

monthdays = {'February':29} and monthdays = {'February':28} overwrite the initial contents of monthdays.

To add a new key to a dictionary, use monthdays['February'] = 29.

This is more a question of mathematics than a Coq question.

there is probably a counter example. Please investigate this: an assignment shuffles the values of registers aux, x1, x2, ..., x7. However, when you look for permutations, you only look at values of x1, x2, ..., x7.

suppose you have one step that stores the value of x1 into aux, duplicates the value of x2 into x1 and x2, and leaves all other registers unchanged. When looking only at the list of values in x1, ..., x7, this step is not a permutation (because of the duplication). Let's call this step s1.

Then consider the step s2 that duplicates the value of aux into aux and x1 and leaves all other values unchanged. Again, when looking only at registers x1, ..., x7, this is not a permutation, because it introduces a value that was not in these registers before.

Now s1::s2::nil is the identity function on registers x1, ..., x7. It is a permutation. But neither s1 nor (s2::nil) are permuting steps or permutation algorithms.

For a Coq counter example, it is enough to prove that s1 is not a permuting step. Here it is:

Your statement appears to be false. Here is a counter example.

In fact your Array works correctly. However, just think about the content of your Array:

It's a matter of it displaying it properly on the device, i suggest you save it to a png or pdf. First save plot as object: