I'm trying to form a propositional logic proof chain for the tautology $\delta \implies \delta$, using only the axioms    $\alpha \implies (\beta \implies \alpha)$ $((\alpha\implies (\beta \implies\gamma))\implies((\alpha\implies\beta)\implies( \alpha\implies\gamma)))$ $((\neg\beta\implies\neg\alpha)\implies(\alpha\implies\beta))$   The only rule of inference is modus ponens.  Seems like it should be easy, but I can't seem to get it.  Help appreciated.
The system L0 is defined as follows:  Axioms: A1 (α → (β → α))  A2 2. (α → (β → γ) → ((α → β) → (α → γ))  A3 ((¬β → ¬α) → ((¬β → α) → β))  In one of my problem sheets, I am told that I am allowed to use the following theorem: if   ⊢ (α → α)    ⊢ (α → α) → (α → α)  My attempt to solve/prove the second one  ⊢ (α → α) → (α → α) choose 2 axioms which match.  α →α   β→α  γ→α  (α → (α → α) → ((α → α) → (α → α))  I'm struggling to see an obvious way of proving the theorem.  Is it provable? And if so, can you point me in the right direction?  Thanks a lot!
What is the best way to give a C# auto-property an initial value? How do you give a C# auto-property an initial value?  I either use the constructor, or revert to the old syntax.    Using the Constructor:  class Person  {     public Person()     {         Name = "Initial Name";     }     public string Name { get; set; } }   Using normal property syntax  (with an initial value)  private string name = "Initial Name"; public string Name  {     get      {         return name;     }     set     {         name = value;     } }   Is there a better way? Can I completely disable Cortana on Windows 10? Windows 10 has Cortana, which I don't like. I disabled it as soon as I could. However, looking in Task Manager, the process for Cortana is still running, and can't be effectively terminated: ending the task simply results in the process respawning a few seconds later. Using the command  taskkill /IM Cortana.exe /F   has the same result: the process respawns.  Is there any way to disable Cortana so that the process doesn't keep running in the background, and doesn't respawn if terminated? How do you apply colormanagement settings to EXR files? How do you apply colormanagement settings to EXR files?  When i do some color adjustment by using colormanagement and try to save exr file and open it in photoshop it looks like colormanagement settings was not applied to it. Which one is correct? (Maybe both are correct.)     He passed away on the morning of March 5.   Or      He passed away in the morning of March 5. How does RewriteBase work in .htaccess pkgloader is not finding l3regex I would like to try a package which seems interesting, but it some dependencies are not satisfied.   Here is my minimal example:  \documentclass{article}  \usepackage{pkgloader} \LoadPackagesNow  \begin{document}     pckloader trial \end{document}   And I get the following error: ! LaTeX Error: File 'l3regex.sty' not found.. Therefore, I tried to update my distribution with the linux commands  tlmgr update --self tlmgr update --all   But the package l3regex is still not foundable. Plus, my computer is not lying since l3regex.sty is not in my folder /texmf-dist/tex/latex/*.  So, is it something normal ? How can I workaround this problem ? I'd like to have Ubuntu installed on a USB stick, that I could plug in (almost) any computer and boot from it, and that I could use (almost) normally, as if it was installed on that computer. I'd also like to have a data partition on it that could be used by (almost) any computer, mostly to store some portable apps, just in case.    I've been told that if I installed Ubuntu in BIOS mode, I wouldn't be able to boot from a computer in UEFI mode, and vice versa.  Since I'd like to be able to boot on any computer, it's a big problem.  What can I do to be able to boot on computers in BIOS or UEFI modes in my case?  I've been searching for solutions but most of them are for specific needs that are different than mine, so it's difficult to extract the minimum requirements from those solutions to apply them to my problem.  Is there a simple, "clean" and quite generic solution to this problem?  I'd appreciate some explainations about the reasons why a specific solution is good or a specific step is needed.    My current setup:   USB stick:  OS: Xubuntu 16.04 i386 installation: full install (from a Live USB built with Rufus) drivers: none proprietary file systems:    partition table:    2: , /boot/efi, 128 Mio, FAT32, [boot,esp] 3: , 128 Mio, HFS+ 4: , 2 Mio, [bios_grub] 5: swap, 512 Mio 6: , /, 24 Gio, ext4, defaults,errors=remount-ro,lazytime,commit=60 1: , /media/data, ~3.6 Gio, FAT32, defaults,lazytime,check=strict  RAM:   /tmp, tmpfs, defaults,nosuid,nodev,noexec,mode=1777   swap:    vm.swappiness=5 swap partition: 512 Mio, for emergency zRAM swap files:    /var/swap/hib.swp: same size as RAM, for hibernation, generated by a script at startup /var/swap/opt.swp: if needed, generated by a script   overprovisioning: ~5.6%   built-in: ~1.2 Gio (apparently) added: 512 Mio  scheduler: deadline How can I remap Caps lock to both Escape and Control? Best way to parse RSS/Atom feeds with PHP Who should own files shared by a group and where should they go How to make the Java.awt.Robot type unicode characters? (Is it possible?) We have a user provided string that may contain unicode characters, and we want the robot to type that string.  How do you convert a string into keyCodes that the robot will use? How do you do it so it is also java version independant (1.3 -> 1.6)?  What we have working for "ascii" chars is  //char c = nextChar(); //char c = 'a'; // this works, and so does 'A' char c = 'á'; // this doesn't, and neither does 'Ă' Robot robot = new Robot(); KeyStroke key = KeyStroke.getKeyStroke("pressed " + Character.toUpperCase(c) ); if( null != key ) {   // should only have to worry about case with standard characters   if (Character.isUpperCase(c))   {     robot.keyPress(KeyEvent.VK_SHIFT);   }    robot.keyPress(key.getKeyCode());   robot.keyRelease(key.getKeyCode());    if (Character.isUpperCase(c))   {     robot.keyRelease(KeyEvent.VK_SHIFT);   } } How to create tagged PDF such that they will be:   good enough for PDF/UA "reflowable" on smaller screens and ebook readers   Any syntax &amp; any engine are of interest, but I'm most interested in using LaTeX with [Xe|pdf]latex engines. Using restroom on airplane while fasten seatbelt light is on? This question about  over on  made me wonder what a passenger is supposed to do when they urgently have to use the restroom while the fasten seat belt light is on. FAA regulations require passengers to obey the seat-belt sign, even if flight attendants do not.  I saw this happen on a flight where an older gentleman asked the flight attendant if he could use the restroom, and she said "not until the light is off". The flight had been free of turbulence for 15 minutes or so by this time, and the flight attendants were walking freely throughout the cabin, collecting post-meal trash, etc. The gentleman who asked to use the restroom appeared to be in quite some discomfort, and practically ran to the restroom when the light went off 10 minutes later.  So what is a passenger supposed to do when he isn't permitted to use the lavatory, when he has to go and can't wait? Just go in their seat? How can I "group by" and sum a column in excel? I'm trying to figure out how to "group by" values in an Excel spreadsheet.  Given the following example (this is an inventory sheet I use at my restaurant:)    At the far right of the spreadsheet there is a column named Cost of Goods Sold.  I would like to be able to come up with a total COGS figure for each Category.  Is this possible in excel? Moreover, I'd like to do it in a separate worksheet so that the first worksheet (what is pictured) isn't affected.  Any ideas?  Thanks! Leading zero in umask 0022 How can I get a reference to a method? Is it possible in Ruby to get a reference to methods of an object ( I would like to know if this can be done without procs/lambdas ) , for example , consider the following code :   class X   def initialize     @map = {}     setup_map   end    private   def setup_map     # @map["a"] = get reference to a method     # @map["b"] = get reference to b method     # @map["c"] = get referebce to c method   end    public   def call(a)     @map["a"](a) if a > 10     @map["b"](a) if a > 20     @map["c"](a) if a > 30   end    def a(arg)      puts "a was called with #{arg}"   end    def b(arg)      puts "b was called with #{arg}"   end    def c(arg)     puts "c was called with #{arg}"   end end   Is it possible to do such thing ? I would like to avoid procs/lambdas because I want to be able to change the behaviour of A,B,C by subclassing . Complex Collision Boxes I'm trying to get a ball to go through in between the cup gear things shown in the screenshot below.   It seems that the collision box does not consider the cups. Yes, the ball fits and they are rotating in the right direction.  If there is already an answer to this, please link it.       If you have any questions about what is going on/ what the project is, feel free to ask. How to spawn a Giant with equipment? I really want to spawn Giant with armor and swords. Because Giants are strong, and great challenge if spawned with enchanted diamond armor and a diamond sword. I am looking at collecting spatial information from a range of stakeholders to identify opportunities within a landscape.  However, the data that we are collecting are not presently mapped and so we wish to use an interactive method to get this information onto a map.  Which online websites/methods would people recommend for circulating to a non-technical audience to try and get them to map what they know?    Where the emphasis is on ease of use by the stakeholder. When a player joins the server, this command block (containing gamemode 2 @a) puts them into gamemode 2/adventure mode. How can I change this so that only non-admin/not "opped" players' gamemode gets changed?