Krivine's notation for lambda-terms Krivine in his book (Lambda-calculus: Types and Models) introduces the grammar of lambda-terms and then abbreviations to denote them. The grammar itself is not ambiguous:  Lambda-terms are obtained by applying, a finite number of times, the following rules :   any variable x is a λ-term ; whenever t and u are λ-terms, then so is (t)u ; whenever t is a λ-term and x is a variable, then λx t is a λ-term.   Then: "the term $(...(((t)u_1)u_2)...)u_k$ will also be written "$(t)u_1 u_2 ...u_k$". But let us consider the two following different $\lambda$-terms:   $v_1 = ((x)\lambda y. (y)z)w$  $v_2 = (x)\lambda y. ((y)z)w$   ($w, x, y, z$ are variables) and now it seems that I have two ways to interpret the notation $(x)\lambda y. (y)z w$:   either $t = x$, $k = 2$, $u_1 =\lambda y. (y)z$ and $u_2 = w$ or $t = x$, $k= 1$ and $u_1 = \lambda y. ((y)z)w$   It seems that the notation "$(t)u_1 u_2 ...u_k$" is ambiguous. Which rule am I violating in my interpretation?
What is the benefit of Krivine's notation? I saw some people uses Krivine's notation for function application when presenting the syntax for the $\lambda$-calculus.  For example, the $\lambda$-term $\lambda f . \lambda x . \lambda y . f\ x\ y$ (with the normal convention that function application associates to the left, so it actually means $\lambda f . \lambda x . \lambda y . ((f\ x)\ y)$) is written $\lambda f . \lambda x . \lambda y . (f)\ x\ y$ (with a similar convention that it actually means $\lambda f . \lambda x . \lambda y . ((f)\ x)\ y$).  I do not see the point of having another pair of parentheses around the innermost $f$.  Why do people use Krivine's notation instead of the usual one?
Which one of these two statements is correct?     Our staff do ...      Our staff does ...   And is staffs ever correct? Applications losing focus for a few seconds in Windows 7 Recently all of my applications keep losing focus every minute (or so) for a few seconds (then the window gets the focus back). It is very annoying, since I'm typing most of the time and get interrupted by this behavior. Things worked fine until a week ago and I have no idea what is causing this error, but I noticed the mouse moves to a certain point on one of my screens when this happens.  I would appreciate any advice: it starts to drive me crazy... How to convert Wubi install into regular install? How do I use filenames that start with a dash "-" as command arguments? Union of uncountable cardinals In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. 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Here is the command I was trying to run: grep -lr -e 'page_path\\' * | xargs sed -i 's/page_path\/page_path//g'  Thanks in advance. Partitions of $n$ into distinct odd and even parts proof Let $p_\text{odd}(n)$ denote the number of partitions of $n$ into an odd number of parts, and let $p_\text{even}(n)$ denote the number of partitions of $n$ into an even number of parts.  How do I prove that    $|p_\text{even}(n) - p_\text{odd}(n)|$ is equal to the partitions of $n$ into distinct odd parts. Show that the number of partitions of $n$ for which no part appears exactly once is equal to the number of partitions of n for which every part is divisible by 2 or 3. Show that the number of partitions of $n$ for which no part appears more than twice is equal to the number of partitions of $n$ for which no part is divisible by 3. Add a transparent image on top of a material I'm designing this animal-type creature, and want to add a transparent .png image to its face with the face features (object has a subsurf). On the left is the image with the features, on the right is the part of the object where I want it to be.    I tried to import the image as a plane, but matching the shape of the plane with the face didn't work, and the lines of the face left shadows on the head, and could be seen as a different object from some angles (although it's the closest I've gotten yet). It looked like this:    So , I tried selecting the faces of the head that the features should go over, and create a new material for it. Unfortunately, since the .png is transparent, the result produced was this:    (where it's impossible to tell the transparency from a black background default color, and where I wouldn't know where to start to adjust size and position).  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I am using the following line in a controller class.      \Drupal::moduleHandler()-&gt;alter('mymodule_myfunction', $plugin_items);   Pareview.sh is giving me this error message.     \Drupal calls should be avoided in classes, use dependency injection instead.   How should I accomplish this? Do I need to create a constructor, create a method, or a service? If so, what code should the service need? I found an example on  for a case where a service already exists.