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lemmas for sum and product groups (permuting and restricting the indexing sets etc.) and a definition of finitely generated groups with some basic lemmas about the concept. New definition: finitely_generated_group and theorems: ABELIAN_SUM_GROUP CARD_EQ_BIJECTIONS_SPECIAL CARD_LE_SUBGROUP_GENERATED COUNTABLE_SUBGROUP_GENERATED CYCLIC_IMP_FINITELY_GENERATED_GROUP FINITELY_GENERATED_ABELIAN_SUBGROUP FINITELY_GENERATED_FINITE_INDEX_SUBGROUP FINITELY_GENERATED_FIXED_INDEX_SUBGROUPS FINITELY_GENERATED_GROUP FINITELY_GENERATED_GROUP_EPIMORPHIC_IMAGE FINITELY_GENERATED_GROUP_QUOTIENT_GROUP FINITELY_GENERATED_IMP_COUNTABLE_GROUP FINITELY_GENERATED_PRODUCT_GROUP FINITELY_GENERATED_PROD_GROUP FINITELY_GENERATED_SUM_GROUP FINITE_GROUP_ACTIONS FINITE_IMAGE_GEN FINITE_IMP_FINITELY_GENERATED_GROUP GROUP_ACTIONS_EQ_ON_GENERATORS GROUP_HOMOMORPHISMS_EQ_ON_GENERATORS GROUP_HOMOMORPHISM_BETWEEN_SUBGROUPS_ALT GROUP_ISOMORPHISMS_BETWEEN_SUBGROUPS_ALT GROUP_ISOMORPHISMS_PRODUCT_GROUP_DISJOINT_UNION GROUP_ISOMORPHISMS_SUM_GROUP_DISJOINT_UNION GROUP_ISOMORPHISM_EPIMORPHISM GROUP_ISOMORPHISM_EPIMORPHISM_ALT GROUP_ISOMORPHISM_PRODUCT_GROUP_DISJOINT_UNION GROUP_ISOMORPHISM_PRODUCT_PROJECTION GROUP_ISOMORPHISM_SUM_GROUP_DISJOINT_UNION GROUP_ISOMORPHISM_SUM_PROJECTION GROUP_PRODUCT_INJECTION GROUP_SUM_INJECTION HAS_SIZE_LEFT_RIGHT_COSETS ISOMORPHIC_COPY_OF_GROUP ISOMORPHIC_GROUP_FINITE_GENERATION ISOMORPHIC_GROUP_TORSION ISOMORPHIC_PRODUCT_GROUP_BIJECTIONS ISOMORPHIC_PRODUCT_GROUP_INSERT ISOMORPHIC_PRODUCT_GROUP_SING ISOMORPHIC_PRODUCT_GROUP_SUPPORT ISOMORPHIC_PRODUCT_GROUP_SYMDIFF ISOMORPHIC_SUM_GROUP_BIJECTIONS ISOMORPHIC_SUM_GROUP_DISJOINT_UNION ISOMORPHIC_SUM_GROUP_INSERT ISOMORPHIC_SUM_GROUP_SING ISOMORPHIC_SUM_GROUP_SUPPORT ISOMORPHIC_SUM_GROUP_SYMDIFF SCHREIER_TRANSVERSAL_LEMMA SUBGROUPS_GENERATED_EQ SUBGROUP_GENERATED_BY_SUBGROUP_GENERATED_IDEMPOT SUBGROUP_GENERATED_INDUCT_ALT SUBGROUP_GENERATED_INDUCT_LEFT SUBGROUP_GENERATED_INDUCT_STRONG SUBGROUP_GENERATED_UNION SUBGROUP_GENERATED_UNION_LEFT SUBGROUP_GENERATED_UNION_RIGHT SUBSET_CARRIER_SUBGROUP_GENERATED TRIVIAL_IMP_FINITELY_GENERATED_GROUP
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HOL LIGHT HOL Light is an interactive theorem prover / proof checker. It is written in Objective CAML (OCaml) and uses the toplevel from OCaml as its front end. This is the HOL Light homepage: http://www.cl.cam.ac.uk/~jrh13/hol-light/index.html and this is the root of the Github code repository: https://github.com/jrh13/hol-light Basic installation instructions are below. For more detailed information on usage, see the Tutorial: http://www.cl.cam.ac.uk/~jrh13/hol-light/tutorial.pdf Refer to the reference manual for more details of individual functions: http://www.cl.cam.ac.uk/~jrh13/hol-light/reference.html (HTML files) http://www.cl.cam.ac.uk/~jrh13/hol-light/reference.pdf (one PDF file) * * * * * * * * INSTALLATION If you use Debian Linux or some other Debian-based Linux distribution (Knoppix, Mint, Ubuntu, etc.), there is actually a "hol-light" package, thanks to Hendrik Tews, so installation of HOL Light and all its prerequisites is as simple as sudo apt-get install hol-light For other OSs, more work is involved. The Objective CAML (OCaml) implementation is a prerequisite for running HOL Light. HOL Light should work with any recent version of OCaml; I've tried it on at least 3.04, 3.06, 3.07+2, 3.08.1, 3.09.3, 3.10.0, 3.11.2 and 4.00. However, for versions >= 3.10 (in 3.10 there was an incompatible change in the camlp4 preprocessor) you will also need to get camlp5 (version >= 4.07). Installing both items of software should not be too difficult, depending on the platform. 1. OCaml: there are packages for many Linux distributions. For example, on a debian derivative like Ubuntu, you may just need to do the following: sudo apt-get install ocaml Alternatively you can download binaries directly, or get sources and build them (which in my experience is usually trouble-free). See the OCaml Web page for downloads and other information. http://caml.inria.fr/ocaml/index.en.html The HOL Light system uses the OCaml "Num" library for rational arithmetic. As of OCaml 4.06, this is no longer included in the core system and will need to be added separately. You can do this using the OCaml package manager "opam" if you use it by opam install num Alternatively you can download the sources from here https://github.com/ocaml/num and build and install them following the instructions on that page, for example git clone https://github.com/ocaml/num mynums cd mynums make all sudo make install [assuming no earlier errors] 2. camlp5: this is needed to run HOL Light under any OCaml >= 3.10. Somtimes you need a recent version of camlp5 to be compatible with your OCaml. I recommend downloading the sources for a recent version from https://camlp5.github.io/ and building it in "strict" mode before installing it, thus: cd software/camlp5-rel701 [or wherever you unpacked sources to] ./configure --strict make sudo make install [assuming no earlier errors] There are also packages for camlp5, so you may be able to get away with just something like sudo apt-get install camlp5 However, you may get a version in "transitional" instead of "strict" mode (do "camlp5 -pmode" to check which you have). Now for HOL Light itself. The instructions below assume a Unix-like environment such as Linux [or Cygwin (see www.cygwin.com) under Windows], but the steps automated by the Makefile are easy enough to invoke manually. There's more detail on doing that in the Tutorial. (0) You can download the HOL Light sources from the Github site. For example, the following will copy the code from the trunk of the Github repository into a new directory 'hol-light': git clone https://github.com/jrh13/hol-light.git The above is now the recommended way of getting HOL Light. There are also gzipped tar files on the HOL Light Web page, but they are only for quite old versions and will probably be difficult to use with recent versions of OCaml. You should next enter the 'hol-light' directory that has been created: cd ./hol-light There are now two alternatives: launch the OCaml toplevel and directly load the HOL Light source files into it, or create a standalone image with all the HOL Light sources pre-loaded. The latter is more convenient, but requires a separate checkpointing program, which may not be available for some platforms. First the basic approach: (1) Do 'make'. This ought to build the appropriate syntax extension file ('pa_j.cmo') for the version of OCaml that you're using. If you have the camlp4 or camlp5 libraries in a non-standard place rather than /usr/local/lib/ocaml/camlp4 or /usr/local/lib/ocaml/camlp5 then you may get an error like this Error while loading "pa_extend.cmo": file not found in path. in which case you should add the right directory to CAMLP4LIB or CAMLP5LIB, e.g. export CAMLP5LIB=$HOME/mylib/ocaml/camlp5 (2) Do 'ocaml'. (Actually for OCaml >= 4.02 I prefer 'ocaml -safe-string' to avoid mutable strings, while you may need something else like 'ocamlnum' on some platforms --- see [*] below.) You should see a prompt, something like: Objective Caml version 4.01.0 # (3) At the OCaml prompt '#', do '#use "hol.ml";;' (the '#' is part of the command, not the prompt) followed by a newline. This should rebuild all the core HOL Light theories, and terminate after a few minutes with the usual OCaml prompt, something like: val search : term list -> (string * thm) list = <fun> - : unit = () File "help.ml" already loaded - : unit = () - : unit = () - : unit = () Camlp5 parsing version 7.03 # HOL Light is now ready for the user to start proving theorems. You can also use the load process (2) and (3) in other directories, but you should either set the environment variable HOLLIGHT_DIR to point to the directory containing the HOL source files, or change the first line of "hol.ml" to give that explicitly, from let hol_dir = ref (try Sys.getenv "HOLLIGHT_DIR" with Not_found -> Sys.getcwd());; to, for example let hol_dir = "/home/johnh/hol-light";; or let hol_dir = "/usr/share/hol";; Now for the alternative approach of building a standalone image. The level of convenience depends on the checkpointing program you have installed. The earlier checkpointing programs in this list are more convenient to use but seem less easy to get going on recent Linux kernel/libc combinations. (1) If you have the 'ckpt' program installed, then the Makefile will conveniently create a HOL Light binary. You can get 'ckpt' here: http://www.cs.wisc.edu/~zandy/ckpt/ Once 'ckpt' is installed, simply type make hol in the 'hol-light' directory, and a standalone HOL Light image called 'hol' should be created. If desired you can move or copy this to some other place such as '~/bin' or '/usr/local/bin'. You then simply type 'hol' (or './hol') to start the system up and start proving theorems. Note that although the HOL binary will work on its own, it does not pre-load all the source files. You will probably want to keep the sources available to be loaded later as needed (if you need additional mathematical theories or tools), so it's better to unpack the HOL distribution somewhere permanent before doing 'make hol'. If you later develop a large body of proofs or tools, you can save the augmented system using the command "self_destruct" (this is the same approach as in the Makefile) rather than re-load each time. For example, the following will create a HOL Light binary (always called 'hol.snapshot'): self_destruct "My version of HOL Light";; (2) Another checkpointing option is CryoPID, which you can get here: http://cryopid.berlios.de/ In this case, the Makefile doesn't have a convenient way of making HOL binaries, but you can make one yourself once HOL Light is loaded and you are sitting in its toplevel loop. (This also works if you have your own extensions loaded, and indeed this is when it's most useful.) Instead of the 'self_destruct' command, use 'checkpoint', which is similar except that the current process is not terminated once the binary (again called hol.snapshot) is created: checkpoint "My version of HOL Light";; (3) A third option which seems to work with recent Linuxes is DMTCP, which you can download from here: http://dmtcp.sourceforge.net/ You may try installing from the packages (e.g. 'sudo dpkg -i dmtcp.deb'), but I found it was better to compile from source. HOL Light does not have convenient commands or scripts to exploit DMTCP, but you can proceed as follows: 1. Start ocaml running under the DMTCP coordinator: dmtcp_checkpoint -n ocaml 2. Use ocaml to load HOL Light as usual, for example: #use "hol.ml";; 3. From another terminal, issue the checkpoint command: dmtcp_command --checkpoint 4. (Don't forget this!) Kill the original ocaml process, e.g. by just typing control-d to the Ocaml prompt. 5. Step 3 created a checkpoint of the OCaml process and a shell script to invoke it, both in the directory in which ocaml was started. Running that should restore the OCaml process with all your state and bindings: ./dmtcp_restart_script.sh (4) If none of these options work, you may find some others on the following Web page. Unfortunately I don't know of any such checkpointing program for either Windows or Mac OS X; I would be glad to hear of one. http://checkpointing.org The directories "Library" and "Examples" may give an idea of the kind of thing that might be done, or may be useful in further work. Thanks to Carl Witty for help with Camlp4 porting and advice on checkpointing programs. * * * * * * * * [*] HOL Light uses the OCaml 'num' library for multiple-precision rationals. On many platforms, including Linux and native Windows, this will be loaded automatically by the HOL root file 'hol.ml'. However, OCaml on some platforms (notably Cygwin) does not support dynamic loading, hence the need to use 'ocamlnum', a toplevel with the 'num' library already installed. You can make your own with: ocamlmktop -o ocamlnum nums.cma