Showing results for levels at canonical
GitHub Repo
https://github.com/heru299/script-copy
heru299/script-copy
-? Print this help message and exit -alertnotify=<cmd> Execute command when a relevant alert is received or we see a really long fork (%s in cmd is replaced by message) -assumevalid=<hex> If this block is in the chain assume that it and its ancestors are valid and potentially skip their script verification (0 to verify all, default: 0000000000000000000b9d2ec5a352ecba0592946514a92f14319dc2b367fc72, testnet: 000000000000006433d1efec504c53ca332b64963c425395515b01977bd7b3b0, signet: 0000002a1de0f46379358c1fd09906f7ac59adf3712323ed90eb59e4c183c020) -blockfilterindex=<type> Maintain an index of compact filters by block (default: 0, values: basic). If <type> is not supplied or if <type> = 1, indexes for all known types are enabled. -blocknotify=<cmd> Execute command when the best block changes (%s in cmd is replaced by block hash) -blockreconstructionextratxn=<n> Extra transactions to keep in memory for compact block reconstructions (default: 100) -blocksdir=<dir> Specify directory to hold blocks subdirectory for *.dat files (default: <datadir>) -blocksonly Whether to reject transactions from network peers. Automatic broadcast and rebroadcast of any transactions from inbound peers is disabled, unless the peer has the 'forcerelay' permission. RPC transactions are not affected. (default: 0) -conf=<file> Specify path to read-only configuration file. Relative paths will be prefixed by datadir location. (default: bitcoin.conf) -daemon Run in the background as a daemon and accept commands -datadir=<dir> Specify data directory -dbcache=<n> Maximum database cache size <n> MiB (4 to 16384, default: 450). In addition, unused mempool memory is shared for this cache (see -maxmempool). -debuglogfile=<file> Specify location of debug log file. Relative paths will be prefixed by a net-specific datadir location. (-nodebuglogfile to disable; default: debug.log) -includeconf=<file> Specify additional configuration file, relative to the -datadir path (only useable from configuration file, not command line) -loadblock=<file> Imports blocks from external file on startup -maxmempool=<n> Keep the transaction memory pool below <n> megabytes (default: 300) -maxorphantx=<n> Keep at most <n> unconnectable transactions in memory (default: 100) -mempoolexpiry=<n> Do not keep transactions in the mempool longer than <n> hours (default: 336) -par=<n> Set the number of script verification threads (-8 to 15, 0 = auto, <0 = leave that many cores free, default: 0) -persistmempool Whether to save the mempool on shutdown and load on restart (default: 1) -pid=<file> Specify pid file. Relative paths will be prefixed by a net-specific datadir location. (default: bitcoind.pid) -prune=<n> Reduce storage requirements by enabling pruning (deleting) of old blocks. This allows the pruneblockchain RPC to be called to delete specific blocks, and enables automatic pruning of old blocks if a target size in MiB is provided. This mode is incompatible with -txindex and -rescan. Warning: Reverting this setting requires re-downloading the entire blockchain. (default: 0 = disable pruning blocks, 1 = allow manual pruning via RPC, >=550 = automatically prune block files to stay under the specified target size in MiB) -reindex Rebuild chain state and block index from the blk*.dat files on disk -reindex-chainstate Rebuild chain state from the currently indexed blocks. When in pruning mode or if blocks on disk might be corrupted, use full -reindex instead. -settings=<file> Specify path to dynamic settings data file. Can be disabled with -nosettings. File is written at runtime and not meant to be edited by users (use bitcoin.conf instead for custom settings). Relative paths will be prefixed by datadir location. (default: settings.json) -startupnotify=<cmd> Execute command on startup. -sysperms Create new files with system default permissions, instead of umask 077 (only effective with disabled wallet functionality) -txindex Maintain a full transaction index, used by the getrawtransaction rpc call (default: 0) -version Print version and exit Connection options: -addnode=<ip> Add a node to connect to and attempt to keep the connection open (see the `addnode` RPC command help for more info). This option can be specified multiple times to add multiple nodes. -asmap=<file> Specify asn mapping used for bucketing of the peers (default: ip_asn.map). Relative paths will be prefixed by the net-specific datadir location. -bantime=<n> Default duration (in seconds) of manually configured bans (default: 86400) -bind=<addr>[:<port>][=onion] Bind to given address and always listen on it (default: 0.0.0.0). Use [host]:port notation for IPv6. Append =onion to tag any incoming connections to that address and port as incoming Tor connections (default: 127.0.0.1:8334=onion, testnet: 127.0.0.1:18334=onion, signet: 127.0.0.1:38334=onion, regtest: 127.0.0.1:18445=onion) -connect=<ip> Connect only to the specified node; -noconnect disables automatic connections (the rules for this peer are the same as for -addnode). This option can be specified multiple times to connect to multiple nodes. -discover Discover own IP addresses (default: 1 when listening and no -externalip or -proxy) -dns Allow DNS lookups for -addnode, -seednode and -connect (default: 1) -dnsseed Query for peer addresses via DNS lookup, if low on addresses (default: 1 unless -connect used) -externalip=<ip> Specify your own public address -forcednsseed Always query for peer addresses via DNS lookup (default: 0) -listen Accept connections from outside (default: 1 if no -proxy or -connect) -listenonion Automatically create Tor onion service (default: 1) -maxconnections=<n> Maintain at most <n> connections to peers (default: 125) -maxreceivebuffer=<n> Maximum per-connection receive buffer, <n>*1000 bytes (default: 5000) -maxsendbuffer=<n> Maximum per-connection send buffer, <n>*1000 bytes (default: 1000) -maxtimeadjustment Maximum allowed median peer time offset adjustment. Local perspective of time may be influenced by peers forward or backward by this amount. (default: 4200 seconds) -maxuploadtarget=<n> Tries to keep outbound traffic under the given target (in MiB per 24h). Limit does not apply to peers with 'download' permission. 0 = no limit (default: 0) -networkactive Enable all P2P network activity (default: 1). Can be changed by the setnetworkactive RPC command -onion=<ip:port> Use separate SOCKS5 proxy to reach peers via Tor onion services, set -noonion to disable (default: -proxy) -onlynet=<net> Make outgoing connections only through network <net> (ipv4, ipv6 or onion). Incoming connections are not affected by this option. This option can be specified multiple times to allow multiple networks. -peerblockfilters Serve compact block filters to peers per BIP 157 (default: 0) -peerbloomfilters Support filtering of blocks and transaction with bloom filters (default: 0) -permitbaremultisig Relay non-P2SH multisig (default: 1) -port=<port> Listen for connections on <port>. Nodes not using the default ports (default: 8333, testnet: 18333, signet: 38333, regtest: 18444) are unlikely to get incoming connections. -proxy=<ip:port> Connect through SOCKS5 proxy, set -noproxy to disable (default: disabled) -proxyrandomize Randomize credentials for every proxy connection. This enables Tor stream isolation (default: 1) -seednode=<ip> Connect to a node to retrieve peer addresses, and disconnect. This option can be specified multiple times to connect to multiple nodes. -timeout=<n> Specify connection timeout in milliseconds (minimum: 1, default: 5000) -torcontrol=<ip>:<port> Tor control port to use if onion listening enabled (default: 127.0.0.1:9051) -torpassword=<pass> Tor control port password (default: empty) -upnp Use UPnP to map the listening port (default: 0) -whitebind=<[permissions@]addr> Bind to the given address and add permission flags to the peers connecting to it. Use [host]:port notation for IPv6. Allowed permissions: bloomfilter (allow requesting BIP37 filtered blocks and transactions), noban (do not ban for misbehavior; implies download), forcerelay (relay transactions that are already in the mempool; implies relay), relay (relay even in -blocksonly mode, and unlimited transaction announcements), mempool (allow requesting BIP35 mempool contents), download (allow getheaders during IBD, no disconnect after maxuploadtarget limit), addr (responses to GETADDR avoid hitting the cache and contain random records with the most up-to-date info). Specify multiple permissions separated by commas (default: download,noban,mempool,relay). Can be specified multiple times. -whitelist=<[permissions@]IP address or network> Add permission flags to the peers connecting from the given IP address (e.g. 1.2.3.4) or CIDR-notated network (e.g. 1.2.3.0/24). Uses the same permissions as -whitebind. Can be specified multiple times. Wallet options: -addresstype What type of addresses to use ("legacy", "p2sh-segwit", or "bech32", default: "bech32") -avoidpartialspends Group outputs by address, selecting all or none, instead of selecting on a per-output basis. Privacy is improved as an address is only used once (unless someone sends to it after spending from it), but may result in slightly higher fees as suboptimal coin selection may result due to the added limitation (default: 0 (always enabled for wallets with "avoid_reuse" enabled)) -changetype What type of change to use ("legacy", "p2sh-segwit", or "bech32"). Default is same as -addresstype, except when -addresstype=p2sh-segwit a native segwit output is used when sending to a native segwit address) -disablewallet Do not load the wallet and disable wallet RPC calls -discardfee=<amt> The fee rate (in BTC/kB) that indicates your tolerance for discarding change by adding it to the fee (default: 0.0001). Note: An output is discarded if it is dust at this rate, but we will always discard up to the dust relay fee and a discard fee above that is limited by the fee estimate for the longest target -fallbackfee=<amt> A fee rate (in BTC/kB) that will be used when fee estimation has insufficient data. 0 to entirely disable the fallbackfee feature. (default: 0.00) -keypool=<n> Set key pool size to <n> (default: 1000). Warning: Smaller sizes may increase the risk of losing funds when restoring from an old backup, if none of the addresses in the original keypool have been used. -maxapsfee=<n> Spend up to this amount in additional (absolute) fees (in BTC) if it allows the use of partial spend avoidance (default: 0.00) -mintxfee=<amt> Fees (in BTC/kB) smaller than this are considered zero fee for transaction creation (default: 0.00001) -paytxfee=<amt> Fee (in BTC/kB) to add to transactions you send (default: 0.00) -rescan Rescan the block chain for missing wallet transactions on startup -spendzeroconfchange Spend unconfirmed change when sending transactions (default: 1) -txconfirmtarget=<n> If paytxfee is not set, include enough fee so transactions begin confirmation on average within n blocks (default: 6) -wallet=<path> Specify wallet path to load at startup. Can be used multiple times to load multiple wallets. Path is to a directory containing wallet data and log files. If the path is not absolute, it is interpreted relative to <walletdir>. This only loads existing wallets and does not create new ones. For backwards compatibility this also accepts names of existing top-level data files in <walletdir>. -walletbroadcast Make the wallet broadcast transactions (default: 1) -walletdir=<dir> Specify directory to hold wallets (default: <datadir>/wallets if it exists, otherwise <datadir>) -walletnotify=<cmd> Execute command when a wallet transaction changes. %s in cmd is replaced by TxID and %w is replaced by wallet name. %w is not currently implemented on windows. On systems where %w is supported, it should NOT be quoted because this would break shell escaping used to invoke the command. -walletrbf Send transactions with full-RBF opt-in enabled (RPC only, default: 0) ZeroMQ notification options: -zmqpubhashblock=<address> Enable publish hash block in <address> -zmqpubhashblockhwm=<n> Set publish hash block outbound message high water mark (default: 1000) -zmqpubhashtx=<address> Enable publish hash transaction in <address> -zmqpubhashtxhwm=<n> Set publish hash transaction outbound message high water mark (default: 1000) -zmqpubrawblock=<address> Enable publish raw block in <address> -zmqpubrawblockhwm=<n> Set publish raw block outbound message high water mark (default: 1000) -zmqpubrawtx=<address> Enable publish raw transaction in <address> -zmqpubrawtxhwm=<n> Set publish raw transaction outbound message high water mark (default: 1000) -zmqpubsequence=<address> Enable publish hash block and tx sequence in <address> -zmqpubsequencehwm=<n> Set publish hash sequence message high water mark (default: 1000) Debugging/Testing options: -debug=<category> Output debugging information (default: -nodebug, supplying <category> is optional). If <category> is not supplied or if <category> = 1, output all debugging information. <category> can be: net, tor, mempool, http, bench, zmq, walletdb, rpc, estimatefee, addrman, selectcoins, reindex, cmpctblock, rand, prune, proxy, mempoolrej, libevent, coindb, qt, leveldb, validation. -debugexclude=<category> Exclude debugging information for a category. Can be used in conjunction with -debug=1 to output debug logs for all categories except one or more specified categories. -help-debug Print help message with debugging options and exit -logips Include IP addresses in debug output (default: 0) -logthreadnames Prepend debug output with name of the originating thread (only available on platforms supporting thread_local) (default: 0) -logtimestamps Prepend debug output with timestamp (default: 1) -maxtxfee=<amt> Maximum total fees (in BTC) to use in a single wallet transaction; setting this too low may abort large transactions (default: 0.10) -printtoconsole Send trace/debug info to console (default: 1 when no -daemon. To disable logging to file, set -nodebuglogfile) -shrinkdebugfile Shrink debug.log file on client startup (default: 1 when no -debug) -uacomment=<cmt> Append comment to the user agent string Chain selection options: -chain=<chain> Use the chain <chain> (default: main). Allowed values: main, test, signet, regtest -signet Use the signet chain. Equivalent to -chain=signet. Note that the network is defined by the -signetchallenge parameter -signetchallenge Blocks must satisfy the given script to be considered valid (only for signet networks; defaults to the global default signet test network challenge) -signetseednode Specify a seed node for the signet network, in the hostname[:port] format, e.g. sig.net:1234 (may be used multiple times to specify multiple seed nodes; defaults to the global default signet test network seed node(s)) -testnet Use the test chain. Equivalent to -chain=test. Node relay options: -bytespersigop Equivalent bytes per sigop in transactions for relay and mining (default: 20) -datacarrier Relay and mine data carrier transactions (default: 1) -datacarriersize Maximum size of data in data carrier transactions we relay and mine (default: 83) -minrelaytxfee=<amt> Fees (in BTC/kB) smaller than this are considered zero fee for relaying, mining and transaction creation (default: 0.00001) -whitelistforcerelay Add 'forcerelay' permission to whitelisted inbound peers with default permissions. This will relay transactions even if the transactions were already in the mempool. (default: 0) -whitelistrelay Add 'relay' permission to whitelisted inbound peers with default permissions. This will accept relayed transactions even when not relaying transactions (default: 1) Block creation options: -blockmaxweight=<n> Set maximum BIP141 block weight (default: 3996000) -blockmintxfee=<amt> Set lowest fee rate (in BTC/kB) for transactions to be included in block creation. (default: 0.00001) RPC server options: -rest Accept public REST requests (default: 0) -rpcallowip=<ip> Allow JSON-RPC connections from specified source. Valid for <ip> are a single IP (e.g. 1.2.3.4), a network/netmask (e.g. 1.2.3.4/255.255.255.0) or a network/CIDR (e.g. 1.2.3.4/24). This option can be specified multiple times -rpcauth=<userpw> Username and HMAC-SHA-256 hashed password for JSON-RPC connections. The field <userpw> comes in the format: <USERNAME>:<SALT>$<HASH>. A canonical python script is included in share/rpcauth. The client then connects normally using the rpcuser=<USERNAME>/rpcpassword=<PASSWORD> pair of arguments. This option can be specified multiple times -rpcbind=<addr>[:port] Bind to given address to listen for JSON-RPC connections. Do not expose the RPC server to untrusted networks such as the public internet! This option is ignored unless -rpcallowip is also passed. Port is optional and overrides -rpcport. Use [host]:port notation for IPv6. This option can be specified multiple times (default: 127.0.0.1 and ::1 i.e., localhost) -rpccookiefile=<loc> Location of the auth cookie. Relative paths will be prefixed by a net-specific datadir location. (default: data dir) -rpcpassword=<pw> Password for JSON-RPC connections -rpcport=<port> Listen for JSON-RPC connections on <port> (default: 8332, testnet: 18332, signet: 38332, regtest: 18443) -rpcserialversion Sets the serialization of raw transaction or block hex returned in non-verbose mode, non-segwit(0) or segwit(1) (default: 1) -rpcthreads=<n> Set the number of threads to service RPC calls (default: 4) -rpcuser=<user> Username for JSON-RPC connections -rpcwhitelist=<whitelist> Set a whitelist to filter incoming RPC calls for a specific user. The field <whitelist> comes in the format: <USERNAME>:<rpc 1>,<rpc 2>,...,<rpc n>. If multiple whitelists are set for a given user, they are set-intersected. See -rpcwhitelistdefault documentation for information on default whitelist behavior. -rpcwhitelistdefault Sets default behavior for rpc whitelisting. Unless rpcwhitelistdefault is set to 0, if any -rpcwhitelist is set, the rpc server acts as if all rpc users are subject to empty-unless-otherwise-specified whitelists. If rpcwhitelistdefault is set to 1 and no -rpcwhitelist is set, rpc server acts as if all rpc users are subject to empty whitelists. -server Accept command line and JSON-RPC commands ~ $
GitHub Repo
https://github.com/kenkinn/cain_conditionalindependence-
kenkinn/cain_conditionalindependence-
(* This file is maintained by Jinfang Wang (wang@math.s.chiba-u.ac.jp). *) Require Import ssreflect ssrfun ssrnat ssrbool ssralg ssrint. Require Import eqtype seq choice bigop fintype finset finfun. (***********************************************************************************************) (* This file contains the scripts related to the following paper, *) (* `Formalization of Probabilistic Conditional Independence Using Coq/SSReflect` *) (* by Jinfang Wang, Manabu Hagiwara and Mitsuharu Yamamoto, *) (* submitted to ITP2015 for possible publication. *) (* *) (* These scripts are compatiable with Coq v8.4 and SSReflect v.1.5. *) (* This file was last updated on 2015/3/12. *) (* *) (* The theory of cain is built upon a bounded lattice. *) (* L == of eqType is a bounded lattice. *) (* bot == bottom of lattice L. *) (* top == top of lattice L. *) (* meet == the binary meet operator, a function of type L->L->L. *) (* join == the binary meet operator, a function of type L->L->L. *) (* ge x y == returns [true] if [x >= y], where x and y are elements of L. *) (* gt x y == returns [true] if [x >y], where x and y are elements of L. *) (* le x y == returns [true] if [x <= y], where x and y are elements of L. *) (* lt x y == returns [true] if [x <y], where x and y are elements of L. *) (* complement_P x y == a proposision if y is a complement of x. *) (* complement_b x y == returns [true] if y is a complement of x. *) (* com x == the complement of x. *) (* *) (* The following are basic components of a cainoid, a pre-form of a cain. *) (* coins == of choiceType is the set of coins. *) (* dot == the dot product, a function of coins. *) (* Mix x y == the Mixed coin . *) (* bob == the unit coin . *) (* up x == the up coin with raising context x. *) (* down x == the down coin with lowering context x . *) (* coin_prod_atomics == an axiom saying that each coin is a product of *) (* finite number of atom coins. *) (* up_injective up == asserting that [up] is an injective function. *) (* down_injective down == asserting that [down] is an injective function. *) (* [up_injective] and [down_injective] are triky and may be removed. *) (* *) (* The following are the five axioms of a cainoid. *) (* dotC == dot product is commutative. *) (* dotA == dot product is associative. *) (* bob_unitL == bob is a unit element from left. *) (* up_down_unitL x == [up x] is the inverse of [down x]; *) (* that is, [dot (up x) (down x) = bob]. *) (* Mix_up_down x y == a property corresponding to the definition of *) (* conditional desity functions in probability theory. *) (* *) (* Definition of independence and conditional independence. *) (* CIP x y z == a proposition for independence of x and y given z. *) (* IP x y == a proposition for independence of x and y. *) (* acinv (x, y) == defines the inverse of the atom coin [Mix x y]. *) (* dot_inv c == defines the inverse of an arbitrary coin c. *) (* *) (* A cainoid satisfies all properties of an abelian group. *) (* These properties can be put together using the record type ZmodMixin. *) (* zmodMixin == ZmodMixin dotA dotC bob_unitL dotV. *) (* zmodType == ZmodType _ zmodMixin. *) (* x \+ y == notation for dot x y. *) (* why not use the notation " _ . _" ? *) (***********************************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. (* (*formalization of the fintary case*) Variable T : finType. Variable D : {set T}. Variable X Y Z : {set T}. (*有限集合*) Definition finjoin (X Y : {set {set T}}) := X :|: Y. Definition finmeet (X Y : {set {set T}}) := X :&: Y. Lemma finjoin_commu : commutative finjoin. Proof. apply/ setUC. Qed. Lemma finjoin_assoc : associative finjoin. Proof. apply/ setUA. Qed. Lemma finjoin_absorp : forall X Y, finmeet X (finjoin X Y) = X. Proof. move=> X Y. rewrite finjoin_commu ; apply/ setKU. Qed. Lemma finmeet_commu : commutative finmeet. Proof. apply/ setIC. Qed. Lemma finmeet_assoc : associative finmeet. Proof. apply/ setIA. Qed. Lemma finmeet_absorp : forall X Y, finjoin X (finmeet X Y) = X. Proof. move=> X Y. rewrite finmeet_commu. apply/ setKI. Qed. Lemma finmeet_distr : forall X Y Z, finmeet X (finjoin Y Z) = finjoin (finmeet X Y) (finmeet X Z). Proof. move=> X Y Z. apply/ setIUr. Qed. Definition fincom (X : {set {set T}}) := ~:X. Lemma fincom_bot : forall X, finmeet X (fincom X) = set0. Proof. move=> X. apply/ setICr. Qed. Lemma fincom_top : forall X, finjoin X (fincom X) = setT. Proof. move=> X. apply/ setUCr. Qed. Record lattice_mixin_of (T : Type) : Type := LatticeMixin { join_op : T -> T -> T; meet_op : T -> T -> T; bot_op : T; top_op : T; _ : commutative join_op; _ : associative join_op; _ : forall x y, meet_op x (join_op x y) = x; _ : commutative meet_op; _ : associative meet_op; _ : forall x y, join_op x (meet_op x y) = x; _ : forall x y z, meet_op x (join_op y z) = join_op (meet_op x y) (meet_op x z); com_op : T -> T; com_op_bot : forall x, meet_op x (com_op x) = bot_op; com_op_top : forall x, join_op x (com_op x) = top_op }. Record lattice : Type := Lattice { carrier :> Type; spec : lattice_mixin_of carrier }. Definition finitary_latticemixin := LatticeMixin finjoin_commu finjoin_assoc finjoin_absorp finmeet_commu finmeet_assoc finmeet_absorp finmeet_distr fincom_bot fincom_top. Canonical fin_lattice := Lattice finitary_latticemixin. (*End of the formalization of the finitary Lattice*) *) (* Basic theories on bounded lattice with a bottom and a top. *) Section Lattice. Parameter L : eqType. Parameters bot top: L. Parameters meet join : L->L->L. Axiom join_commutative : commutative join. Axiom join_associative : associative join. Axiom join_absorption : forall x y, meet x (join x y) = x. Axiom meet_commutative : commutative meet. Axiom meet_associative : associative meet. Axiom meet_absorption : forall x y, join x (meet x y) = x. Axiom meet_distributive : forall x y z, meet x (join y z) = join (meet x y) (meet x z). Lemma join_distributive x y z : join x (meet y z) = meet (join x y) (join x z). Proof. by rewrite [in RHS]meet_distributive [meet _ x]meet_commutative join_absorption [in RHS]meet_commutative [in RHS]meet_distributive join_associative [meet z x]meet_commutative meet_absorption [in RHS]meet_commutative. Qed. Lemma join_idempotent(*冪等性*) : idempotent join. Proof. move=>x. move: (join_absorption x x) => Haj. move: (meet_absorption x (join x x)) => Ham. by rewrite -{2}[in LHS ]Haj. Qed. Lemma meet_idempotent : idempotent meet. Proof. move=>x. move: (meet_absorption x x) => Ham. move: (join_absorption x (meet x x)) => Haj. by rewrite -{2}[in LHS ]Ham. Qed. (* Introduce partial order in L *) Definition ge x y := join x y == x. (* <=> (y <= x) *) Definition gt x y := (ge x y) && (x !=y). (* <=> (y < x) *) Definition le x y := ge y x. (* <=> (x <= y)*) Definition lt x y := (le x y) && (x !=y). (* <=> (x < y)*) Axiom bot_minimum : forall x, ge x bot. Axiom top_maximum : forall x, ge top x. Lemma ge_reflexive : reflexive ge. Proof. by move => x; rewrite /ge join_idempotent. Qed. Lemma ge_transitive : transitive ge. Proof. rewrite /ge => x y z /eqP Hyx /eqP Hxz. apply/eqP. by rewrite -Hyx -[in RHS]Hxz join_associative. Qed. Lemma ge_antisymmetric : antisymmetric ge. (*反対称性、x~y & y~x => x=y*) Proof. rewrite /ge => x y /andP [] /eqP Hxy /eqP <-. by rewrite join_commutative. Qed. Lemma join_ge x y : ge (join x y) x. Proof. rewrite /ge. by rewrite -!join_associative [join y x] join_commutative join_associative join_idempotent. Qed. (* [complement_P x y] is a relation between x and y. *) Definition complement_P x y := (meet x y = bot) /\ (join x y = top). (* [complement_b x y] is [true] if y is a complement of x. *) Definition complement_b x y := (meet x y == bot) && (join x y == top). (* Given x of L, [com x] returns the complement of x. *) Parameter com : L -> L. Axiom com_bot : forall x, meet x (com x) = bot. Axiom com_top : forall x, join x (com x) = top. Lemma complement_eqP x y : complement_P x y <-> (complement_b x y) = true. Proof. apply conj. rewrite /complement_P /complement_b => c. apply /andP. destruct c. by rewrite H H0. rewrite /complement_P /complement_b. case /andP => h1 h2. by apply conj; apply/eqP. Qed. Lemma com_P x : complement_P x (com x). Proof. rewrite /complement_P. apply conj. apply com_bot. apply com_top. Restart. rewrite /complement_P. apply conj. by apply com_bot. by apply com_top. Qed. (* Like [ge], [gt] is also transitive. *) Lemma gt_transitive : transitive gt. Proof. rewrite /gt /transitive. move=>y x z. move /andP. case. move=>gxy nxy. move /andP. case. move=>gyz nyz. rewrite (ge_transitive gxy gyz) /=. (*?*) apply /eqP. case. move=>xz. move:nxy. move /eqP. case. apply:ge_antisymmetric. rewrite gxy. rewrite xz. rewrite gyz. by[]. Restart. rewrite /gt => y x z /andP [] gxy nxy /andP [] gyz nyz. rewrite (ge_transitive gxy gyz) /=. apply/eqP => xz. move: nxy. move/eqP. case. apply: ge_antisymmetric. by rewrite gxy xz gyz. Qed. (* an alternative proof *) Goal transitive gt. Proof. rewrite /gt => y x z /andP [] gxy nxy /andP [] gyz nyz. apply /andP; apply conj. by move: gyz; exact: ge_transitive. apply /negP. move=> /eqP xz. rewrite xz in gxy. move: nyz; move /negP. case. apply/eqP; apply: ge_antisymmetric. by apply /andP. Qed. Lemma top_unique y : (forall x, ge y x) -> y == top. Proof. move => Hge; apply /eqP; apply: ge_antisymmetric. by rewrite Hge top_maximum. Qed. Lemma bot_unique y : (forall x, ge x y) -> y == bot. Proof. move => Hge; apply /eqP; apply: ge_antisymmetric. by rewrite Hge bot_minimum. Qed. Lemma join_bot_unitL x : join bot x = x. Proof. by rewrite join_commutative; move: (bot_minimum x); rewrite/ge=> H; apply /eqP. Qed. Lemma join_bot_unitR x : join x bot = x. Proof. by rewrite join_commutative join_bot_unitL. Qed. Lemma meet_bot_unitL x : meet bot x = bot. Proof. move: (join_bot_unitL x). move=> H. by rewrite -H join_absorption. Qed. Lemma meet_bot_unitR x : meet x bot = bot. Proof. by rewrite meet_commutative meet_bot_unitL. Qed. Lemma join_bot : join bot bot = bot. Proof. by move: (join_bot_unitL bot). Qed. Lemma join_top_unitL x : join top x = top. Proof. by rewrite -(com_top x) join_commutative join_associative join_idempotent. Qed. Lemma join_top_unitR x : join x top = top. Proof. by rewrite join_commutative join_top_unitL. Qed. Lemma meet_top_unitL x : meet top x = x. Proof. move: (top_maximum x); rewrite/ge; move/eqP => join_top. by rewrite -[in LHS]join_top meet_commutative join_commutative join_absorption. Qed. Lemma meet_top_unitR x : meet x top = x. Proof. by rewrite meet_commutative meet_top_unitL. Qed. Lemma complement_unique x y z : (complement_P x y) /\ (complement_P x z) -> y = z. Proof. rewrite/complement_P. case => yCx zCx. move:yCx; case=> xy_bot xy_top. move:zCx; case=>xz_bot xz_top. have Hyxz: join (meet x z) y = y. - rewrite xz_bot. by apply: join_bot_unitL. rewrite -Hyxz. rewrite join_commutative join_distributive [join y x] join_commutative xy_top. move: (meet_top_unitL (join y z)) ; move => Hmeet_top. rewrite Hmeet_top. have Hzxy: join z (meet x y) = z. - by rewrite xy_bot join_commutative join_bot_unitL. rewrite -[in RHS]Hzxy join_distributive [join z x]join_commutative xz_top. move: (meet_top_unitL (join z y)); move => Hmeet_top'. by rewrite Hmeet_top' join_commutative. Qed. Lemma join_eq_bot x y : join x y == bot -> (x == bot) && (y == bot). Proof. suff H: forall x y, join x y == bot -> x == bot. by move => Hjoinbot; apply /andP; split; move: Hjoinbot; [| rewrite join_commutative]; exact: H. move => x0 y0 /eqP H. apply /eqP. apply: ge_antisymmetric. rewrite bot_minimum /ge /=. by rewrite -H -join_associative [join y0 x0]join_commutative join_associative join_idempotent. Qed. Theorem ge_bot : forall x, ge bot x -> x == bot. Proof. by move => x /join_eq_bot /andP []. Qed. Lemma minimum_unique: forall y, (forall x, ge x y) -> y == bot. Proof. move => y Hge. apply /eqP. apply: ge_antisymmetric. by rewrite Hge bot_minimum. Qed. End Lattice. (* >>> Begin formalization of the cainoid. *) Section Cainoid. (* Define coins, the dot product, and atom coins. *) Parameter coins : choiceType. Parameter dot : coins -> coins -> coins. Parameter mix : L -> L -> coins. Definition bob := mix bot bot. Definition up x := mix x bot. Definition down x := mix bot x. (* A coin is built as a product of mixed coins. *) Axiom coin_prod_atomics : forall c:coins, {acs:seq (L*L) | c = \big[dot/bob]_(ac <- acs) mix ac.1 ac.2}. (* [up] and [down] are both injective functions. *) Axiom up_injective : injective up. Axiom down_injective : injective down. (* Axioms of a cainoid. *) (* These axioms determine the properties of the dot product. *) Axiom dotC : commutative dot. Axiom dotA : associative dot. Axiom bob_unitL : left_id bob dot. Axiom up_down_unitL : forall x, dot (up x) (down x) = bob. Axiom mix_up_down : forall x y, x != bot -> mix x y = dot (up (join x y)) (down y). (* Basic properties on [bob], the unit of a cainoid. *) Lemma bob_unitR : right_id bob dot. Proof. by move=>x; rewrite dotC bob_unitL. Qed. Lemma down_up_unitL : forall x, dot (down x) (up x) = bob. Proof. by move=> x; rewrite dotC; apply up_down_unitL. Qed. Lemma up_bot_bob : up bot = bob. Proof. done. Qed. Lemma down_bot_bob : down bot = bob. Proof. by[]. Qed. Lemma mix_bot_bot_bob : mix bot bot = bob. Proof. by[]. Qed. Lemma lemma1_1 x : mix x x = bob. Proof. move: (altP (x =P bot)). (*move: (altP (x =P bot)) + caseで場合分け*) case. move=> H. rewrite H. by[]. move=> Hxneqbob. rewrite mix_up_down. rewrite join_idempotent. rewrite up_down_unitL. by[]. Restart. case: (altP (x =P bot)) => [-> | Hxneqbob] //. by rewrite mix_up_down // join_idempotent up_down_unitL. Qed. Lemma lemma1_2 x y : ge x y -> mix x y = dot (up x) (down y). Proof. have [->| Hxneqbob] := altP (x =P bot). (*altP~をスタックのトップに追加し[]があるためcaseを行う。さらに[]の中の->で１サブゴールで出てくるトップを代入する。|の後ろは2サブゴールのトップの名前。*) - move/ge_bot => /eqP ->. (*-は意味なし*) (*by rewrite up_bot_bob down_bot_bob bob_unitL.*) by rewrite up_down_unitL. - by rewrite mix_up_down // /ge => /eqP ->. Qed. Lemma lemma1_3 x y w : x != bot -> ge (join x y) w -> mix x y = mix (join x w) y. Proof. move => Hxneqbot. rewrite /ge !mix_up_down //. - move/eqP => <-. by rewrite -!join_associative [join y w]join_commutative. - apply/eqP => /eqP /join_eq_bot /andP [] /eqP. by move/eqP: Hxneqbot. Qed. (* This lemma concerns transforming a coin identity to another coin identity, by dividing an [up] or [down] coin, or equivalently, by multypling a corresponding [down] or [up] coin. *) Lemma div_up_down : forall (A B : coins) (x : L), dot A (down x) = B <-> A = dot B (up x). Proof. split. - by move=> h; rewrite -[in RHS]h -dotA [dot (_ x) (_ x)]dotC up_down_unitL bob_unitR. - by move=> h; rewrite h -dotA up_down_unitL bob_unitR. Qed. (* Representation of lattice order by a coin identity. *) (* [x <= y] if and only if [mix x y = 1]. *) (*prop1*) Lemma le_mix_bob_eq : forall x y, x != bot -> (le x y <-> mix x y = bob). Proof. split. - rewrite /le /ge mix_up_down; last by[]. rewrite div_up_down bob_unitL join_commutative. by move /eqP ->. - rewrite mix_up_down; last by[]. rewrite div_up_down bob_unitL. move=> Hup. apply up_injective in Hup. move /eqP in Hup. by rewrite /le /ge join_commutative. Qed. Lemma mix_up_ge_down: forall x y, ge x y -> mix x y = dot (up x) (down y). Proof. move => x y. have [-> | Hxneqbob] := altP (x =P bot). - move/ge_bot => /eqP ->. by rewrite up_bot_bob down_bot_bob bob_unitL. - by rewrite mix_up_down // /ge => /eqP ->. Qed. Lemma mix_join_ge_down: forall x y w, x != bot -> ge (join x y) w -> mix x y = mix (join x w) y. Proof. move => x y w Hxneqbot. rewrite /ge !mix_up_down //. - move/eqP => <-. by rewrite -!join_associative [join y w]join_commutative. - apply/eqP => /eqP /join_eq_bot /andP [] /eqP. by move/eqP: Hxneqbot. Qed. Theorem Bayes_Theorem: forall x y, x != bot -> y != bot -> mix x y = dot (mix y x) (dot (up x) (down y)). Proof. move=> x y Hx Hy. rewrite [in RHS]mix_up_down; last exact. by rewrite dotA -![dot (dot (up _) _) _]dotA [dot (_ x) (_ x)]dotC up_down_unitL bob_unitL join_commutative mix_up_down. Qed. Lemma div_down_right: forall A B x, dot A (down x) = B -> A = dot B (up x). Proof. move=> A B x <-. by rewrite -dotA [dot (_ x) _]dotC up_down_unitL dotC bob_unitL. Qed. (* >>> inverse coins*) (* [acinv (x, y)] defines the inverse of the atom coin [mix x y]. *) Definition acinv (ac : L * L) := let: (x, y) := ac in if x == bot then up y else dot (up y) (down (join x y)). (* Now show that [acinv (x, y)] is indeed the inverse of [mix x y]. *) Lemma acinvN : forall ac : L * L, dot (acinv ac) (mix ac.1 ac.2) = bob. Proof. move => [x y] /=. case: (altP (x =P bot)) => [-> | xneqbot]; first by exact: up_down_unitL. rewrite mix_up_down // -dotA [x in dot _ x]dotA. by rewrite [dot (down _) (up _)]dotC !(up_down_unitL, bob_unitL). Qed. (* A cainoid is an abelian group. To show this fact we first show that there exists an inverse coin for an arbitrary coin in a cainoid. *) Lemma dot_inv_sig: forall c:coins, {d | dot d c = bob}. Proof. move => c. case: (coin_prod_atomics c) => acs ->. elim: acs => [| ac acs [d Hd]] ; first by exists bob; rewrite big_nil bob_unitL. exists (dot d (acinv ac)). by rewrite big_cons -dotA [dot (acinv _) (dot _ _)]dotA acinvN bob_unitL. Qed. (* The following function [dot_inv], of type [coins -> coins], defines an inverse for an arbitrary coin of a cainoid. *) Definition dot_inv (c:coins) := sval (dot_inv_sig c). (* The definition for [dot_inv] is indeed a right one. *) Lemma dotV : left_inverse bob dot_inv dot. Proof. move => c. exact: (svalP (dot_inv_sig c)). Qed. (* A cainoid has all properties of an Abelian group. *) (* Now we put all these relevant properties together. *) Definition zmodMixin := ZmodMixin dotA dotC bob_unitL dotV. Canonical zmodType := ZmodType _ zmodMixin. (* Now we introduce the usual plus notation for abelian groups. *) Notation "x \+ y" := (dot x y)(at level 50, left associativity). (* All the properties known to abelian groups can now be exported to a cainoid directly. *) Lemma dotrC: forall a b, a \+ b = b \+ a. Proof. by apply: GRing.addrC. Qed. Lemma dotrA: forall x y z, x \+ (y \+ z) = (x \+ y) \+ z. Proof. by apply: GRing.addrA. Qed. Example dotrA': forall x y z, (x \+ y) \+ z = x \+ (y \+ z). Proof. move=> x y z; apply: Logic.eq_sym; exact: GRing.addrA. Qed. Example dotrA'': forall x y z, (x \+ y) \+ z = x \+ (y \+ z). Proof. move=> x y z; by rewrite dotrA. Qed. Lemma dotrAC: forall x y z, x \+ y \+ z = x \+ z \+ y. Proof. move=> x y z. by rewrite -dotrA [y \+ z]dotrC dotrA. Qed. Lemma dotrACA :forall x y z t, (x \+ y) \+ (z \+ t) = (x \+ z) \+ (y \+ t). Proof. exact: GRing.addrACA. Qed. Example dotr_perm :forall a b c d, a \+ b \+ c \+ d = d \+ c \+ b \+ a. Proof. move=> a b c d. by rewrite [_ \+ d]dotrC [(_ \+ _) \+ c]dotrC [a \+ b]dotrC [d \+ (c \+ _)]dotrA [_ \+ (b \+ a)]dotrA. Qed. (* In an abelian group, the unit element is 0. *) (* So in the ring scope (%R), bob corresponds to 0. *) Lemma bob0 : bob = 0%R. Proof. done. Qed. Lemma dot0r: forall x, bob \+ x = x. Proof. exact: GRing.add0r. Qed. Lemma dotr0: forall x, x \+ bob = x. Proof. exact: GRing.addr0. Qed. Lemma dotNr :forall x, (dot_inv x) \+ x = bob. Proof. exact: GRing.addNr. Qed. End Cainoid. (* The following are experimental *) (* updated 2015/3/26 *) (* Axioms from classic logic *) Axiom peirce_axiom : forall P Q: Prop, ((P->Q)->P)->P. Axiom classic_axiom : forall P:Prop, ~ ~P -> P. Axiom excluded_middle_axiom : forall P:Prop, P \/ ~P. Axiom de_morgan_not_and_not_axiom : forall P Q:Prop, ~(~P /\ ~Q) -> P\/Q. Axiom implies_to_or_axiom : forall P Q:Prop, (P->Q) -> (~P\/Q). Definition peirce := forall P Q: Prop, ((P->Q)->P)->P. Definition classic := forall P:Prop, ~ ~P -> P. Definition excluded_middle := forall P:Prop, P \/ ~P. Definition de_morgan_not_and_not := forall P Q:Prop, ~(~P/\~Q) -> P\/Q. Definition implies_to_or := forall P Q:Prop, (P->Q) -> (~P\/Q). Lemma or_imply_and: forall A B C:Prop, (A \/ B -> C) -> (A->C) /\ (B->C). Proof. move=> A B C H. apply conj. by move=>HA; apply: H; left . by move=>HB; apply: H; right . Qed. (* de Morgran's law: part 1 *) (* previously named not_a_or_b_implies_not_a_and_not_b *) Lemma morgan_not_or: forall (A B : Prop), ~(A \/ B) -> ~A /\ ~B. Proof. move=> A B. rewrite/not. by apply or_imply_and. Qed. (* an alternative proof *) Theorem not_a_or_b_implies_not_a_and_not_b : forall A B, ~ (A \/ B) -> (~ A) /\ (~ B). Proof. move=>A B nAB. split. by move=> HA; apply: nAB; left. by move=> HB; apply: nAB; right. Qed. Theorem peirce_implies_classic : peirce -> classic. Proof. rewrite /peirce /classic. move=> p P nnP; rewrite /not in nnP. by apply p with (Q := False). Qed. (* alternativ coq-proof *) Theorem peirce_implies_classic'': peirce -> classic. Proof. unfold peirce. unfold classic. intros p P nnP. unfold not in nnP. apply p with (Q := False). intros h. apply nnP in h. inversion h. Qed. Theorem classic_implies_peirce : classic -> peirce. Proof. rewrite/classic /peirce. move => nnp P Q HPQ. apply: nnp => np. by apply np; apply: HPQ. Qed. Lemma classic_eq_peirce : classic <-> peirce. Proof. apply conj. by apply classic_implies_peirce. by apply peirce_implies_classic. Qed. Example classic_implies_peirce': forall P Q: Prop, (~ ~P -> P) -> (((P->Q)->P)->P). Proof. move=> P Q nnP H. apply: nnP => nP. by apply nP; apply: H. Qed. Theorem classic_implies_excluded_middle : classic -> excluded_middle. Proof. rewrite /classic /excluded_middle => c P. apply: c. move=> h. apply not_a_or_b_implies_not_a_and_not_b in h. by destruct h as [nh nnh]. Qed. Example classic_implies_excluded_middle': forall P, (P \/ ~ P). Proof. move=>P; move:classic_axiom => c. move: (c (P\/~P)) => HPoP; apply: HPoP. move=> nnPoP. apply not_a_or_b_implies_not_a_and_not_b in nnPoP. by destruct nnPoP. Qed. Theorem excluded_middle_implies_classic : excluded_middle -> classic. Proof. rewrite /excluded_middle /classic. move=> em P nnP. by destruct (em P). Qed. (* an alternative proof *) Theorem excluded_middle_implies_classic' : excluded_middle -> classic. Proof. rewrite /excluded_middle /classic. move=> em P nnP. move: (em (P\/~P))=> H. destruct H as [PnP | nPnP]. by destruct PnP as [HP | HnP]. apply not_a_or_b_implies_not_a_and_not_b in nPnP. by destruct nPnP. Qed. Theorem classic_implies_de_morgan_not_and_not : classic -> de_morgan_not_and_not. Proof. rewrite /classic /de_morgan_not_and_not. move=> c P Q H; move: (c (P\/Q)). apply. move=> H1. by apply not_a_or_b_implies_not_a_and_not_b in H1. Qed. (* an alternative proof *) Theorem classic_implies_de_morgan_not_and_not' : classic -> de_morgan_not_and_not. Proof. rewrite /classic /de_morgan_not_and_not. move=> c P Q H; apply: c; move=> h. by apply not_a_or_b_implies_not_a_and_not_b in h. Qed. Theorem de_morgan_not_and_not_implies_excluded_middle : de_morgan_not_and_not -> excluded_middle. Proof. rewrite /de_morgan_not_and_not /excluded_middle. move=>dm P. apply: dm =>H. by destruct H. Qed. Theorem de_morgan_not_and_not_implies_classic : de_morgan_not_and_not -> classic. Proof. rewrite /de_morgan_not_and_not /classic. move=> dm P nnP. move: (dm P P)=> H. have t: P \/ P -> P. by case. apply:t; apply H; move=> h. by destruct h. Qed. (* an alternative proof using what have already been proved.*) Goal de_morgan_not_and_not -> classic. Proof. move=> dm. by apply excluded_middle_implies_classic; apply de_morgan_not_and_not_implies_excluded_middle; apply dm. Qed. Theorem excluded_middle_implies_implies_to_or: excluded_middle -> implies_to_or. Proof. rewrite /excluded_middle /implies_to_or. move=> em P Q h. destruct (em P). by right; apply: h. by left. Qed. Theorem implies_to_or_implies_excluded_middle: implies_to_or -> excluded_middle. Proof. rewrite /implies_to_or /excluded_middle. move=> to P. destruct (to P P) . by[]. by right. by left. Qed. (* beginner's proof *) Theorem implies_to_or_implies_excluded_middle' : implies_to_or -> excluded_middle. Proof. rewrite /implies_to_or /excluded_middle. move=> to P. move: (to P P) => H. suff: ~ P \/ P -> P \/ ~ P. by move=> h; apply h; apply H. move=> PnP; destruct PnP. by right. by left. Qed. (* previously known as not_a_and_b_implies_not_a_or_not_b *) Theorem morgan_not_and : forall A B, ~ (A /\ B) -> (~ A) \/ (~ B). Proof. move=>A B nAB. move: not_a_or_b_implies_not_a_and_not_b => dm. move: (dm (~A) (~B)) => Hdm. move: excluded_middle_axiom => em. move: (em (~A \/ ~B)) => Hem. destruct Hem. by[]. apply Hdm in H; destruct H as [ h1 h2]. apply classic_axiom in h1. apply classic_axiom in h2. by have t: A /\ B. Qed. (* proof by contradiction *) Lemma contradiction : forall P Q : Prop, (~Q -> ~P) <-> (P -> Q). Proof. move=> P Q. apply conj. move=> nQnP HP. move: excluded_middle_axiom => em. move: (em Q) => emQ. destruct emQ as [q | nq]. by[]. by apply nQnP in nq. move=> H Hq. move/H. by[]. Qed. (* an example using contradiction Lemma *) Example de_morgan_not_and : forall A B, ~ (A /\ B) -> (~ A) \/ (~ B). Proof. move=>A B; apply contradiction. move=> H. apply not_a_or_b_implies_not_a_and_not_b in H. move=>h; apply: h. by destruct H as [HA HB]; apply classic_axiom in HA; apply classic_axiom in HB. Qed. Lemma not_not_and_double_not: forall P Q: Prop, ~(~P /\ ~Q) -> ~ ~ P \/ ~ ~ Q . Proof. by move=>P Q H; apply de_morgan_not_and in H. Qed. Goal forall P Q:Prop, ~(~P /\ ~Q) -> P\/Q. Proof. move=> P Q H. apply de_morgan_not_and in H. destruct H as [HP | HQ]. apply classic_axiom in HP; by left. apply classic_axiom in HQ; by right. Qed. (* prove properties for [gt] *) (* Students should fill in here. *) (* prove the same properties for [le] *) (* Students should fill in here. *) (* prove the same properties for [lt] *) (* Students should fill in here. *) (* prove properties for [le] and [lt] *) (* Students should fill in here. *) (* [remove_last] removes the last element of a list. *) Fixpoint remove_last (T:Type) (l: seq T) := let l' := rev l in match l' with | nil => nil | h::t => rev t end. (* [tail] removes the first element of a list. *) Fixpoint tail (T:Type) (l: seq T) := match l with | [::] => [::] | _::t => t end. (* [big_cons] defines the rule for recursive computation. *) (* [big_cons_dot] (probably not needed) is a special case of [big_cons], with dot operator of coins: c1...cn = c1.(c2...cn) *) Lemma big_cons_dot: forall (cs: seq coins), \big[dot/bob]_(c <- cs) c = let first := head bob cs in dot first (\big[dot/bob]_(c <- tail cs) c). Proof. move=> cs. case: cs. rewrite big_nil /=. by rewrite bob_unitL. move=> a l. by rewrite big_cons /=. Qed. (* Power of a coin, [ncoins n c =c...c]. *) Definition ncoins n c := \big[dot/bob]_(i <- nseq n c) i. (* alternative definition of power using [iter]. *) Definition ncoins' n c := if (n==0) then bob else (iter n.-1 (dot c) c). (* power of a raising/lowering coin *) Definition nups n x := ncoins n (up x). Definition ndowns n x := ncoins n (down x). Lemma updownnK : forall n x, dot (nups n x) (ndowns n x) = bob. Proof. (* FILL IN HERE *) Admitted. Lemma ncoins1': forall n c, ncoins' n.+1 c = dot (ncoins' n c) c. Proof. case. move=>c. by rewrite /ncoins' /= bob_unitL. rewrite /ncoins'/=; move=> n c; by rewrite dotC. Qed. Lemma ncoins0_bob : forall x, ncoins 0 x = bob. Proof. by move => x; rewrite /ncoins big_nil. Qed. Lemma up_power0_bob : forall x, nups 0 x = bob. Proof. move=> x. rewrite /nups /ncoins big_cons_dot/=; by rewrite big_nil bob_unitR. Qed. Lemma down_power0_bob : forall x, ndowns 0 x = bob. Proof. move=> x. rewrite /ndowns /ncoins big_cons_dot /=; by rewrite big_nil bob_unitR. Qed. Lemma up_power1 : forall x, nups 1 x = up x. Proof. move=> x. rewrite /nups /ncoins big_cons_dot /=. by rewrite big_nil bob_unitR. Qed. Lemma down_power1 : forall x, ndowns 1 x = down x. Proof. move=> x. rewrite /ndowns /ncoins big_cons_dot /=; by rewrite big_nil bob_unitR. Qed. Lemma updown1K : forall x, dot (nups 1 x) (ndowns 1 x) = bob. Proof. by move=> x; rewrite up_power1 down_power1 up_down_unitL. Qed. Example up3 : forall x, nups 3 x = ncoins' 3 (up x). Proof. move=>x. rewrite /nups /ncoins /ncoins'. do 3! rewrite big_cons_dot /=. (* big_cons is also ok. *) by rewrite big_nil bob_unitR. Qed. Example down3 : forall x, ndowns 3 x = ncoins' 3 (down x). Proof. move=>x. rewrite /ndowns /ncoins /ncoins'. do 3! rewrite big_cons/=. by rewrite big_nil bob_unitR. Qed. Lemma nups_eq : forall n x, nups n x = ncoins' n (up x). Proof. move=> n x. rewrite /nups /ncoins /ncoins' big_cons_dot /=. elim n. by rewrite big_nil bob_unitL /=. move=> m iH. rewrite big_cons_dot /=. rewrite iH /=. case m=> /=. by rewrite bob_unitR. by[]. Qed. Lemma ndowns_eq : forall n x, ndowns n x = ncoins' n (down x). Proof. move=> n x. rewrite /ndowns /ncoins /ncoins' big_cons_dot /=. elim n => /=. by rewrite big_nil bob_unitL /=. move=> m iH. rewrite big_cons_dot /= iH. case m =>/=. by rewrite bob_unitR /=. done. Qed. Lemma ncoins2_eq: forall n x, ncoins n x = ncoins' n x. Proof. move=> n x. rewrite /ncoins /ncoins' big_cons_dot /=. elim n =>/=. by rewrite big_nil bob_unitL /=. move=> m iH. rewrite big_cons_dot /= iH. case m =>/=. by rewrite bob_unitR. done. Qed. Lemma nupdownK : forall n x, dot (nups n x) (ndowns n x) = bob. Proof. move=> n x; rewrite nups_eq ndowns_eq. elim n. by rewrite /ncoins' /=; rewrite bob_unitL. move=> m iH. do 2! rewrite ncoins1'. have ac: forall a b, dot a b = bob -> dot (dot a (up x)) (dot b (down x)) = bob. - move=> a b ih. - by rewrite -!dotA [ dot _ (dot b _)] dotA [dot _ b] dotC -![dot (dot b _) _]dotA up_down_unitL bob_unitR ih. by rewrite ac. Qed. (* work from here *) (* 2015/3/27 *) (* working with integers using library [ssrint]. *) Require Import ssrint. (* addition of two integers *) Definition addz := intZmod.addz. (* negative numbers are denoted by [Negz n] == -(n+1). *) Eval compute in addz 1 2. Eval compute in addz 3 (-1). Eval compute in addz 1 (Negz 0). Eval compute in addz 1 (Negz 1). (* Opposite of an integer n, [oppz n] == -n. For instance, oppz Posz 0 == Posz 0, oppz Posz n.+1 == Neg n, oppz Negz 0 == Posz 1, oppz Negz n == Posz n.+1. *) Definition oppz := intZmod.oppz. Eval compute in addz 1 (oppz 1). Eval compute in addz (Negz 3) (oppz (Negz 3)). Goal forall n, addz n (oppz n) = Posz 0. Proof. (* FILL IN LINES HERE *) Admitted. (* Notations for integers, addition, and subtraction. (1) 1,2,3, etc. also stand for int, when coerced, eg. 1+1. (2) - 1%:Z, - 2%:Z, -3%:Z, stand for negative integers, -1, -2, -3, etc.. (3) m+n, integer addition for any m, n integers, e.g., 1+2, - 5%:Z + - 7%:Z, etc.. (4) m-n, integer subtraction for any m, n integers, e.g., 1-7%:Z, - 5%:Z - 7%:Z, etc.. *) (* 2015/3/27 *) Local Open Scope int_scope. Local Notation "0" := (Posz 0) : int_scope. Local Notation "-%Z" := (@oppz) : int_scope. Local Notation "- x" := (oppz x) : int_scope. Local Notation "+%Z" := (@addz) : int_scope. Local Notation "x + y" := (addz x y) : int_scope. Local Notation "x - y" := (x + - y) : int_scope. (* addition of two integers *) Eval compute in 0 + 0. Eval compute in 3+0. Eval compute in Negz 3 + 4. Eval compute in 4 + Negz 3. Eval compute in Negz 3 + Negz 3. Goal forall m n:nat, addz m n = addn m n. Proof. (* FILL IN LINES HERE *) Admitted. (* to confirm the useage of the notations *) (* the oppsite numbers *) Goal -(-0)=0. Proof. by[]. Qed. Goal -(-1) = 1. Proof. by[]. Qed. Goal -(-2%:Z) = 2%:Z. Proof. by[]. Qed. Goal -(-2%:Z) = 2. Proof. by[]. Qed. Goal - -2%:Z = 2. Proof. by[]. Qed. Goal --2%:Z = 2. Proof. by[]. Qed. Goal -------2%:Z = -2%:Z. Proof. by[]. Qed. Goal forall n, n - n =0. Proof. move => n; elim n. by[]. by case. by case. Qed. Goal forall n, oppz (oppz n) = n. Proof. by do 2! case. Qed. Goal forall n: int, - - n = n. Proof. by do 2! case. Qed. Goal forall n, - Negz n = n.+1. Proof. by []. Qed. (* Define integer powers of coins. *) Definition nupsz n x := match n with | Posz n' => nups n' x | Negz n' => ncoins n'.+1 (down x) end. Definition ndownsz n x := match n with | Posz n' => ndowns n' x | Negz n' => ncoins n'.+1 (up x) end. Lemma upz_power0_bob: forall x, nupsz 0 x = bob. Proof. move=> x /=. by rewrite up_power0_bob. Qed. Lemma upz_power1: forall x, nupsz 1 x = up x. Proof. move=> x /=. by rewrite up_power1. Qed. Lemma downz_power0_bob: forall x, ndownsz 0 x = bob. Proof. move=> x /=. by rewrite down_power0_bob. Qed. Lemma downz_power1: forall x, ndownsz 1 x = down x. Proof. move=> x /=. by rewrite down_power1. Qed. Lemma upz_power_m1: forall x, nupsz (-1%:Z) x = down x. Proof. move=> x /=. rewrite /ncoins big_cons_dot /=. by rewrite big_nil bob_unitR. Qed. Lemma downz_power_m1: forall x, ndownsz (-1%:Z) x = up x. Proof. move=> x /=. rewrite /ncoins big_cons_dot /=. by rewrite big_nil bob_unitR. Qed. Lemma nupsz_eq: forall (n:nat) x, nupsz n x = nups n x. Proof. by []. Qed. Lemma ndownsz_eq: forall (n:nat) x, ndownsz n x = ndowns n x. Proof. by []. Qed. Lemma nupszdown_eq: forall (n:nat) x, nupsz (-n%:Z) x = ndowns n x. Proof. case. by[]. by[]. Qed. Lemma ndownszup_eq: forall (n:nat) x, ndownsz (-n%:Z) x = nups n x. Proof. case. by[]. by[]. Qed. Lemma updownz_eq: forall n x, nupsz n x = ndownsz (-n) x. Proof. case; move=> n x /=. by rewrite /nups ndownszup_eq /= /ndowns. by[]. Qed. Lemma downupz_eq: forall n x, ndownsz n x = nupsz (-n) x. Proof. case; move=> n x. by rewrite nupszdown_eq. by[]. Qed. Lemma upzK: forall n x, dot (nupsz n x) (nupsz (-n) x) = bob. Proof. case; move=> n x. by rewrite nupsz_eq nupszdown_eq nupdownK /=. move=> /=. have tem: ndowns n.+1 x = ncoins n.+1 (down x). by[]. by rewrite -!tem dotC nupdownK. Qed. Lemma downzK: forall n x, dot (ndownsz n x) (ndownsz (-n) x) = bob. Proof. move=> n x; by rewrite -!updownz_eq downupz_eq dotC upzK. Qed. Lemma mix_up: forall x y, x != bot -> mix x y = dot (nupsz 1 (join x y)) (nupsz (-1) y). Proof. move=> x y iH; rewrite mix_up_down. by rewrite -!upz_power1 -!downz_power1 downupz_eq. by[]. Qed. Lemma atomic_up: forall x y, mix x y = if x==bot then down y else dot (nupsz 1 (join x y)) (nupsz (-1) y). Proof. move=> x y. case: ifPn; first by move/eqP => ->. exact: mix_up. Qed. (* Representation of coins in terms of raising coins. *) (* Lower coins can be expressed as powers of raising coins. *) Lemma down_up_ex: forall x, {n : int | down x = nupsz n x}. Proof. by exists (-1%:Z); rewrite upz_power_m1. Qed. (* Mixed coins can be decomposed as the product of powers of raising coins *) Lemma mix_up_ex: forall x y, x != bot -> {a: L*L | mix x y = dot (nupsz 1 a.1) (nupsz (-1) a.2)}. Proof. by move=> x y; exists (join x y, y); rewrite mix_up. Qed. (* >>> An example using dependent type *) Example ex_imp_ex A (f g: A -> Prop): (exists a: A, f a) -> (forall x: A, f x -> g x) -> exists b: A, g b. Proof. case => a Sa SiT. by exists a; apply: SiT. Qed. (* <<< An example using dependent type *) (* An important theorem says that any coin can be decomposed as the product of powers of raising coins. *) (* By definition, any coin is a product of mixed coins. *) Print coin_prod_atomics. (* ==== forall c : coins, {acs : seq (L * L) | c = \big[dot/bob]_(ac <- acs) mix ac.1 ac.2} ] *) (* Define coin products without using bigop *) Fixpoint coin_prod (l:seq coins) : coins := match l with | nil => bob | h :: t => dot h (coin_prod t) end. (* Parameters c d:coins. Eval compute in (coin_prod [::]). Eval compute in (coin_prod [::bob]). Eval compute in (coin_prod [:: c]). Eval compute in (coin_prod [:: c; d]). Eval compute in (coin_prod [::c; d; d]). Eval compute in (coin_prod (c :: [::d; d])). Eval compute in (coin_prod [:: c; c; d; d]). *) Lemma coin_prod_rec : forall a l, coin_prod (a :: l) = dot a (coin_prod l). Proof. move=> a l. case l. by rewrite /coin_prod bob_unitR. by rewrite /coin_prod //. Qed. (* Define coin products using bigop *) Definition coin_prod_bigop (l:seq coins) : coins := \big[dot/bob]_(c <- l) c. Lemma coin_prod_eq : forall l, coin_prod l = coin_prod_bigop l. Proof. elim. by rewrite /coin_prod /coin_prod_bigop big_nil //. move=>a l. rewrite /coin_prod_bigop =>h. by rewrite coin_prod_rec big_cons h. Qed. (* Define a list of mixed coins corresponding to a list of type [L*L]. *) Fixpoint seqL_to_seqC (l : seq (L * L)) : seq coins := match l with | nil => nil | h :: t => (mix h.1 h.2) :: (seqL_to_seqC t) end. Lemma seqL_to_seqC_rec : forall (a:L*L) (l: seq (L*L)), seqL_to_seqC (a :: l) = (mix a.1 a.2) :: (seqL_to_seqC l). Proof. by move=> a. Qed. (* Eval compute in (seqL_to_seqC [::]). Eval compute in (seqL_to_seqC [:: (bot,bot)]). Eval compute in (seqL_to_seqC [:: (bot,bot); (top, bot)]). *) Lemma seqL_to_seqC_eq : forall l : seq (L * L), coin_prod (seqL_to_seqC l) = \big[dot/bob]_(x <- l) mix x.1 x.2. Proof. - elim. by rewrite big_nil. - move=>a l h. by rewrite seqL_to_seqC_rec big_cons coin_prod_rec h. Qed. (* A list of type [seq (L * L)] produces a product of mixed coins, which can be rewritten as a product of interger powers. *) Fixpoint seqL_to_seqUp (l : seq (L * L)) : coins := match l with | nil => bob | h :: t => let c:= if (h.1==bot) then (down h.2) else dot (up (join h.1 h.2)) (down h.2) in dot c (seqL_to_seqUp t) end. Eval compute in (seqL_to_seqUp [::]). Eval compute in (seqL_to_seqUp [:: (bot, top)]). Lemma ref_eq (T: eqType) : forall x:T, (x == x) = true. Proof. move=> x. by apply /eqP. Qed. Goal forall x, seqL_to_seqUp [:: (bot, x)] = down x. Proof. by move=> x /=; rewrite ref_eq bob_unitR. Qed. Lemma double_not : forall P:Prop, ~ ~ P <-> P. Proof. move=> P. split. apply contradiction. by rewrite /not . by rewrite /not . Qed. Lemma negb_false_iff : forall (T:eqType) (x y: T), x != y <-> ((x == y) = false). Proof. split. rewrite /negb. destruct (x==y). by[]. by[]. move=> xny. rewrite /negb. by move: xny ->. Qed. Goal forall x y, x != bot -> seqL_to_seqUp [:: (x, y)] = mix x y. Proof. move=> x y h /=. rewrite mix_up_down; last by[]. move:h. rewrite negb_false_iff => ->. by rewrite bob_unitR //. Qed. Goal forall x y : L, {xns : seq (int*L) | seqL_to_seqUp [:: (x, y)] = \big[dot/bob]_(xn <- xns) nupsz xn.1 xn.2}. Proof. move=> x y. case: (altP (x =P bot)). exists [:: (-1%:Z, y)] =>/=. move: p; move /eqP ->. by rewrite bob_unitR /= big_cons /= big_nil bob_unitR ncoins2_eq /ncoins' . - exists [:: (1%:Z, join x y); (-1%:Z, y)] => /= . move:i; rewrite negb_false_iff =>->. do 2! rewrite big_cons //=. rewrite big_nil up_power1 /ncoins big_cons big_nil //=. by do 3! rewrite bob_unitR. Qed. (* <<<< selective induction *) Fixpoint evenb n := if n is n'.+2 then evenb n' else n == 0%N. Lemma evenb_plus n m : evenb n -> evenb m -> evenb (n + m). Proof. move: (leqnn n) . elim: n {1 3 4}n. (* --- among the 4 n's, do induction on the second one, the same as [elim: n {-2}n]. *) by case=>//. move =>n Hn. case=>//. case=>// n0. by move/ltnW /Hn=>//. Qed. (* Any coin can be written as the product of powers of raising coins. *) Theorem coin_prod_up : forall c : coins, {xns:seq (int*L) | c = \big[dot/bob]_(xn <- xns) nupsz xn.1 xn.2}. Proof. move=>c. move: (coin_prod_atomics c). case=> x. move:c. (* essential step for induction! *) elim: x. move=> c ca; exists nil; rewrite big_nil; by rewrite big_nil in ca. (* the second subgoal *) move=> a l H c. rewrite big_cons. rewrite atomic_up. case: ifPn => _. rewrite -upz_power_m1. set c1:= \big[dot/bob]_(j <- l) mix j.1 j.2. case: (H c1) => // il Hc1 Hc. exists ((-1%Z, a.2) :: il). by rewrite big_cons Hc Hc1. -set c1:= \big[dot/bob]_(j <- l) mix j.1 j.2. case: (H c1) => // il Hc1 Hc. exists ((1%:Z, (join a.1 a.2)) :: (-1%Z, a.2) :: il). do 2! rewrite big_cons. by rewrite Hc Hc1 dotA. Qed. (* return [true] if [seq L] has mutually distinct elements *) Fixpoint neqs (l:seq L) : bool := match l with | nil => true | h :: t => match t with | nil => true | h' :: t' => (h != h') && (neqs t') end end. Eval compute in neqs [::]. Eval compute in neqs [:: bot]. Eval compute in neqs [:: bot;bot]. Lemma ref: forall (T:eqType) (x:T), x == x. Proof. by[]. Qed. Lemma nnref: forall (T:eqType) (x:T), ~(x != x). Proof. move=> T x; by rewrite /not /negb ref. Qed. Lemma nref: forall (T:eqType) (x:T), (x != x) = false. Proof. by move=> T x; rewrite /negb ref. Qed. (* Return true if each element of [l:seq L] is smaller than [x:L]. *) Fixpoint seq_le (l:seq L) (x:L) : bool := match l with | nil => true | h :: t => if (ge x h) then (seq_le t x) else false end. (* this definition is equivalent to [all (ge x) l]. *) Goal all (ge bot) [::]. Proof. by[]. Qed. Eval compute in all (ge bot) [::]. Goal seq_le [::] bot. Proof. by[]. Qed. Eval compute in seq_le [::] bot. Lemma seq_le_all: forall x l, seq_le l x == all (ge x) l. Proof. move=> x l. elim l. by[]. move=>a l1 h. rewrite /seq_le /all. by case: ifP. Qed. (* Return true if each element of [l:seq L] is strictly smaller than [x:L]. *) Fixpoint seq_lt (l:seq L) (x:L) : bool := match l with | nil => true | h :: t => if gt x h then (seq_lt t x) else false end. (* this definition is equivalent to [all (gt x) l]. *) Lemma seq_lt_all: forall x l, seq_lt l x == all (gt x) l. Proof. move=> x l. elim l. by[]. move=>a l1 h. rewrite /seq_lt /all. by case: ifP. Qed. Goal seq_lt [::] bot. Proof. by[]. Qed. Eval compute in seq_lt [::] bot. (* Return true if each natural of a list is strictly smaller than a given natural. *) Fixpoint seqn_lt (l:seq nat) (x:nat) : bool := match l with | nil => true | h :: t => if (h < x) then (seqn_lt t x) else false end. (* this definition is equivalent to [all (fun z => z< x) l]. *) Lemma seqn_lt_all: forall (x:nat) l, seqn_lt l x == all (fun z => z< x) l. Proof. move=> x l. elim l. by[]. move=>a l1 h. rewrite /seqn_lt /all. by case: ifP. Qed. (* pick the first elements from [seq (A*B)] *) Fixpoint seq_1 {A B: Type} (l:seq (A*B)) : seq A:= match l with | nil => nil | h :: t => h.1 :: seq_1 t end. (* the same as [map (fun x => x.1) l] *) (* pick the second elements from [seq (A*B)] *) Fixpoint seq_2 {A B: Type} (l:seq (A*B)) : seq B:= match l with | nil => nil | h :: t => h.2 :: seq_2 t end. (* the same as [map (fun x => x.2) l] *) Lemma seq_1_map {A B: Type}: forall l:seq (A*B), seq_1 l = map (fun x => x.1) l. Proof. elim. by[]. move=> a l h. have t1: seq_1 (a :: l) = a.1::seq_1 l. by[]. have t2: [seq x.1 | x <- a :: l] = a.1::[seq x.1 | x <- l]. by[]. by rewrite t1 t2 h. Qed. Lemma seq_2_map {A B: Type}: forall l:seq (A*B), seq_2 l = map (fun x => x.2) l. Proof. elim. by[]. move=> a l h. have t1: seq_2 (a :: l) = a.2::seq_2 l. by[]. have t2: [seq x.2 | x <- a :: l] = a.2::[seq x.2 | x <- l]. by[]. by rewrite t1 t2 h. Qed. (*New fomalization of prime coins by Shimoyama.*) Lemma coin_decomp : forall x : L, exists y c xns, x != bot -> ge x y /\ c = \big[dot/bob]_(xn <- xns) nupsz xn.1 xn.2 /\ seq_lt (seq_2 xns) x /\ up x == dot c (up y). Proof. move=> x; exists x; exists bob; exists [::(1%:Z, bot)]; move=> H. apply/and_assoc. apply/(and_assoc _ _ (up x == dot bob (up x))). apply:conj. apply:conj. apply:conj. by apply:ge_reflexive. by rewrite big_cons big_nil upz_power1 up_bot_bob dot0r. rewrite /seq_2 /snd /seq_lt /gt; case:ifP. - done. - move/ andP; case; apply: conj. apply/ bot_minimum. apply/ H. by rewrite dot0r. Qed. (*formalization of cain*) (* Define ctext *) Parameter ctext : coins -> L. Fixpoint neq0_seq (l:seq int) : bool := match l with | nil => true | h :: t => match t with | nil => (h !=0) | h' :: t' => (h != h') && (neq0_seq t') end end. Axiom weak_ctext: forall (c:coins) (nx: seq(int*L)), c = \big[dot/bob]_(i <- nx) nupsz i.1 i.2 -> uniq [seq i.2 | i <- nx](*neqs (seq_2 nx)*) -> 0 \notin [seq i.1 | i <- nx] (*neq0_seq (seq_1 nx)*) -> ctext c = \big[join/bot]_(i <- nx) i.2. Lemma addnzE(n m:nat) : (addn n m)%:Z = addz n m. Proof. elim:n => //. Qed. Theorem addncoins n m c: dot (ncoins n c)(ncoins m c) = ncoins (n + m) c. Proof. elim: n => [|n Hn]. by rewrite add0n ncoins0_bob bob_unitL. by rewrite addSn !ncoins2_eq !ncoins1' dotrC dotrA -!ncoins2_eq [dot _ (ncoins n c)]dotrC Hn. Qed. (* move=> [[|n]|n] [[|m]|m] /=; rewrite ?NegzE ?oppzK ?addnS ?addn0 ?subn0 //; *) (* rewrite ?ltnS[m <= n]leqNgt [n <= m]leqNgt; case: ltngtP=> hmn /=; *) (* by rewrite ?hmn ?subnn // ?oppzK ?subSS ?subnS ?prednK // ?subn_gt0. *) Theorem neg_nupsz n x : ncoins n (down x) = nupsz (-n%:Z) x. Proof. by elim:n =>[|n Hn] /=;rewrite /nups ?ncoins0_bob. Qed. Theorem poz_nupsz n x : ncoins n (up x) = nupsz (n%:Z) x. Proof. by elim:n. Qed. Theorem nupszp_eq m x: (0 <= m) -> (nupsz m%:Z x) = nups m x. Proof. by elim:m => [h|m h1 h2] => //. Qed. Theorem nupszn_eq m x: (0 <= m) -> (ndownsz m%:Z x) = ndowns m x. Proof. by elim:m => [h|m h1 h2] => //. Qed. Lemma ncoins_subup n m x: m <= n -> ncoins n (up x) = dot (ncoins (n - m) (up x)) (ncoins m (up x)). Proof. move => h. by rewrite addncoins (subnK h). Qed. Lemma ncoins_subdown n m x: m <= n -> ncoins n (down x) = dot (ncoins (n - m) (down x)) (ncoins m (down x)). Proof. move => h. by rewrite addncoins (subnK h). Qed. Theorem addnupsz n m x: dot (nupsz n x)(nupsz m x) = nupsz (n + m) x. Proof. move: n m. move=> [[|n]|n] [[|m]|m] /= ; rewrite ?NegzE ?addn0 ?subn0 //; rewrite /nups ?ncoins0_bob ?bob_unitL ?bob_unitR; rewrite ?addncoins ?addSn ?addnS //; rewrite !ncoins2_eq !ncoins1' dotrACA ?up_down_unitL; rewrite ?down_up_unitL bob_unitR -!ncoins2_eq. rewrite subzSS. move:(leq_total m n) => /orP [] H. rewrite (subzn H) (@ncoins_subup n m x H) -dotA nupdownK bob_unitR. move/ (leq_sub2r m) in H; rewrite subnn in H. by rewrite (@nupszp_eq (n - m) x H). rewrite (@ncoins_subdown m n x H) dotC -dotA [dot (ncoins n (down x)) _]dotC; rewrite nupdownK bob_unitR updownz_eq oppz_add GRing.addrC GRing.opprK; rewrite (subzn H). move/ (leq_sub2r n) in H; rewrite subnn in H. by rewrite (@nupszn_eq (m - n) x H). rewrite GRing.addrC. rewrite subzSS. move:(leq_total n m) => /orP [] H. rewrite (subzn H). rewrite (@ncoins_subup m n x H) dotC -dotA nupdownK bob_unitR. move/ (leq_sub2r n) in H; rewrite subnn in H. by rewrite (@nupszp_eq (m - n) x H). rewrite (@ncoins_subdown n m x H) -dotA [dot (ncoins m (down x)) _]dotC nupdownK bob_unitR; rewrite updownz_eq oppz_add GRing.addrC GRing.opprK; rewrite (subzn H). move/ (leq_sub2r m) in H; rewrite subnn in H. by rewrite (@nupszn_eq (n - m) x H). Qed. (* 2016/07/20 *) Fixpoint head_in_tail (a:(int*L))(l:seq(int*L)):= match l with | nil => [::a] | h :: t => if a.2 == h.2 then ((a.1 + h.1),a.2)::t else h :: (head_in_tail a t) end. Fixpoint remove0L (xns:seq(int*L)):= match xns with | nil => nil | h::t => if h.1 == 0 then remove0L t else h ::(remove0L t) end. Theorem coin_prod_up_eq_non0 (xns:seq(int*L)): \big[dot/bob]_(xn <- xns) nupsz xn.1 xn.2 = \big[dot/bob]_(xn <- (remove0L xns)) nupsz xn.1 xn.2 . Proof. elim: xns => /=. -by[]. -move => a l H. case: ifP. move/eqP => a0. by rewrite big_cons a0 upz_power0_bob bob_unitL H. move => a0f. by rewrite !big_cons H. Qed. Theorem coin_prod_up_eq_hit (a:(int*L))(xns:seq(int*L)): \big[dot/bob]_(xn <- a :: xns) nupsz xn.1 xn.2 = \big[dot/bob]_(xn <- (head_in_tail a xns)) nupsz xn.1 xn.2 . Proof. elim: xns => /=. -by[]. -move => a0 l H. case: ifP; move/eqP => ha. rewrite !big_cons ha => /=. by rewrite -addnupsz dotA. by rewrite [in RHS] big_cons -H !big_cons dotC dotrAC dotA. Qed. Theorem zero_notin_rem (xns:seq (int*L)): 0 \in [seq i.1 | i <- (remove0L xns)] = false. Proof. elim:xns. by[]. move => a l H /=. case:ifP => a0. exact: H. rewrite in_cons eq_sym a0 Bool.orb_false_l. apply: H. Qed. Theorem hit_map_eq a xns: a.2 \in [seq i.2 | i <- xns] -> [seq i.2 | i <- xns] = [seq i.2 | i <- (head_in_tail a xns)]. Proof. elim: xns => /=. -by[ ]. -move => a0 l h1. rewrite in_cons => /orP [] /eqP h2. case:ifP => /eqP. by rewrite map_cons h2 => /=. by rewrite h2. move/eqP in h2. case:ifP => /eqP h3. by rewrite map_cons h3 => /=. rewrite map_cons. rewrite h2 in h1. by rewrite h1. Qed. Theorem coin_prod_seq_uniq xns: {uxns:seq (int*L) | \big[dot/bob]_(i <- xns) nupsz i.1 i.2 = \big[dot/bob]_(ui <- uxns)nupsz ui.1 ui.2 /\ uniq ([seq i.2 | i <- uxns]) }. Proof. elim: xns => /=. exists nil. by[]. move => a l. case => yms [] h1 h2. case H :(a.2 \notin [seq i.2 | i <- yms]). exists (a :: yms). split. by rewrite !big_cons -h1. apply/andP. exact: conj; last first => //. move:H;rewrite/negb. case:ifP => H _ //. exists (head_in_tail a yms). split. by rewrite -coin_prod_up_eq_hit !big_cons h1. by rewrite -(hit_map_eq H) h2. Qed. Lemma notin_rem_notin (a:(int*L))(l:seq(int*L)): a.2 \notin [seq i.2 | i <- l] -> a.2 \notin [seq i.2 | i <- (remove0L l)]. Proof. elim: l => /=. by[]. move => a' l h1 h2; case: ifP => // a'10. apply: h1. move:h2; rewrite in_cons /negb. case: ifP => // ;rewrite Bool.orb_false_iff. case => h2 h3 _ ; case:ifP => // h4. by rewrite -h3. move: h2. rewrite map_cons in_cons /negb. case: ifP => //. rewrite Bool.orb_false_iff. case => h2 h3 _. case:ifP => //. rewrite in_cons h2 Bool.orb_false_l. move:h1. rewrite /negb. case: ifP => h4; case:ifP => h5 _ // _. by rewrite -h3. Qed. Lemma rem_uniq (xns:seq (int*L)) : uniq ([seq i.2 | i <- xns]) -> uniq ([seq i.2 | i <- (remove0L xns)]). Proof. elim:xns => /=. by[]. move => a l h1. move/andP =>[] h2 h3. case:ifP => a0 /=. exact: h1. apply/andP. split; last first. exact: h1. exact: notin_rem_notin. Qed. Theorem coin_prod_up_strong c: {xns:seq (int*L) | c = \big[dot/bob]_(xn <- xns) nupsz xn.1 xn.2 /\ uniq ([seq i.2 | i <- xns]) /\ 0 \notin [seq i.1 | i <- xns] }. Proof. move: (@coin_prod_up c) => [] xns Hc. move: (@coin_prod_seq_uniq xns) => [] uxns [] uHc1 uHc2. exists (remove0L uxns). split. -by rewrite uHc1 in Hc ;rewrite -coin_prod_up_eq_non0. rewrite (coin_prod_up_eq_non0 uxns) in uHc1. split. by apply:rem_uniq. move : Hc uHc1 uHc2. elim: uxns => /=. by[]. move => a l h1 h2. case:ifP. move/eqP => a0 h3. move/andP => [] h4 h5. apply: h1 => //. move => a0f; rewrite big_cons /= => hc1; rewrite in_cons /negb. by case :ifP ;rewrite eq_sym a0f Bool.orb_false_l zero_notin_rem. Qed. (* 2016/07/13 *) Axiom ctext1 : forall x, ctext (up x) = x. Axiom ctext2 : forall x y, x != bot /\ ge y x -> ctext (mix x y) = bot. Axiom ctext3 : forall x y, x != bot /\ ~~ ge y x -> ctext (mix x y) = join x y. Axiom ctext4 : forall x, ctext (down x) = x. Lemma ctext1_1 x: {xns| up x = \big[dot/bob]_(xn <- xns) nupsz xn.1 xn.2 /\ uniq ([seq i.2 | i <- xns]) /\ 0 \notin ([seq i.1 | i <- xns]) /\ \big[join/bot]_(i <- xns) i.2 = x}. Proof. exists [::(1%:Z,x)] => //=. split. by rewrite big_cons big_nil bob_unitR -up_power1 => //=. rewrite big_cons big_nil join_commutative => //=. do 2 split => //. exact:join_bot_unitL. Qed. Lemma ctext1' x: ctext (up x) = x. Proof. move:(@ctext1_1 x). case => x0 [] h1 [] h2 [] h3 h4. move: (@weak_ctext (up x) x0)=> h5. rewrite h4 in h5. by rewrite (h5 h1 h2 h3). Qed. Lemma dot_inv_prod_up c nx: c = \big[dot/bob]_(i <- nx) nupsz i.1 i.2 -> dot_inv c = \big[dot/bob]_(i <- nx) nupsz (- i.1) i.2. Proof. Admitted. Theorem Th3_1 c: ctext c = ctext (dot_inv c). Proof. Admitted. Theorem Th3_2 c1 c2: ge (join (ctext c1) (ctext c2)) (ctext (dot c1 c2)). Proof. Admitted. Lemma not_bot_gt_bot x : x != bot = gt x bot. Proof. rewrite /gt; apply/eqP. case: ifP. case /andP => H1 H2; exact/eqP. rewrite (@bot_minimum x) /= ;by move/negbFE /eqP. Qed. Definition cintegrable x c := ge (ctext c) x. Parameter mcoin : L -> coins. Axiom mcoin_axiom : forall y, ge y (ctext (mcoin y)). Axiom mcoin_axiom':forall x c, ge x (ctext c)-> c = mcoin x. Lemma gt_joinL(*used in lemma5's proof*) a b : gt (join a b) b -> gt a bot. Proof. case: (altP (b =P bot)) => [-> | nbbot]. move: (@join_bot_unitL a)=> H. by rewrite join_commutative H. rewrite /gt. move /andP => [] gjabb njabb. apply /andP; apply conj. exact:bot_minimum. apply/negP. move=> /eqP abot. move: njabb; move /negP. case. by rewrite abot join_bot_unitL. Qed. Lemma ge_meet_def(*used in lemma5's proof*) a b: ge a b = (meet a b == b). Proof. rewrite /ge; apply/eqP;case: ifPn; move/eqP => H. by rewrite -{2}(@join_absorption a b) meet_distributive meet_idempotent H. rewrite /not. move => H1; move:H. rewrite /not => /eqP. by rewrite -H1 meet_commutative join_commutative join_absorption nref. Qed. Lemma meet_not_bot (*lemma5*)a b x: ge (join a b) x -> gt x b -> meet a x != bot. Proof. rewrite ge_meet_def meet_commutative meet_distributive;move /eqP => H1. rewrite /gt. move/andP =>[]; rewrite ge_meet_def;move/eqP => H2 H3. rewrite not_bot_gt_bot. apply: (@gt_joinL (meet a x) b). rewrite -{1}H2 [meet _ x] meet_commutative H1. rewrite /gt. apply/andP. split. by rewrite ge_meet_def;apply /eqP. by[]. Qed. Lemma also_integ x y c cy : cintegrable x c /\ meet y x = bot /\ ge y (ctext cy) -> cintegrable x (dot cy c). Proof. Admitted. Parameter coin_margin : L -> coins -> coins. Notation "c /d x" := (coin_margin x c) (at level 50). Axiom margin0 : forall x c, ctext (c /d x) = meet (ctext c) (com x). Axiom margin1 : forall x y, ge y x -> (up y) /d x = up (meet y (com x)). Axiom margin2 : forall x1 x2 c1 c2, meet x1 x2 = bot /\ meet x1 (ctext c2) = bot /\ meet x2 (ctext c1) = bot /\ ge (ctext c1) x1 /\ ge (ctext c2) x2 /\ cintegrable (join x1 x2) (dot c1 c2) -> (dot c1 c2) /d (join x1 x2) = dot (c1 /d x1) (c2 /d x2). Axiom margin3 : forall c, c /d bot = c. (*we define cain by this cainoid satisfying these properties*) (*Th4*) Theorem const_margin' x y c cx : cintegrable x cx /\ meet x y = bot /\ ge y (ctext c) -> dot c cx /d x = dot c (cx /d x). Proof. rewrite /cintegrable; move=> []=> H1 []=> H2 H3. rewrite -[in RHS](@margin3 c). rewrite -[in RHS]margin2; last first. apply :conj; first by rewrite meet_bot_unitL. apply :conj; first by rewrite meet_bot_unitL. apply :conj; last first. apply :conj; first by rewrite /ge join_bot_unitR. apply :conj; first by[]. apply /also_integ; first by[]. apply :conj; first by rewrite /cintegrable join_bot_unitL H1. apply :conj; first by rewrite join_bot_unitL meet_commutative. by[]. rewrite ge_meet_def in H3. move :H3; move /eqP => H3. by rewrite -H3 meet_associative H2 meet_bot_unitL. by rewrite join_bot_unitL. Qed. Theorem const_margin x c cx : cintegrable x cx /\ meet x (ctext c) = bot -> dot c cx /d x = dot c (cx /d x). Proof. rewrite /cintegrable; move=> []=> H1=> H2. rewrite -[in LHS](@join_bot_unitL x) margin2. rewrite margin3; first by[]. apply :conj; first by rewrite meet_bot_unitL. apply :conj; first by rewrite meet_bot_unitL. apply :conj; first by[]. apply :conj; first by rewrite /ge join_bot_unitR. apply :conj; first by[]. rewrite (@also_integ (join bot x) (ctext c) cx c); first by[]. apply :conj; first by rewrite /cintegrable join_bot_unitL H1. apply :conj; first by rewrite join_bot_unitL meet_commutative. by rewrite /ge join_idempotent. Qed. Lemma com_join_P x y : complement_P (join x y) (meet (com x) (com y)). Proof. rewrite/ complement_P; apply: conj. -rewrite meet_commutative meet_distributive [_ (com y)]meet_commutative -meet_associative [_ x]meet_commutative com_bot meet_bot_unitR [_ (com x)]meet_commutative -meet_associative [_ y]meet_commutative com_bot meet_bot_unitR; apply: join_bot_unitL. -by rewrite join_distributive {1}[_ y]join_commutative -join_associative com_top -join_associative com_top 2!join_top_unitR meet_idempotent. Qed. Lemma com_meet_P x y : complement_P (meet x y) (join (com x) (com y)). Proof. rewrite/ complement_P; apply: conj. -by rewrite meet_distributive {1}[_ y]meet_commutative -!meet_associative !com_bot !meet_bot_unitR join_bot. -by rewrite join_commutative join_distributive join_commutative !join_associative com_top -join_associative [_ _ y]join_commutative com_top join_top_unitL join_top_unitR meet_idempotent. Qed. Lemma com_join x y : com (join x y) = meet (com x) (com y). Proof. apply: (@complement_unique (join x y)). apply: conj. -apply: com_P. -apply: com_join_P. Qed. Lemma com_meet x y : com (meet x y) = join (com x) (com y). Proof. apply: (@complement_unique (meet x y)). apply: conj. -apply: com_P. -apply: com_meet_P. Qed. Lemma double_com x : com (com x) = x. Proof. apply: (@complement_unique (com x)). apply: conj. -apply: com_P. -rewrite /complement_P meet_commutative join_commutative; apply/ com_P. Qed. Lemma Th5_1 x y z : x != bot /\ mix x y = mix x z -> up (join x y) = dot (mix x z) (up y). Proof. case=> H1 H2. rewrite -H2 mix_up_down. by rewrite -dotA down_up_unitL dotr0. by[]. Qed. Lemma Th5_2 x y z : x != bot /\ mix x y = mix x z -> up (meet (join x y) (com (meet (meet y (com x)) (com z)))) = dot (mix x z) (up (meet y (com (meet (meet y (com x)) (com z))))). Proof. case=> H1 H2; rewrite -margin1. rewrite (@Th5_1 x y z). rewrite const_margin. rewrite margin1. by[]. rewrite/ ge. by rewrite -meet_associative meet_absorption. apply: conj. rewrite/ cintegrable ctext1. rewrite /ge; by rewrite -meet_associative meet_absorption. move: (@orbN (ge z x)). move/ orP. case. move=> H3. rewrite ctext2. by rewrite meet_bot_unitR. by split. move=> H3. rewrite ctext3. by rewrite meet_distributive meet_commutative meet_associative [_ y _]meet_commutative meet_associative com_bot !meet_bot_unitL -meet_associative [_ _ z]meet_commutative com_bot meet_bot_unitR join_bot. by split. by split. by rewrite/ ge 2!join_distributive -join_associative join_idempotent !join_absorption. Qed. Lemma Th5_3 x y z : meet (join x y) (com (meet (meet y (com x)) (com z))) = join x (meet y z). Proof. rewrite !com_meet !double_com !meet_distributive ![_ (join x y) _]meet_commutative 2!meet_distributive [_ _ y]meet_commutative com_bot. by rewrite [_ bot]join_commutative join_bot_unitL meet_idempotent meet_absorption [_ x]join_commutative [_ _ x]meet_commutative meet_absorption join_distributive join_associative join_idempotent meet_commutative -join_distributive. Qed. Lemma Th5_4 x y z : x != bot /\ mix x y = mix x z -> up (meet y (com (meet (meet y (com x)) (com z)))) = up (meet z (com (meet (meet z (com x)) (com y)))). Proof. move=> H1; move: (H1); move /Th5_2; rewrite Th5_3=> H2. have H3: dot (mix x z) (up (meet y (com (meet (meet y (com x)) (com z))))) = dot (mix x y) (up (meet z (com (meet (meet z (com x)) (com y))))). -by rewrite -H2 meet_commutative -Th5_3; apply: Th5_2; apply: conj; case: H1. case: H1=> H1' H1''; move: H3; rewrite H1'' mix_up_down. by rewrite -dotA [_ (down z) _]dotC dotA; move/ div_down_right; rewrite -[RHS]dotA [_ _ (up z)]dotC dotA -[_ _ (up z)]dotA down_up_unitL bob_unitR [RHS]dotC; move/ div_up_down; rewrite dotC dotA down_up_unitL bob_unitL. by[]. Qed. Lemma Th5_5 x y z : meet (com (meet (meet y (com x)) (com z))) (com (meet (meet z (com x)) (com y))) = join (join x (meet y z)) (meet (com z) (com y)). Proof. rewrite !com_meet !double_com; rewrite [_ _ y]join_commutative [_ (com z) _]join_commutative join_associative meet_distributive [_ _ (com z)]meet_commutative [_ (com z) _]meet_distributive [_ (com z) z]meet_commutative com_bot [_ _ bot]join_commutative join_bot_unitL meet_commutative [_ y x]join_commutative. have H: com (meet (meet y (com x)) (com z)) = join (join (com y) x) z. -by rewrite !com_meet !double_com. by rewrite -H Th5_3 meet_distributive join_associative -[_ (join x (meet y z)) (meet (com z) (com y))]join_associative join_commutative join_associative [_ _ x]join_commutative meet_commutative meet_absorption join_associative. Qed. Theorem Th5 x y z : mix x y = mix x z -> mix x y = mix x (meet y z). Proof. move=> H1. move: (excluded_middle_axiom (x = bot)). case. -by move=> h1; move: H1; rewrite h1; move/ down_injective=> h2; rewrite -h2 meet_idempotent. -move/ eqP=> H2. have H3: (meet y (com (meet (meet y (com x)) (com z)))) = (meet z (com (meet (meet z (com x)) (com y)))). +by apply /up_injective /Th5_4. have H4: meet y (com (meet (meet y (com x)) (com z))) = meet y z. +by rewrite -[_ y _]meet_idempotent {2}H3 meet_associative -[meet (meet y (com (meet (meet y (com x)) (com z)))) z]meet_associative [_ _ z]meet_commutative meet_associative -meet_associative Th5_5 meet_distributive [_ x _]join_commutative join_absorption meet_absorption. rewrite [in RHS]mix_up_down; first 1 [apply/ eqP|]. by rewrite eq_sym; apply/ eqP; rewrite div_up_down H1 -[in LHS]Th5_3 -H4 Th5_2. by[]. Qed. Lemma com_bot_top_P : complement_P bot top. Proof. rewrite/ complement_P; split. -apply /meet_bot_unitL. -apply /join_bot_unitL. Qed. Lemma com_bot_top : com bot = top. Proof. apply /complement_unique. exact: bot. split. -apply /com_P. -apply /com_bot_top_P. Qed. Lemma com_top_bot : com top = bot. Proof. apply /complement_unique. exact: top. split. -apply /com_P. -move: com_bot_top_P => H; rewrite/ complement_P meet_commutative join_commutative; apply /H. Qed. Lemma meet_eq_ge x y : meet x y = bot <-> ge (com x) y. Proof. split. -move=> H; rewrite/ ge; apply/ eqP; move: (com_meet x y); rewrite H com_bot_top; move=> H1; by rewrite -[in RHS](join_bot_unitL (com x)) -com_top_bot H1 com_join !double_com [in RHS]join_commutative join_distributive [_ _ x]join_commutative com_top meet_top_unitL. -by rewrite/ ge; move /eqP => H1; rewrite -(double_com x) -(double_com y) -com_join -H1 -join_associative com_top join_top_unitR com_top_bot. Qed. Theorem Th6 x y z : gt x z /\ meet y z = bot -> mix x y /d z = mix (meet x (com z)) y. Proof. case; rewrite /gt /ge; move /andP =>[] /eqP => H1 H2; move: (excluded_middle_axiom (x = bot)); case. -by move=> H3; rewrite H3; rewrite H3 join_bot_unitL in H1; rewrite H1 margin3 meet_bot_unitL. -move/ eqP=> H3 H4; rewrite mix_up_down ; last by[]; rewrite dotC const_margin; last first. -rewrite /cintegrable /ge ctext1 ctext4. apply: conj. -by rewrite [_ _ y]join_commutative -join_associative H1. -by rewrite meet_commutative. rewrite margin1; last first. -by rewrite /ge [_ _ y]join_commutative -join_associative H1. rewrite meet_commutative meet_distributive. rewrite meet_commutative in H4. move: H4; move/ meet_eq_ge; rewrite ge_meet_def; move/ eqP=> H4; rewrite H4. rewrite dotC -mix_up_down ; first by rewrite meet_commutative. apply: meet_not_bot. exact z. by rewrite /ge [_ _ z]join_commutative com_top join_top_unitL. by rewrite /gt; apply /andP; split; first by rewrite /ge; apply /eqP. Qed. Theorem Th7_1 x : up x /d x = bob. Proof. rewrite margin1; first by rewrite com_bot up_bot_bob. by rewrite /ge join_idempotent. Qed. Theorem Th7_2 x y : x != bot -> mix x y /d meet x (com y) = bob. Proof. move=> H; rewrite mix_up_down; last by[]. rewrite -(meet_top_unitR (join x y)) -(com_top y) join_commutative -join_distributive dotC const_margin; first rewrite margin1; first by rewrite com_meet double_com join_distributive com_top meet_top_unitR [_ _ y]join_commutative -join_distributive com_bot join_commutative join_bot_unitL dotC up_down_unitL. by rewrite /ge -join_associative join_idempotent. apply: conj. by rewrite /cintegrable ctext1 /ge -join_associative join_idempotent. by rewrite ctext4 -meet_associative [_ _ y]meet_commutative com_bot meet_bot_unitR. Qed. Corollary Co1_1 c y : meet (ctext c) y = bot -> dot c (up y) /d y = c. Proof. move=> H; rewrite const_margin; first rewrite margin1; first by rewrite com_bot up_bot_bob bob_unitR. apply: ge_reflexive. apply: conj; first rewrite/ cintegrable ctext1; first apply: ge_reflexive. by rewrite meet_commutative. Qed. Corollary Co1_2 y z c : meet (ctext c) z = bot /\ meet y z = bot -> dot c (up (join y z)) /d z = dot c (up y). Proof. case=> H1 H2; rewrite const_margin; first rewrite margin1; first rewrite meet_commutative meet_distributive [_ _ z]meet_commutative com_bot join_bot_unitR; move: H2; rewrite meet_commutative =>/meet_eq_ge; rewrite ge_meet_def=>/ eqP=> H2; first by rewrite H2. by rewrite /ge -join_associative join_idempotent. apply: conj; first by rewrite /cintegrable ctext1 /ge -join_associative join_idempotent. by rewrite meet_commutative H1. Qed. Corollary Co1_3 x y z c : meet x z = bot /\ meet y z = bot /\ meet (ctext c) z = bot -> dot c (up (join x (join y z))) /d z = dot c (up (join x y)). Proof. case=> H1 [] H2 H3; rewrite const_margin. rewrite margin1. rewrite join_associative meet_commutative meet_distributive [_ _ z]meet_commutative com_bot join_bot_unitR meet_commutative. have H: meet z (join x y) = bot. -by rewrite meet_distributive meet_commutative [_ _ y]meet_commutative H1 H2 join_bot. by move: H; move /meet_eq_ge; rewrite ge_meet_def; move /eqP; rewrite meet_commutative=> H4; rewrite H4. by rewrite join_associative /ge -join_associative join_idempotent. rewrite meet_commutative in H3; apply: conj; last by[]. by rewrite /cintegrable ctext1 join_associative /ge -join_associative join_idempotent. Qed. Corollary Co1_4 x y z : x != bot /\ meet x y = bot /\ meet y z = bot -> mix (join x y) z /d y = mix x z. Proof. case => H1 [] H2 H3; rewrite mix_up_down; first rewrite dotC const_margin. rewrite margin1. rewrite -join_associative [_ _ z]join_commutative join_associative meet_commutative meet_distributive [_ _ y]meet_commutative com_bot join_bot_unitR meet_commutative. have H: meet y (join x z) = bot. -by rewrite meet_distributive meet_commutative H2 H3 join_bot. by move: H; move /meet_eq_ge; rewrite ge_meet_def; move /eqP; rewrite meet_commutative=> H4; rewrite H4 dotC -mix_up_down. by rewrite -join_associative [_ y _]join_commutative join_associative /ge -join_associative join_idempotent. apply: conj. -by rewrite /cintegrable ctext1 -join_associative [_ y _]join_commutative join_associative /ge -join_associative join_idempotent. -by rewrite ctext4. have H: (x != bot) || (y != bot) -> join x y != bot. -rewrite -negb_and; apply: contra; apply/join_eq_bot. by apply /H /orP /or_introl. Qed. Lemma meet_ge x y:ge x (meet x y). Proof. rewrite/ge. rewrite join_distributive. rewrite join_idempotent. apply/eqP. by apply:join_absorption. Qed. Lemma meet_ge' x y:ge x y <->meet x y = y. Proof. rewrite/ge. split. move=>H. move/eqP in H. rewrite meet_commutative -H meet_distributive. rewrite meet_idempotent join_commutative. apply/eqP. apply:meet_ge. move=>H. apply/eqP. by rewrite -H meet_absorption. Qed. Lemma ge_w_L x y z:ge x (join y z)-> ge x y. Proof. move=>H. apply:(@ge_transitive (join y z) x y)=>//. apply meet_ge'. by rewrite meet_commutative join_absorption. Qed. Lemma meet_ge_meet x y z:ge x y -> ge (meet x z) (meet y z). Proof. move=>H. rewrite meet_ge'=>//. move/meet_ge' in H. by rewrite (meet_commutative y _) meet_associative -(meet_associative x z z) meet_idempotent (meet_commutative x z) -meet_associative H. Qed. Theorem N_law1 x y z : x != bot /\ ge (com z) (ctext (mix x (join y z))) /\ meet x (com z)!=bot -> mix x (join y z) = mix (meet x (com z)) (meet y (com z)). Proof. case => H1 [] H2 H3. -move:(excluded_middle_axiom (ge (join y z) x));case=>Hyz. move:(le_mix_bob_eq)(Hyz)=>H Hyz'. rewrite/le in H. move/H in Hyz. rewrite Hyz=>//. move/(@meet_ge_meet _ _ (com z)) in Hyz'. rewrite meet_commutative meet_distributive (meet_commutative (com z) z) com_bot meet_commutative join_bot_unitR in Hyz'. move/H in Hyz'. by rewrite Hyz'. -rewrite ctext3 in H2=>//. move/negP in Hyz. rewrite [in RHS]mix_up_down=>//. move:(H2)=>H2';move:(H2)=>h2. move/ge_w_L in H2. move/meet_eq_ge in H2. rewrite join_associative (join_commutative x _) -join_associative in h2; move/ge_w_L in h2. move/meet_eq_ge in h2. rewrite meet_commutative {1}[(meet y (com z))]meet_commutative -meet_distributive meet_commutative -(join_bot_unitR (meet (join x y) (com z))) -(com_bot z) meet_commutative [_ z _]meet_commutative -meet_distributive meet_commutative -margin1. --have H: up (join (join x y) z) = dot (mix x (join y z)) (up (join y z)). -by rewrite mix_up_down; first rewrite -dotA down_up_unitL bob_unitR join_associative. rewrite H. --have h: ge (join y z) z. by rewrite join_commutative join_ge. move/margin1 in h. rewrite const_margin. rewrite meet_commutative meet_distributive (meet_commutative _ y) (meet_commutative _ z) com_bot join_bot_unitR in h. by rewrite h -dotrA up_down_unitL dotr0. apply:conj=>//. by rewrite /cintegrable ctext1 /ge -join_associative join_idempotent. rewrite ctext3=>//. move/meet_ge' in H2'. by rewrite -H2' meet_associative com_bot meet_bot_unitL. by rewrite join_commutative join_ge. by move/negP in Hyz. Qed. Lemma ctext_eq_ge x c : meet (ctext c) (com x) = bot <-> ge x (ctext c). Proof. split. by move=> H; apply /eqP; rewrite /ge -[in RHS](@join_bot_unitR x) -H join_distributive com_top meet_top_unitR. by rewrite ge_meet_def; move /eqP =>H; rewrite -H [_ x _]meet_commutative -meet_associative com_bot meet_bot_unitR. Qed. Lemma N_law2_1 x y z cx cy : meet (ctext cx) x = bot /\ meet (ctext cy) y = bot /\ up (join (join x y) z) = dot cy cx -> up (meet (join y z) (com x)) = dot cx (cy /d x). Proof. case=> H1 [] H2 H3. rewrite -(@join_bot_unitR (meet (join y z) (com x))) -(@com_bot x) meet_commutative [_ x _]meet_commutative -meet_distributive meet_commutative join_commutative join_associative -margin1; last first; first by rewrite /ge -join_associative join_commutative -join_associative join_commutative join_associative join_idempotent. rewrite H3 dotC const_margin; first by[]. apply: conj; last first; first by rewrite meet_commutative. rewrite /cintegrable /ge; move: (@Th3_2 cy cx); rewrite -H3 ctext1=> H4. move: (@join_ge x y) (@join_ge (join x y) z)=> H5 H6. have H: ge (join (join x y) z) x. -apply/ (@ge_transitive (join x y)); by[]. have H': ge (join (ctext cy) (ctext cx)) x. -apply/ (@ge_transitive (join (join x y) z)); by[]. move: H1; rewrite -{1}(double_com x) ctext_eq_ge ge_meet_def; move /eqP=> H1. rewrite ge_meet_def in H'; move /eqP in H'. by rewrite -H' -H1 meet_commutative meet_distributive meet_associative com_bot meet_bot_unitL join_bot_unitR meet_commutative meet_absorption. Qed. Lemma dot_eq c1 c2 c3 : c1 = c2 <-> dot c1 c3 = dot c2 c3. Proof. split. by move=> H; rewrite H. by move=> H; rewrite -(dotr0 c1) -(dotr0 c2) -(dotNr c3) [_ _ c3]dotC !dotA H. Qed. Lemma N_law2_2 x y z cx cy : ge (com x) y /\ meet (ctext cx) x = bot /\ meet (ctext cy) y = bot /\ up (join (join x y) z) = dot cy cx -> up (meet z (meet (com x) (com y))) = dot (cx /d y) (cy /d x). Proof. case=> H1 [] H2 [] H3 H4. rewrite (@dot_eq _ _ (up (meet (join x z) (com y)))) -(@join_bot_unitL (meet z (meet (com x) (com y)))) -(@meet_bot_unitL (com x)) -(com_bot y) -meet_associative [_ (com y) (com x)]meet_commutative meet_commutative [_ z _]meet_commutative -meet_distributive meet_commutative meet_associative -margin1; first rewrite dotC -[in LHS]const_margin; first rewrite (@N_law2_1 x y z cx cy); first rewrite [in LHS](@N_law2_1 y x z cy cx); first rewrite dotA [_ _ cx]dotC dotA [_ cx _]dotC -H4 -dotA dotC const_margin; first rewrite margin1; first by rewrite [_ _ y]join_commutative -join_associative meet_commutative meet_distributive [_ _ y]meet_commutative com_bot join_bot_unitL meet_commutative. by rewrite /ge join_commutative join_associative [_ x _]join_commutative join_associative join_idempotent. split. by rewrite /cintegrable /ge ctext1 join_commutative join_associative [_ x _]join_commutative join_associative join_idempotent; first last. move: (@Th3_2 (cx /d y) (cy /d x)); rewrite ge_meet_def; move /eqP=> H5. by rewrite -H5 meet_associative meet_distributive !margin0 [_ _ (com y)]meet_commutative meet_associative com_bot meet_bot_unitL meet_associative [_ y _]meet_commutative H3 meet_bot_unitL join_idempotent meet_bot_unitL. by rewrite [_ y _]join_commutative dotC. by[]. split; first rewrite /cintegrable /ge join_commutative ctext1 join_distributive join_associative join_idempotent; rewrite /ge in H1; move: H1; move /eqP=> H1. by rewrite [_ _ (com x)]join_commutative H1. by rewrite ctext1 [_ _ (com y)]meet_commutative meet_associative com_bot meet_bot_unitL. by rewrite /ge in H1; move: H1; move /eqP=> H1; rewrite /ge join_commutative join_distributive join_associative join_idempotent [_ _ (com x)]join_commutative H1. Qed. Theorem N_law2 x y z cx cy : ge (com x) y /\ meet (ctext cx) x = bot /\ meet (ctext cy) y = bot /\ up (join (join x y) z) = dot cy cx -> up (join (join x y) z) = dot (dot (up (meet (join x z) (com y))) (up (meet (join y z) (com x)))) (down (meet (meet z (com x)) (com y))). Proof. case=> H1 [] H2 [] H3 H4. rewrite (@N_law2_1 y x z cy cx); last by rewrite [_ y _]join_commutative dotC. rewrite (@N_law2_1 x y z cx cy); last by[]. rewrite [_ cy _]dotC dotA -[_ _ cx]dotA [_ (cx /d y) _]dotC -2!dotA -H4 [_ (cx /d y) _]dotA -meet_associative -(@N_law2_2 x y z cx cy); last by[]. by rewrite (@up_down_unitL (meet z (meet (com x) (com y)))) dotr0. Qed. Section conditional_independence. (*def 10*) Definition co_ind x y z:= (mix x (join y z))=(mix x z). Definition ind x y:= (co_ind x y bot). (*Notation "x._y|z":=(@co_ind _ _ _). Notation "x._y":=(@ind _ _).*) Lemma join_bot_bot x y: x!=bot-> y!=bot -> (join x y)!=bot. Proof. move=>XH YH. apply/eqP. case. move/eqP=>H0. move:(@join_eq_bot x y H0). move/andP. case. move=>X. move/eqP. by apply/eqP. Qed. Theorem co_ind_eq_1 x y z: x!=bot -> y!=bot -> (co_ind x y z <-> (mix (join x y) z)=dot (mix x z) (mix y z)). Proof. move=> XH YH . split. -rewrite/ co_ind=>H. rewrite -H. rewrite[in RHS] mix_up_down=>//. rewrite[in RHS] mix_up_down=>//. rewrite -dotrAC dotrA -dotrA dotrACA dotrA dotr_perm down_up_unitL dot0r dotC join_associative mix_up_down=>//. by apply join_bot_bot=>//. -rewrite/co_ind mix_up_down. rewrite div_up_down. move=>H. rewrite mix_up_down=>//. rewrite join_associative H. rewrite [mix y _]mix_up_down=>//. rewrite dotrA div_up_down dotr_perm up_down_unitL dot0r dotrC=>//. by apply join_bot_bot=>//. Qed. Theorem co_ind_eq_2 x y z: x!=bot -> y!=bot -> (co_ind x y z <-> (up (join (join x y) z)=dot (up (join y z)) (mix x z))). Proof. move=>XH YH. rewrite/co_ind. rewrite mix_up_down=>//. by rewrite div_up_down -join_associative dotrC. Qed. Theorem ind_eq x y: x!=bot -> y!=bot -> (ind x y <-> up (join x y) = dot (up x) (up y)). Proof. move=>XH YH. rewrite/ind/co_ind join_commutative. move:(join_bot_unitL y)=>->. rewrite mix_up_down=>//. by rewrite div_up_down. Qed. (*co_ind is symmetry*) Theorem co_sym x y z:x!=bot->y!=bot-> (co_ind x y z <-> co_ind y x z). Proof. move=>X Y. by split;move=>H; apply co_ind_eq_1 in H=>//; rewrite co_ind_eq_1=>//; rewrite join_commutative dotrC=>//. Qed. (*Prop6*) Proposition eq_co_ge x y: x!=bot -> y!=bot ->(co_ind x x y <-> ge y x). Proof. move=>X Y. rewrite (@co_ind_eq_1 x x y)=>//. rewrite join_idempotent. rewrite mix_up_down=>//. rewrite div_up_down. rewrite dotrA. rewrite dotr_perm. rewrite up_down_unitL dot0r. rewrite dotrC -div_up_down=>//. rewrite up_down_unitL. rewrite dotrC -div_up_down bob_unitL. split. move=>H. apply down_injective in H. rewrite/ge. rewrite join_commutative. by apply/eqP. rewrite/ge=>H. rewrite join_commutative. move/eqP in H. by rewrite H. Qed. Proposition ge_co_1 x y z : ge z y -> co_ind x y z. Proof. rewrite/ge/co_ind. move=>H. move/eqP in H. rewrite join_commutative H. by[]. Qed. Proposition ge_co_2 x y z : x!=bot -> ge z x -> co_ind x y z. Proof. rewrite/ge/co_ind. move=>X H. rewrite mix_up_down=>//. rewrite join_commutative. rewrite -join_associative. move/eqP in H. rewrite H up_down_unitL. rewrite mix_up_down=>//. by rewrite join_commutative H up_down_unitL. Qed. Lemma co_mc0 x y b:meet x y = bot->meet y b!=bot-> x = meet x (join (com y) b). Proof. move=>A H. rewrite meet_commutative in A;apply meet_eq_ge in A=>//. apply meet_ge' in A;rewrite meet_commutative in A. by rewrite meet_distributive A meet_absorption. Qed. Lemma co_mc00 x y b:meet x y = bot->meet y b!=bot-> (com x)=join (com x) (meet y (com b)). Proof. move=>A B. rewrite [in LHS](@co_mc0 x y b)=>//. by rewrite com_meet com_join double_com. Qed. Lemma gt_0 x y z:gt x y -> gt x (meet y z). Proof. rewrite/gt. move/andP=>[h1 h2]. move:(h1)=>h. apply/andP;split. move/eqP in h1. apply/eqP. by rewrite join_distributive h1 join_absorption. apply/negP; move=> /eqP h3. move/eqP in h1;apply meet_ge' in h. rewrite h3 join_commutative join_distributive join_idempotent -h3 join_absorption in h1. move:h2;move/negP. case. by rewrite h1. Qed. Lemma co_mc1 x y z b:x!=bot->y!=bot->meet x y = bot->meet y b!=bot-> (mix (join x y) z) /d (meet y (meet (com b) (com z)))=mix (join x (meet y b)) z. Proof. move=>X Y A H. rewrite Th6. rewrite com_meet -com_join double_com (join_associative (com y) _ _). rewrite meet_distributive. rewrite meet_commutative meet_distributive meet_commutative -co_mc0=>//. rewrite meet_commutative meet_distributive com_bot join_bot_unitL. apply/eqP;rewrite eq_sym;apply/eqP. apply:(@lemma1_3 (join x (meet y b)) z (meet (join x y) z)). apply:join_bot_bot=>//. -apply:(@ge_transitive z (join (join x (meet y b)) z) (meet (join x y) z)). rewrite join_commutative join_ge=>//. rewrite meet_commutative meet_ge=>//. split. -rewrite meet_associative. apply:gt_0. rewrite/gt;apply/andP;split. apply:(@ge_transitive y (join x y) (meet y (com b))). rewrite join_commutative join_ge=>//. apply meet_ge. apply/negP. move=>/eqP h. move:(A)=>a. apply meet_eq_ge in a. apply meet_ge' in a. move:(A)=>c. apply meet_eq_ge in c;move/eqP in c. move:H;move/negP. case. apply/eqP. rewrite meet_commutative. apply (@meet_eq_ge b y). apply meet_ge'. rewrite meet_commutative -h. rewrite-[in RHS](@join_bot_unitL y). rewrite -(@com_bot x) (join_commutative (meet _ _) _) join_distributive. rewrite meet_commutative (join_commutative y x) h (join_commutative y _) c. by rewrite meet_associative a. --rewrite meet_commutative -meet_associative -(meet_associative (com b) _ _) (meet_commutative (com _) z) com_bot. by rewrite meet_commutative (meet_commutative _ bot) meet_bot_unitL meet_bot_unitL. Qed. Lemma meet_join x y:(join x y = x)<->(meet x y = y). Proof. split=>H. apply meet_ge'. by apply/eqP. apply meet_ge' in H. by move/eqP in H. Qed. Lemma ge_mix_bob_eq x y:x!=bot-> (ge y x <-> (mix x y = bob)). Proof. move=>X. move:(@le_mix_bob_eq x y)=>H. rewrite/le in H. by apply H. Qed. Lemma co_mc2 x y z b:x!=bot->y!=bot->meet x y = bot->meet y b!=bot->~~(ge z y)->~~ (ge z x)-> (dot (mix x z) (mix y z))/d (meet y (meet (com b) (com z)))= dot (mix x z) (mix (meet y b) z). Proof. move=>X Y A B C D. rewrite const_margin. rewrite Th6. rewrite com_meet -com_join double_com meet_distributive. rewrite com_bot join_bot_unitL meet_distributive. rewrite -(@lemma1_3 (meet y b) z (meet y z))=>//. -apply:(@ge_transitive z (join (meet y b) z) (meet y z)). by rewrite join_commutative join_ge. by rewrite meet_commutative meet_ge. split. -rewrite meet_associative. apply:gt_0=>//. rewrite/gt;apply/andP;split. by rewrite meet_ge. apply/negP; move=> /eqP H. move:B;move/negP. case. by rewrite H -meet_associative (meet_commutative _ b) com_bot meet_commutative meet_bot_unitL. --rewrite meet_commutative -meet_associative -(meet_associative (com b) _ _) (meet_commutative (com _) z) com_bot. by rewrite meet_commutative (meet_commutative _ bot) meet_bot_unitL meet_bot_unitL. ---split. +rewrite/cintegrable. rewrite ctext3=>//. apply/eqP. rewrite join_distributive (join_commutative _ y) join_associative join_idempotent. by rewrite join_distributive -(join_associative y z (com z)) com_top join_top_unitR meet_top_unitR join_absorption. rewrite ctext3=>//. rewrite meet_distributive (meet_commutative _ x) meet_associative A meet_bot_unitL -meet_associative -meet_associative (meet_commutative _ z) com_bot meet_associative meet_bot_unitR join_idempotent=>//. Qed. Lemma margin_eq x y z:(x = y)-> (x /d z) = (y /d z). Proof. by move=>->. Qed. Theorem M_law x y z a b: (meet x y)=bot ->(meet x a)!=bot ->(meet y b)!=bot -> co_ind x y z ->co_ind (meet x a) (meet y b) z. Proof. move=> H1 H2 H3 H. move:(excluded_middle_axiom (x = bot));case=>Hx. rewrite co_sym=>//. rewrite Hx meet_bot_unitL. apply: ge_co_1. by apply bot_minimum. move:(excluded_middle_axiom (y = bot));case=>Hy. rewrite Hy meet_bot_unitL. by apply:ge_co_1; apply bot_minimum. move/eqP in Hx; move/eqP in Hy. move:(excluded_middle_axiom (ge z x));case=>Hzx. apply:ge_co_2=>//. apply:(@ge_transitive x z (meet x a))=>//. by rewrite meet_ge. move:(excluded_middle_axiom (ge z y));case=>Hzy. apply:ge_co_1. apply:(@ge_transitive y z (meet y b))=>//. by rewrite meet_ge. move:(excluded_middle_axiom (ge z (meet y b)));case=>Hzy'. by apply:ge_co_1. move/negP in Hzx;move/negP in Hzy;move/negP in Hzy'. -apply co_ind_eq_1 in H=>//. rewrite co_ind_eq_1=>//. apply (@margin_eq _ _ (meet y (meet (com b) (com z)))) in H. rewrite co_mc1 in H=>//. rewrite co_mc2 in H=>//. rewrite join_commutative dotrC in H. apply (@margin_eq _ _ (meet x (meet (com a) (com z)))) in H. rewrite co_mc1 in H=>//. rewrite co_mc2 in H=>//. by rewrite join_commutative dotrC. by rewrite meet_commutative meet_associative H1 meet_bot_unitL. by rewrite meet_commutative meet_associative H1 meet_bot_unitL. Qed. Theorem N_law1' x y z: meet x y = bot -> meet y z = bot -> meet x z = bot -> x != bot /\ ge (com z) (ctext (mix x (join y z)))/\ meet x (com z)!=bot-> mix x (join y z) = mix x y. Proof. move=> A B C H. apply N_law1 in H. rewrite meet_commutative in B;rewrite meet_commutative in C. apply meet_eq_ge in B;apply meet_eq_ge in C. apply meet_ge' in B;apply meet_ge' in C. rewrite meet_commutative in B;rewrite meet_commutative in C. by rewrite B C in H. Qed. Theorem N_law2' x y z cx cy : meet x y = bot -> meet y z = bot -> meet x z = bot -> ge (com x) y /\ meet (ctext cx) x = bot /\ meet (ctext cy) y = bot /\ up (join (join x y) z) = dot cy cx -> up (join (join x y) z) = dot (dot (up (join x z)) (up (join y z))) (down z). Proof. move=>a b c H. move:(a)(b)(c)=>A B C. apply N_law2 in H. rewrite meet_commutative in A. apply meet_eq_ge in A;apply meet_eq_ge in B;apply meet_eq_ge in C. apply meet_ge' in A;apply meet_ge' in B;apply meet_ge' in C. apply meet_eq_ge in a;apply meet_ge' in a. rewrite meet_commutative meet_distributive A B in H. rewrite meet_commutative meet_distributive a C in H. by rewrite (meet_commutative z _) C meet_commutative B in H. Qed. Theorem co_fact x y z cx cy:x!=bot->y!=bot->z!=bot-> meet x y = bot -> meet y z = bot -> meet x z = bot -> ge (com x) y /\ meet (ctext cx) x = bot /\ meet (ctext cy) y = bot /\ up (join x (join y z)) = dot cx cy -> co_ind x y z . Proof. move=>X Y Z A B C. case=>Ha [] Hb [] Hc Hd. apply co_ind_eq_2=>//. rewrite mix_up_down//. rewrite dotrA (dotrC (up (join y z)) _). move:(A)=>a;apply meet_eq_ge in A. apply (@N_law2' x y z cx cy)=>//. split=>//. split=>//. split=>//. by rewrite -join_associative dotrC. Qed. Theorem co_fact' x y z:x!=bot->y!=bot->z!=bot-> meet x y = bot -> meet y z = bot -> meet x z = bot -> co_ind x y z -> exists cx cy,ge (com x) (ctext cx) /\ ge (com y) (ctext cy) /\ up (join x (join y z)) = dot cx cy. Proof. move=>X Y Z A B C H. apply co_ind_eq_2 in H=>//. exists (up (join y z)), (mix x z). split. apply/meet_eq_ge. by rewrite ctext1 meet_distributive A C join_idempotent. split. apply/meet_eq_ge. move:(excluded_middle_axiom (ge z x));case=>Hzx. apply ge_mix_bob_eq in Hzx=>//. by rewrite Hzx ctext1 meet_bot_unitR. move/negP in Hzx. rewrite ctext3=>//. by rewrite meet_distributive B meet_commutative A join_idempotent. by rewrite join_associative. Qed. Theorem co_decomp x y z a b: a!=bot -> b!=bot -> meet x y = bot -> co_ind x y z -> ge x a -> ge y b -> co_ind a b z. Proof. move=> A B H0 H1 Ha Hb. apply meet_ge' in Ha. apply meet_ge' in Hb. rewrite -Ha -Hb. apply:M_law=>//. rewrite Ha=>//. rewrite Hb=>//. Qed. Theorem co_w_u x y z w:x!=bot -> y!=bot -> z!=bot -> w!=bot-> meet x y = bot -> meet x z = bot -> meet x w = bot -> meet y z = bot -> meet y w = bot -> meet w z = bot -> co_ind x (join y z) w -> co_ind x y (join z w). Proof. move=>X Y Z W A B C D E F H. move:(H)=>H'. apply co_ind_eq_2 in H=>//. rewrite co_ind_eq_2=>//. rewrite join_associative in H. rewrite join_associative H. rewrite[in RHS]dotrC -div_up_down. rewrite dotrC join_associative. rewrite dotrA (@dotrC (down (join (join y z) w)) _). rewrite up_down_unitL dot0r=>//. apply/eqP. rewrite eq_sym. apply/eqP. apply (@M_law x (join y z) w x (join z w)) in H'=>//. rewrite/co_ind in H'. rewrite meet_idempotent in H'. rewrite meet_distributive in H'. rewrite (join_commutative y _) in H'. rewrite (meet_commutative _ z) join_absorption in H'. rewrite (meet_commutative _ w) in H'. rewrite meet_distributive in H'. rewrite (meet_commutative _ y) F E in H'. rewrite join_idempotent in H'. move:(bot_minimum z)=>I. rewrite/ge in I. move/eqP in I. rewrite I in H'=>//. rewrite meet_distributive A B join_idempotent=>//. rewrite meet_idempotent=>//. rewrite (join_commutative y _) -join_distributive E. move:(bot_minimum z)=>I;rewrite/ge in I;move/eqP in I. rewrite I=>//. apply join_bot_bot=>//. Qed. Corollary M_l x y z w:x!=bot -> y!=bot -> z!=bot -> w!=bot-> meet x y = bot -> meet x z = bot -> meet x w = bot -> meet y z = bot -> meet y w = bot -> meet w z = bot -> co_ind x (join y z) w -> co_ind x z w. Proof. move=>X Y Z W A B C D E F H'. rewrite/co_ind. apply (@M_law x (join y z) w x (join z w)) in H'=>//. rewrite/co_ind in H'. rewrite meet_idempotent in H'. rewrite meet_distributive in H'. rewrite (join_commutative y _) in H'. rewrite (meet_commutative _ z) join_absorption in H'. rewrite (meet_commutative _ w) in H'. rewrite meet_distributive in H'. rewrite (meet_commutative _ y) F E in H'. rewrite join_idempotent in H'. move:(bot_minimum z)=>I. rewrite/ge in I. move/eqP in I. rewrite I in H'=>//. rewrite meet_distributive A B join_idempotent=>//. rewrite meet_idempotent=>//. rewrite (join_commutative y _) -join_distributive E. move:(bot_minimum z)=>I;rewrite/ge in I;move/eqP in I. rewrite I=>//. Qed. Theorem co_contr x y z w:x!=bot -> y!=bot -> z!=bot -> w!=bot -> meet x y = bot -> meet x z = bot -> meet x w = bot -> meet y z = bot -> meet y w = bot -> meet w z = bot -> (co_ind x y (join z w) /\ co_ind x z w <-> co_ind x (join y z) w). Proof. move=>X Y Z W A B C D E F. split. move=>[Ha Hb]. apply co_sym in Ha=>//. apply co_sym in Hb=>//. apply co_ind_eq_2 in Ha=>//. apply co_ind_eq_2 in Hb=>//. rewrite (join_commutative z x) -join_associative in Hb. rewrite Hb in Ha. rewrite co_sym=>//. rewrite co_ind_eq_2=>//. rewrite (join_commutative _ x) join_associative (join_commutative x y). rewrite -join_associative Ha. rewrite[in RHS]dotrC. rewrite -div_up_down. rewrite dotrC -(dotrA (up (join x w)) _ _). rewrite dotrA (dotrC (down (join x w)) (up (join x w))). rewrite up_down_unitL dot0r. rewrite mix_up_down=>//. rewrite mix_up_down=>//. rewrite dotrA. rewrite (dotrC (dot (dot (up (join z w)) (down w)) (up (join y (join z w))))). rewrite dotrA dotrA. rewrite (dotrC _ (up (join z w))). rewrite up_down_unitL dot0r. rewrite dotrC join_associative mix_up_down=>//. apply:join_bot_bot=>//. apply:join_bot_bot=>//. apply:join_bot_bot=>//. -move=>H. split. apply:co_w_u=>//. apply: (@M_l x y z w)=>//. Qed. Corollary co_con x y z:x!=bot -> y!=bot -> z!=bot -> meet x y = bot -> meet x z = bot -> meet y z = bot->((co_ind x y z /\ ind x z)<->ind x (join y z)). Proof. move=>X Y Z A B C. split. move=>[Ha Hb]. rewrite ind_eq=>//. apply ind_eq in Hb=>//. apply co_sym in Ha=>//. apply co_ind_eq_2 in Ha=>//. rewrite Hb in Ha. rewrite (join_commutative y x) -join_associative in Ha. rewrite Ha. apply/eqP. rewrite eq_sym. apply/eqP. rewrite[in RHS] dotrC (dotrC (up x) (up z)) dotrA. rewrite dotrC -div_up_down. rewrite -dotrA dotrC up_down_unitL bob_unitL -div_up_down=>//. rewrite -mix_up_down=>//. apply:join_bot_bot=>//. move=>H. move:(H)=>H'. split. apply ind_eq in H=>//. rewrite co_ind_eq_2=>//. rewrite -join_associative H. rewrite[in RHS]dotrC. rewrite -div_up_down. rewrite -dotrA (dotrC (up x) _). rewrite up_down_unitL bob_unitL. rewrite/ind in H'. apply (@M_law x (join y z) bot x z) in H'. rewrite meet_idempotent in H'. rewrite meet_commutative join_commutative join_absorption in H'. rewrite/co_ind in H'. rewrite join_commutative join_bot_unitL in H'. rewrite H'=>//. rewrite meet_distributive. rewrite A B join_bot=>//. rewrite meet_idempotent=>//. rewrite meet_commutative join_commutative join_absorption=>//. apply:join_bot_bot=>//. apply (@M_law x (join y z) bot x z) in H'. rewrite meet_idempotent in H'. rewrite meet_commutative join_commutative join_absorption in H'. rewrite/ind=>//. rewrite meet_distributive. rewrite A B join_bot=>//. rewrite meet_idempotent=>//. rewrite meet_commutative join_commutative join_absorption=>//. Qed. Theorem co_inter x y z w:x!=bot -> y!=bot -> z!=bot -> w!=bot->meet x (com z)!=bot-> meet x y = bot -> meet x z = bot -> meet x w = bot -> meet y z = bot -> meet y w = bot -> meet w z = bot -> (co_ind x (join y z) w <-> (co_ind x y (join z w)/\co_ind x z (join y w))). Proof. move=>X Y Z W XZ A B C D E F. split. -move=>H. apply:conj. apply:co_w_u=>//. rewrite join_commutative in H;apply:co_w_u=>//. by rewrite meet_commutative. by rewrite meet_commutative. by rewrite meet_commutative. -move=>[HA HB]. move:(HA)(HB)=>Ha Hb. move/co_ind_eq_2 in Ha=>//; move/co_ind_eq_2 in Hb=>//. rewrite dotrC in Ha; rewrite dotrC in Hb. rewrite (join_associative y _ _) (join_commutative y z) -(join_associative z _ _)in Ha. apply div_up_down in Ha=>//;apply div_up_down in Hb=>//. rewrite -join_associative (join_associative y z w) (join_commutative y z) (join_associative x _ w) (join_associative x z y) -join_associative in Ha. rewrite Ha in Hb. ---have Hm: mix x (join z w)=mix x w. rewrite join_commutative. apply:N_law1'=>//. split;first by[]. split;last by[]. rewrite join_commutative Hb. move: (excluded_middle_axiom (ge (join y w) x)); case=>Hyw. rewrite ctext2=>//. by rewrite bot_minimum. move/negP in Hyw. rewrite ctext3=>//. by rewrite -meet_eq_ge meet_distributive meet_distributive meet_commutative B meet_commutative D meet_commutative F join_bot_unitL join_bot_unitL. apply div_up_down in Ha. rewrite dotrC Hm in Ha. rewrite co_ind_eq_2=>//. by rewrite (join_commutative y z) -join_associative -join_associative join_associative. by apply:join_bot_bot. Qed. End conditional_independence. Section separoid. Parameter sep:L -> L -> L -> bool. Axiom sep_id :forall x y, sep x y x. Axiom sep_co : forall x y z, sep x y z -> sep y x z. Axiom sep_decomp : forall x y z w, sep x y z -> ge y w -> sep x w z. Axiom sep_w_u : forall x y z w, sep x y z -> ge y w -> sep x y (join z w). Axiom sep_contr : forall x y z w, sep x y z -> sep x w (join y z) -> sep x (join y w) z. Axiom sep_strong : forall x y z w, ge y z -> ge y w -> sep x y z -> sep x y w -> sep x y (meet z w). Lemma sep_idL x y: sep x y y. Proof. apply:sep_co. by apply:sep_id. Qed. (* Lemma sep_xx x y: sep x x y -> sep x (join x y) y. Proof. move:(@sep_idL x y). move=>Ha Hb ;move:Ha Hb. move:(@sep_contr x y y x)=>H. by rewrite join_idempotent join_commutative in H. Qed. *) Lemma ge_sep_eq x y: ge y x -> sep x x y. Proof. (*move: (sep_idL x y).*) apply:sep_decomp. by apply:sep_idL. Qed. Proposition ge_sep_1 x y z: ge z y -> sep x y z. Proof. move:(@sep_idL x z). apply:(@sep_decomp x z z y). Qed. Proposition ge_sep_2 x y z: x!=bot -> ge z x -> sep x y z. Proof. move=>H h. apply sep_co. by apply:(@ge_sep_1 y x z). Qed. Lemma sep_de x y z w: sep x (join y w) z -> sep x y z. Proof. move=>H. move:H (@join_ge y w). apply:(@sep_decomp x (join y w) z y). Qed. Lemma sep_join x y z:sep x y z -> sep x (join y z) z. Proof. move=>H. move:(sep_idL x z)=>Ha. rewrite -(@join_idempotent z) in H. rewrite join_commutative. apply:sep_contr=>//. Qed. Lemma sep_join' x y z:sep x y z -> sep x y (join y z). Proof. move=>H. rewrite join_commutative. apply:(@sep_w_u x y z y)=>//. by rewrite ge_reflexive. Qed. Lemma sep_join'2 x y z:sep x y z -> sep x y (join x z). Proof. move=>H. apply:sep_co. apply:sep_join'. by apply sep_co. Qed. Lemma sep_join'3 x y z:sep x y z -> sep x y (join x (join y z)). Proof. move=>H. move:(sep_join' H)=>H'. apply:sep_join'2=>//. Qed. Theorem M_law_sep x y z a b: (meet x y)=bot ->(meet x a)!=bot ->(meet y b)!=bot -> sep x y z ->sep (meet x a) (meet y b) z. Proof. move=>H1 H2 H3 H. rewrite(@sep_decomp (meet x a) y z (meet y b))=>//. rewrite sep_co=>//. rewrite (@sep_decomp y x z (meet x a))=>//. rewrite sep_co=>//. by apply meet_ge. by apply meet_ge. Qed. Proposition mar_sep x1 x2 x3 x4 x5: sep x3 x1 x2 -> sep x4 (join x1 x2) x3 -> sep x5 (join (join x1 x2) x3) x4 -> sep x3 (join x1 x5) (join x2 x4). Proof. move=>Ha Hb Hc. move:(Hc)=>Hc'. apply (@sep_w_u x5 (join (join x1 x2) x3) x4 x2) in Hc'. rewrite (join_commutative x4 _) join_commutative in Hc'. apply sep_de in Hc'. rewrite join_commutative sep_contr=>//. rewrite sep_co=>//. -rewrite join_commutative -join_associative. rewrite sep_co=>//. rewrite(@sep_decomp x1 (join x3 (join x4 x5)) (join x2 (join x4 x5)) x3)=>//. rewrite sep_w_u=>//. rewrite sep_contr=>//. rewrite sep_co=>//. rewrite sep_contr=>//. -move:(Hb)=>Hb'. apply(@sep_w_u x4 (join x1 x2) x3 x2) in Hb'. rewrite sep_co=>//. rewrite (@sep_decomp x4 (join x1 x2) (join x3 x2) x1)=>//. rewrite join_ge=>//. rewrite join_commutative join_ge=>//. -rewrite sep_co=>//. move:(Hc)=>HC. rewrite -join_associative in HC. apply(@sep_w_u x5 (join x1 (join x2 x3)) x4 (join x2 x3)) in HC. rewrite (@sep_decomp x5 (join x1 (join x2 x3)) (join x4 (join x3 x2)) x1)=>//. rewrite (join_commutative x3 _)=>//. rewrite join_ge=>//. rewrite join_commutative join_ge=>//. rewrite join_commutative join_ge=>//. rewrite join_ge=>//. rewrite (join_commutative x1 _) -join_associative join_ge=>//. Qed. (*conditional independence => separoid*) Theorem co_id x y:x!=bot -> co_ind x y x. Proof. move=>X. apply ge_co_2=>//. Print ge_co_2. by rewrite/le ge_reflexive. Qed. Theorem co_ind_sep x y z w:x!=bot -> y!=bot -> z!=bot -> w!=bot -> meet x y = bot -> meet x z = bot -> meet x w = bot -> meet y z = bot -> meet y w = bot -> meet w z = bot -> (co_ind x y x /\ (co_ind x y z -> co_ind y x z) /\(co_ind x y z -> ge y w -> co_ind x w z) /\(co_ind x y z -> ge y w -> co_ind x y (join z w)) /\ (co_ind x y z -> co_ind x w (join y z) -> co_ind x (join y w) z) /\(ge y z -> ge y w -> co_ind x y z -> co_ind x y w -> co_ind x y (meet z w))). Proof. move=>X Y Z W A B C D E F. split. -by apply co_id. split. --by apply co_sym. split. ---move=>H H'. apply:(@co_decomp x y z x w)=>//. by rewrite ge_reflexive. split. ----move=>H H'. rewrite/ge/le in H'. move/eqP in H'. rewrite -H' in H. move/co_w_u in H=>//. rewrite join_commutative. apply:H=>//. by rewrite meet_commutative. split. -----move=>H1 H2. rewrite join_commutative. apply (@co_contr x w y z)=>//. by rewrite meet_commutative. by rewrite meet_commutative. ------rewrite/co_ind. move=>H1 H2 H H'. move:( H1)=>H3;move:(H2)=>H4. apply meet_ge' in H3;apply meet_ge' in H4. move/eqP in H1;move/eqP in H2. rewrite join_distributive H1 H2 meet_idempotent. rewrite H1 in H; rewrite H2 in H'. rewrite H in H'. rewrite H. by apply Th5. Qed. End separoid. Section graphoid. Parameter grap: L -> L -> L -> bool. Axiom grap_sym:forall x y z, grap x y z -> grap y x z. Axiom grap_decomp:forall x y z w, grap x (join y w) z -> grap x y z. Axiom grap_w_u:forall x y z w, grap x (join y z) w -> grap x y (join z w). Axiom grap_contr:forall x y z w, grap x y (join z w) /\ grap x z w -> grap x (join y z) w. Axiom grap_int:forall x y z w, grap x y (join z w) /\ grap x z (join y w) -> grap x (join y z) w.
GitHub Repo
https://github.com/VeritasVox/The-Unanimous-Testimony-Divine-Sovereignty-and-Monergistic-Grace-Across-All-66-Books-of-Scripture
VeritasVox/The-Unanimous-Testimony-Divine-Sovereignty-and-Monergistic-Grace-Across-All-66-Books-of-Scripture
66 books. Zero exceptions. An exegetical proof that every canonical book teaches God's absolute sovereignty and monergistic grace. Addresses all synergist proof-texts at the pericope level. The burden now rests on those who deny it. Soli Deo Gloria.
GitHub Repo
https://github.com/hetpatel-web/atmabodh-texts
hetpatel-web/atmabodh-texts
Canonical, source-verified Hindu scriptures and philosophical texts compiled from public-domain and freely licensed sources. This repository serves as the immutable source of truth for Atmabodh, maintaining verse-level structure, attribution, licensing metadata, and auditability for use across web, mobile, and AI-powered applications.
GitHub Repo
https://github.com/Ronald106/Surviv.io
Ronald106/Surviv.io
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class='btns-game-double-row'> <div class='btn-game-container'> <a id='btn-game-settings' class='btn-game-tab-select btn-game-menu btn-game-menu-selected btn-darken' data-tab='settings'></a> <div class='btn-double-row game-menu-icon-static settings-icon'></div> </div> <div class='btn-game-container'> <a id='btn-game-keybinds' class='btn-game-tab-select btn-game-menu btn-darken' data-tab='keybinds'></a> <div class='btn-double-row game-menu-icon-static keybind-icon'></div> </div> </div> <div id='ui-game-tab-settings' class='ui-game-tab'> <a id='btn-game-fullscreen' class='btn-fullscreen-toggle btn-game-menu btn-darken' data-l10n='game-full-screen'>Full Screen</a> <div id='btn-touch-styles' class='btns-game-double-row'> <div class='btn-game-container'> <a id='btn-game-move-style' class='btn-double-row btn-game-touch-style btn-darken'></a> <div class='btn-double-row game-menu-icon-toggle movement-icon'></div> </div> <div class='btn-game-container'> <a id='btn-game-aim-style' class='btn-double-row btn-game-touch-style btn-darken'></a> <div class='btn-double-row game-menu-icon-toggle target-icon'></div> </div> </div> <a id='btn-game-aim-line' class='btn-game-menu btn-darken locked-on-icon' data-l10n='game-aim-line'>Aim Line</a> <a id='btn-game-sound' class='btn-sound-toggle btn-game-menu btn-darken audio-on-icon' data-l10n='game-sound'>Sound</a> <div class="slider-container ui-slider-container"> <p class='slider-text' data-l10n='index-master-volume'>Master Volume</p> <input type="range" min="0" max="100" value="100" class="slider sl-master-volume"> </div> <div class="slider-container ui-slider-container"> <p class='slider-text' data-l10n='index-sfx-volume'>SFX Volume</p> <input type="range" min="0" max="100" value="100" class="slider sl-sound-volume"> </div> <div class="slider-container ui-slider-container"> <p class='slider-text' data-l10n='index-music-volume'>Music Volume</p> <input type="range" min="0" max="100" value="100" class="slider sl-music-volume"> 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id='ui-team-pings' class='ui-emote-wheel'> <div id='ui-team-ping-middle' class='ui-emote-middle ui-emote-circle ui-emote-parent' data-id='0'> <div class='ui-emote ui-emote-bg-circle'></div> <div class='ui-emote ui-emote-hl'></div> <div class='ui-emote-image ui-emote-image-large'></div> </div> <div id='ui-team-ping-top' class='ui-emote-top ui-emote-quarter ui-emote-parent' data-id='1'> <div class='ui-emote ui-emote-bg-quarter'></div> <div class='ui-emote ui-emote-hl'></div> <div class='ui-emote-image ui-emote-image-large'></div> </div> <div id='ui-team-ping-right' class='ui-emote-right ui-emote-quarter ui-emote-parent' data-id='2'> <div class='ui-emote ui-emote-bg-quarter'></div> <div class='ui-emote ui-emote-hl'></div> <div class='ui-emote-image ui-emote-image-large'></div> </div> <div id='ui-team-ping-bottom' class='ui-emote-bottom ui-emote-quarter ui-emote-parent' data-id='3'> <div class='ui-emote ui-emote-bg-quarter'></div> <div class='ui-emote ui-emote-hl'></div> <div class='ui-emote-image ui-emote-image-large'></div> </div> <div id='ui-team-ping-bottom-left' class='ui-emote-bottom-left ui-emote-eighth ui-emote-parent' data-id='4'> <div class='ui-emote ui-emote-bg-eighth'></div> <div class='ui-emote ui-emote-hl'></div> <div class='ui-emote-image ui-emote-image-small'></div> </div> <div id='ui-team-ping-top-left' class='ui-emote-top-left ui-emote-eighth ui-emote-parent' data-id='5'> <div class='ui-emote ui-emote-bg-eighth'></div> <div class='ui-emote ui-emote-hl'></div> <div class='ui-emote-image ui-emote-image-small'></div> </div> </div> <div id='ui-team-indicators' class='js-ui-map-hidden js-desktop-ui-hud-show'> <div class='ui-team-indicator ui-indicator-main' data-id='0'> <div class='ui-team-indicator-pos ui-background-yellow'></div> </div> <div class='ui-team-indicator ui-indicator-main' data-id='1'> <div class='ui-team-indicator-pos ui-background-purple'></div> </div> <div class='ui-team-indicator ui-indicator-main' data-id='2'> <div class='ui-team-indicator-pos ui-background-cyan'></div> </div> <div class='ui-team-indicator ui-indicator-main' data-id='3'> <div class='ui-team-indicator-pos ui-background-orange'></div> </div> <div class='ui-team-indicator ui-indicator-main' data-id='4'> <div class='ui-team-indicator-pos ui-background-yellow'></div> </div> <div class='ui-team-indicator ui-indicator-ping' data-id='0'> <div class='ui-team-indicator-pos ui-team-indicator-image'></div> </div> <div class='ui-indicator-ping-border' data-id='0'></div> <div class='ui-team-indicator ui-indicator-ping' data-id='1'> <div class='ui-team-indicator-pos ui-team-indicator-image'></div> </div> <div class='ui-indicator-ping-border' data-id='1'></div> <div class='ui-team-indicator ui-indicator-ping' data-id='2'> <div class='ui-team-indicator-pos ui-team-indicator-image'></div> </div> <div class='ui-indicator-ping-border' data-id='2'></div> <div class='ui-team-indicator ui-indicator-ping' data-id='3'> <div class='ui-team-indicator-pos ui-team-indicator-image'></div> </div> <div class='ui-indicator-ping-border' data-id='3'></div> <div class='ui-team-indicator ui-indicator-ping' data-id='4'> <div class='ui-team-indicator-pos ui-team-indicator-image'></div> </div> <div class='ui-indicator-ping-border' data-id='4'></div> </div> <div id='ui-right-center' class='ui-right-center-desktop js-ui-map-hidden'> <div id='ui-medical-interactive'> <div id='ui-loot-bandage' class='ui-loot ui-outline-hover ui-scale-hover ui-medical tooltip'> <div class='tooltip-text'> <div class='tooltip-title' data-l10n='game-bandage'>Bandage</div> <div class='tooltip-description'><span data-l10n='game-bandage-tooltip'>Left-click to restore 15 health.</span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> </div> <div id='ui-loot-healthkit' class='ui-loot ui-outline-hover ui-scale-hover ui-medical tooltip'> <div class='tooltip-text'> <div class='tooltip-title' data-l10n='game-healthkit'>Med Kit</div> <div class='tooltip-description'><span data-l10n='game-healthkit-tooltip'>Left-click to restore 100 health.</div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> </div> <div id='ui-loot-soda' class='ui-loot ui-outline-hover ui-scale-hover ui-medical tooltip'> <div class='tooltip-text'> <div class='tooltip-title' data-l10n='game-soda'>Soda</div> <div class='tooltip-description'><span data-l10n='game-soda-tooltip'>Left-click to boost adrenaline by 25.</span><br><span data-l10n='game-adrenaline-tooltip'>Adrenaline restores health over time.</span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> </div> <div id='ui-loot-chocolateBox' class='ui-loot ui-outline-hover ui-scale-hover ui-medical tooltip'> <div class='tooltip-text'> <div class='tooltip-title' data-l10n='game-chocolateBox'>ChocolateBox</div> <div class='tooltip-description'><span data-l10n='game-soda-tooltip'>Left-click to boost adrenaline by 25.</span><br><span data-l10n='game-adrenaline-tooltip'>Adrenaline restores health over time.</span><br><span data-l10n='game-chocolateBox-tooltip'>Inmune to frenemy effect for 20s.</span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> </div> <div id='ui-loot-painkiller' class='ui-loot ui-outline-hover ui-scale-hover ui-medical tooltip'> <div class='tooltip-text'> <div class='tooltip-title' data-l10n='game-painkiller'>Pills</div> <div class='tooltip-description'><span data-l10n='game-painkiller-tooltip'>Left-click to boost adrenaline by 50.</div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> </div> <div id='ui-loot-flask' class='ui-loot ui-outline-hover ui-scale-hover ui-medical tooltip'> <div class='tooltip-text'> <div class='tooltip-title' data-l10n='game-flask'>Flask</div> <div class='tooltip-description'><span data-l10n='game-flask-tooltip'>Left-click for freeze damage immunity for 20 seconds.</span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> </div> </div> <div id='ui-ammo-interactive'> <div id='ui-loot-50AE' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-50AE'></div> <div class='tooltip-description'><span data-l10n='game-50AE-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(30,30,30,0.75)'></div> </div> <div id='ui-loot-9mm' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-9mm'></div> <div class='tooltip-description'><span data-l10n='game-9mm-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(255,153,0,0.75)'></div> </div> <div id='ui-loot-308sub' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-308sub'></div> <div class='tooltip-description'><span data-l10n='game-308sub-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(49,56,0,0.75)'></div> </div> <div id='ui-loot-12gauge' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-12gauge'></div> <div class='tooltip-description'><span data-l10n='game-12gauge-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(255,0,0,0.75)'></div> </div> <div id='ui-loot-flare' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-flare'></div> <div class='tooltip-description'><span data-l10n='game-flare-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(255,85,0,0.75)'></div> </div> <div id='ui-loot-762mm' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-762mm'></div> <div class='tooltip-description'><span data-l10n='game-762mm-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(0,102,255,0.75)'></div> </div> <div id='ui-loot-45acp' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-45acp'></div> <div class='tooltip-description'><span data-l10n='game-45acp-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(121,0,255,0.75)'></div> </div> <div id='ui-loot-556mm' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-556mm'></div> <div class='tooltip-description'><span data-l10n='game-556mm-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(3,123,0,0.75)'></div> </div> <div id='ui-loot-40mm' class='ui-outline-hover ui-ammo tooltip'> <div class='tooltip-text click-through'> <div class='tooltip-title' data-l10n='game-40mm'></div> <div class='tooltip-description'><span data-l10n='game-40mm-tooltip'></span></div> </div> <div class='ui-loot-count'>0</div> <img class='ui-loot-image' src=''></img> <div class='ui-loot-overlay' style='background: rgba(12,221,171,0.75)'></div> </div> </div> <div id='ui-emote-button'></div> </div> <div id='ui-top-left' class='js-ui-hud-show click-through'> <div id='ui-team' class='js-ui-mobile-map-hidden'> <div class='ui-team-member ui-bg-standard' data-id='0'> <div class='ui-team-member-name'></div> <div class='ui-team-member-color ui-background-yellow'></div> <div class='ui-team-member-status'></div> <div class='ui-team-member-health'> <div class='ui-bar-inner ui-health-actual'></div> </div> </div> <div class='ui-team-member ui-bg-standard' data-id='1'> <div class='ui-team-member-name'></div> <div class='ui-team-member-color ui-background-purple'></div> <div class='ui-team-member-status'></div> <div class='ui-team-member-health'> <div class='ui-bar-inner ui-health-actual'></div> </div> </div> <div class='ui-team-member ui-bg-standard' data-id='2'> <div class='ui-team-member-name'></div> <div class='ui-team-member-color ui-background-cyan'></div> <div class='ui-team-member-status'></div> <div class='ui-team-member-health'> <div class='ui-bar-inner ui-health-actual'></div> </div> </div> <div class='ui-team-member ui-bg-standard' data-id='3'> <div class='ui-team-member-name'></div> <div class='ui-team-member-color ui-background-orange'></div> <div class='ui-team-member-status'></div> <div class='ui-team-member-health'> <div class='ui-bar-inner ui-health-actual'></div> </div> </div> </div> </div> <div id='ui-map-wrapper' class='ui-map-wrapper-desktop click-through'> <div id='ui-map-container'> <div id='ui-map-info' class='js-ui-map-hidden js-ui-hud-show'> <div id='ui-gas-icon' class='gas-icon'></div> <div id='ui-gas-timer'></div> </div> <div id='ui-spec-counter' class='js-ui-map-hidden js-ui-hud-show'> <div id='ui-spec-counter-icon'></div> <div id='ui-spec-counter-number'></div> </div> <div id='ui-settings-container-desktop' class='js-ui-desktop-map-hidden'> <div id='ui-map-expand-desktop' class='ui-map-expand ui-settings-button'> <img id='mag-glass-white' src=''></img> </div> <div id='ui-map-minimize' class='ui-settings-button js-ui-map-hidden'> <img id='ui-minimize-img' src=''></img> </div> </div> <div id='ui-settings-container-mobile' class='js-ui-mobile-map-hidden'> <div id='ui-alive-info'> <div id='ui-alive-icon' class='ui-map-icon alive-icon'></div> <div id='ui-map-counter-default' class='js-ui-players-alive'>0</div> <div id='ui-map-counter-faction'> <span class='ui-players-alive-red js-ui-players-alive-red'>0</span>:<span class='ui-players-alive-blue js-ui-players-alive-blue'>0</span> </div> </div> <div id='ui-map-expand-mobile' class='ui-map-expand ui-settings-button'></div> </div> </div> </div> <div id='ui-menu-display' class='ui-settings-button prop-event'></div> <div id='ui-kill-leader-container'> <div id='ui-kill-leader-wrapper' class='js-ui-map-hidden js-ui-hud-show hide-on-mobile'> <div id='ui-kill-leader-name'>Waiting for new leader</div> <div id='ui-kill-leader-icon'></div> <div id='ui-kill-leader-count'>0</div> </div> </div> <div id='ui-killfeed-wrapper'> <div id='ui-killfeed' class='js-ui-map-hidden js-ui-hud-show'> <div id='ui-killfeed-contents'></div> </div> </div> <div id='ui-leaderboard-wrapper'> <div id='ui-leaderboard' class='js-ui-hud-show'> <div id='ui-leaderboard-alive'> <span class='ui-players-alive js-ui-players-alive'>0</span> </div> <div id='ui-leaderboard-alive-faction'> <span class='ui-players-alive-red js-ui-players-alive-red'>0</span> <span class='ui-players-alive-blue js-ui-players-alive-blue'>0</span> </div> <div class='ui-leaderboard-header' data-l10n='game-alive'>Alive</div> </div> <div id='ui-kill-counter-wrapper' class='js-ui-map-show'> <div id='ui-kill-counter'> <span class='ui-player-kills js-ui-player-kills'>0</span> </div> <div class='ui-kill-counter-header' data-l10n='game-kills'>Kills</div> </div> </div> <div id='ui-top-center-scopes-wrapper'> <div id='ui-top-center-scopes' class='click-through js-ui-map-hidden'> <div id='ui-scope-1xscope' class='ui-scope ui-zoom'> <div class='ui-zoom-text'> <div class='ui-zoom-level'>1<span class='ui-zoom-append'>x</span></div> </div> </div> <div id='ui-scope-2xscope' class='ui-scope ui-zoom ui-outline-hover'> <div class='ui-zoom-text'> <div class='ui-zoom-level'>2<span class='ui-zoom-append'>x</span></div> </div> </div> <div id='ui-scope-4xscope' class='ui-scope ui-zoom ui-outline-hover'> <div class='ui-zoom-text'> <div class='ui-zoom-level'>4<span class='ui-zoom-append'>x</span></div> </div> </div> <div id='ui-scope-8xscope' class='ui-scope ui-zoom ui-outline-hover'> <div class='ui-zoom-text'> <div class='ui-zoom-level'>8<span class='ui-zoom-append'>x</span></div> </div> </div> <div id='ui-scope-15xscope' class='ui-scope ui-zoom ui-outline-hover'> <div class='ui-zoom-text'> <div class='ui-zoom-level'>15<span class='ui-zoom-append'>x</span></div> </div> </div> </div> </div> <div id='ui-top-center' class='js-ui-map-hidden js-ui-hud-show click-through'> <div id='ui-waiting-text' class='top-center-text'><span data-l10n='game-waiting-for-players'>Waiting for players</span>...</div> <div id='ui-spectate-text' class='top-center-text ui-spectate-mode'> <div class='spectate-text spectate-desc' data-l10n='game-spectating'>Spectating</div> <div id='spectate-player' class='spectate-text'></div> </div> </div> <div id='ui-upper-center' class='click-through'> <div id='ui-announcement'></div> </div> <div id='ui-upper-center-1' class='click-through'> <div id='ui-pickup-notification'> <div id='ui-pickup-description'>Item picked up</div> </div> </div> <div id='ui-lower-center' class='js-ui-map-hidden js-ui-hud-show click-through'> <div id='ui-interaction'> <div id='ui-interaction-press'></div> <div id='ui-interaction-outer'> <div id='ui-interaction-description'></div> </div> </div> </div> <div id='ui-bottom-center-2' class='click-through'> <div id='ui-pickup-message' class='notify-message'></div> <div id='ui-perk-message-wrapper' class='notify-message'> <div id='ui-perk-message-image-wrapper'> <div id='ui-perk-message-image-icon'></div> </div> <div id='ui-perk-message-name' class='notify-message'></div> <div id='ui-perk-message-acquired' class='notify-message'></div> </div> </div> <div id='ui-equipped-ammo-wrapper' class='js-ui-map-hidden js-ui-hud-show'> <div id='ui-equipped-ammo'> <div id='ui-bullet-counter'> <div id='ui-current-clip'>0</div> <div id='ui-remaining-ammo'>0</div> <div id='ui-reload-button-container'> <div id='ui-reload-button'></div> </div> </div> </div> </div> <div id='ui-bottom-center-0' class='click-through js-ui-map-hidden js-ui-hud-show'> <div id='ui-boost-counter'> <div class='ui-boost-base' id='ui-boost-counter-0'> <div class='ui-bar-inner'></div> </div> <div class='ui-boost-base' id='ui-boost-counter-1'> <div class='ui-bar-inner'></div> </div> <div class='ui-boost-base' id='ui-boost-counter-2'> <div class='ui-bar-inner'></div> </div> <div class='ui-boost-base' id='ui-boost-counter-3'> <div class='ui-bar-inner'></div> </div> </div> <div id='ui-health-counter'> <div id='ui-health-flair-left' class='ui-health-flair'></div> <div id='ui-health-flair-right' class='ui-health-flair'></div> <div id='ui-health-container'> <div class='ui-bar-inner' id='ui-health-actual'></div> <div class='ui-bar-inner' id='ui-health-depleted'></div> </div> </div> </div> <div id='ui-bottom-center-left' class='js-ui-map-hidden'> <div id='ui-perk-0' class='ui-armor-counter tooltip-perk ui-outline-hover'> <div class='tooltip-text'> <div class='tooltip-title'></div> <div class='tooltip-desc'></div> </div> <img class='ui-armor-image ui-loot-image' src=''></img> </div> <div id='ui-perk-1' class='ui-armor-counter tooltip-perk ui-outline-hover'> <div class='tooltip-text'> <div class='tooltip-title'></div> <div class='tooltip-desc'></div> </div> <img class='ui-armor-image ui-loot-image' src=''></img> </div> <div id='ui-perk-2' class='ui-armor-counter tooltip-perk ui-outline-hover'> <div class='tooltip-text'> <div class='tooltip-title'></div> <div class='tooltip-desc'></div> </div> <img class='ui-armor-image ui-loot-image' src=''></img> </div> </div> <div id='ui-bottom-center-right' class='js-ui-map-hidden'> <div id='ui-armor-helmet' class='ui-armor-counter ui-outline-hover'> <div class='ui-armor-counter-inner'></div> <div class='ui-armor-level'></div> <img class='ui-armor-image ui-loot-image' src=''></img> </div> <div id='ui-armor-chest' class='ui-armor-counter ui-outline-hover'> <div class='ui-armor-counter-inner'></div> <div class='ui-armor-level'></div> <img class='ui-armor-image ui-loot-image' src=''></img> </div> <div id='ui-armor-backpack' class='ui-armor-counter'> <div class='ui-armor-counter-inner'></div> <div class='ui-armor-level'></div> <img class='ui-armor-image ui-loot-image' src=''></img> </div> </div> <div id='ui-bottom-right' class='js-ui-map-hidden'> <div id='ui-weapon-container'> <div class='ui-weapon-info'> <div class='ui-weapon-switch ui-outline-hover' id='ui-weapon-id-1' data-slot='1'> <div class='ui-weapon-name'></div> <div class='ui-weapon-number'>1</div> <img class='ui-weapon-image' src=''></img> </div> </div> <div class='ui-weapon-info'> <div class='ui-weapon-switch ui-outline-hover' id='ui-weapon-id-2' data-slot='2'> <div class='ui-weapon-name'></div> <div class='ui-weapon-number'>2</div> <img class='ui-weapon-image' src=''></img> </div> </div> <div class='ui-weapon-info'> <div class='ui-weapon-switch ui-outline-hover' id='ui-weapon-id-3'> <div class='ui-weapon-name'></div> <div class='ui-weapon-number'>3</div> <img class='ui-weapon-image' src=''></img> </div> </div> <div class='ui-weapon-info'> <div class='ui-weapon-switch ui-outline-hover' id='ui-weapon-id-4'> <div class='ui-weapon-name'></div> <div class='ui-weapon-number'>4</div> <div class='ui-weapon-exp ui-weapon-ammo-counter'>0</div> <img class='ui-weapon-image' src=''></img> </div> </div> </div> </div> <div id='ui-spectate-options-wrapper'> <div id='ui-spectate-options' class='ui-spectate-mode js-ui-hud-show click-through'> <div id='ui-spectate-buttons' class='ui-bg-standard'> <a class='menu-option btn-darken' id='btn-spectate-next-player' data-l10n='game-next-teammate'>Next Teammate</a> <a class='menu-option btn-darken' id='btn-spectate-prev-player' data-l10n='game-previous-teammate'>Previous Teammate</a> <a class='menu-option btn-darken' id='btn-spectate-view-stats' data-l10n='game-view-match-stats'>View Match Stats</a> <a class='menu-option btn-darken btn-quit' id='btn-spectate-quit' data-l10n='game-leave-game'>Leave Game</a> </div> <div id='ui-spectate-stats' class='ui-bg-standard'> <div id='ui-spectate-stats-header' data-l10n='game-your-results'>Your Results</div> <table id='ui-spectate-stats-table'> <tbody id='ui-spectate-stats-data'> </tbody> </table> </div> </div> </div> <div id='ui-bottom-center-1' class='click-through'> <div id='ui-kills'> <div id='ui-kill-text'></div> <div id='ui-kill-count'></div> </div> </div> </div> <div id='ui-stats'> <div id='ui-stats-bg'></div> <div id='ui-stats-contents'> <div id='ui-stats-contents-inner'> <div id='ui-stats-header'></div> <div id='ui-stats-info-box'></div> <div id='ui-stats-options'></div> <div id='ui-stats-logo'></div> </div> <div id='ui-stats-ad-container-desktop' class='ui-stats-ad-container'> <div id='surviv-io_300x250_2'></div> </div> <div id='ui-stats-ad-container-mobile' class='ui-stats-ad-container'> <div id='surviv-io_300x250_mobile_2'></div> </div> </div> </div> <div class='ui-stats-adblock surviv-shirts'> <a class='surviv-shirts-link' href='/~/https://www.amazon.com/s?rh=n%3A7141123011%2Cp_4%3Asurviv.io&ref=w_bl_sl_s_ap_web_7141123011' target="_blank"></a> </div> <div class='ui-stats-adblock adblock-plea'> <span>Please consider supporting us by disabling your adblocker.</span> </div> </div> <div id='start-menu-wrapper'> <div id='safari-margin'></div> <div id='background'> <div id='start-overlay'></div> </div> <div id='event-modal'> <div id='modal-container'> <p id='modal-header' data-l10n='index-modal-test'>(HEADER TEXT)</p> <img id='modal-img-1'> <div id='modal-content'> <img id='modal-img-2'> <p id='modal-text' data-l10n='event-desc'>Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Tincidunt dui ut ornare lectus sit amet est.</p> </div> <button class='close' id='modal-btn' data-l10n='modal-btn-txt' type='button'>(BUTTON TEXT)</button> </div> </div> <div id='start-main'> <div id='start-top-left'> <div id='btn-hamburger' class='icon-hamburger'></div> <div id='start-top-left-desktop'> <div id='featured-streamers'> <div class='streaming-header'> <div id='streaming-header-title' data-l10n='index-streaming-live'>Streaming Live!</div> <div id='streaming-icon'></div> </div> <div class='streamer-list'></div> <div id='featured-streamer-template'> <a href='' target='_blank' class='btn-streamer btn-darken'></a> </div> </div> <div id='featured-youtuber'> <div class='youtube-header'> <div id='youtube-header-title' data-l10n='index-featured-youtuber'>Featured YouTuber</div> </div> <a href='' target='_blank' class='btn-youtuber btn-darken'></a> </div> </div> </div> <!-- Accounts Modals --> <div id='modal-account-name-change' class='modal modal-account'> <div class='modal-content modal-close'> <div class='modal-header modal-header-name'> <span class='close close-corner'></span> <h2 id='modal-account-name-title' data-l10n='index-create-account'>Create Account</h2> </div> <div class='modal-body modal-body-name'> <div class='modal-settings-item'> <p id='modal-account-name-desc' class='modal-body-text' data-l10n='index-set-account-name'>Set your account name:</p> <div id='modal-body-warning'></div> <input type='text' class='menu-option player-name-input' tabindex='0' placeholder='Enter name' id='modal-account-name-input' maxlength='16' /> </div> </div> <div class='modal-footer modal-footer-name modal-footer-round'> <h3 id='modal-account-name-finish' class='finish' data-l10n='index-finish'>Finish</h3> </div> </div> </div> <div id='modal-account-reset-stats' class='modal modal-account'> <div class='modal-content modal-close'> <div class='modal-header modal-header-reset-stats'> <span class='close close-corner'></span> <h2 id='modal-account-name-title' data-l10n='index-reset-stats'>Reset Stats</h2> </div> <div class='modal-body modal-body-reset-stats'> <div class='modal-settings-item'> <p id='modal-account-reset-stats-desc' class='modal-body-text' data-l10n='index-reset-stats-desc'>Enter "RESET STATS" to reset your stats:</p> <p id='modal-account-reset-stats-desc-2' class='modal-body-text' data-l10n='index-reset-stats-desc-2'>(This does not reset pass progress.)</p> <input type='text' class='menu-option player-name-input' tabindex='0' placeholder='' id='modal-account-reset-stats-input' maxlength='16' /> </div> </div> <div class='modal-footer modal-footer-reset-stats modal-footer-round'> <h3 id='modal-account-reset-stats-finish' class='finish' data-l10n='index-confirm'>Confirm</h3> </div> </div> </div> <div id='modal-account-delete' class='modal modal-account'> <div class='modal-content modal-close'> <div class='modal-header modal-header-delete'> <span class='close close-corner'></span> <h2 id='modal-account-name-title' data-l10n='index-delete-account'>Delete Account</h2> </div> <div class='modal-body modal-body-delete'> <div class='modal-settings-item'> <p id='modal-account-delete-desc' class='modal-body-text' data-l10n='index-delete-account-desc'>Enter "DELETE" to delete your account:</p> <input type='text' class='menu-option player-name-input' tabindex='0' placeholder='' id='modal-account-delete-input' maxlength='16' /> </div> </div> <div class='modal-footer modal-footer-delete modal-footer-round'> <h3 id='modal-account-delete-finish' class='finish' data-l10n='index-confirm'>Confirm</h3> </div> </div> </div> <!-- Accounts Mobile --> <div id='modal-mobile-account' class='modal'> <div class='modal-content modal-close'> <div class='modal-header'> <span class='close close-corner'></span> <h2 data-l10n='index-account' data-l10n='index-account'>Account</h2> </div> <div id='modal-mobile-account-body'> <div class='account-buttons-settings account-buttons-wrapper account-block-arrow modal-close'> <div class='account-buttons'> <div class='account-stats-link btn-player-stats-link menu-option btn-darken btn-standard ' data-l10n='index-my-stats'>My Stats</div> <div class='btn-account-link btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-link-account'>Link Account</div> <div class='btn-account-change-name btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-change-account-name'>Change Account Name</div> <div class='btn-account-reset-stats btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-reset-stats'>Reset Stats</div> <div class='btn-account-delete btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-delete-account'>Delete Account</div> <div class='btn-account-logout btn-account-grey menu-option btn-darken btn-standard' data-l10n='index-log-out'>Log Out</div> </div> </div> <div id='account-login-options-mobile' class='account-buttons-link-account account-buttons-wrapper account-block-arrow modal-close'> <div class='login-options-content'></div> </div> </div> </div> </div> <!-- Accounts --> <div id='start-top-right'> <div class='account-block'> <div class='account-details-top'> <div class="account-details-top-buttons"> <div class="account-details-button-wrapper account-details-button-loadout menu-option btn-darken-alt"> <div class='account-details-button'> <div id='loadout-alert-main' class='account-alert-main account-alert'></div> <div class='account-link account-loadout-link'> <span data-l10n='index-loadout'>Loadout</span> </div> </div> </div> <div class="account-details-button-wrapper account-details-link-out menu-option btn-darken hide-on-mobile"> <div class='account-details-button'> <div class='account-link account-stats-link'><span data-l10n='index-my-stats'>My Stats</span></div> </div> </div> <div class="account-details-button-wrapper account-details-link-out menu-option btn-darken hide-on-mobile"> <div class='account-details-button'> <div class='account-link account-leaderboard-link'><span data-l10n='index-leaderboards'>Leaderboards</span></div> </div> </div> </div> </div> <div class='account-overview'> <div class="account-details-user account-details-block"> <div class="account-details"> <div class="account-avatar"></div> <div id="account-login" class="account-player-name account-name account-name-user" data-l10n='index-log-in-desc'>Log In / Create Account</div> <div id="account-player-name" class="account-player-name account-name account-name-user" style="display:none"></div> <div class="account-loading-container"> <div class="account-loading"></div> </div> </div> </div> </div> <div class='account-buttons-settings account-buttons-wrapper account-block-arrow modal-close hide-on-mobile'> <div class='account-buttons'> <div class='btn-account-link btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-link-account'>Link Account</div> <div class='btn-account-change-name btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-change-account-name'>Change Account Name</div> <div class='btn-account-reset-stats btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-reset-stats'>Reset Stats</div> <div class='btn-account-delete btn-account-turq menu-option btn-darken btn-standard' data-l10n='index-delete-account'>Delete Account</div> <div class='btn-account-logout btn-account-grey menu-option btn-darken btn-standard' data-l10n='index-log-out'>Log Out</div> </div> </div> <div id='account-login-options' class='account-buttons-link-account account-buttons-wrapper account-block-arrow modal-close hide-on-mobile'> <div class='login-options-content'></div> </div> </div> </div> <div id='start-main-center'> <div id='start-row-header'> <div id='server-warning'></div> </div> <div id='start-row-top'> <div id='start-rotate-reminder' class='menu-column menu-block'> <span>Rotate to landscape for a better experience.</span> </div> <div id='left-column' class='menu-column'> <div id='ad-block-left'> <div class='surviv-shirts'> <a href='/~/https://www.amazon.com/s?rh=n%3A7141123011%2Cp_4%3Asurviv.io&ref=w_bl_sl_s_ap_web_7141123011' target="_blank"></a> </div> <div class='adblock-plea'> <span>Please consider supporting us by disabling your adblocker.</span> </div> <div class='ad-block-med-rect' id='surviv-io_300x250'> </div> </div> <div id='social-share-block-wrapper'> <div id='social-share-block' class='menu-block'> <div class='btn-social-wrapper'> <a href='/~/https://facebook.com/surviviogame' target='_blank' class='btn-social btn-darken btn-facebook'></a> <a href='/~/https://twitter.com/survivio' target='_blank' class='btn-social btn-darken btn-twitter'></a> <a href='/~/https://www.instagram.com/surviviogame/' target='_blank' class='btn-social btn-darken btn-instagram'></a> <a href='/~/https://discord.gg/survivio' target='_blank' class='btn-social btn-darken btn-discord'></a> <a href='/~/https://www.youtube.com/c/survivio?sub_confirmation=1' target='_blank' class='btn-social btn-darken btn-youtube'></a> </div> </div> </div> </div> <div id='start-menu' class='menu-column menu-block'> <div class='play-loading-outer'> <div class='play-loading-inner'> <div class='play-loading-spinner'></div> </div> </div> <div class='play-button-container'> <div id='player-options'> <input type='text' class='menu-option player-name-input' tabindex='0' placeholder='Enter your name here' id='player-name-input-solo' maxlength='16' /> <a class='btn-darken menu-option player-options-btn' id='btn-customize'></a> </div> <select id='server-select-main' class='server-select menu-option btn-hollow btn-hollow-selected'> <optgroup id='server-opts' label='Region' data-l10n='index-region'> <option value='na' data-label='North America' data-l10n='index-north-america'>North America</option> <option value='sa' data-label='South America' data-l10n='index-south-america'>South America</option> <option value='eu' data-label='Europe' data-l10n='index-europe'>Europe</option> <option value='as' data-label='Asia' data-l10n='index-asia'>Asia</option> <option value='kr' data-label='Korea' data-l10n='index-korea'>South Korea</option> </optgroup> </select> <a class='btn-green btn-darken menu-option' id='btn-start-mode-0' data-l10n='index-play-solo'>Play Solo</a> <div id='btns-quick-start'> <a class='btn-green btn-darken menu-option' id='btn-start-mode-1' data-l10n='index-play-duo'>Play Duo</a> <a class='btn-green btn-darken menu-option' id='btn-start-mode-2' data-l10n='index-play-squad'>Play Squad</a> </div> <div class='btns-double-row'> <a class='btn-darken menu-option btn-team-option' id='btn-join-team' data-l10n='index-join-team'>Join Team</a> <a class='btn-darken menu-option btn-team-option' id='btn-create-team' data-l10n='index-create-team'>Create Team</a> </div> <div id='btn-help' class='menu-option btn-darken' data-l10n='index-how-to-play'>How to Play</div> <div id='start-help'> <h1 data-l10n='index-controls'>Controls</h1> <p><span class='help-action' data-l10n='index-movement'></span>: <span class='help-control' data-l10n='index-movement-ctrl'></span></p> <p><span class='help-action' data-l10n='index-aim'></span>: <span class='help-control' data-l10n='index-aim-ctrl'></span></p> <p><span class='help-action' data-l10n='index-punch'></span>/<span class='help-action' data-l10n='index-shoot'></span>: <span class='help-control' data-l10n='index-shoot-ctrl'></span></p> <p><span class='help-action' data-l10n='index-change-weapons'></span>: <span class='help-control' data-l10n='index-change-weapons-ctrl'></span></p> <p class='hide-on-mobile'><span class='help-action' data-l10n='index-stow-weapons'></span>: <span class='help-control' data-l10n='index-stow-weapons-ctrl'></span></p> <p class='hide-on-mobile'><span class='help-action' data-l10n='index-swap-weapons'></span>: <span class='help-control' data-l10n='index-swap-weapons-ctrl'></span></p> <p class='hide-on-mobile'><span class='help-action' data-l10n='index-swap-weapon-slots'></span>: <span class='help-control' data-l10n='index-swap-weapon-slots-ctrl'></span></p> <p><span class='help-action' data-l10n='index-reload'></span>: <span class='help-control' data-l10n='index-reload-ctrl'></span></p> <p><span class='help-action' data-l10n='index-scope-zoom'></span>: <span class='help-control' data-l10n='index-scope-zoom-ctrl'></span></p> <p><span class='help-action' data-l10n='index-pickup'></span>/<span class='help-action' data-l10n='index-loot'></span>/<span class='help-action' data-l10n='index-revive'></span>: <span class='help-control' data-l10n='index-pickup-ctrl'></span></p> <p><span class='help-action' data-l10n='index-use-medical'></span>: <span class='help-control' data-l10n='index-use-medical-ctrl'></span></p> <p><span class='help-action' data-l10n='index-drop-item'></span>: <span class='help-control' data-l10n='index-drop-item-ctrl'></span></p> <p><span class='help-action' data-l10n='index-cancel-action'></span>: <span class='help-control' data-l10n='index-cancel-action-ctrl'></span></p> <p><span class='help-action' data-l10n='index-view-map'></span>: <span class='help-control' data-l10n='index-view-map-ctrl'></span></p> <p class='hide-on-mobile'><span class='help-action' data-l10n='index-toggle-minimap'></span>: <span class='help-control' data-l10n='index-toggle-minimap-ctrl'></span></p> <p><span class='help-action' data-l10n='index-use-ping'></span>: <span class='help-control' data-l10n='index-use-ping-ctrl'></span></p> <p><span class='help-action' data-l10n='index-use-emote'></span>: <span class='help-control' data-l10n='index-use-emote-ctrl'></span></p> <h1 data-l10n='index-how-to-play'>How to Play</h1> <p data-l10n='index-tips-1-desc'>The goal of surviv.io is to be the last player standing. You only live once per game - there is no respawn!</p> <h1 data-l10n='index-tips-2'>2D PUBG</h1> <p data-l10n='index-tips-2-desc'>If you've played other battle royale games like PUBG, Fortnite or H1Z1, then you're already halfway there! Think of surviv.io as 2D PUBG (with slightly less desync and more chicken).</p> <h1 data-l10n='index-tips-3'>Loot and Kill</h1> <p data-l10n='index-tips-3-desc'>You'll begin the game with no items other than a simple backpack. Move around the map to find loot: weapons, ammo, scopes, and medical items. Eliminate other players and you can take their loot!</p> <h1 data-l10n='index-tips-4'>Red = Bad!</h1> <p data-l10n='index-tips-4-desc'>Players aren't the only thing that can hurt you. The deadly red zone will move in from the sides of the map and deal increasingly greater damage if you stand in it. Keep an eye on the map and stay safe.</p> </div> </div> </div> <div id='team-menu' class='menu-column menu-block'> <div class='play-loading-outer'> <div class='play-loading-inner'> <div class='play-loading-spinner'></div> </div> </div> <div class='play-button-container'> <a class='btn-darken menu-option' id='btn-team-leave' data-l10n='index-leave-team'>Leave Team</a> <div id='team-menu-connecting'> <div class='ui-spinner'></div> <div class='team-menu-connecting-text' id='team-menu-joining-text'><span data-l10n='index-joining-team'>Joining team</span> ...</div> <div class='team-menu-connecting-text' id='team-menu-creating-text'><span data-l10n='index-creating-team'>Creating team</span> ...</div> </div> <div id='team-menu-contents'> <div id='team-desc'> <div id='team-desc-text'><span id='invite-link-text' data-l10n='index-invite-link'>Invite link</span>: <span id='team-url'></span> </div> <a id='team-copy-url' class='copy-item btn-darken'></a> <a id='team-hide-url' class='hide-item btn-darken'></a> <div id='team-code-text'><span id='invite-code-text' data-l10n='index-invite-code'>Invite code</span>: <span id='team-code'></span></div> </div> <div id='team-menu-columns'> <div id='team-menu-members' class='team-menu-block'> <div id='team-menu-member-list'></div> </div> <div id='team-menu-options' class='team-menu-block'> <select id='team-server-select' class='btn-hollow btn-hollow-selected server-select menu-option'> <optgroup id='team-server-opts' label='Region'> <option value='na' data-label='North America' data-l10n='index-north-america'>North America</option> <option value='sa' data-label='South America' data-l10n='index-south-america'>South America</option> <option value='eu' data-label='Europe' data-l10n='index-europe'>Europe</option> <option value='as' data-label='Asia' data-l10n='index-asia'>Asia</option> <option value='kr' data-label='Korea' data-l10n='index-korea'>South Korea</option> </optgroup> </select> <div class='team-menu-options-buttons'> <a class='btn-hollow btn-hollow-selected btn-darken team-menu-option btn-team-queue' id='btn-team-queue-mode-1' data-l10n='index-duo'>Duo</a> <a class='btn-hollow btn-hollow-selected btn-darken team-menu-option btn-team-queue' id='btn-team-queue-mode-2' data-l10n='index-squad'>Squad</a> </div> <div class='team-menu-options-buttons'> <a class='btn-hollow btn-hollow-selected btn-darken team-menu-option btn-team-fill' id='btn-team-fill-auto' data-l10n='index-auto-fill'>Auto Fill</a> <a class='btn-hollow btn-darken team-menu-option btn-team-fill' id='btn-team-fill-none' data-l10n='index-no-fill'>No Fill</a> </div> <div id='team-menu-options-start'> <a class='btn-green btn-darken menu-option' id='btn-start-team' data-label='Play' data-l10n='index-play'>Play</a> <div id='msg-wait-reason'></div> </div> </div> </div> </div> </div> </div> <div id='team-mobile-link' class='menu-column menu-block'> <div id='team-mobile-link-desc' class='team-mobile-link-text' data-l10n='index-join-team-help'>Got a team link or code? Paste it here:</div> <div id='team-mobile-link-warning' class='team-mobile-link-text'>Invalid link!</div> <input type='text' class='menu-option' contenteditable="false" tabindex='0' autofocus placeholder='Paste team link or enter code here' id='team-link-input' /> <a class='btn-darken menu-option btn-team-option' id='btn-team-mobile-link-join' data-l10n='index-join-team'>Join Team</a> <a class='btn-darken menu-option' id='btn-team-mobile-link-leave' data-l10n='index-back-to-main'>Back to Main Menu</a> </div> <div id='right-column' class='menu-column'> <div id='pass-wrapper'> <div id='pass-unlock-tooltip'> <div class='tooltip-pass-title'></div> <div class='tooltip-pass-desc'></div> </div> <div class='right-column-toggle news-toggle'> <div class='account-alert'></div> </div> <div id='pass-loading' class='pass-column menu-block'> <div id='pass-loading-spinner'></div> </div> <div id='pass-block' class='pass-column menu-block'> <div id='pass-name'> <span id='pass-name-text'></span> </div> <div id='pass-progress' class=''> <div id='pass-progress-level' class=''></div> <div id='pass-progress-bar' class=''> <div id='pass-progress-bar-fill'></div> <div id='pass-progress-xp'> <span id='pass-progress-xp-current'></span> <span>/</span> <span id='pass-progress-xp-target'></span> <span>XP</span> </div> </div> <div id='pass-progress-unlock-wrapper'> <div id='pass-progress-unlock' class=''> <div id='pass-progress-unlock-image' class=''></div> </div> <div id='pass-progress-unlock-type-wrapper' class=''> <div id='pass-progress-unlock-type-border' class=''> <div id='pass-progress-unlock-type-image' class=''></div> </div> <div id='pass-progress-unlock-overlay' class=''></div> </div> </div> </div> <div id='pass-quest-wrapper' class=''> <div id='pass-locked'> <div id='btn-pass-locked' class='menu-option btn-darken' data-l10n='quest-login-prompt'>Log in to get quests!</div> </div> <div id='pass-quest-0' class='pass-quest'> <div class='pass-quest-timer'></div> <div class='pass-quest-spinner-container'> <div class='pass-quest-spinner'></div> </div> <div class='pass-quest-refresh-prompt'> <div class='pass-quest-refresh-prompt-text' data-l10n='quest-refresh-prompt'>Get a new quest?</div> <div class='pass-quest-refresh-confirm btn-pass-quest-refresh'></div> <div class='pass-quest-refresh-cancel btn-pass-quest-refresh'></div> </div> <div class='pass-quest-info'> <div class='pass-quest-desc'></div> <div class='pass-quest-xp'></div> <div class='pass-quest-refresh'></div> <div class='pass-quest-progress'> <div class='pass-quest-bar'> <div class='pass-quest-bar-fill'></div> </div> <div class='pass-quest-counter'> <span class='pass-quest-counter-current'></span> <span>/</span> <span class='pass-quest-counter-target'></span> </div> </div> </div> </div> <div id='pass-quest-1' class='pass-quest'> <div class='pass-quest-timer'></div> <div class='pass-quest-spinner-container'> <div class='pass-quest-spinner'></div> </div> <div class='pass-quest-refresh-prompt'> <div class='pass-quest-refresh-prompt-text' data-l10n='quest-refresh-prompt'>Get a new quest?</div> <div class='pass-quest-refresh-confirm btn-pass-quest-refresh'></div> <div class='pass-quest-refresh-cancel btn-pass-quest-refresh'></div> </div> <div class='pass-quest-info'> <div class='pass-quest-desc'></div> <div class='pass-quest-xp'></div> <div class='pass-quest-refresh'></div> <div class='pass-quest-progress'> <div class='pass-quest-bar'> <div class='pass-quest-bar-fill'></div> </div> <div class='pass-quest-counter'> <span class='pass-quest-counter-current'></span> <span>/</span> <span class='pass-quest-counter-target'></span> </div> </div> </div> </div> </div> </div> </div> <div id='news-wrapper'> <div class='right-column-toggle pass-toggle'></div> <div id='news-block' class='menu-block'> <div id='news'> <h3 class='news-header'>What's New!</h3> <div id='news-current' data-date='2020-2-24'> <small class='news-date'>February 24, 2020</small> <p class='news-paragraph'><strong>Curveball</strong></p> <p class='news-paragraph'>The <span class='highlight'>Savannah</span> has returned with a new perk to help flush snipers out of position.</p> <p class='news-paragraph'>The <span class='highlight'>Closer Perk</span> throws grenades much farther, allowing you to put Snipers on the move.</p> </div> <div data-date='2020-2-10'> <small class='news-date'>February 10, 2020</small> <p class='news-paragraph'><strong>Keep Your Enemies Closer</strong></p> <p class='news-paragraph'>PARMA recognizes this season of togetherness and generously offers to help rivals resolve their differences. The all new <span class='highlight'>Heart Cannon</span> diffuses damage taken from your frenemies. For groups of future friends, the same effect is achieved with the all-new <span class='highlight'>Heart Grenade</span>.</p> <p class='news-paragraph'>If you do not trust the intentions of your new friends, eat an entire <span class='highlight'>box of chocolates</span> to counter the effect of <span class='highlight'>Heart Weapons</span>.</p> </div> <div data-date='2020-1-27'> <small class='news-date'>January 27, 2020</small> <p class='news-paragraph'><strong>Dodge This</strong></p> <p class='news-paragraph'><span class='highlight'>Woods</span> Mode is back with two new weapons and things are bound to get explosive. The <span class='highlight'>PKM</span> and <span class='highlight'>Hawk 12G</span> join the Woods Map arsenal, along with increased grenade capacity.</p> <p class='news-paragraph'>The <span class='highlight'>PKM</span> packs a punch while balancing mobility and range. Based on the M870 with an added magazine, the <span class='highlight'>Hawk 12G</span>'s ammo capacity of 5 is the same as its parent weapon, the fire delay is faster and fires a tighter spread. </p> </div> <div data-date='2020-1-13'> <small class='news-date'>January 13, 2020</small> <p class='news-paragraph'><strong>Stay frosty</strong></p> <p class='news-paragraph'>Winter has arrived on the Island with new features and is more dangerous than ever, including PARMA's latest item, the <span class='highlight'>Flask</span>.</p> <p class='news-paragraph'>Track your enemy's footprints and use the <span class='highlight'>Snow Fox</span> perk to cover up your footprints. Be sure to keep moving or take freezing damage. Flasks and the <span class='highlight'>Polar Bear</span> perk keep you from freezing. And watch out for ice. It's slippery.</p> </div> <div data-date='2019-12-30'> <small class='news-date'>December 30, 2019</small> <p class='news-paragraph'><strong>Free Fryer</strong></p> <p class='news-paragraph'>PARMA's FSTMS division is pleased to introduce the next generation in starch-based modern warfare: the <span class='highlight'>spud gun</span>.</p> <p class='news-paragraph'>Officially designated the SMG-8 (Spud Missile Generator), the spud gun uses a proprietary breech-to-muzzle heat expander to fry and propel wedge-shaped projectiles at tremendous speeds. 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We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information that you've provided to them or that they've collected from your use of their services. You consent to our cookies if you continue to use our website. Click <a href='#' id='btn-cookie-opt-out' class='close'>here</a> to opt out of using cookies.</p> </div> <div class='modal-footer modal-footer-round'> <h3 class='close close-footer'><div class='btn-cookies-close btn-darken'>Got it!</div></h3> </div> </div> </div> </div> <script type="text/javascript" src="js/manifest.8e62061f.js"></script><script type="text/javascript" src="js/vendor.bd007df0.js"></script><script type="text/javascript" src="js/app.55f8f3a9.
GitHub Repo
https://github.com/ekrim/canonical-cuda-gemm
ekrim/canonical-cuda-gemm
My canonical CUDA GEMM, carefully designed to illustrate vectorized loads, tiling at the block and register level, padding for bank conflicts, and more
GitHub Repo
https://github.com/frostworks/nodebb-plugin-aumtag
frostworks/nodebb-plugin-aumtag
[ALPHA] Counts hashtag mentions within posts and aggregates them at the topic level. Duhsigned for canonical vs community nbb object tagging. Implemented by Gemini.
GitHub Repo
https://github.com/vohidjon123/google
vohidjon123/google
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GitHub Repo
https://github.com/pushchain/push-chain-pusd
pushchain/push-chain-pusd
Canonical PUSD contracts and integrations for Push Chain. PUSD is designed to target a 1:1 USD peg, backed by pooled USDC and USDT liquidity aggregated across chains, with mint and burn enforced at the protocol level.
GitHub Repo
https://github.com/MateusNobreSilva/app_send_mail