I need help about proving the following sequent:  $$ \vdash Q \rightarrow (P \rightarrow Q) $$  I tried to prove it by trying to introduce a conditional proof, but somehow can't get the suppositions correctly in the beginning. I would appreciate some help.   Additionally, here is another similar sequent I've supposedly proved, but not completely sure:   $$ \vdash Q \rightarrow (P \rightarrow P) $$    Explanations for the proof:  CP: Conditional Proof Introduction  A: Assumption  Leftmost Column: Score-board for keeping track of suppositions.  I'd appreciate any help and suggestions.
Firstly, how do I read it? Is below right? With braces   $$p \vdash (q \Rightarrow p)$$  The given proof is:    ${p}$, premise ${q}$, assumption  $p$ by (1) // what???  ${q} \Rightarrow p$ by implication introduction with 2 and 2.1 QED...    I can't really get the link between $p$ and $q$ in 2.1 ...
Centering figure captions in IEEEtran I am working on a paper using the IEEEtran template in LyX.  I have run into a problem: While the contents (in this case, a table) of the figure float are centered, the caption is aligned to the left (see screenshot below). How can I center the caption as well?    This is generated by the following LyX document:    What I've tried:   Center the table (the table is centered) Center all the contents in the float (only the table is centered) Select the caption and center it (no effect)   Technical details:   LyX 2.0.5.1 Mac OS X 10.7.5 (Lion) All centering was performed by Right Click>Paragraph Settings>Align: Center If a blocking creature leaves the battlefield, is the attacker still blocked, or will it damage the opponent? Suppose Player 1 has a Dragon on the table and  in hand, while Player 2 has two Spiders on the table.  Player 1 chooses to attack with the Dragon and Player 2 chooses to block with one of the Spiders.  Before combat damage is to be dealt, Player 1 uses Unsummon on the blocking Spider.    Does Player 2 take any life damage from the attacking Dragon? I already searched the internet but could not find a solution to this. Maybe it is the same question as  but there was no answer.  I want to render a video with tracking, but the output "loses" color information and gets kind of greyish. I first thought that it is related to the renderer, but it happens in Cycles and Eevee. Therefore i made a sample image, added it to the image sequence editor and rendered it as 1 frame. Resulting in the same problem.  I placed the original image beside blenders output so that you can see the difference. The only idea I came up with is that something with the lighting is not correct but I can't find a setting for that.  I am using Blender 2.80 with the default settings. Only set the output resolution to match the input resolution. What determines speciation at a molecular level?  At what point does a scientist determine two lineages are different enough to be considered separate species?  Does it have a margin of error? R Difference between rbinom and sample I'm trying to model a Roulette game where if you bet on red (out of red(18/37), black(18/37), green(1/37)) and if you the ball lands on red you win 1 dollar and if it doesn't you lose $1. So in this case, is it a binomial distribution? Since there are only 2 outcomes from the gamblers perspective, red or not red. If thats the case are these 2 R functions the same?  set.seed(3) sample(c(0,1),100,replace=T,prob=c(19/37,18/37))     rbinom(n = 100, size = 1, prob = 18/37) Sometimes when I'm making a sauce or soup that needs to be a bit spicy, I have no troubles spicing how I want to.  But if the dinner party is a bit late, or I made the sauce in advance to really draw out all the flavors of the spices, vegetables and meat, sometimes I end up with a way too spicy/hot dish.  You can always pour a bit water and try to dilute it a bit, but that also takes away all of the good flavor. So is there some trick that I'm missing I can use, when this happens to me (other than being more careful or using the spices later)? How to run 32-bit app in Ubuntu 64-bit? Duplicate explanation should encourage improvement of original Fast way of finding lines in one file that are not in another? Deleting multiple elements from a list Should "move-to-chat" be provided on demand (as a button perhaps)? Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I'm using Ubuntu 12.04 &amp; everything about it is great except that it just won't resume after suspend. I've tried the suggestions listed in this thread  but none worked. I'm using Lenovo Y510P. I'm new to Ubuntu, or unix for that matter, so please let me know if there's anymore  information that I should be posting. Thanks.  update  I tried the suggestion by Dima on this post . The machine wakes up from hibernation but immediately the screen gives a screen like this     Wondering if this is an issue with the graphics card or something. Prepositions used with "inquire" out file with multiple criteria in python? How can I execute a file without giving myself execute permissions (with chmod u+x) for it?   If I try and tick the 'Allow executing file as program' checkmark, the checkmark is immediately removed. python get only class attribute no superclasses Does D&amp;D 5E have a mechanic similar to  or  as it exists in D&amp;D 3.5?  To put it simply, in D&amp;D 3.5, touch AC excludes armor bonuses when armor can't help against an attack, and flatfooted AC excludes your Dexterity bonus when you haven't acted yet in combat. MSO Review Queue Privileges are Misleading When I visit Stack Overflow, I see the review option in the bar at the top of the screen.    However when on MSO, I don't see it. I assumed this was because I didn't have enough reputation to access the queue. I was curious what the requirement was on meta, so I moseyed on over to the privileges page. To my surprise, I saw that the privilege is listed as only requiring 500 rep and informs me I already have it:    Confused, I manually typed in meta.stackoverflow.com/review to see what would happen. The page did in fact load, but as you can see meta has no "First Posts" or "Late Answers" queue.    Seeing on that page that a minimum of 2k reputation is required to access any queues on MSO, I report the following bug:  1) On the privileges list on MSO, the 500 rep privilege of "Access first posts and late answers review queues" should not appear.  I also have the following feature request:  2) On the privileges list on MSO, it would be useful if instead there was a 2k privilege that informed you that you have access to the "Suggested Edits" and "Low Quality Posts" queues. Cannot get grub menu to timeout (or go away)