Classical logic is somewhat surprising! Prove that given two arbitrary propositions a and b, either a implies b or the converse
Enrico is a lazy student. When asked to reverse and filter a list he comes up with the following ugly code.
The teacher asks him to prove that his ugly code is equivalent to the nicer code [seq x <- rev l | p x] he could have written by reusing the seq library.
Such proof is not trivial:
Relevant keywords for Search are: rev cons cat filter rcons Hint: it is perfectly fine to state intermediate lemmas
Prove that if (s1 :|: s2) is disjoint from (s1 :|: s3) then s1 is empty
Prove the equivalence of these two sums. E.g. (n=8)
1 + 3 + 5 + 7 = 7-0 + 7-2 + 7-4 + 7-6
Now, some algebra.
We remind what Gauss integer are and recall some theory about it.
Prove these two facts. (Hint: conjugation is a morphism)
paper proofhere and then do it in Coq:
Definition and properties of lagrange polynomials.
You can prove only one of the following lemmas
The endomorphisms of a space E of finite dimension n, such that u o v = 0 and v + u is invertible are exactly the endomorphisms such that Ker u ⊕ Im u = E.