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| 1 | ·75d1ad25912d6802e8cf65b69b6de413·11112740·debug·optional·coq-dbgsym_8.18.0+dfsg-1_amd64.deb | 1 | ·75d1ad25912d6802e8cf65b69b6de413·11112740·debug·optional·coq-dbgsym_8.18.0+dfsg-1_amd64.deb |
| 2 | · | 2 | ·56495492a24fce0f18c6a9bcd2c94aff·81717960·math·optional·coq_8.18.0+dfsg-1_amd64.deb |
| 3 | ·885282477212bd5be1877fbc72e9a7f4·738548·debug·optional·coqide-dbgsym_8.18.0+dfsg-1_amd64.deb | 3 | ·885282477212bd5be1877fbc72e9a7f4·738548·debug·optional·coqide-dbgsym_8.18.0+dfsg-1_amd64.deb |
| 4 | ·28547edabb42d4f3148830b274809c8c·2005448·math·optional·coqide_8.18.0+dfsg-1_amd64.deb | 4 | ·28547edabb42d4f3148830b274809c8c·2005448·math·optional·coqide_8.18.0+dfsg-1_amd64.deb |
| 5 | ·b1fe305e13425de10dfdb98828b91117·2260004·debug·optional·libcoq-core-ocaml-dbgsym_8.18.0+dfsg-1_amd64.deb | 5 | ·b1fe305e13425de10dfdb98828b91117·2260004·debug·optional·libcoq-core-ocaml-dbgsym_8.18.0+dfsg-1_amd64.deb |
| 6 | ·37915300a6f7c0811a242c52a51a6793·44435736·ocaml·optional·libcoq-core-ocaml-dev_8.18.0+dfsg-1_amd64.deb | ||
| 7 | ·804ced76df9c34cacb771f20774f77c3·23217984·ocaml·optional·libcoq-core-ocaml_8.18.0+dfsg-1_amd64.deb | ||
| 8 | · | 6 | ·dd92fe4e087661d6fd2310dbe34f0f69·44430860·ocaml·optional·libcoq-core-ocaml-dev_8.18.0+dfsg-1_amd64.deb |
| 7 | ·9dac3fa16f24c421d3e86106fdf911c9·23212132·ocaml·optional·libcoq-core-ocaml_8.18.0+dfsg-1_amd64.deb | ||
| 8 | ·e6a1002c81d45dac58c55c8b3b7f23cc·23235284·math·optional·libcoq-stdlib_8.18.0+dfsg-1_amd64.deb | ||
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| 1 | -rw-r--r--···0········0········0········4·2023-12-21·13:34:31.000000·debian-binary | 1 | -rw-r--r--···0········0········0········4·2023-12-21·13:34:31.000000·debian-binary |
| 2 | -rw-r--r--···0········0········0·····2284·2023-12-21·13:34:31.000000·control.tar.xz | 2 | -rw-r--r--···0········0········0·····2284·2023-12-21·13:34:31.000000·control.tar.xz |
| 3 | -rw-r--r--···0········0········0·8171 | 3 | -rw-r--r--···0········0········0·81715484·2023-12-21·13:34:31.000000·data.tar.xz |
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| 3 | drwxr-xr-x···0·root·········(0)·root·········(0)········0·2023-12-21·13:34:31.000000·./usr/bin/ | 3 | drwxr-xr-x···0·root·········(0)·root·········(0)········0·2023-12-21·13:34:31.000000·./usr/bin/ |
| 4 | -rwxr-xr-x···0·root·········(0)·root·········(0)······891·2023-12-21·13:34:31.000000·./usr/bin/coq-makefile-make-both-single-timing-files | 4 | -rwxr-xr-x···0·root·········(0)·root·········(0)······891·2023-12-21·13:34:31.000000·./usr/bin/coq-makefile-make-both-single-timing-files |
| 5 | -rwxr-xr-x···0·root·········(0)·root·········(0)······931·2023-12-21·13:34:31.000000·./usr/bin/coq-makefile-make-both-time-files | 5 | -rwxr-xr-x···0·root·········(0)·root·········(0)······931·2023-12-21·13:34:31.000000·./usr/bin/coq-makefile-make-both-time-files |
| 6 | -rwxr-xr-x···0·root·········(0)·root·········(0)······697·2023-12-21·13:34:31.000000·./usr/bin/coq-makefile-make-one-time-file | 6 | -rwxr-xr-x···0·root·········(0)·root·········(0)······697·2023-12-21·13:34:31.000000·./usr/bin/coq-makefile-make-one-time-file |
| 7 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1088480·2023-12-21·13:34:31.000000·./usr/bin/coq-tex | 7 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1088480·2023-12-21·13:34:31.000000·./usr/bin/coq-tex |
| 8 | -rwxr-xr-x···0·root·········(0)·root·········(0)··3202920·2023-12-21·13:34:31.000000·./usr/bin/coq_makefile | 8 | -rwxr-xr-x···0·root·········(0)·root·········(0)··3202920·2023-12-21·13:34:31.000000·./usr/bin/coq_makefile |
| 9 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628072·2023-12-21·13:34:31.000000·./usr/bin/coqc | 9 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628072·2023-12-21·13:34:31.000000·./usr/bin/coqc |
| 10 | -rwxr-xr-x···0·root·········(0)·root·········(0)·416111 | 10 | -rwxr-xr-x···0·root·········(0)·root·········(0)·41611160·2023-12-21·13:34:31.000000·./usr/bin/coqc.byte |
| 11 | -rwxr-xr-x···0·root·········(0)·root·········(0)··8089736·2023-12-21·13:34:31.000000·./usr/bin/coqchk | 11 | -rwxr-xr-x···0·root·········(0)·root·········(0)··8089736·2023-12-21·13:34:31.000000·./usr/bin/coqchk |
| 12 | -rwxr-xr-x···0·root·········(0)·root·········(0)··3472744·2023-12-21·13:34:31.000000·./usr/bin/coqdep | 12 | -rwxr-xr-x···0·root·········(0)·root·········(0)··3472744·2023-12-21·13:34:31.000000·./usr/bin/coqdep |
| 13 | -rwxr-xr-x···0·root·········(0)·root·········(0)··3234024·2023-12-21·13:34:31.000000·./usr/bin/coqdoc | 13 | -rwxr-xr-x···0·root·········(0)·root·········(0)··3234024·2023-12-21·13:34:31.000000·./usr/bin/coqdoc |
| 14 | -rwxr-xr-x···0·root·········(0)·root·········(0)·42154 | 14 | -rwxr-xr-x···0·root·········(0)·root·········(0)·42154635·2023-12-21·13:34:31.000000·./usr/bin/coqidetop.byte |
| 15 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21288904·2023-12-21·13:34:31.000000·./usr/bin/coqidetop.opt | 15 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21288904·2023-12-21·13:34:31.000000·./usr/bin/coqidetop.opt |
| 16 | -rwxr-xr-x···0·root·········(0)·root·········(0)··8323688·2023-12-21·13:34:31.000000·./usr/bin/coqnative | 16 | -rwxr-xr-x···0·root·········(0)·root·········(0)··8323688·2023-12-21·13:34:31.000000·./usr/bin/coqnative |
| 17 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1385664·2023-12-21·13:34:31.000000·./usr/bin/coqpp | 17 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1385664·2023-12-21·13:34:31.000000·./usr/bin/coqpp |
| 18 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1785832·2023-12-21·13:34:31.000000·./usr/bin/coqtimelog2html | 18 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1785832·2023-12-21·13:34:31.000000·./usr/bin/coqtimelog2html |
| 19 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628200·2023-12-21·13:34:31.000000·./usr/bin/coqtop | 19 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628200·2023-12-21·13:34:31.000000·./usr/bin/coqtop |
| 20 | -rwxr-xr-x···0·root·········(0)·root·········(0)·598959 | 20 | -rwxr-xr-x···0·root·········(0)·root·········(0)·59895986·2023-12-21·13:34:31.000000·./usr/bin/coqtop.byte |
| 21 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628200·2023-12-21·13:34:31.000000·./usr/bin/coqtop.opt | 21 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628200·2023-12-21·13:34:31.000000·./usr/bin/coqtop.opt |
| 22 | -rwxr-xr-x···0·root·········(0)·root·········(0)···898336·2023-12-21·13:34:31.000000·./usr/bin/coqwc | 22 | -rwxr-xr-x···0·root·········(0)·root·········(0)···898336·2023-12-21·13:34:31.000000·./usr/bin/coqwc |
| 23 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628936·2023-12-21·13:34:31.000000·./usr/bin/coqworker.opt | 23 | -rwxr-xr-x···0·root·········(0)·root·········(0)·21628936·2023-12-21·13:34:31.000000·./usr/bin/coqworker.opt |
| 24 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1299144·2023-12-21·13:34:31.000000·./usr/bin/coqworkmgr | 24 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1299144·2023-12-21·13:34:31.000000·./usr/bin/coqworkmgr |
| 25 | -rwxr-xr-x···0·root·········(0)·root·········(0)·24433608·2023-12-21·13:34:31.000000·./usr/bin/csdpcert | 25 | -rwxr-xr-x···0·root·········(0)·root·········(0)·24433608·2023-12-21·13:34:31.000000·./usr/bin/csdpcert |
| 26 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1307656·2023-12-21·13:34:31.000000·./usr/bin/ocamllibdep | 26 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1307656·2023-12-21·13:34:31.000000·./usr/bin/ocamllibdep |
| 27 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1543528·2023-12-21·13:34:31.000000·./usr/bin/votour | 27 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1543528·2023-12-21·13:34:31.000000·./usr/bin/votour |
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| 023bef40:·2e41 | 023bef40:·a32e·4173·7365·7274·5f66·6169·6c75·7265··..Assert_failure | ||
| 023bef50:·0800 | 023bef50:·5c08·0000·2000·9004·6d40·90a0·c092·a0c0··\...·...m@...... | ||
| 023bef60:·9004 | 023bef60:·b390·041b·4090·4002·05f5·e100·404a·a0c0··....@.@.....@J.. | ||
| 023bef70:·9004 | 023bef70:·b390·046a·4090·4002·05f5·e100·4040·a004··...j@.@.....@@.. | ||
| 023bef80:·4002 | 023bef80:·0540·0205·f5e1·0040·5640·4041·04dc·a0b0··.@.....@V@@A.... | ||
| 023bef90:·3d6f | 023bef90:·a03d·6f63·616d·6c2e·7761·726e·5f6f·6e5f··.=ocaml.warn_on_ | ||
| 023befa0:·6974 | 023befa0:·6c69·7465·7261·6c5f·7061·7474·6572·6e04··literal_pattern. | ||
| 023befb0:·04 | 023befb0:·e004·9904·e040·9204·17a3·3044·6976·6973··.....@....0Divis | ||
| 023befc0:·6f6e | 023befc0:·696f·6e5f·6279·5f7a·6572·6f59·0800·0020··ion_by_zeroY...· | ||
| 023befd0:·0418 | 023befd0:·0004·1840·9040·4041·04e5·a0b0·a004·0904··...@.@@A........ | ||
| 023befe0:·04a1 | 023befe0:·e804·a104·e840·9204·07a3·2b45·6e64·5f6f··.....@....+End_o | ||
| 023beff0:·5f | 023beff0:·665f·6669·6c65·5808·0000·2000·0420·4090··f_fileX...·..·@. | ||
| 023bf000:·404 | 023bf000:·4040·4104·eda0·b0a0·0411·04f0·04a9·04f0··@@A............. | ||
| 023bf010:·9204 | 023bf010:·4092·0407·a327·4661·696c·7572·6555·0800··@....'FailureU.. | ||
| 023bf020:·20 | 023bf020:·0020·0004·2840·90a0·0424·4040·4104·f6a0··.·..(@...$@@A... | ||
| 023bf030:·a004 | 023bf030:·b0a0·041a·04f9·04b2·04f9·4092·0408·a330··..........@....0 | ||
| 023bf040:·6e76 | 023bf040:·496e·7661·6c69·645f·6172·6775·6d65·6e74··Invalid_argument | ||
| 023bf050:· | 023bf050:·5408·0000·2000·0431·4090·a004·2d40·4041··T...·..1@...-@@A | ||
| 023bf060:·f | 023bf060:·04ff·a0b0·a004·2305·0102·04bb·0501·0240··......#........@ | ||
| 023bf070:·0408 | 023bf070:·9204·08a3·2d4d·6174·6368·5f66·6169·6c75··....-Match_failu | ||
| 023bf080:·6552 | 023bf080:·7265·5208·0000·2000·043a·4090·a0c0·92a0··reR...·..:@..... | ||
| 023bf090:·39a0 | 023bf090:·0439·a004·35a0·0436·4002·05f5·e100·405c··.9..5..6@.....@\ | ||
| 023bf0a0:·404 | 023bf0a0:·4040·4105·010d·a0b0·a004·3105·0110·04c9··@@A.......1..... | ||
| 023bf0b0:·0 | 023bf0b0:·0501·1040·9204·0da3·294e·6f74·5f66·6f75··...@....)Not_fou | ||
| 023bf0c0:·6456 | 023bf0c0:·6e64·5608·0000·2000·0448·4090·4040·4105··ndV...·..H@.@@A. | ||
| 023bf0d0:·1 | 023bf0d0:·0115·a0b0·a004·3905·0118·04d1·0501·1840··......9........@ | ||
| 023bf0e0:·0407 | 023bf0e0:·9204·07a3·2d4f·7574·5f6f·665f·6d65·6d6f··....-Out_of_memo | ||
| 023bf0f0:·7 | 023bf0f0:·7279·5308·0000·2000·0450·4090·4040·4105··ryS...·..P@.@@A. | ||
| 023bf100:·1 | 023bf100:·011d·a0b0·a004·4105·0120·04d9·0501·2040··......A..·....·@ | ||
| 023bf110:·0407 | 023bf110:·9204·07a3·2e53·7461·636b·5f6f·7665·7266··.....Stack_overf | ||
| 023bf120:·6f | 023bf120:·6c6f·775a·0800·0020·0004·5840·9040·4041··lowZ...·..X@.@@A | ||
| 023bf130:·01 | 023bf130:·0501·25a0·b0a0·0449·0501·2804·e105·0128··..%....I..(....( | ||
| 023bf140:·9204 | 023bf140:·4092·0407·a32e·5379·735f·626c·6f63·6b65··@.....Sys_blocke | ||
| 023bf150:·5f69 | 023bf150:·645f·696f·5b08·0000·2000·0460·4090·4040··d_io[...·..`@.@@ | ||
| 023bf160:·0501 | 023bf160:·4105·012d·a0b0·a004·5105·0130·04e9·0501··A..-....Q..0.... | ||
| 023bf170:·4092 | 023bf170:·3040·9204·07a3·2953·7973·5f65·7272·6f72··0@....)Sys_error | ||
| 023bf180:· | 023bf180:·5708·0000·2000·0468·4090·a004·6440·4041··W...·..h@...d@@A | ||
| 023bf190:·01 | 023bf190:·0501·36a0·b0a0·045a·0501·3904·f205·0139··..6....Z..9....9 | ||
| 023bf1a0:·9204 | 023bf1a0:·4092·0408·a33a·556e·6465·6669·6e65·645f··@....:Undefined_ | ||
| 023bf1b0:·6563 | 023bf1b0:·7265·6375·7273·6976·655f·6d6f·6475·6c65··recursive_module | ||
| 023bf1c0:· | 023bf1c0:·5d08·0000·2000·0471·4090·a0c0·92a0·0470··]...·..q@......p | ||
| 023bf1d0:·046c | 023bf1d0:·a004·6ca0·046d·4002·05f5·e100·4063·4040··..l..m@.....@c@@ | ||
| 023bf1e0:·0501 | 023bf1e0:·4105·0144·a0b0·a004·6805·0147·0501·0005··A..D....h..G.... | ||
| 023bf1f0:·4 | 023bf1f0:·0147·4092·040d·a325·6279·7465·7343·0800··.G@....%bytesC.. | ||
| 023bf200:· | 023bf200:·0038·0040·4040·4140·4040·4040·0501·4b40··.8.@@@A@@@@@..K@ | ||
| 023bf210:·4092 | 023bf210:·4040·9204·0390·9226·5374·646c·6962·9223··@@.....&Stdlib.# | ||
| 023bf220:·7074·696f·6eb1·2553·746f | 023bf220:·496e·7492·264f·7074·696f·6eb1·2553·746f··Int.&Option.%Sto | ||
| 023bf230:·40c0·90a1·9092·2553·746f | 023bf230:·7265·0101·0d41·40c0·90a1·9092·2553·746f··re...A@.....%Sto | ||
| 023bf240:·c02f·636c·6962·2f65·78 | 023bf240:·7265·2153·40b0·c02f·636c·6962·2f65·786e··re!S@../clib/exn | ||
| 023bf250:·6 | 023bf250:·696e·666f·2e6d·6c4f·0103·8701·0387·c004··info.mlO........ | ||
| 023bf260:·0 | 023bf260:·024f·0103·8701·03a3·40a1·0501·6240·b121··.O......@...b@.! | ||
| 023bf270:· | 023bf270:·7401·010e·4208·0000·3800·a0c0·9090·2161··t...B...8.....!a | ||
| 023bf280:·0 | 023bf280:·0205·f5e1·0040·0105·a740·4140·4190·c0b3··.....@...@A@A... | ||
| 023bf290:·6965·6c64·a0c0·9604·0d | 023bf290:·a190·041b·2566·6965·6c64·a0c0·9604·0d44··....%field.....D | ||
| 023bf2a0:·400 | 023bf2a0:·4001·05df·4090·4002·05f5·e100·4001·05e1··@...@.@.....@... | ||
| 023bf2b0:· | 023bf2b0:·a047·40a0·4240·4040·b0c0·0419·5101·03a5··.G@.B@@@....Q... | ||
| 023bf2c0:· | 023bf2c0:·0103·a5c0·041a·5101·03a5·0103·bf40·4040··......Q......@@@ | ||
| 023bf2d0:·b124·696e·666f·0101·1743 | 023bf2d0:·40a1·0501·7a41·b124·696e·666f·0101·1743··@...zA.$info...C | ||
| 023bf2e0:·4040·4190·c0b3·a190·042e | 023bf2e0:·0800·0038·0040·4040·4190·c0b3·a190·042e··...8.@@@A....... | ||
| 023bf2f0:· | 023bf2f0:·2174·4090·4002·05f5·e100·4001·05f2·4040··!t@.@.....@...@@ | ||
| 023bf300:· | 023bf300:·4040·b0c0·0427·5301·03c1·0103·c1c0·0428··@@...'S........( | ||
| 023bf310:·d440·4040·40a1·0501·8 | 023bf310:·5301·03c1·0103·d440·4040·40a1·0501·8842··S......@@@@....B | ||
| 023bf320:·0101·1844·0800·0038· | 023bf320:·b124·6965·786e·0101·1844·0800·0038·0040··.$iexn...D...8.@ | ||
| 023bf330:· | 023bf330:·4040·4190·c092·a0c0·b390·0501·3140·9040··@@A.........1@.@ | ||
| 023bf340:·0 | 023bf340:·0205·f5e1·0040·0105·fba0·c0b3·9004·1d40··.....@.........@ | ||
| Max diff block lines reached; 34774953/35616584 bytes (97.64%) of diff not shown. | |||
| Offset 2376596, 258067 lines modified | Offset 2376596, 258070 lines modified | ||
| 02443930:·c040·b004·4f42·4040·4140·4240·4084·95a6··.@..OB@@A@B@@... | 02443930:·c040·b004·4f42·4040·4140·4240·4084·95a6··.@..OB@@A@B@@... | ||
| 02443940:·be00·0000·5200·0000·0600·0000·2100·0000··....R.......!... | 02443940:·be00·0000·5200·0000·0600·0000·2100·0000··....R.......!... | ||
| 02443950:·17a0·302e·2f5f·6275·696c·642f·6465·6661··..0./_build/defa | 02443950:·17a0·302e·2f5f·6275·696c·642f·6465·6661··..0./_build/defa | ||
| 02443960:·756c·74a0·352e·2f5f·6275·696c·642f·6465··ult.5./_build/de | 02443960:·756c·74a0·352e·2f5f·6275·696c·642f·6465··ult.5./_build/de | ||
| 02443970:·6661·756c·742f·636c·6962·a009·252e·2f5f··fault/clib..%./_ | 02443970:·6661·756c·742f·636c·6962·a009·252e·2f5f··fault/clib..%./_ | ||
| 02443980:·6275·696c·642f·6465·6661·756c·742f·636c··build/default/cl | 02443980:·6275·696c·642f·6465·6661·756c·742f·636c··build/default/cl | ||
| 02443990:·6962·2f2e·636c·6962·2e6f·626a·732f·6279··ib/.clib.objs/by | 02443990:·6962·2f2e·636c·6962·2e6f·626a·732f·6279··ib/.clib.objs/by | ||
| 024439a0:·7465·4000·08c5·1084·95a6·be00·002f·d | 024439a0:·7465·4000·08c5·1084·95a6·be00·002f·df00··te@........../.. | ||
| 024439b0:·0008·4 | 024439b0:·0008·4300·0023·6700·0022·d3a0·0800·002c··..C..#g.."....., | ||
| 024439c0:·0001·06ac·2745·786e·696e·666f·b0c0·265f··....'Exninfo..&_ | 024439c0:·0001·06ac·2745·786e·696e·666f·b0c0·265f··....'Exninfo..&_ | ||
| 024439d0:·6e6f·6e65·5f40·4000·ff04·0241·4122·3f3f··none_@@....AA"?? | 024439d0:·6e6f·6e65·5f40·4000·ff04·0241·4122·3f3f··none_@@....AA"?? | ||
| 024439e0:·4140·d040·4040·4040·b040·4040·4040·a008··A@.@@@@@.@@@@@.. | 024439e0:·4140·d040·4040·4040·b040·4040·4040·a008··A@.@@@@@.@@@@@.. | ||
| 024439f0:·0000·2c00·0106·5404·0904·0841·223f·3f41··..,...T....A"??A | 024439f0:·0000·2c00·0106·5404·0904·0841·223f·3f41··..,...T....A"??A | Diff chunk too large, falling back to line-by-line diff (4380 lines added, 4380 lines removed) | |
| 02443a00:·b0b0·b0b0·b0b0·b0b0·b0b0·b1b0·b0b0·b0b0··................ | 02443a00:·b0b0·b0b0·b0b0·b0b0·b0b0·b1b0·b0b0·b0b0··................ | ||
| 02443a10:·b2b0·b0b0·b0b1·b1b1·c3ab·a | 02443a10:·b2b0·b0b0·b0b1·b1b1·c3ab·aba7·b1b2·b2b2··................ | ||
| 02443a20:·b2b2·b2b2·b2b2·b2b2·b | 02443a20:·b2b2·b2b2·b2b2·b2b2·b2b1·b1b1·b1b1·b1b1··................ | ||
| 02443a30:·b1b1·b1b1·b1b1·b1b1·40a3 | 02443a30:·b1b1·b1b1·b1b1·b1b1·b140·a325·6172·7261··.........@.%arra | ||
| 02443a40:·4808 | 02443a40:·7948·0800·0038·00a0·c090·4002·05f5·e100··yH...8....@..... | ||
| 02443a50:·4 | 02443a50:·404d·4041·4041·40a0·007f·40a0·4040·4040··@M@A@A@...@.@@@@ | ||
| 02443a60:· | 02443a60:·044a·4040·4092·0408·a324·626f·6f6c·4508··.J@@@....$boolE. | ||
| 02443a70:·0038 | 02443a70:·0000·3800·4040·a1a0·e0a3·2566·616c·7365··..8.@@....%false | ||
| 02443a80:·9040 | 02443a80:·5e90·4040·0454·4092·0403·a0e0·a324·7472··^.@@.T@......$tr | ||
| 02443a90:·655f | 02443a90:·7565·5f90·4040·045a·4092·0403·4040·4140··ue_.@@.Z@...@@A@ | ||
| 02443aa0:·4040·4004 | 02443aa0:·4040·4040·045b·4041·4092·0410·a324·6368··@@@@.[@A@....$ch | ||
| 02443ab0:·7242 | 02443ab0:·6172·4208·0000·3800·4040·4041·4040·4040··arB...8.@@@A@@@@ | ||
| 02443ac0:·04 | 02443ac0:·4004·5f40·4140·9204·03a3·2365·786e·4708··@._@A@....#exnG. | ||
| 02443ad0:·00 | 02443ad0:·0000·3800·4040·4141·4040·4040·4004·6340··..8.@@AA@@@@@.c@ | ||
| 02443ae0:·4092 | 02443ae0:·4040·9204·03a3·3565·7874·656e·7369·6f6e··@@....5extension | ||
| 02443af0:·636f | 02443af0:·5f63·6f6e·7374·7275·6374·6f72·5008·0000··_constructorP... | ||
| 02443b00:· | 02443b00:·3800·4040·4041·4040·4040·4004·6740·4040··8.@@@A@@@@@.g@@@ | ||
| 02443b10:·0403 | 02443b10:·9204·03a3·2566·6c6f·6174·4408·0000·3800··....%floatD...8. | ||
| 02443b20:·4040·4 | 02443b20:·4040·4041·4040·4040·4004·6b40·4040·9204··@@@A@@@@@.k@@@.. | ||
| 02443b30:·a32a | 02443b30:·03a3·2a66·6c6f·6174·6172·7261·7951·0800··..*floatarrayQ.. | ||
| 02443b40:· | 02443b40:·0038·0040·4040·4140·4040·4040·046f·4040··.8.@@@A@@@@@.o@@ | ||
| 02443b50:·9204 | 02443b50:·4092·0403·a323·696e·7441·0800·0038·0040··@....#intA...8.@ | ||
| 02443b60:·404 | 02443b60:·4040·4140·4040·4040·0473·4041·4092·0403··@@A@@@@@.s@A@... | ||
| 02443b70:·2569 | 02443b70:·a325·696e·7433·324c·0800·0038·0040·4040··.%int32L...8.@@@ | ||
| 02443b80:·4 | 02443b80:·4140·4040·4040·0477·4040·4092·0403·a325··A@@@@@.w@@@....% | ||
| 02443b90:·6e74 | 02443b90:·696e·7436·344d·0800·0038·0040·4040·4140··int64M...8.@@@A@ | ||
| 02443ba0:·4040·40 | 02443ba0:·4040·4040·047b·4040·4092·0403·a326·6c61··@@@@.{@@@....&la | ||
| 02443bb0:·795f | 02443bb0:·7a79·5f74·4e08·0000·3800·a0c0·043a·0205··zy_tN...8....:.. | ||
| 02443bc0:·e1 | 02443bc0:·f5e1·0040·4f40·4140·4140·a059·40a0·4040··...@O@A@A@.Y@.@@ | ||
| 02443bd0:·40 | 02443bd0:·4040·0483·4040·4092·0407·a324·6c69·7374··@@..@@@....$list | ||
| 02443be0:·0800 | 02443be0:·4908·0000·3800·a0c0·0442·0205·f5e1·0040··I...8....B.....@ | ||
| 02443bf0:·4041 | 02443bf0:·5040·41a1·a0e0·a322·5b5d·6190·4040·048f··P@A...."[]a.@@.. | ||
| 02443c00:·9204 | 02443c00:·4092·0403·a0e0·a322·3a3a·6290·a004·0ea0··@......"::b..... | ||
| 02443c10:·b390 | 02443c10:·c0b3·9004·16a0·0413·4090·4002·05f5·e100··........@.@..... | ||
| 02443c20:·5140 | 02443c20:·4051·4040·049c·4092·040a·4040·4140·a059··@Q@@..@...@@A@.Y | ||
| 02443c30:· | 02443c30:·40a0·4040·4040·049f·4040·4092·041b·a329··@.@@@@..@@@....) | ||
| 02443c40:·6174 | 02443c40:·6e61·7469·7665·696e·744b·0800·0038·0040··nativeintK...8.@ | ||
| 02443c50:·404 | 02443c50:·4040·4140·4040·4040·04a3·4040·4092·0403··@@A@@@@@..@@@... | ||
| 02443c60:·266f | 02443c60:·a326·6f70·7469·6f6e·4a08·0000·3800·a0c0··.&optionJ...8... | ||
| 02443c70:·6202 | 02443c70:·0462·0205·f5e1·0040·5340·41a1·a0e0·a324··.b.....@S@A....$ | ||
| 02443c80:·6f6e | 02443c80:·4e6f·6e65·6390·4040·04af·4092·0403·a0e0··Nonec.@@..@..... | ||
| 02443c90:·2453 | 02443c90:·a324·536f·6d65·6490·a004·0e40·4004·b640··.$Somed....@@..@ | ||
| 02443ca0:· | 02443ca0:·9204·0440·4041·40a0·5940·a040·4040·4004··...@@A@.Y@.@@@@. | ||
| 02443cb0:·4040 | 02443cb0:·b940·4040·9204·15a3·2673·7472·696e·674f··.@@@....&stringO | ||
| 02443cc0:·0000 | 02443cc0:·0800·0038·0040·4040·4140·4040·4040·04bd··...8.@@@A@@@@@.. | ||
| 02443cd0:·4040·9204 | 02443cd0:·4040·4092·0403·a324·756e·6974·4608·0000··@@@....$unitF... | ||
| 02443ce0:·0040 | 02443ce0:·3800·4040·a1a0·e0a3·2228·2960·9040·4004··8.@@...."()`.@@. | ||
| 02443cf0:·4092·040 | 02443cf0:·c740·9204·0340·4041·4040·4040·4004·c840··.@...@@A@@@@@..@ | ||
| 02443d00:·4092 | 02443d00:·4140·9204·0aa3·2e41·7373·6572·745f·6661··A@.....Assert_fa | ||
| 02443d10:·6c75 | 02443d10:·696c·7572·655c·0800·0020·0090·046d·4090··ilure\...·...m@. | ||
| 02443d20:·c092 | 02443d20:·a0c0·92a0·c0b3·9004·1b40·9040·0205·f5e1··.........@.@.... | ||
| 02443d30:·404a | 02443d30:·0040·4aa0·c0b3·9004·6a40·9040·0205·f5e1··.@J.....j@.@.... | ||
| 02443d40:· | 02443d40:·0040·40a0·0405·4002·05f5·e100·4056·4040··.@@...@.....@V@@ | ||
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| 00001ea0:· | 00001ea0:·0424·4040·4092·0403·a32a·666c·6f61·7461··.$@@@....*floata | ||
| 00001eb0:· | 00001eb0:·7272·6179·5108·0000·3800·4040·4041·4040··rrayQ...8.@@@A@@ | ||
| 00001ec0:·4040·40 | 00001ec0:·4040·4004·2840·4040·9204·03a3·2369·6e74··@@@.(@@@....#int | ||
| 00001ed0:· | 00001ed0:·4108·0000·3800·4040·4041·4040·4040·4004··A...8.@@@A@@@@@. | ||
| 00001ee0:· | 00001ee0:·2c40·4140·9204·03a3·2569·6e74·3332·4c08··,@A@....%int32L. | ||
| 00001ef0:·0 | 00001ef0:·0000·3800·4040·4041·4040·4040·4004·3040··..8.@@@A@@@@@.0@ | ||
| 00001f00:·4040· | 00001f00:·4040·9204·03a3·2569·6e74·3634·4d08·0000··@@....%int64M... | ||
| 00001f10:· | 00001f10:·3800·4040·4041·4040·4040·4004·3440·4040··8.@@@A@@@@@.4@@@ | ||
| 00001f20:· | 00001f20:·9204·03a3·266c·617a·795f·744e·0800·0038··....&lazy_tN...8 | ||
| 00001f30:· | 00001f30:·00a0·c004·3d02·05f5·e100·404f·4041·4041··....=.....@O@A@A | ||
| 00001f40:·4 | 00001f40:·40a0·5940·a040·4040·4004·3c40·4040·9204··@.Y@.@@@@.<@@@.. | ||
| 00001f50:·0 | 00001f50:·07a3·246c·6973·7449·0800·0038·00a0·c004··..$listI...8.... | ||
| 00001f60:· | 00001f60:·4502·05f5·e100·4050·4041·a1a0·e0a3·225b··E.....@P@A...."[ | ||
| 00001f70:·5 | 00001f70:·5d61·9040·4004·4840·9204·03a0·e0a3·223a··]a.@@.H@......": | ||
| 00001f80:·3a | 00001f80:·3a62·90a0·040e·a0c0·b390·0416·a004·1340··:b.............@ | ||
| 00001f90:· | 00001f90:·9040·0205·f5e1·0040·5140·4004·5540·9204··.@.....@Q@@.U@.. | ||
| 00001fa0:·040 | 00001fa0:·0a40·4041·40a0·5940·a040·4040·4004·5840··.@@A@.Y@.@@@@.X@ | ||
| 00001fb0:·4040· | 00001fb0:·4040·9204·1ba3·296e·6174·6976·6569·6e74··@@....)nativeint | ||
| 00001fc0:· | 00001fc0:·4b08·0000·3800·4040·4041·4040·4040·4004··K...8.@@@A@@@@@. | ||
| 00001fd0:· | 00001fd0:·5c40·4040·9204·03a3·266f·7074·696f·6e4a··\@@@....&optionJ | ||
| 00001fe0:· | 00001fe0:·0800·0038·00a0·c004·6502·05f5·e100·4053··...8....e.....@S | ||
| 00001ff0:· | 00001ff0:·4041·a1a0·e0a3·244e·6f6e·6563·9040·4004··@A....$Nonec.@@. | ||
| 00002000:· | 00002000:·6840·9204·03a0·e0a3·2453·6f6d·6564·90a0··h@......$Somed.. | ||
| 00002010:· | 00002010:·040e·4040·046f·4092·0404·4040·4140·a059··..@@.o@...@@A@.Y | ||
| 00002020:· | 00002020:·40a0·4040·4040·0472·4040·4092·0415·a326··@.@@@@.r@@@....& | ||
| 00002030:· | 00002030:·7374·7269·6e67·4f08·0000·3800·4040·4041··stringO...8.@@@A | ||
| 00002040:·4 | 00002040:·4040·4040·4004·7640·4040·9204·03a3·2475··@@@@@.v@@@....$u | ||
| 00002050:· | 00002050:·6e69·7446·0800·0038·0040·40a1·a0e0·a322··nitF...8.@@...." | ||
| 00002060:·2 | 00002060:·2829·6090·4040·0480·4092·0403·4040·4140··()`.@@..@...@@A@ | ||
| 00002070:·4040·4040· | 00002070:·4040·4040·0481·4041·4092·040a·a32e·4173··@@@@..@A@.....As | ||
| 00002080:·73 | 00002080:·7365·7274·5f66·6169·6c75·7265·5c08·0000··sert_failure\... | ||
| 00002090:· | 00002090:·2000·9004·6d40·90a0·c092·a0c0·b390·041b···...m@.......... | ||
| 000020a0:· | 000020a0:·4090·4002·05f5·e100·404a·a0c0·b390·046a··@.@.....@J.....j | ||
| 000020b0:· | 000020b0:·4090·4002·05f5·e100·4040·a004·0540·0205··@.@.....@@...@.. | ||
| 000020c0:· | 000020c0:·f5e1·0040·5640·4041·0495·a0b0·a03d·6f63··...@V@@A.....=oc | ||
| 000020d0:·6 | 000020d0:·616d·6c2e·7761·726e·5f6f·6e5f·6c69·7465··aml.warn_on_lite | ||
| 000020e0:· | 000020e0:·7261·6c5f·7061·7474·6572·6e04·9904·9c04··ral_pattern..... | ||
| 000020f0:· | 000020f0:·9940·9204·17a3·3044·6976·6973·696f·6e5f··.@....0Division_ | ||
| 00002100:· | 00002100:·6279·5f7a·6572·6f59·0800·0020·0004·1840··by_zeroY...·...@ | ||
| 00002110:· | 00002110:·9040·4041·049e·a0b0·a004·0904·a104·a404··.@@A............ | ||
| 00002120:· | 00002120:·a140·9204·07a3·2b45·6e64·5f6f·665f·6669··.@....+End_of_fi | ||
| 00002130:·6 | 00002130:·6c65·5808·0000·2000·0420·4090·4040·4104··leX...·..·@.@@A. | ||
| 00002140:· | 00002140:·a6a0·b0a0·0411·04a9·04ac·04a9·4092·0407··............@... | ||
| 00002150:· | 00002150:·a327·4661·696c·7572·6555·0800·0020·0004··.'FailureU...·.. | ||
| 00002160:· | 00002160:·2840·90a0·0424·4040·4104·afa0·b0a0·041a··(@...$@@A....... | ||
| 00002170:· | 00002170:·04b2·04b5·04b2·4092·0408·a330·496e·7661··......@....0Inva | ||
| 00002180:·6 | 00002180:·6c69·645f·6172·6775·6d65·6e74·5408·0000··lid_argumentT... | ||
| 00002190:· | 00002190:·2000·0431·4090·a004·2d40·4041·04b8·a0b0···..1@...-@@A.... | ||
| 000021a0:· | 000021a0:·a004·2304·bb04·be04·bb40·9204·08a3·2d4d··..#......@....-M | ||
| 000021b0:· | 000021b0:·6174·6368·5f66·6169·6c75·7265·5208·0000··atch_failureR... | ||
| 000021c0:· | 000021c0:·2000·043a·4090·a0c0·92a0·0439·a004·35a0···..:@......9..5. | ||
| 000021d0:· | 000021d0:·0436·4002·05f5·e100·405c·4040·4104·c6a0··.6@.....@\@@A... | ||
| 000021e0:· | 000021e0:·b0a0·0431·04c9·04cc·04c9·4092·040d·a329··...1......@....) | ||
| 000021f0:· | 000021f0:·4e6f·745f·666f·756e·6456·0800·0020·0004··Not_foundV...·.. | ||
| 00002200:· | 00002200:·4840·9040·4041·04ce·a0b0·a004·3904·d104··H@.@@A......9... | ||
| 00002210:· | 00002210:·d404·d140·9204·07a3·2d4f·7574·5f6f·665f··...@....-Out_of_ | ||
| 00002220:· | 00002220:·6d65·6d6f·7279·5308·0000·2000·0450·4090··memoryS...·..P@. | ||
| 00002230:· | 00002230:·4040·4104·d6a0·b0a0·0441·04d9·04dc·04d9··@@A......A...... | ||
| 00002240:· | 00002240:·4092·0407·a32e·5374·6163·6b5f·6f76·6572··@.....Stack_over | ||
| 00002250:· | 00002250:·666c·6f77·5a08·0000·2000·0458·4090·4040··flowZ...·..X@.@@ | ||
| 00002260:·404 | 00002260:·4104·dea0·b0a0·0449·04e1·04e4·04e1·4092··A......I......@. | ||
| 00002270:· | 00002270:·0407·a32e·5379·735f·626c·6f63·6b65·645f··....Sys_blocked_ | ||
| 00002280:· | 00002280:·696f·5b08·0000·2000·0460·4090·4040·4104··io[...·..`@.@@A. | ||
| 00002290:· | 00002290:·e6a0·b0a0·0451·04e9·04ec·04e9·4092·0407··.....Q......@... | ||
| 000022a0:· | 000022a0:·a329·5379·735f·6572·726f·7257·0800·0020··.)Sys_errorW...· | ||
| 000022b0:· | 000022b0:·0004·6840·90a0·0464·4040·4104·efa0·b0a0··..h@...d@@A..... | ||
| 000022c0:· | 000022c0:·045a·04f2·04f5·04f2·4092·0408·a33a·556e··.Z......@....:Un | ||
| 000022d0:·6 | 000022d0:·6465·6669·6e65·645f·7265·6375·7273·6976··defined_recursiv | ||
| 000022e0:· | 000022e0:·655f·6d6f·6475·6c65·5d08·0000·2000·0471··e_module]...·..q | ||
| 000022f0:· | 000022f0:·4090·a0c0·92a0·0470·a004·6ca0·046d·4002··@......p..l..m@. | ||
| 00002300:·0 | 00002300:·05f5·e100·4063·4040·4104·fda0·b0a0·0468··....@c@@A......h | ||
| 00002310:· | 00002310:·0501·0005·0103·0501·0040·9204·0da3·2562··.........@....%b | ||
| 00002320:· | 00002320:·7974·6573·4308·0000·3800·4040·4041·4040··ytesC...8.@@@A@@ | ||
| 00002330:·4040·40 | 00002330:·4040·4005·0104·4040·4092·0403·9092·2653··@@@...@@@.....&S | ||
| 00002340:· | 00002340:·7464·6c69·6292·264f·7074·696f·6e40·41b0··tdlib.&Option@A. | ||
| 00002350:· | 00002350:·0501·3605·0139·40a0·b090·f0a0·2763·6f6d··..6..9@.....'com | ||
| 00002360:· | 00002360:·7061·7265·0101·0ca0·0402·b0c0·0501·524e··pare..........RN | ||
| 00002370:· | 00002370:·0102·cb01·02d1·c005·0153·4e01·02cb·0102··.........SN..... | ||
| 00002380:· | 00002380:·d840·d0b1·40d0·b390·0501·4da0·9021·74b0··.@..@.....M..!t. | ||
| 00002390:· | 00002390:·c005·015d·4e01·02cb·0102·dbc0·0501·5e4e··...]N.........^N | ||
| 000023a0:· | 000023a0:·0102·cb01·02dc·4040·c0b3·0409·4090·4002··......@@....@.@. | ||
| 000023b0:·0 | 000023b0:·05f5·e100·4001·05a5·0800·0030·0005·014c··....@......0...L | ||
| 000023c0:·0 | 000023c0:·0501·4c05·014c·0501·4c05·014c·0501·4c05··..L..L..L..L..L. | ||
| 000023d0:·014 | 000023d0:·014c·0501·4c40·b105·014c·0501·5805·0152··.L..L@...L..X..R | ||
| 000023e0:· | 000023e0:·4041·b004·0804·0740·40d0·b140·d0b3·9005··@A.....@@..@.... | ||
| 000023f0:· | 000023f0:·015e·a090·2174·b0c0·0501·6e4e·0102·cb01··.^..!t....nN.... | ||
| 00002400:· | 00002400:·02e0·c005·016f·4e01·02cb·0102·e140·40c0··.....oN......@@. | ||
| 00002410:· | 00002410:·b304·0940·9040·0205·f5e1·0040·0105·a604··...@.@.....@.... | ||
| Max diff block lines reached; 14060/120827 bytes (11.64%) of diff not shown. | |||
| Offset 211, 16 lines modified | Offset 211, 16 lines modified | ||
| 00000d20:·a026·5374·646c·6962·9030·79b0·e9d3·b6f7··.&Stdlib.0y..... | 00000d20:·a026·5374·646c·6962·9030·79b0·e9d3·b6f7··.&Stdlib.0y..... | ||
| 00000d30:·fed0·7eb3·cc2a·bb96·1b91·a0a0·3843·616d··..~..*......8Cam | 00000d30:·fed0·7eb3·cc2a·bb96·1b91·a0a0·3843·616d··..~..*......8Cam | ||
| 00000d40:·6c69·6e74·6572·6e61·6c46·6f72·6d61·7442··linternalFormatB | 00000d40:·6c69·6e74·6572·6e61·6c46·6f72·6d61·7442··linternalFormatB | ||
| 00000d50:·6173·6963·7390·308f·8f63·4558·798e·e408··asics.0..cEXy... | 00000d50:·6173·6963·7390·308f·8f63·4558·798e·e408··asics.0..cEXy... | ||
| 00000d60:·df3c·50a5·539b·1540·8495·a6be·0000·0006··.<P.S..@........ | 00000d60:·df3c·50a5·539b·1540·8495·a6be·0000·0006··.<P.S..@........ | ||
| 00000d70:·0000·0003·0000·0008·0000·0008·a041·a090··.............A.. | 00000d70:·0000·0003·0000·0008·0000·0008·a041·a090··.............A.. | ||
| 00000d80:·4040·4361·6d6c·3139·3939·5430·3331·8495··@@Caml1999T031.. | 00000d80:·4040·4361·6d6c·3139·3939·5430·3331·8495··@@Caml1999T031.. | ||
| 00000d90:·a6be·0000·3fa | 00000d90:·a6be·0000·3fa6·0000·0900·0000·2830·0000··....?.......(0.. | ||
| 00000da0:·25 | 00000da0:·2589·0800·003c·0025·4469·6666·3292·b0a0··%....<.%Diff2... | ||
| 00000db0:·b09e·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | 00000db0:·b09e·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | ||
| 00000dc0:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | 00000dc0:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | ||
| 00000dd0:·a0a0·a0c0·91b2·0a00·0001·400a·2020·2041··..........@.···A | 00000dd0:·a0a0·a0c0·91b2·0a00·0001·400a·2020·2041··..........@.···A | ||
| 00000de0:·6e20·696d·706c·656d·656e·7461·7469·6f6e··n·implementation | 00000de0:·6e20·696d·706c·656d·656e·7461·7469·6f6e··n·implementation | ||
| 00000df0:·206f·6620·4575·6765·6e65·204d·7965·7273···of·Eugene·Myers | 00000df0:·206f·6620·4575·6765·6e65·204d·7965·7273···of·Eugene·Myers | ||
| 00000e00:·2720·4f28·4e44·2920·4469·6666·6572·656e··'·O(ND)·Differen | 00000e00:·2720·4f28·4e44·2920·4469·6666·6572·656e··'·O(ND)·Differen | ||
| 00000e10:·6365·2041·6c67·6f72·6974·686d·5c5b·315c··ce·Algorithm\[1\ | 00000e10:·6365·2041·6c67·6f72·6974·686d·5c5b·315c··ce·Algorithm\[1\ |
| Offset 1, 9 lines modified | Offset 1, 9 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
| 00000010:·0000·7d5 | 00000010:·0000·7d5e·0000·1645·0000·5bcc·0000·5966··..}^...E..[...Yf | ||
| 00000020:·0800·003c·0027·4578·6e69·6e66·6f91·b0a0··...<.'Exninfo... | 00000020:·0800·003c·0027·4578·6e69·6e66·6f91·b0a0··...<.'Exninfo... | ||
| 00000030:·b09d·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | 00000030:·b09d·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | ||
| 00000040:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | 00000040:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | ||
| 00000050:·a0a0·a0c0·91b2·09dc·2045·6e72·6963·6865··........·Enriche | 00000050:·a0a0·a0c0·91b2·09dc·2045·6e72·6963·6865··........·Enriche | ||
| 00000060:·6420·6578·6365·7074·696f·6e73·2068·6176··d·exceptions·hav | 00000060:·6420·6578·6365·7074·696f·6e73·2068·6176··d·exceptions·hav | ||
| 00000070:·6520·616e·2061·6464·6974·696f·6e61·6c20··e·an·additional· | 00000070:·6520·616e·2061·6464·6974·696f·6e61·6c20··e·an·additional· | ||
| 00000080:·6669·656c·6420·6174·2074·6865·2065·6e64··field·at·the·end | 00000080:·6669·656c·6420·6174·2074·6865·2065·6e64··field·at·the·end |
| Offset 95, 675 lines modified | Offset 95, 675 lines modified | ||
| 000005e0:·6c69·6290·3079·b0e9·d3b6·f7fe·d07e·b3cc··lib.0y.......~.. | 000005e0:·6c69·6290·3079·b0e9·d3b6·f7fe·d07e·b3cc··lib.0y.......~.. | ||
| 000005f0:·2abb·961b·91a0·a038·4361·6d6c·696e·7465··*......8Camlinte | 000005f0:·2abb·961b·91a0·a038·4361·6d6c·696e·7465··*......8Camlinte | ||
| 00000600:·726e·616c·466f·726d·6174·4261·7369·6373··rnalFormatBasics | 00000600:·726e·616c·466f·726d·6174·4261·7369·6373··rnalFormatBasics | ||
| 00000610:·9030·8f8f·6345·5879·8ee4·08df·3c50·a553··.0..cEXy....<P.S | 00000610:·9030·8f8f·6345·5879·8ee4·08df·3c50·a553··.0..cEXy....<P.S | ||
| 00000620:·9b15·4084·95a6·be00·0000·0600·0000·0300··..@............. | 00000620:·9b15·4084·95a6·be00·0000·0600·0000·0300··..@............. | ||
| 00000630:·0000·0800·0000·08a0·41a0·9040·4043·616d··........A..@@Cam | 00000630:·0000·0800·0000·08a0·41a0·9040·4043·616d··........A..@@Cam | ||
| 00000640:·6c31·3939·3954·3033·3184·95a6·be00·0029··l1999T031......) | 00000640:·6c31·3939·3954·3033·3184·95a6·be00·0029··l1999T031......) | ||
| 00000650:·a | 00000650:·a400·0005·9900·0019·a900·0017·a708·0000··................ | ||
| 00000660:·3c00·2745·786e·696e·666f·92b0·a0b0·9eb0··<.'Exninfo...... | 00000660:·3c00·2745·786e·696e·666f·92b0·a0b0·9eb0··<.'Exninfo...... | ||
| 00000670:·a02a·6f63·616d·6c2e·7465·7874·b0c0·265f··.*ocaml.text..&_ | 00000670:·a02a·6f63·616d·6c2e·7465·7874·b0c0·265f··.*ocaml.text..&_ | ||
| 00000680:·6e6f·6e65·5f40·4000·ff04·0241·90a0·a0a0··none_@@....A.... | 00000680:·6e6f·6e65·5f40·4000·ff04·0241·90a0·a0a0··none_@@....A.... | ||
| 00000690:·c091·b209·2c20·4164·6469·7469·6f6e·616c··....,·Additional | 00000690:·c091·b209·2c20·4164·6469·7469·6f6e·616c··....,·Additional | ||
| 000006a0:·2069·6e66·6f72·6d61·7469·6f6e·2077·6f72···information·wor | 000006a0:·2069·6e66·6f72·6d61·7469·6f6e·2077·6f72···information·wor | ||
| 000006b0:·6e20·6279·2065·7863·6570·7469·6f6e·732e··n·by·exceptions. | 000006b0:·6e20·6279·2065·7863·6570·7469·6f6e·732e··n·by·exceptions. | ||
| 000006c0:·20b0·c030·636c·6962·2f65·786e·696e·666f···..0clib/exninfo | 000006c0:·20b0·c030·636c·6962·2f65·786e·696e·666f···..0clib/exninfo | ||
| 000006d0:·2e6d·6c69·4b01·02a4·0102·a4c0·0402·4b01··.mliK.........K. | 000006d0:·2e6d·6c69·4b01·02a4·0102·a4c0·0402·4b01··.mliK.........K. | ||
| 000006e0:·02a4·0102·d540·4004·0440·4040·0404·4004··.....@@..@@@..@. | 000006e0:·02a4·0102·d540·4004·0440·4040·0404·4004··.....@@..@@@..@. | Diff chunk too large, falling back to line-by-line diff (484 lines added, 484 lines removed) | |
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| Offset 157, 782 lines modified | Offset 157, 781 lines modified | ||
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| Offset 141, 16 lines modified | Offset 141, 16 lines modified | ||
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| Offset 202, 1000 lines modified | Offset 202, 1000 lines modified | ||
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| 00000ed0:· | 00000ed0:·0403·a323·696e·7441·0800·0038·0040·4040··...#intA...8.@@@ | ||
| 00000ee0:·4 | 00000ee0:·4140·4040·4040·0461·4041·4092·0403·a325··A@@@@@.a@A@....% | ||
| 00000ef0:· | 00000ef0:·696e·7433·324c·0800·0038·0040·4040·4140··int32L...8.@@@A@ | ||
| 00000f00:·4040·4040· | 00000f00:·4040·4040·0465·4040·4092·0403·a325·696e··@@@@.e@@@....%in | ||
| 00000f10:· | 00000f10:·7436·344d·0800·0038·0040·4040·4140·4040··t64M...8.@@@A@@@ | ||
| 00000f20:·4040· | 00000f20:·4040·0469·4040·4092·0403·a326·6c61·7a79··@@.i@@@....&lazy | ||
| 00000f30:· | 00000f30:·5f74·4e08·0000·3800·a0c0·043a·0205·f5e1··_tN...8....:.... | ||
| 00000f40:· | 00000f40:·0040·4f40·4140·4140·a059·40a0·4040·4040··.@O@A@A@.Y@.@@@@ | ||
| 00000f50:· | 00000f50:·0471·4040·4092·0407·a324·6c69·7374·4908··.q@@@....$listI. | ||
| 00000f60:·0 | 00000f60:·0000·3800·a0c0·0442·0205·f5e1·0040·5040··..8....B.....@P@ | ||
| 00000f70:·4 | 00000f70:·41a1·a0e0·a322·5b5d·6190·4040·047d·4092··A...."[]a.@@.}@. | ||
| 00000f80:· | 00000f80:·0403·a0e0·a322·3a3a·6290·a004·0ea0·c0b3··....."::b....... | ||
| 00000f90:· | 00000f90:·9004·16a0·0413·4090·4002·05f5·e100·4051··......@.@.....@Q | ||
| 00000fa0:· | 00000fa0:·4040·048a·4092·040a·4040·4140·a059·40a0··@@..@...@@A@.Y@. | ||
| 00000fb0:· | 00000fb0:·4040·4040·048d·4040·4092·041b·a329·6e61··@@@@..@@@....)na | ||
| 00000fc0:· | 00000fc0:·7469·7665·696e·744b·0800·0038·0040·4040··tiveintK...8.@@@ | ||
| 00000fd0:·4 | 00000fd0:·4140·4040·4040·0491·4040·4092·0403·a326··A@@@@@..@@@....& | ||
| 00000fe0:· | 00000fe0:·6f70·7469·6f6e·4a08·0000·3800·a0c0·0462··optionJ...8....b | ||
| 00000ff0:· | 00000ff0:·0205·f5e1·0040·5340·41a1·a0e0·a324·4e6f··.....@S@A....$No | ||
| 00001000:·6 | 00001000:·6e65·6390·4040·049d·4092·0403·a0e0·a324··nec.@@..@......$ | ||
| 00001010:· | 00001010:·536f·6d65·6490·a004·0e40·4004·a440·9204··Somed....@@..@.. | ||
| 00001020:·04 | 00001020:·0440·4041·40a0·5940·a040·4040·4004·a740··.@@A@.Y@.@@@@..@ | ||
| 00001030:·4040· | 00001030:·4040·9204·15a3·2673·7472·696e·674f·0800··@@....&stringO.. | ||
| 00001040:·00 | 00001040:·0038·0040·4040·4140·4040·4040·04ab·4040··.8.@@@A@@@@@..@@ | ||
| 00001050:·40 | 00001050:·4092·0403·a324·756e·6974·4608·0000·3800··@....$unitF...8. | ||
| 00001060:· | 00001060:·4040·a1a0·e0a3·2228·2960·9040·4004·b540··@@...."()`.@@..@ | ||
| 00001070:· | 00001070:·9204·0340·4041·4040·4040·4004·b640·4140··...@@A@@@@@..@A@ | ||
| 00001080:· | 00001080:·9204·0aa3·2e41·7373·6572·745f·6661·696c··.....Assert_fail | ||
| 00001090:· | 00001090:·7572·655c·0800·0020·0090·046d·4090·a0c0··ure\...·...m@... | ||
| 000010a0:· | 000010a0:·92a0·c0b3·9004·1b40·9040·0205·f5e1·0040··.......@.@.....@ | ||
| 000010b0:·4 | 000010b0:·4aa0·c0b3·9004·6a40·9040·0205·f5e1·0040··J.....j@.@.....@ | ||
| 000010c0:·40 | 000010c0:·40a0·0405·4002·05f5·e100·4056·4040·4104··@...@.....@V@@A. | ||
| 000010d0:· | 000010d0:·caa0·b0a0·3d6f·6361·6d6c·2e77·6172·6e5f··....=ocaml.warn_ | ||
| 000010e0:· | 000010e0:·6f6e·5f6c·6974·6572·616c·5f70·6174·7465··on_literal_patte | ||
| 000010f0:· | 000010f0:·726e·04ce·0499·04ce·4092·0417·a330·4469··rn......@....0Di | ||
| 00001100:· | 00001100:·7669·7369·6f6e·5f62·795f·7a65·726f·5908··vision_by_zeroY. | ||
| 00001110:·0 | 00001110:·0000·2000·0418·4090·4040·4104·d3a0·b0a0··..·...@.@@A..... | ||
| 00001120:· | 00001120:·0409·04d6·04a1·04d6·4092·0407·a32b·456e··........@....+En | ||
| 00001130:·6 | 00001130:·645f·6f66·5f66·696c·6558·0800·0020·0004··d_of_fileX...·.. | ||
| 00001140:·04 | 00001140:·2040·9040·4041·04db·a0b0·a004·1104·de04···@.@@A.......... | ||
| 00001150:· | 00001150:·a904·de40·9204·07a3·2746·6169·6c75·7265··...@....'Failure | ||
| 00001160:· | 00001160:·5508·0000·2000·0428·4090·a004·2440·4041··U...·..(@...$@@A | ||
| 00001170:· | 00001170:·04e4·a0b0·a004·1a04·e704·b204·e740·9204··.............@.. | ||
| 00001180:·0 | 00001180:·08a3·3049·6e76·616c·6964·5f61·7267·756d··..0Invalid_argum | ||
| 00001190:·6 | 00001190:·656e·7454·0800·0020·0004·3140·90a0·042d··entT...·..1@...- | ||
| 000011a0:· | 000011a0:·4040·4104·eda0·b0a0·0423·04f0·04bb·04f0··@@A......#...... | ||
| 000011b0:· | 000011b0:·4092·0408·a32d·4d61·7463·685f·6661·696c··@....-Match_fail | ||
| 000011c0:· | 000011c0:·7572·6552·0800·0020·0004·3a40·90a0·c092··ureR...·..:@.... | ||
| 000011d0:· | 000011d0:·a004·39a0·0435·a004·3640·0205·f5e1·0040··..9..5..6@.....@ | ||
| 000011e0:· | 000011e0:·5c40·4041·04fb·a0b0·a004·3104·fe04·c904··\@@A......1..... | ||
| 000011f0:· | 000011f0:·fe40·9204·0da3·294e·6f74·5f66·6f75·6e64··.@....)Not_found | ||
| 00001200:· | 00001200:·5608·0000·2000·0448·4090·4040·4105·0103··V...·..H@.@@A... | ||
| 00001210:· | 00001210:·a0b0·a004·3905·0106·04d1·0501·0640·9204··....9........@.. | ||
| 00001220:·0 | 00001220:·07a3·2d4f·7574·5f6f·665f·6d65·6d6f·7279··..-Out_of_memory | ||
| 00001230:· | 00001230:·5308·0000·2000·0450·4090·4040·4105·010b··S...·..P@.@@A... | ||
| 00001240:· | 00001240:·a0b0·a004·4105·010e·04d9·0501·0e40·9204··....A........@.. | ||
| 00001250:·0 | 00001250:·07a3·2e53·7461·636b·5f6f·7665·7266·6c6f··...Stack_overflo | ||
| 00001260:· | 00001260:·775a·0800·0020·0004·5840·9040·4041·0501··wZ...·..X@.@@A.. | ||
| 00001270:· | 00001270:·13a0·b0a0·0449·0501·1604·e105·0116·4092··.....I........@. | ||
| 00001280:· | 00001280:·0407·a32e·5379·735f·626c·6f63·6b65·645f··....Sys_blocked_ | ||
| 00001290:· | 00001290:·696f·5b08·0000·2000·0460·4090·4040·4105··io[...·..`@.@@A. | ||
| 000012a0:·0 | 000012a0:·011b·a0b0·a004·5105·011e·04e9·0501·1e40··......Q........@ | ||
| 000012b0:· | 000012b0:·9204·07a3·2953·7973·5f65·7272·6f72·5708··....)Sys_errorW. | ||
| 000012c0:·0 | 000012c0:·0000·2000·0468·4090·a004·6440·4041·0501··..·..h@...d@@A.. | ||
| 000012d0:· | 000012d0:·24a0·b0a0·045a·0501·2704·f205·0127·4092··$....Z..'....'@. | ||
| 000012e0:· | 000012e0:·0408·a33a·556e·6465·6669·6e65·645f·7265··...:Undefined_re | ||
| 000012f0:·6 | 000012f0:·6375·7273·6976·655f·6d6f·6475·6c65·5d08··cursive_module]. | ||
| 00001300:·0 | 00001300:·0000·2000·0471·4090·a0c0·92a0·0470·a004··..·..q@......p.. | ||
| 00001310:· | 00001310:·6ca0·046d·4002·05f5·e100·4063·4040·4105··l..m@.....@c@@A. | ||
| 00001320:·0 | 00001320:·0132·a0b0·a004·6805·0135·0501·0005·0135··.2....h..5.....5 | ||
| 00001330:· | 00001330:·4092·040d·a325·6279·7465·7343·0800·0038··@....%bytesC...8 | ||
| 00001340:· | 00001340:·0040·4040·4140·4040·4040·0501·3940·4040··.@@@A@@@@@..9@@@ | ||
| 00001350:· | 00001350:·9204·0390·9226·5374·646c·6962·9223·496e··.....&Stdlib.#In | ||
| 00001360:· | 00001360:·7440·4105·0134·a0b0·9eb0·0501·4590·a0a0··t@A..4......E... | ||
| 00001370:· | 00001370:·a0c0·91b2·3220·7b35·2043·6f6e·7374·7275··....2·{5·Constru | ||
| 00001380:· | 00001380:·6374·6f72·737d·20b0·c005·0140·4d01·02ba··ctors}·....@M... | ||
| 00001390:·b | 00001390:·0102·bac0·0501·414d·0102·ba01·02d1·4040··......AM......@@ | ||
| 000013a0:· | 000013a0:·0403·4040·4004·0340·0403·0501·3f04·03a0··..@@@..@....?... | ||
| 000013b0:· | 000013b0:·b0a1·41a0·0800·0028·00b1·2174·0101·0b41··..A....(..!t...A | ||
| 000013c0:· | 000013c0:·a004·02b0·c005·014b·4f01·02d3·0102·dcc0··.......KO....... | ||
| 000013d0:·4 | 000013d0:·0501·4c4f·0102·d301·02dd·40a0·a0d0·9021··..LO......@....! | ||
| 000013e0:· | 000013e0:·61c0·9090·0404·0205·f5e1·0040·0105·a608··a..........@.... | ||
| 000013f0:·0 | 000013f0:·0000·3000·0501·5205·0152·0501·5205·0152··..0...R..R..R..R | ||
| 00001400:·0501·5 | 00001400:·0501·5205·0152·0501·5205·0152·40b1·0501··..R..R..R..R@... | ||
| 00001410:· | 00001410:·5204·1008·0000·3800·a0c0·9040·0205·f5e1··R.....8....@.... | ||
| 00001420:·00 | 00001420:·0040·0105·a540·4140·4040·a047·40a0·4240··.@...@A@@@.G@.B@ | ||
| 00001430:·4040· | 00001430:·4040·b0c0·0501·5e4f·0102·d301·02d3·c005··@@....^O........ | ||
| 00001440:· | 00001440:·015f·5201·0302·0103·1940·4040·40a1·0501··._R......@@@@... | ||
| 00001450:· | 00001450:·7640·4041·b0c0·0501·624f·0102·d301·02d9··v@@A....bO...... | ||
| 00001460:·016 | 00001460:·c005·0163·4f01·02d3·0102·db40·40a0·4041··...cO......@@.@A | ||
| Max diff block lines reached; 17371/136353 bytes (12.74%) of diff not shown. | |||
| Offset 1, 9 lines modified | Offset 1, 9 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
| 00000010:·0000·a35 | 00000010:·0000·a358·0000·1c27·0000·7590·0000·7248··...X...'..u...rH | ||
| 00000020:·0800·003c·002b·5365·676d·656e·7474·7265··...<.+Segmenttre | 00000020:·0800·003c·002b·5365·676d·656e·7474·7265··...<.+Segmenttre | ||
| 00000030:·6591·b0a0·b09d·b0a0·2a6f·6361·6d6c·2e74··e.......*ocaml.t | 00000030:·6591·b0a0·b09d·b0a0·2a6f·6361·6d6c·2e74··e.......*ocaml.t | ||
| 00000040:·6578·74b0·c026·5f6e·6f6e·655f·4040·00ff··ext..&_none_@@.. | 00000040:·6578·74b0·c026·5f6e·6f6e·655f·4040·00ff··ext..&_none_@@.. | ||
| 00000050:·0402·4190·a0a0·a0c0·91b2·09ba·2054·6869··..A.........·Thi | 00000050:·0402·4190·a0a0·a0c0·91b2·09ba·2054·6869··..A.........·Thi | ||
| 00000060:·7320·6d6f·6475·6c65·2069·7320·6120·7665··s·module·is·a·ve | 00000060:·7320·6d6f·6475·6c65·2069·7320·6120·7665··s·module·is·a·ve | ||
| 00000070:·7279·2073·696d·706c·6520·696d·706c·656d··ry·simple·implem | 00000070:·7279·2073·696d·706c·6520·696d·706c·656d··ry·simple·implem | ||
| 00000080:·656e·7461·7469·6f6e·206f·6620·2273·6567··entation·of·"seg | 00000080:·656e·7461·7469·6f6e·206f·6620·2273·6567··entation·of·"seg |
| Offset 1, 9 lines modified | Offset 1, 9 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
| 00000010:·0001·8b7 | 00000010:·0001·8b7c·0000·4b75·0001·2d7e·0001·2a41··...|..Ku..-~..*A | ||
| 00000020:·0800·003c·0028·5465·726d·696e·616c·91b0··...<.(Terminal.. | 00000020:·0800·003c·0028·5465·726d·696e·616c·91b0··...<.(Terminal.. | ||
| 00000030:·a0b0·a341·a008·0000·2800·b125·636f·6c6f··...A....(..%colo | 00000030:·a0b0·a341·a008·0000·2800·b125·636f·6c6f··...A....(..%colo | ||
| 00000040:·7201·010b·41a0·0402·b0c0·3063·6c69·622f··r...A.....0clib/ | 00000040:·7201·010b·41a0·0402·b0c0·3063·6c69·622f··r...A.....0clib/ | ||
| 00000050:·7465·726d·696e·616c·2e6d·6c4b·0102·a401··terminal.mlK.... | 00000050:·7465·726d·696e·616c·2e6d·6c4b·0102·a401··terminal.mlK.... | ||
| 00000060:·02a9·c004·024b·0102·a401·02ae·4040·0800··.....K......@@.. | 00000060:·02a9·c004·024b·0102·a401·02ae·4040·0800··.....K......@@.. | ||
| 00000070:·0038·0040·4040·4190·c098·d0a0·a02a·4c49··.8.@@@A......*LI | 00000070:·0038·0040·4040·4190·c098·d0a0·a02a·4c49··.8.@@@A......*LI | ||
| 00000080:·4748·545f·4359·414e·9040·a0a0·2549·4e44··GHT_CYAN.@..%IND | 00000080:·4748·545f·4359·414e·9040·a0a0·2549·4e44··GHT_CYAN.@..%IND |
| Offset 1, 2906 lines modified | Offset 1, 2906 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
| 00000010:·0000·b57 | 00000010:·0000·b579·0000·2128·0000·866f·0000·84e4··...y..!(...o.... | ||
| 00000020:·0800·003c·0024·5472·6965·91b0·a0b0·98d0··...<.$Trie...... | 00000020:·0800·003c·0024·5472·6965·91b0·a0b0·98d0··...<.$Trie...... | ||
| 00000030:·b121·5301·0115·44a0·0402·b0c0·2c63·6c69··.!S...D.....,cli | 00000030:·b121·5301·0115·44a0·0402·b0c0·2c63·6c69··.!S...D.....,cli | ||
| 00000040:·622f·7472·6965·2e6d·6c4b·0102·a401·02b0··b/trie.mlK...... | 00000040:·622f·7472·6965·2e6d·6c4b·0102·a401·02b0··b/trie.mlK...... | ||
| 00000050:·c004·024b·0102·a401·02b1·4090·d091·b0a0··...K......@..... | 00000050:·c004·024b·0102·a401·02b1·4090·d091·b0a0··...K......@..... | ||
| 00000060:·b0a1·41a0·0800·0028·00b1·256c·6162·656c··..A....(..%label | 00000060:·b0a1·41a0·0800·0028·00b1·256c·6162·656c··..A....(..%label | ||
| 00000070:·0101·0b41·a004·02b0·c004·104d·0102·b801··...A.......M.... | 00000070:·0101·0b41·a004·02b0·c004·104d·0102·b801··...A.......M.... | ||
| 00000080:·02bf·c004·114d·0102·b801·02c4·4040·0800··.....M......@@.. | 00000080:·02bf·c004·114d·0102·b801·02c4·4040·0800··.....M......@@.. | ||
| 00000090:·0038·0040·4040·4140·4040·4040·b0c0·0414··.8.@@@A@@@@@.... | 00000090:·0038·0040·4040·4140·4040·4040·b0c0·0414··.8.@@@A@@@@@.... | ||
| 000000a0:·4d01·02b8·0102·ba04·0440·4040·40a1·0421··M........@@@@..! | 000000a0:·4d01·02b8·0102·ba04·0440·4040·40a1·0421··M........@@@@..! | ||
| 000000b0:·4040·4041·4004·0340·4008·0000·3000·a040··@@@A@..@@...0..@ | 000000b0:·4040·4041·4004·0340·4008·0000·3000·a040··@@@A@..@@...0..@ | Diff chunk too large, falling back to line-by-line diff (2763 lines added, 2763 lines removed) | |
| 000000c0:·4004·0104·0104·0104·0104·0104·0104·0140··@..............@ | 000000c0:·4004·0104·0104·0104·0104·0104·0104·0140··@..............@ | ||
| 000000d0:·aba | 000000d0:·abab·a7b1·b2b2·b2b2·b2b2·b2b2·b2b2·b2b2··................ | ||
| 000000e0:·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1 | 000000e0:·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1··................ | ||
| 000000f0:·a325 | 000000f0:·40a3·2561·7272·6179·4808·0000·3800·a0c0··@.%arrayH...8... | ||
| 00000100:·4002 | 00000100:·9040·0205·f5e1·0040·4d40·4140·4140·a000··.@.....@M@A@A@.. | ||
| 00000110:· | 00000110:·7f40·a040·4040·40b0·c026·5f6e·6f6e·655f··.@.@@@@..&_none_ | ||
| 00000120:·4000 | 00000120:·4040·00ff·0402·4140·4040·9204·0ba3·2462··@@....A@@@....$b | ||
| 00000130:·6f6 | 00000130:·6f6f·6c45·0800·0038·0040·40a1·a0e0·a325··oolE...8.@@....% | ||
| 00000140:·616c | 00000140:·6661·6c73·655e·9040·4004·0d40·9204·03a0··false^.@@..@.... | ||
| 00000150:·a324 | 00000150:·e0a3·2474·7275·655f·9040·4004·1340·9204··..$true_.@@..@.. | ||
| 00000160:· | 00000160:·0340·4041·4040·4040·4004·1440·4140·9204··.@@A@@@@@..@A@.. | ||
| 00000170:·a324 | 00000170:·10a3·2463·6861·7242·0800·0038·0040·4040··..$charB...8.@@@ | ||
| 00000180:·4 | 00000180:·4140·4040·4040·0418·4041·4092·0403·a323··A@@@@@..@A@....# | ||
| 00000190:·786e | 00000190:·6578·6e47·0800·0038·0040·4041·4140·4040··exnG...8.@@AA@@@ | ||
| 000001a0:·400 | 000001a0:·4040·041c·4040·4092·0403·a335·6578·7465··@@..@@@....5exte | ||
| 000001b0:·7369 | 000001b0:·6e73·696f·6e5f·636f·6e73·7472·7563·746f··nsion_constructo | ||
| 000001c0:·5008 | 000001c0:·7250·0800·0038·0040·4040·4140·4040·4040··rP...8.@@@A@@@@@ | ||
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| Max diff block lines reached; 17230/399581 bytes (4.31%) of diff not shown. | |||
| Offset 1, 13201 lines modified | Offset 1, 13202 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
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| 00000030:·b09d·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | 00000030:·b09d·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | ||
| 00000040:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | 00000040:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | ||
| 00000050:·a0a0·a0c0·91b2·3320·556e·6963·6f64·6520··......3·Unicode· | 00000050:·a0a0·a0c0·91b2·3320·556e·6963·6f64·6520··......3·Unicode· | ||
| 00000060:·7574·696c·6974·6965·7320·b0c0·2f63·6c69··utilities·../cli | 00000060:·7574·696c·6974·6965·7320·b0c0·2f63·6c69··utilities·../cli | ||
| 00000070:·622f·756e·6963·6f64·652e·6d6c·4b01·02a4··b/unicode.mlK... | 00000070:·622f·756e·6963·6f64·652e·6d6c·4b01·02a4··b/unicode.mlK... | ||
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| 000002d0:·4040 | 000002d0:·8e40·4040·9204·1ba3·296e·6174·6976·6569··.@@@....)nativei | ||
| 000002e0:·744b | 000002e0:·6e74·4b08·0000·3800·4040·4041·4040·4040··ntK...8.@@@A@@@@ | ||
| 000002f0:·049 | 000002f0:·4004·9240·4040·9204·03a3·266f·7074·696f··@..@@@....&optio | ||
| 00000300:·4a08 | 00000300:·6e4a·0800·0038·00a0·c004·6202·05f5·e100··nJ...8....b..... | ||
| 00000310:·5340 | 00000310:·4053·4041·a1a0·e0a3·244e·6f6e·6563·9040··@S@A....$Nonec.@ | ||
| 00000320:·04 | 00000320:·4004·9e40·9204·03a0·e0a3·2453·6f6d·6564··@..@......$Somed | ||
| 00000330:·a004 | 00000330:·90a0·040e·4040·04a5·4092·0404·4040·4140··....@@..@...@@A@ | ||
| 00000340:·5940· | 00000340:·a059·40a0·4040·4040·04a8·4040·4092·0415··.Y@.@@@@..@@@... | ||
| 00000350:·2673 | 00000350:·a326·7374·7269·6e67·4f08·0000·3800·4040··.&stringO...8.@@ | ||
| 00000360:·4 | 00000360:·4041·4040·4040·4004·ac40·4040·9204·03a3··@A@@@@@..@@@.... | ||
| 00000370:·756e | 00000370:·2475·6e69·7446·0800·0038·0040·40a1·a0e0··$unitF...8.@@... | ||
| 00000380:·2228 | 00000380:·a322·2829·6090·4040·04b6·4092·0403·4040··."()`.@@..@...@@ | ||
| 00000390:·4 | 00000390:·4140·4040·4040·04b7·4041·4092·040a·a32e··A@@@@@..@A@..... | ||
| 000003a0:·7373 | 000003a0:·4173·7365·7274·5f66·6169·6c75·7265·5c08··Assert_failure\. | ||
| 000003b0:·00 | 000003b0:·0000·2000·9004·6d40·90a0·c092·a0c0·b390··..·...m@........ | ||
| 000003c0:·1b40 | 000003c0:·041b·4090·4002·05f5·e100·404a·a0c0·b390··..@.@.....@J.... | ||
| 000003d0:·6a40 | 000003d0:·046a·4090·4002·05f5·e100·4040·a004·0540··.j@.@.....@@...@ | ||
| 000003e0:·05f5 | 000003e0:·0205·f5e1·0040·5640·4041·04cb·a0b0·a03d··.....@V@@A.....= | ||
| 000003f0:·6361 | 000003f0:·6f63·616d·6c2e·7761·726e·5f6f·6e5f·6c69··ocaml.warn_on_li | ||
| 00000400:·6572 | 00000400:·7465·7261·6c5f·7061·7474·6572·6e04·cf04··teral_pattern... | ||
| 00000410:·04 | 00000410:·9904·cf40·9204·17a3·3044·6976·6973·696f··...@....0Divisio | ||
| 00000420:·5f62 | 00000420:·6e5f·6279·5f7a·6572·6f59·0800·0020·0004··n_by_zeroY...·.. | ||
| 00000430:·4090 | 00000430:·1840·9040·4041·04d4·a0b0·a004·0904·d704··.@.@@A.......... | ||
| 00000440:·04 | 00000440:·a104·d740·9204·07a3·2b45·6e64·5f6f·665f··...@....+End_of_ | ||
| 00000450:·6 | 00000450:·6669·6c65·5808·0000·2000·0420·4090·4040··fileX...·..·@.@@ | ||
| 00000460:·04 | 00000460:·4104·dca0·b0a0·0411·04df·04a9·04df·4092··A.............@. | ||
| 00000470:·07a3 | 00000470:·0407·a327·4661·696c·7572·6555·0800·0020··...'FailureU...· | ||
| 00000480:·0428 | 00000480:·0004·2840·90a0·0424·4040·4104·e5a0·b0a0··..(@...$@@A..... | ||
| 00000490:·1a04 | 00000490:·041a·04e8·04b2·04e8·4092·0408·a330·496e··........@....0In | ||
| 000004a0:·616c | 000004a0:·7661·6c69·645f·6172·6775·6d65·6e74·5408··valid_argumentT. | ||
| 000004b0:·00 | 000004b0:·0000·2000·0431·4090·a004·2d40·4041·04ee··..·..1@...-@@A.. | ||
| 000004c0:·b0a0 | 000004c0:·a0b0·a004·2304·f104·bb04·f140·9204·08a3··....#......@.... | ||
| 000004d0:·4d61 | 000004d0:·2d4d·6174·6368·5f66·6169·6c75·7265·5208··-Match_failureR. | ||
| 000004e0:·00 | 000004e0:·0000·2000·043a·4090·a0c0·92a0·0439·a004··..·..:@......9.. | ||
| 000004f0:·a004 | 000004f0:·35a0·0436·4002·05f5·e100·405c·4040·4104··5..6@.....@\@@A. | ||
| 00000500:·a0b0 | 00000500:·fca0·b0a0·0431·04ff·04c9·04ff·4092·040d··.....1......@... | ||
| 00000510:·294e | 00000510:·a329·4e6f·745f·666f·756e·6456·0800·0020··.)Not_foundV...· | ||
| 00000520:·0448 | 00000520:·0004·4840·9040·4041·0501·04a0·b0a0·0439··..H@.@@A.......9 | ||
| 00000530:·0 | 00000530:·0501·0704·d105·0107·4092·0407·a32d·4f75··........@....-Ou | ||
| 00000540:·5f6f | 00000540:·745f·6f66·5f6d·656d·6f72·7953·0800·0020··t_of_memoryS...· | ||
| 00000550:·0450 | 00000550:·0004·5040·9040·4041·0501·0ca0·b0a0·0441··..P@.@@A.......A | ||
| 00000560:·0 | 00000560:·0501·0f04·d905·010f·4092·0407·a32e·5374··........@.....St | ||
| 00000570:·636b | 00000570:·6163·6b5f·6f76·6572·666c·6f77·5a08·0000··ack_overflowZ... | ||
| 00000580:·000 | 00000580:·2000·0458·4090·4040·4105·0114·a0b0·a004···..X@.@@A....... | ||
| 00000590:·0501 | 00000590:·4905·0117·04e1·0501·1740·9204·07a3·2e53··I........@.....S | ||
| 000005a0:·735f | 000005a0:·7973·5f62·6c6f·636b·6564·5f69·6f5b·0800··ys_blocked_io[.. | ||
| 000005b0:·20 | 000005b0:·0020·0004·6040·9040·4041·0501·1ca0·b0a0··.·..`@.@@A...... | ||
| 000005c0:·5105 | 000005c0:·0451·0501·1f04·e905·011f·4092·0407·a329··.Q........@....) | ||
| 000005d0:·7973 | 000005d0:·5379·735f·6572·726f·7257·0800·0020·0004··Sys_errorW...·.. | ||
| 000005e0:·4090 | 000005e0:·6840·90a0·0464·4040·4105·0125·a0b0·a004··h@...d@@A..%.... | ||
| 000005f0:·0501 | 000005f0:·5a05·0128·04f2·0501·2840·9204·08a3·3a55··Z..(....(@....:U | ||
| 00000600:·6 | 00000600:·6e64·6566·696e·6564·5f72·6563·7572·7369··ndefined_recursi | ||
| 00000610:·655f | 00000610:·7665·5f6d·6f64·756c·655d·0800·0020·0004··ve_module]...·.. | ||
| 00000620:·4090 | 00000620:·7140·90a0·c092·a004·70a0·046c·a004·6d40··q@......p..l..m@ | ||
| 00000630:·05f5 | 00000630:·0205·f5e1·0040·6340·4041·0501·33a0·b0a0··.....@c@@A..3... | ||
| 00000640:·6805 | 00000640:·0468·0501·3605·0100·0501·3640·9204·0da3··.h..6.....6@.... | ||
| 00000650:·6279 | 00000650:·2562·7974·6573·4308·0000·3800·4040·4041··%bytesC...8.@@@A | ||
| 00000660:·4040·4040·0501 | 00000660:·4040·4040·4005·013a·4040·4092·0403·9092··@@@@@..:@@@..... | ||
| 00000670:·5374 | 00000670:·2653·7464·6c69·6292·2349·6e74·9226·4f70··&Stdlib.#Int.&Op | ||
| 00000680:·a0b0·a341·a008·0000·2800 | 00000680:·7469·6f6e·4040·a0b0·a341·a008·0000·2800··tion@@...A....(. | ||
| 00000690:·7573·0101·0b41·a004·02b | 00000690:·b126·7374·6174·7573·0101·0b41·a004·02b0··.&status...A.... | ||
| 000006a0:· | 000006a0:·c005·013f·4d01·02be·0102·c3c0·0501·404d··...?M.........@M | ||
| 000006b0:· | 000006b0:·0102·be01·02c9·4040·0800·0038·0040·40a1··......@@...8.@@. | ||
| 000006c0:·7474·6572·0101·0c90·404 | 000006c0:·a0e0·a026·4c65·7474·6572·0101·0c90·4040··...&Letter....@@ | ||
| 000006d0:·0 | 000006d0:·b0c0·0501·494d·0102·be01·02cc·c005·014a··....IM.........J | ||
| 000006e0:· | 000006e0:·4d01·02be·0102·d240·40a1·0501·6141·a0e0··M......@@...aA.. | ||
| 000006f0:·7450·6172·7401·010d·9 | 000006f0:·a029·4964·656e·7450·6172·7401·010d·9040··.)IdentPart....@ | ||
| 00000700:·4 | 00000700:·40b0·c005·0152·4d01·02be·0102·d3c0·0501··@....RM......... | ||
| 00000710:· | 00000710:·534d·0102·be01·02de·4040·a105·016a·42a0··SM......@@...jB. | ||
| 00000720:·626f·6c01·010e·9040·40b | 00000720:·e0a0·2653·796d·626f·6c01·010e·9040·40b0··..&Symbol....@@. | ||
| 00000730:· | 00000730:·c005·015b·4d01·02be·0102·dfc0·0501·5c4d··...[M.........\M | ||
| 00000740:· | 00000740:·0102·be01·02e7·4040·a105·0173·43a0·e0a0··......@@...sC... | ||
| 00000750:·5365·7001·010f·9040·40b0· | 00000750:·2849·6465·6e74·5365·7001·010f·9040·40b0··(IdentSep....@@. | ||
| 00000760:· | 00000760:·c005·0164·4d01·02be·0102·e8c0·0501·654d··...dM.........eM | ||
| 00000770:· | 00000770:·0102·be01·02f2·4040·a105·017c·44a0·e0a0··......@@...|D... | ||
| 00000780:·6 | 00000780:·2953·6570·6172·6174·6f72·0101·1090·4040··)Separator....@@ | ||
| 00000790:·0 | 00000790:·b0c0·0501·6d4d·0102·be01·02f3·c005·016e··....mM.........n | ||
| 000007a0:·fe40·40a1·0501·8 | 000007a0:·4d01·02be·0102·fe40·40a1·0501·8545·a0e0··M......@@....E.. | ||
| 000007b0:·726f·6c01·0111·9040·40b | 000007b0:·a027·436f·6e74·726f·6c01·0111·9040·40b0··.'Control....@@. | ||
| 000007c0:· | 000007c0:·c005·0176·4d01·02be·0102·ffc0·0501·774d··...vM.........wM | ||
| 000007d0:· | 000007d0:·0102·be01·0308·4040·a105·018e·46a0·e0a0··......@@....F... | ||
| Max diff block lines reached; 72438/1820439 bytes (3.98%) of diff not shown. | |||
| Offset 1, 153202 lines modified | Offset 1, 153203 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
| 00000010:·0025·6 | 00000010:·0025·6702·0005·d1e0·0016·a11c·0016·99bf··.%g............. | ||
| 00000020:·0800·003c·002c·556e·6963·6f64·6574·6162··...<.,Unicodetab | 00000020:·0800·003c·002c·556e·6963·6f64·6574·6162··...<.,Unicodetab | ||
| 00000030:·6c65·91b0·a0b0·9db0·a02a·6f63·616d·6c2e··le.......*ocaml. | 00000030:·6c65·91b0·a0b0·9db0·a02a·6f63·616d·6c2e··le.......*ocaml. | ||
| 00000040:·7465·7874·b0c0·265f·6e6f·6e65·5f40·4000··text..&_none_@@. | 00000040:·7465·7874·b0c0·265f·6e6f·6e65·5f40·4000··text..&_none_@@. | ||
| 00000050:·ff04·0241·90a0·a0a0·c091·b209·2620·556e··...A........&·Un | 00000050:·ff04·0241·90a0·a0a0·c091·b209·2620·556e··...A........&·Un | ||
| 00000060:·6963·6f64·6520·7461·626c·6573·2067·656e··icode·tables·gen | 00000060:·6963·6f64·6520·7461·626c·6573·2067·656e··icode·tables·gen | ||
| 00000070:·6572·6174·6564·2075·7369·6e67·2055·5543··erated·using·UUC | 00000070:·6572·6174·6564·2075·7369·6e67·2055·5543··erated·using·UUC | ||
| 00000080:·442e·20b0·c034·636c·6962·2f75·6e69·636f··D.·..4clib/unico | 00000080:·442e·20b0·c034·636c·6962·2f75·6e69·636f··D.·..4clib/unico | ||
| 00000090:·6465·7461·626c·652e·6d6c·4140·40c0·0402··detable.mlA@@... | 00000090:·6465·7461·626c·652e·6d6c·4140·40c0·0402··detable.mlA@@... | ||
| 000000a0:·4140·6b40·4004·0440·4040·0404·4004·0404··A@k@@..@@@..@... | 000000a0:·4140·6b40·4004·0440·4040·0404·4004·0404··A@k@@..@@@..@... | Diff chunk too large, falling back to line-by-line diff (153191 lines added, 153190 lines removed) | |
| 000000b0:·0408·0000·3000·a040·4004·0104·0104·0104··....0..@@....... | 000000b0:·0408·0000·3000·a040·4004·0104·0104·0104··....0..@@....... | ||
| 000000c0:·0104·0104·0104·0140·aba | 000000c0:·0104·0104·0104·0140·abab·a7b1·b2b2·b2b2··.......@........ | ||
| 000000d0:·b2b2·b2b2·b2b2·b2b | 000000d0:·b2b2·b2b2·b2b2·b2b2·b1b1·b1b1·b1b1·b1b1··................ | ||
| 000000e0:·b1b1·b1b1·b1b1·b140 | 000000e0:·b1b1·b1b1·b1b1·b1b1·40a3·2561·7272·6179··........@.%array | ||
| 000000f0:·0800 | 000000f0:·4808·0000·3800·a0c0·9040·0205·f5e1·0040··H...8....@.....@ | ||
| 00000100:·4 | 00000100:·4d40·4140·4140·a000·7f40·a040·4040·4004··M@A@A@...@.@@@@. | ||
| 00000110:·4040 | 00000110:·3940·4040·9204·08a3·2462·6f6f·6c45·0800··9@@@....$boolE.. | ||
| 00000120:·38 | 00000120:·0038·0040·40a1·a0e0·a325·6661·6c73·655e··.8.@@....%false^ | ||
| 00000130:· | 00000130:·9040·4004·4340·9204·03a0·e0a3·2474·7275··.@@.C@......$tru | ||
| 00000140:·5f90 | 00000140:·655f·9040·4004·4940·9204·0340·4041·4040··e_.@@.I@...@@A@@ | ||
| 00000150:·4040·04 | 00000150:·4040·4004·4a40·4140·9204·10a3·2463·6861··@@@.J@A@....$cha | ||
| 00000160:· | 00000160:·7242·0800·0038·0040·4040·4140·4040·4040··rB...8.@@@A@@@@@ | ||
| 00000170:·4 | 00000170:·044e·4041·4092·0403·a323·6578·6e47·0800··.N@A@....#exnG.. | ||
| 00000180:· | 00000180:·0038·0040·4041·4140·4040·4040·0452·4040··.8.@@AA@@@@@.R@@ | ||
| 00000190:·9204 | 00000190:·4092·0403·a335·6578·7465·6e73·696f·6e5f··@....5extension_ | ||
| 000001a0:·6f6e | 000001a0:·636f·6e73·7472·7563·746f·7250·0800·0038··constructorP...8 | ||
| 000001b0:· | 000001b0:·0040·4040·4140·4040·4040·0456·4040·4092··.@@@A@@@@@.V@@@. | ||
| 000001c0:·03a3 | 000001c0:·0403·a325·666c·6f61·7444·0800·0038·0040··...%floatD...8.@ | ||
| 000001d0:·404 | 000001d0:·4040·4140·4040·4040·045a·4040·4092·0403··@@A@@@@@.Z@@@... | ||
| 000001e0:·2a | 000001e0:·a32a·666c·6f61·7461·7272·6179·5108·0000··.*floatarrayQ... | ||
| 000001f0:· | 000001f0:·3800·4040·4041·4040·4040·4004·5e40·4040··8.@@@A@@@@@.^@@@ | ||
| 00000200:·0403 | 00000200:·9204·03a3·2369·6e74·4108·0000·3800·4040··....#intA...8.@@ | ||
| 00000210:·4 | 00000210:·4041·4040·4040·4004·6240·4140·9204·03a3··@A@@@@@.b@A@.... | ||
| 00000220:·696e | 00000220:·2569·6e74·3332·4c08·0000·3800·4040·4041··%int32L...8.@@@A | ||
| 00000230:·4040·4040·046 | 00000230:·4040·4040·4004·6640·4040·9204·03a3·2569··@@@@@.f@@@....%i | ||
| 00000240:·7436 | 00000240:·6e74·3634·4d08·0000·3800·4040·4041·4040··nt64M...8.@@@A@@ | ||
| 00000250:·4040·046 | 00000250:·4040·4004·6a40·4040·9204·03a3·266c·617a··@@@.j@@@....&laz | ||
| 00000260:·5f74 | 00000260:·795f·744e·0800·0038·00a0·c004·3a02·05f5··y_tN...8....:... | ||
| 00000270:·0040 | 00000270:·e100·404f·4041·4041·40a0·5940·a040·4040··..@O@A@A@.Y@.@@@ | ||
| 00000280:·047 | 00000280:·4004·7240·4040·9204·07a3·246c·6973·7449··@.r@@@....$listI | ||
| 00000290:·0000 | 00000290:·0800·0038·00a0·c004·4202·05f5·e100·4050··...8....B.....@P | ||
| 000002a0:·41a1 | 000002a0:·4041·a1a0·e0a3·225b·5d61·9040·4004·7e40··@A...."[]a.@@.~@ | ||
| 000002b0:·0403 | 000002b0:·9204·03a0·e0a3·223a·3a62·90a0·040e·a0c0··......"::b...... | ||
| 000002c0:·9004 | 000002c0:·b390·0416·a004·1340·9040·0205·f5e1·0040··.......@.@.....@ | ||
| 000002d0:·4040 | 000002d0:·5140·4004·8b40·9204·0a40·4041·40a0·5940··Q@@..@...@@A@.Y@ | ||
| 000002e0:· | 000002e0:·a040·4040·4004·8e40·4040·9204·1ba3·296e··.@@@@..@@@....)n | ||
| 000002f0:·7469 | 000002f0:·6174·6976·6569·6e74·4b08·0000·3800·4040··ativeintK...8.@@ | ||
| 00000300:·4 | 00000300:·4041·4040·4040·4004·9240·4040·9204·03a3··@A@@@@@..@@@.... | ||
| 00000310:·6f70 | 00000310:·266f·7074·696f·6e4a·0800·0038·00a0·c004··&optionJ...8.... | ||
| 00000320:·0205 | 00000320:·6202·05f5·e100·4053·4041·a1a0·e0a3·244e··b.....@S@A....$N | ||
| 00000330:·6e65 | 00000330:·6f6e·6563·9040·4004·9e40·9204·03a0·e0a3··onec.@@..@...... | ||
| 00000340:·536f | 00000340:·2453·6f6d·6564·90a0·040e·4040·04a5·4092··$Somed....@@..@. | ||
| 00000350:·04 | 00000350:·0404·4040·4140·a059·40a0·4040·4040·04a8··..@@A@.Y@.@@@@.. | ||
| 00000360:·4040·9204 | 00000360:·4040·4092·0415·a326·7374·7269·6e67·4f08··@@@....&stringO. | ||
| 00000370:·00 | 00000370:·0000·3800·4040·4041·4040·4040·4004·ac40··..8.@@@A@@@@@..@ | ||
| 00000380:·4092 | 00000380:·4040·9204·03a3·2475·6e69·7446·0800·0038··@@....$unitF...8 | ||
| 00000390:· | 00000390:·0040·40a1·a0e0·a322·2829·6090·4040·04b6··.@@...."()`.@@.. | ||
| 000003a0:·9204 | 000003a0:·4092·0403·4040·4140·4040·4040·04b7·4041··@...@@A@@@@@..@A | ||
| 000003b0:·9204 | 000003b0:·4092·040a·a32e·4173·7365·7274·5f66·6169··@.....Assert_fai | ||
| 000003c0:·7572 | 000003c0:·6c75·7265·5c08·0000·2000·9004·6d40·90a0··lure\...·...m@.. | ||
| 000003d0:·92a0 | 000003d0:·c092·a0c0·b390·041b·4090·4002·05f5·e100··........@.@..... | ||
| 000003e0:·4aa0 | 000003e0:·404a·a0c0·b390·046a·4090·4002·05f5·e100··@J.....j@.@..... | ||
| 000003f0:·40 | 000003f0:·4040·a004·0540·0205·f5e1·0040·5640·4041··@@...@.....@V@@A | ||
| 00000400:·c | 00000400:·04cb·a0b0·a03d·6f63·616d·6c2e·7761·726e··.....=ocaml.warn | ||
| 00000410:·6f6e | 00000410:·5f6f·6e5f·6c69·7465·7261·6c5f·7061·7474··_on_literal_patt | ||
| 00000420:·726e | 00000420:·6572·6e04·cf04·9904·cf40·9204·17a3·3044··ern......@....0D | ||
| 00000430:·7669 | 00000430:·6976·6973·696f·6e5f·6279·5f7a·6572·6f59··ivision_by_zeroY | ||
| 00000440:·0000 | 00000440:·0800·0020·0004·1840·9040·4041·04d4·a0b0··...·...@.@@A.... | ||
| 00000450:·0409 | 00000450:·a004·0904·d704·a104·d740·9204·07a3·2b45··.........@....+E | ||
| 00000460:·645f | 00000460:·6e64·5f6f·665f·6669·6c65·5808·0000·2000··nd_of_fileX...·. | ||
| 00000470:· | 00000470:·0420·4090·4040·4104·dca0·b0a0·0411·04df··.·@.@@A......... | ||
| 00000480:·a904 | 00000480:·04a9·04df·4092·0407·a327·4661·696c·7572··....@....'Failur | ||
| 00000490:·5508 | 00000490:·6555·0800·0020·0004·2840·90a0·0424·4040··eU...·..(@...$@@ | ||
| 000004a0:·04 | 000004a0:·4104·e5a0·b0a0·041a·04e8·04b2·04e8·4092··A.............@. | ||
| 000004b0:·08a3 | 000004b0:·0408·a330·496e·7661·6c69·645f·6172·6775··...0Invalid_argu | ||
| 000004c0:·656e | 000004c0:·6d65·6e74·5408·0000·2000·0431·4090·a004··mentT...·..1@... | ||
| 000004d0:·4040 | 000004d0:·2d40·4041·04ee·a0b0·a004·2304·f104·bb04··-@@A......#..... | ||
| 000004e0:·4092 | 000004e0:·f140·9204·08a3·2d4d·6174·6368·5f66·6169··.@....-Match_fai | ||
| 000004f0:·7572 | 000004f0:·6c75·7265·5208·0000·2000·043a·4090·a0c0··lureR...·..:@... | ||
| 00000500:·a004 | 00000500:·92a0·0439·a004·35a0·0436·4002·05f5·e100··...9..5..6@..... | ||
| 00000510:·5c40 | 00000510:·405c·4040·4104·fca0·b0a0·0431·04ff·04c9··@\@@A......1.... | ||
| 00000520:·f | 00000520:·04ff·4092·040d·a329·4e6f·745f·666f·756e··..@....)Not_foun | ||
| 00000530:·5608 | 00000530:·6456·0800·0020·0004·4840·9040·4041·0501··dV...·..H@.@@A.. | ||
| 00000540:·a0b0 | 00000540:·04a0·b0a0·0439·0501·0704·d105·0107·4092··.....9........@. | ||
| 00000550:·07a3 | 00000550:·0407·a32d·4f75·745f·6f66·5f6d·656d·6f72··...-Out_of_memor | ||
| 00000560:·5308·00 | 00000560:·7953·0800·0020·0004·5040·9040·4041·0501··yS...·..P@.@@A.. | ||
| 00000570:·a0b0 | 00000570:·0ca0·b0a0·0441·0501·0f04·d905·010f·4092··.....A........@. | ||
| 00000580:·07a3 | 00000580:·0407·a32e·5374·6163·6b5f·6f76·6572·666c··....Stack_overfl | ||
| 00000590:·775a | 00000590:·6f77·5a08·0000·2000·0458·4090·4040·4105··owZ...·..X@.@@A. | ||
| 000005a0:·1 | 000005a0:·0114·a0b0·a004·4905·0117·04e1·0501·1740··......I........@ | ||
| 000005b0:·0407 | 000005b0:·9204·07a3·2e53·7973·5f62·6c6f·636b·6564··.....Sys_blocked | ||
| 000005c0:·696f | 000005c0:·5f69·6f5b·0800·0020·0004·6040·9040·4041··_io[...·..`@.@@A | ||
| 000005d0:·01 | 000005d0:·0501·1ca0·b0a0·0451·0501·1f04·e905·011f··.......Q........ | ||
| 000005e0:·9204 | 000005e0:·4092·0407·a329·5379·735f·6572·726f·7257··@....)Sys_errorW | ||
| 000005f0:·0000 | 000005f0:·0800·0020·0004·6840·90a0·0464·4040·4105··...·..h@...d@@A. | ||
| 00000600:·2 | 00000600:·0125·a0b0·a004·5a05·0128·04f2·0501·2840··.%....Z..(....(@ | ||
| 00000610:·0408 | 00000610:·9204·08a3·3a55·6e64·6566·696e·6564·5f72··....:Undefined_r | ||
| 00000620:·6375 | 00000620:·6563·7572·7369·7665·5f6d·6f64·756c·655d··ecursive_module] | ||
| 00000630:·0000 | 00000630:·0800·0020·0004·7140·90a0·c092·a004·70a0··...·..q@......p. | ||
| 00000640:·6ca0 | 00000640:·046c·a004·6d40·0205·f5e1·0040·6340·4041··.l..m@.....@c@@A | ||
| 00000650:·01 | 00000650:·0501·33a0·b0a0·0468·0501·3605·0100·0501··..3....h..6..... | ||
| 00000660:·4092 | 00000660:·3640·9204·0da3·2562·7974·6573·4308·0000··6@....%bytesC... | ||
| 00000670:· | 00000670:·3800·4040·4041·4040·4040·4005·013a·4040··8.@@@A@@@@@..:@@ | ||
| 00000680:·9204 | 00000680:·4092·0403·9092·2653·7464·6c69·6292·2349··@.....&Stdlib.#I | ||
| 00000690:·74 | 00000690:·6e74·9226·4f70·7469·6f6e·4040·a0b0·a140··nt.&Option@@...@ | ||
| 000006a0:· | 000006a0:·a0c0·e0a0·a022·6c75·0101·0ba0·0402·b0c0··....."lu........ | ||
| 000006b0:·004 | 000006b0:·0501·4144·0045·0049·c005·0142·4400·4500··..AD.E.I...BD.E. | ||
| 000006c0:· | 000006c0:·4b40·b004·0304·0240·40c0·96c0·b390·04e2··K@.....@@....... | ||
| 000006d0:·c096 | 000006d0:·a0c0·96c0·92a0·c096·c0b3·0497·4090·4002··............@.@. | ||
| 000006e0:·05 | 000006e0:·05f5·e100·4001·05ac·4140·0105·a9a0·c096··....@...A@...... | ||
| 000006f0:· | 000006f0:·c0b3·049d·4090·4002·05f5·e100·4001·05ad··....@.@.....@... | ||
| 00000700:· | 00000700:·4140·0105·aa40·0205·f5e1·0040·0105·ab41··A@...@.....@...A | ||
| 00000710:·400 | 00000710:·4001·05a7·4090·4002·05f5·e100·4001·05a6··@...@.@.....@... | ||
| 00000720:· | 00000720:·4140·0105·a505·0158·40e0·b8a0·9022·3a3a··A@.....X@....":: | ||
| 00000730:· | 00000730:·b0c0·0501·6145·0050·0052·c005·0162·0102··....aE.P.R...b.. | ||
| 00000740:· | 00000740:·7701·33a9·0133·aa41·0800·0038·0004·eec0··w.3..3.A...8.... | ||
| 00000750:· | 00000750:·b304·1e04·fc90·4002·05f5·e100·4052·4004··......@.....@R@. | ||
| 00000760:· | 00000760:·ef42·9140·4141·4041·0501·7440·4004·e9a0··.B.@AA@A..t@@... | ||
| 00000770:· | 00000770:·e097·a0e0·9190·0041·b0c0·0501·7045·0050··.......A....pE.P | ||
| 00000780:· | 00000780:·0053·c005·0171·4500·5000·5a40·4004·2105··.S...qE.P.Z@@.!. | ||
| 00000790:· | 00000790:·016f·40a0·e091·9000·5ab0·c005·0177·4500··.o@.....Z....wE. | ||
| 000007a0:· | 000007a0:·5000·5bc0·0501·7845·0050·0062·4040·0422··P.[...xE.P.b@@." | ||
| 000007b0:·0 | 000007b0:·0501·7640·40b0·0419·c005·017a·4500·5000··..v@@......zE.P. | ||
| 000007c0:· | 000007c0:·6340·40c0·9604·3141·4001·05ae·0501·7a40··c@@...1A@.....z@ | ||
| 000007d0:· | 000007d0:·a0e0·b8a0·0423·b0c0·0501·8246·0065·0067··.....#.....F.e.g | ||
| Max diff block lines reached; -1/21141194 bytes (-0.00%) of diff not shown. | |||
| Offset 1, 2575 lines modified | Offset 1, 2575 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | 00000000:·4361·6d6c·3139·3939·5430·3331·8495·a6be··Caml1999T031.... | ||
| 00000010:·0000·a0c | 00000010:·0000·a0cc·0000·1bbf·0000·7403·0000·71df··..........t...q. | ||
| 00000020:·0800·003c·0029·556e·696f·6e66·696e·6491··...<.)Unionfind. | 00000020:·0800·003c·0029·556e·696f·6e66·696e·6491··...<.)Unionfind. | ||
| 00000030:·b0a0·b09d·b0a0·2a6f·6361·6d6c·2e74·6578··......*ocaml.tex | 00000030:·b0a0·b09d·b0a0·2a6f·6361·6d6c·2e74·6578··......*ocaml.tex | ||
| 00000040:·74b0·c026·5f6e·6f6e·655f·4040·00ff·0402··t..&_none_@@.... | 00000040:·74b0·c026·5f6e·6f6e·655f·4040·00ff·0402··t..&_none_@@.... | ||
| 00000050:·4190·a0a0·a0c0·91b2·093b·2041·6e20·696d··A........;·An·im | 00000050:·4190·a0a0·a0c0·91b2·093b·2041·6e20·696d··A........;·An·im | ||
| 00000060:·7065·7261·7469·7665·2069·6d70·6c65·6d65··perative·impleme | 00000060:·7065·7261·7469·7665·2069·6d70·6c65·6d65··perative·impleme | ||
| 00000070:·6e74·6174·696f·6e20·6f66·2070·6172·7469··ntation·of·parti | 00000070:·6e74·6174·696f·6e20·6f66·2070·6172·7469··ntation·of·parti | ||
| 00000080:·7469·6f6e·7320·7669·6120·556e·696f·6e2d··tions·via·Union- | 00000080:·7469·6f6e·7320·7669·6120·556e·696f·6e2d··tions·via·Union- | ||
| 00000090:·4669·6e64·20b0·c031·636c·6962·2f75·6e69··Find·..1clib/uni | 00000090:·4669·6e64·20b0·c031·636c·6962·2f75·6e69··Find·..1clib/uni | ||
| 000000a0:·6f6e·6669·6e64·2e6d·6c4b·0102·a401·02a4··onfind.mlK...... | 000000a0:·6f6e·6669·6e64·2e6d·6c4b·0102·a401·02a4··onfind.mlK...... | ||
| 000000b0:·c004·024b·0102·a401·02e4·4040·0404·4040··...K......@@..@@ | 000000b0:·c004·024b·0102·a401·02e4·4040·0404·4040··...K......@@..@@ | ||
| 000000c0:·4004·0440·0404·0404·0800·0030·00a0·4040··@..@.......0..@@ | 000000c0:·4004·0440·0404·0404·0800·0030·00a0·4040··@..@.......0..@@ | Diff chunk too large, falling back to line-by-line diff (2374 lines added, 2374 lines removed) | |
| 000000d0:·0401·0401·0401·0401·0401·0401·0401·40ab··..............@. | 000000d0:·0401·0401·0401·0401·0401·0401·0401·40ab··..............@. | ||
| 000000e0:·a | 000000e0:·aba7·b1b2·b2b2·b2b2·b2b2·b2b2·b2b2·b2b1··................ | ||
| 000000f0:·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1· | 000000f0:·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b140··...............@ | ||
| 00000100:·2561 | 00000100:·a325·6172·7261·7948·0800·0038·00a0·c090··.%arrayH...8.... | ||
| 00000110:·0205 | 00000110:·4002·05f5·e100·404d·4041·4041·40a0·007f··@.....@M@A@A@... | ||
| 00000120:· | 00000120:·40a0·4040·4040·0439·4040·4092·0408·a324··@.@@@@.9@@@....$ | ||
| 00000130:·6 | 00000130:·626f·6f6c·4508·0000·3800·4040·a1a0·e0a3··boolE...8.@@.... | ||
| 00000140:·6661 | 00000140:·2566·616c·7365·5e90·4040·0443·4092·0403··%false^.@@.C@... | ||
| 00000150:·e0a3 | 00000150:·a0e0·a324·7472·7565·5f90·4040·0449·4092··...$true_.@@.I@. | ||
| 00000160:·0 | 00000160:·0403·4040·4140·4040·4040·044a·4041·4092··..@@A@@@@@.J@A@. | ||
| 00000170:·10a3 | 00000170:·0410·a324·6368·6172·4208·0000·3800·4040··...$charB...8.@@ | ||
| 00000180:·4 | 00000180:·4041·4040·4040·4004·4e40·4140·9204·03a3··@A@@@@@.N@A@.... | ||
| 00000190:·6578 | 00000190:·2365·786e·4708·0000·3800·4040·4141·4040··#exnG...8.@@AA@@ | ||
| 000001a0:·4040·045 | 000001a0:·4040·4004·5240·4040·9204·03a3·3565·7874··@@@.R@@@....5ext | ||
| 000001b0:·6e73 | 000001b0:·656e·7369·6f6e·5f63·6f6e·7374·7275·6374··ension_construct | ||
| 000001c0:·7250 | 000001c0:·6f72·5008·0000·3800·4040·4041·4040·4040··orP...8.@@@A@@@@ | ||
| 000001d0:·04 | 000001d0:·4004·5640·4040·9204·03a3·2566·6c6f·6174··@.V@@@....%float | ||
| 000001e0:· | 000001e0:·4408·0000·3800·4040·4041·4040·4040·4004··D...8.@@@A@@@@@. | ||
| 000001f0:·4040 | 000001f0:·5a40·4040·9204·03a3·2a66·6c6f·6174·6172··Z@@@....*floatar | ||
| 00000200:·6179 | 00000200:·7261·7951·0800·0038·0040·4040·4140·4040··rayQ...8.@@@A@@@ | ||
| 00000210:·40 | 00000210:·4040·045e·4040·4092·0403·a323·696e·7441··@@.^@@@....#intA | ||
| 00000220:·0000 | 00000220:·0800·0038·0040·4040·4140·4040·4040·0462··...8.@@@A@@@@@.b | ||
| 00000230:·4140 | 00000230:·4041·4092·0403·a325·696e·7433·324c·0800··@A@....%int32L.. | ||
| 00000240:· | 00000240:·0038·0040·4040·4140·4040·4040·0466·4040··.8.@@@A@@@@@.f@@ | ||
| 00000250:·9204 | 00000250:·4092·0403·a325·696e·7436·344d·0800·0038··@....%int64M...8 | ||
| 00000260:· | 00000260:·0040·4040·4140·4040·4040·046a·4040·4092··.@@@A@@@@@.j@@@. | ||
| 00000270:·03a3 | 00000270:·0403·a326·6c61·7a79·5f74·4e08·0000·3800··...&lazy_tN...8. | ||
| 00000280:·c004 | 00000280:·a0c0·043a·0205·f5e1·0040·4f40·4140·4140··...:.....@O@A@A@ | ||
| 00000290:·5940· | 00000290:·a059·40a0·4040·4040·0472·4040·4092·0407··.Y@.@@@@.r@@@... | ||
| 000002a0:·246c | 000002a0:·a324·6c69·7374·4908·0000·3800·a0c0·0442··.$listI...8....B | ||
| 000002b0:·05f5 | 000002b0:·0205·f5e1·0040·5040·41a1·a0e0·a322·5b5d··.....@P@A...."[] | ||
| 000002c0:·9040 | 000002c0:·6190·4040·047e·4092·0403·a0e0·a322·3a3a··a.@@.~@......":: | ||
| 000002d0:·90a0 | 000002d0:·6290·a004·0ea0·c0b3·9004·16a0·0413·4090··b.............@. | ||
| 000002e0:·0205 | 000002e0:·4002·05f5·e100·4051·4040·048b·4092·040a··@.....@Q@@..@... | ||
| 000002f0:·404 | 000002f0:·4040·4140·a059·40a0·4040·4040·048e·4040··@@A@.Y@.@@@@..@@ | ||
| 00000300:·9204 | 00000300:·4092·041b·a329·6e61·7469·7665·696e·744b··@....)nativeintK | ||
| 00000310:·0000 | 00000310:·0800·0038·0040·4040·4140·4040·4040·0492··...8.@@@A@@@@@.. | ||
| 00000320:·4040·9204 | 00000320:·4040·4092·0403·a326·6f70·7469·6f6e·4a08··@@@....&optionJ. | ||
| 00000330:·0038 | 00000330:·0000·3800·a0c0·0462·0205·f5e1·0040·5340··..8....b.....@S@ | ||
| 00000340:·a1a0 | 00000340:·41a1·a0e0·a324·4e6f·6e65·6390·4040·049e··A....$Nonec.@@.. | ||
| 00000350:·9204 | 00000350:·4092·0403·a0e0·a324·536f·6d65·6490·a004··@......$Somed... | ||
| 00000360:·4040 | 00000360:·0e40·4004·a540·9204·0440·4041·40a0·5940··.@@..@...@@A@.Y@ | ||
| 00000370:· | 00000370:·a040·4040·4004·a840·4040·9204·15a3·2673··.@@@@..@@@....&s | ||
| 00000380:·7269 | 00000380:·7472·696e·674f·0800·0038·0040·4040·4140··tringO...8.@@@A@ | ||
| 00000390:·4040·40 | 00000390:·4040·4040·04ac·4040·4092·0403·a324·756e··@@@@..@@@....$un | ||
| 000003a0:·7446 | 000003a0:·6974·4608·0000·3800·4040·a1a0·e0a3·2228··itF...8.@@...."( | ||
| 000003b0:·6090 | 000003b0:·2960·9040·4004·b640·9204·0340·4041·4040··)`.@@..@...@@A@@ | ||
| 000003c0:·4040·04 | 000003c0:·4040·4004·b740·4140·9204·0aa3·2e41·7373··@@@..@A@.....Ass | ||
| 000003d0:·7274 | 000003d0:·6572·745f·6661·696c·7572·655c·0800·0020··ert_failure\...· | ||
| 000003e0:·9004 | 000003e0:·0090·046d·4090·a0c0·92a0·c0b3·9004·1b40··...m@..........@ | ||
| 000003f0:·4002 | 000003f0:·9040·0205·f5e1·0040·4aa0·c0b3·9004·6a40··.@.....@J.....j@ | ||
| 00000400:·4002 | 00000400:·9040·0205·f5e1·0040·40a0·0405·4002·05f5··.@.....@@...@... | ||
| 00000410:·0040 | 00000410:·e100·4056·4040·4104·cba0·b0a0·3d6f·6361··..@V@@A.....=oca | ||
| 00000420:·6c2e | 00000420:·6d6c·2e77·6172·6e5f·6f6e·5f6c·6974·6572··ml.warn_on_liter | ||
| 00000430:·6c5f | 00000430:·616c·5f70·6174·7465·726e·04cf·0499·04cf··al_pattern...... | ||
| 00000440:·9204 | 00000440:·4092·0417·a330·4469·7669·7369·6f6e·5f62··@....0Division_b | ||
| 00000450:·5f7a | 00000450:·795f·7a65·726f·5908·0000·2000·0418·4090··y_zeroY...·...@. | ||
| 00000460:·404 | 00000460:·4040·4104·d4a0·b0a0·0409·04d7·04a1·04d7··@@A............. | ||
| 00000470:·9204 | 00000470:·4092·0407·a32b·456e·645f·6f66·5f66·696c··@....+End_of_fil | ||
| 00000480:· | 00000480:·6558·0800·0020·0004·2040·9040·4041·04dc··eX...·..·@.@@A.. | ||
| 00000490:·b0a0 | 00000490:·a0b0·a004·1104·df04·a904·df40·9204·07a3··...........@.... | ||
| 000004a0:·4661 | 000004a0:·2746·6169·6c75·7265·5508·0000·2000·0428··'FailureU...·..( | ||
| 000004b0:·90a0 | 000004b0:·4090·a004·2440·4041·04e5·a0b0·a004·1a04··@...$@@A........ | ||
| 000004c0:·04b2 | 000004c0:·e804·b204·e840·9204·08a3·3049·6e76·616c··.....@....0Inval | ||
| 000004d0:·645f | 000004d0:·6964·5f61·7267·756d·656e·7454·0800·0020··id_argumentT...· | ||
| 000004e0:·0431 | 000004e0:·0004·3140·90a0·042d·4040·4104·eea0·b0a0··..1@...-@@A..... | ||
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| 1 | Package:·libcoq-stdlib | 1 | Package:·libcoq-stdlib |
| 2 | Source:·coq | 2 | Source:·coq |
| 3 | Version:·8.18.0+dfsg-1 | 3 | Version:·8.18.0+dfsg-1 |
| 4 | Architecture:·amd64 | 4 | Architecture:·amd64 |
| 5 | Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org> | 5 | Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org> |
| 6 | Installed-Size:·15641 | 6 | Installed-Size:·156419 |
| 7 | Recommends:·coq | 7 | Recommends:·coq |
| 8 | Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1),·coq-theories | 8 | Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1),·coq-theories |
| 9 | Replaces:·coq-libs·(<<·8.2.pl1),·coq-theories | 9 | Replaces:·coq-libs·(<<·8.2.pl1),·coq-theories |
| 10 | Provides:·coq-theories·(=·8.18.0+dfsg-1),·libcoq-stdlib-w1rn2 | 10 | Provides:·coq-theories·(=·8.18.0+dfsg-1),·libcoq-stdlib-w1rn2 |
| 11 | Section:·math | 11 | Section:·math |
| 12 | Priority:·optional | 12 | Priority:·optional |
| 13 | Homepage:·http://coq.inria.fr/ | 13 | Homepage:·http://coq.inria.fr/ |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab3 | 49 | <a·id="lab334"></a><h1·class="section">DO·NOT·REQUIRE·OR·IMPORT!·This·is·a·temporary·file·preparing·deprecation</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>.<br/> |
| 54 | <br/> | 54 | <br/> |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="doc"> | 56 | <div·class="doc"> |
| 57 | <a·id="lab3 | 57 | <a·id="lab335"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">arith</span></span>·hint·database</h1> |
| 58 | <div·class="paragraph">·</div> | 58 | <div·class="paragraph">·</div> |
| 59 | <a·id="lab3 | 59 | <a·id="lab336"></a><h2·class="section"><span·class="inlinecode"><span·class="id"·title="var">lt</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h2> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | #[<span·class="id"·title="var">global</span>]<br/> | 62 | #[<span·class="id"·title="var">global</span>]<br/> |
| 63 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_trans</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/> | 63 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_trans</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/> |
| 64 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">Nat.le_antisymm</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/> | 64 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">Nat.le_antisymm</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/> |
| 65 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_0_l</span>·<span·class="id"·title="var">Nat.nle_succ_0</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/> | 65 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_0_l</span>·<span·class="id"·title="var">Nat.nle_succ_0</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/> |
| Offset 52, 15 lines modified | Offset 52, 15 lines modified | ||
| 52 | <br/> | 52 | <br/> |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | Equality·on·natural·numbers· | 55 | Equality·on·natural·numbers· |
| 56 | <div·class="paragraph">·</div> | 56 | <div·class="paragraph">·</div> |
| 57 | <a·id="lab | 57 | <a·id="lab278"></a><h1·class="section">Propositional·equality</h1> |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="code"> | 59 | <div·class="code"> |
| 60 | <br/> | 60 | <br/> |
| 61 | <span·class="id"·title="keyword">Fixpoint</span>·<a·id="eq_nat"·class="idref"·href="#eq_nat"><span·class="id"·title="definition">eq_nat</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<span·class="id"·title="keyword">Prop</span>·:=<br/> | 61 | <span·class="id"·title="keyword">Fixpoint</span>·<a·id="eq_nat"·class="idref"·href="#eq_nat"><span·class="id"·title="definition">eq_nat</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<span·class="id"·title="keyword">Prop</span>·:=<br/> |
| 62 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#n:1"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Arith.EqNat.html#m:2"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/> | 62 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#n:1"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Arith.EqNat.html#m:2"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/> |
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| 69 | <br/> | 69 | <br/> |
| 70 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·<span·class="id"·title="var">n</span>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/> | 70 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·<span·class="id"·title="var">n</span>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/> |
| 71 | <br/> | 71 | <br/> |
| 72 | </div> | 72 | </div> |
| 73 | <div·class="doc"> | 73 | <div·class="doc"> |
| 74 | <a·id="lab | 74 | <a·id="lab289"></a><h1·class="section">Inductive·definition·of·<span·class="inlinecode"><span·class="id"·title="var">even</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">odd</span></span></h1> |
| 75 | </div> | 75 | </div> |
| 76 | <div·class="code"> | 76 | <div·class="code"> |
| 77 | <br/> | 77 | <br/> |
| 78 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Even·file·is·obsolete.·Use·Nat.Even·or·Nat.Even_alt·instead.")]<br/> | 78 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Even·file·is·obsolete.·Use·Nat.Even·or·Nat.Even_alt·instead.")]<br/> |
| 79 | <span·class="id"·title="keyword">Notation</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="abbreviation">even</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.Even_alt"><span·class="id"·title="inductive">Nat.Even_alt</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 79 | <span·class="id"·title="keyword">Notation</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="abbreviation">even</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.Even_alt"><span·class="id"·title="inductive">Nat.Even_alt</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | Theorems·about·<span·class="inlinecode"><span·class="id"·title="var">gt</span></span>·in·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>.· | 51 | Theorems·about·<span·class="inlinecode"><span·class="id"·title="var">gt</span></span>·in·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab | 53 | <a·id="lab281"></a><h1·class="section">This·file·is·OBSOLETE,·see·<span·class="inlinecode"><span·class="id"·title="var">Arith_base</span></span>·instead.</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> | 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> | 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab282"></a><h1·class="section">Order·and·successor</h1> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="code"> | 65 | <div·class="code"> |
| 66 | <br/> | 66 | <br/> |
| 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Gt·file·is·obsolete.·Use·Nat.lt_0_succ·instead.")]<br/> | 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Gt·file·is·obsolete.·Use·Nat.lt_0_succ·instead.")]<br/> |
| 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="gt_Sn_O"·class="idref"·href="#gt_Sn_O"><span·class="id"·title="abbreviation">gt_Sn_O</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#gt_Sn_O_stt"><span·class="id"·title="definition">Arith_prebase.gt_Sn_O_stt</span></a>.<br/> | 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="gt_Sn_O"·class="idref"·href="#gt_Sn_O"><span·class="id"·title="abbreviation">gt_Sn_O</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#gt_Sn_O_stt"><span·class="id"·title="definition">Arith_prebase.gt_Sn_O_stt</span></a>.<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | Order·on·natural·numbers.· | 51 | Order·on·natural·numbers.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab3 | 53 | <a·id="lab337"></a><h1·class="section">This·file·is·OBSOLETE,·see·<span·class="inlinecode"><span·class="id"·title="var">Arith_base</span></span>·instead.</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> | 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="doc"> | 60 | <div·class="doc"> |
| 61 | <a·id="lab3 | 61 | <a·id="lab338"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·an·order·on·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span></h1> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="code"> | 63 | <div·class="code"> |
| 64 | <br/> | 64 | <br/> |
| 65 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Le·file·is·obsolete.·Use·Nat.le_refl·instead.")]<br/> | 65 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Le·file·is·obsolete.·Use·Nat.le_refl·instead.")]<br/> |
| 66 | <span·class="id"·title="keyword">Notation</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="abbreviation">le_refl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.le_refl"><span·class="id"·title="lemma">Nat.le_refl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 66 | <span·class="id"·title="keyword">Notation</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="abbreviation">le_refl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.le_refl"><span·class="id"·title="lemma">Nat.le_refl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | Strict·order·on·natural·numbers.· | 51 | Strict·order·on·natural·numbers.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab30 | 53 | <a·id="lab300"></a><h1·class="section">This·file·is·OBSOLETE,·see·<span·class="inlinecode"><span·class="id"·title="var">Arith_base</span></span>·instead.</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> | 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="doc"> | 60 | <div·class="doc"> |
| 61 | <a·id="lab30 | 61 | <a·id="lab301"></a><h1·class="section">Irreflexivity</h1> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="code"> | 63 | <div·class="code"> |
| 64 | <br/> | 64 | <br/> |
| 65 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Lt·file·is·obsolete.·Use·Nat.lt_irrefl·instead.")]<br/> | 65 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Lt·file·is·obsolete.·Use·Nat.lt_irrefl·instead.")]<br/> |
| 66 | <span·class="id"·title="keyword">Notation</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="abbreviation">lt_irrefl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.lt_irrefl"><span·class="id"·title="lemma">Nat.lt_irrefl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· | 66 | <span·class="id"·title="keyword">Notation</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="abbreviation">lt_irrefl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.lt_irrefl"><span·class="id"·title="lemma">Nat.lt_irrefl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· |
| 67 | <br/> | 67 | <br/> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="doc"> | 69 | <div·class="doc"> |
| 70 | <a·id="lab30 | 70 | <a·id="lab302"></a><h1·class="section">Relationship·between·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1> |
| 71 | </div> | 71 | </div> |
| 72 | <div·class="code"> | 72 | <div·class="code"> |
| 73 | <br/> | 73 | <br/> |
| 74 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Lt·file·is·obsolete.·Use·the·bidirectional·version·Nat.le_succ_l·instead.")]<br/> | 74 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Lt·file·is·obsolete.·Use·the·bidirectional·version·Nat.le_succ_l·instead.")]<br/> |
| 75 | <span·class="id"·title="keyword">Notation</span>·<a·id="lt_le_S"·class="idref"·href="#lt_le_S"><span·class="id"·title="abbreviation">lt_le_S</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#lt_le_S_stt"><span·class="id"·title="definition">Arith_prebase.lt_le_S_stt</span></a>.<br/> | 75 | <span·class="id"·title="keyword">Notation</span>·<a·id="lt_le_S"·class="idref"·href="#lt_le_S"><span·class="id"·title="abbreviation">lt_le_S</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#lt_le_S_stt"><span·class="id"·title="definition">Arith_prebase.lt_le_S_stt</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab280"></a><h1·class="section">THIS·FILE·IS·DEPRECATED.·Use·<span·class="inlinecode"><span·class="id"·title="var">PeanoNat.Nat</span></span>·or·<span·class="inlinecode"><span·class="id"·title="var">Arith.Arith_base</span></span>·instead.</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab3 | 49 | <a·id="lab310"></a><h1·class="section">THIS·FILE·IS·DEPRECATED.·Use·<span·class="inlinecode"><span·class="id"·title="var">PeanoNat.Nat</span></span>·or·<span·class="inlinecode"><span·class="id"·title="var">Arith.Arith_base</span></span>·instead.</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>.<br/> |
| Offset 48, 43 lines modified | Offset 48, 43 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | Properties·of·subtraction·between·natural·numbers.· | 51 | Properties·of·subtraction·between·natural·numbers.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab29 | 53 | <a·id="lab293"></a><h1·class="section">This·file·is·OBSOLETE,·see·<span·class="inlinecode"><span·class="id"·title="var">Arith_base</span></span>·instead.</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> | 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> | 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab294"></a><h1·class="section">0·is·right·neutral</h1> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="code"> | 65 | <div·class="code"> |
| 66 | <br/> | 66 | <br/> |
| 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Minus·file·is·obsolete.·Use·Nat.sub_0_r·(and·symmetry·of·equality)·instead.")]<br/> | 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Minus·file·is·obsolete.·Use·Nat.sub_0_r·(and·symmetry·of·equality)·instead.")]<br/> |
| 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="minus_n_O"·class="idref"·href="#minus_n_O"><span·class="id"·title="abbreviation">minus_n_O</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#minus_n_O_stt"><span·class="id"·title="definition">Arith_prebase.minus_n_O_stt</span></a>.<br/> | 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="minus_n_O"·class="idref"·href="#minus_n_O"><span·class="id"·title="abbreviation">minus_n_O</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#minus_n_O_stt"><span·class="id"·title="definition">Arith_prebase.minus_n_O_stt</span></a>.<br/> |
| 69 | <br/> | 69 | <br/> |
| 70 | </div> | 70 | </div> |
| 71 | <div·class="doc"> | 71 | <div·class="doc"> |
| 72 | <a·id="lab | 72 | <a·id="lab295"></a><h1·class="section">Permutation·with·successor</h1> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="code"> | 74 | <div·class="code"> |
| 75 | <br/> | 75 | <br/> |
| 76 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Minus·file·is·obsolete.·Use·Nat.sub_succ_l·(and·symmetry·of·equality)·instead.")]<br/> | 76 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Minus·file·is·obsolete.·Use·Nat.sub_succ_l·(and·symmetry·of·equality)·instead.")]<br/> |
| 77 | <span·class="id"·title="keyword">Notation</span>·<a·id="minus_Sn_m"·class="idref"·href="#minus_Sn_m"><span·class="id"·title="abbreviation">minus_Sn_m</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#minus_Sn_m_stt"><span·class="id"·title="definition">Arith_prebase.minus_Sn_m_stt</span></a>.<br/> | 77 | <span·class="id"·title="keyword">Notation</span>·<a·id="minus_Sn_m"·class="idref"·href="#minus_Sn_m"><span·class="id"·title="abbreviation">minus_Sn_m</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#minus_Sn_m_stt"><span·class="id"·title="definition">Arith_prebase.minus_Sn_m_stt</span></a>.<br/> |
| Offset 45, 76 lines modified | Offset 45, 76 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab3 | 49 | <a·id="lab343"></a><h1·class="section">Properties·of·multiplication.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | <a·id="lab3 | 51 | <a·id="lab344"></a><h1·class="section">This·file·is·OBSOLETE,·see·<span·class="inlinecode"><span·class="id"·title="var">Arith_base</span></span>·instead.</h1> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
| 54 | <br/> | 54 | <br/> |
| 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> | 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> | 57 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="doc"> | 60 | <div·class="doc"> |
| 61 | <a·id="lab3 | 61 | <a·id="lab345"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·is·a·semi-ring</h1> |
| 62 | <div·class="paragraph">·</div> | 62 | <div·class="paragraph">·</div> |
| 63 | <a·id="lab34 | 63 | <a·id="lab346"></a><h2·class="section">Zero·property</h2> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="code"> | 65 | <div·class="code"> |
| 66 | <br/> | 66 | <br/> |
| 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_0_l·instead.")]<br/> | 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_0_l·instead.")]<br/> |
| 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_0_l"·class="idref"·href="#mult_0_l"><span·class="id"·title="abbreviation">mult_0_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_0_l"><span·class="id"·title="lemma">Nat.mul_0_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).·#[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_0_r·instead.")]<br/> | 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_0_l"·class="idref"·href="#mult_0_l"><span·class="id"·title="abbreviation">mult_0_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_0_l"><span·class="id"·title="lemma">Nat.mul_0_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).·#[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_0_r·instead.")]<br/> |
| 69 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_0_r"·class="idref"·href="#mult_0_r"><span·class="id"·title="abbreviation">mult_0_r</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_0_r"><span·class="id"·title="lemma">Nat.mul_0_r</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· | 69 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_0_r"·class="idref"·href="#mult_0_r"><span·class="id"·title="abbreviation">mult_0_r</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_0_r"><span·class="id"·title="lemma">Nat.mul_0_r</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· |
| 70 | <br/> | 70 | <br/> |
| 71 | </div> | 71 | </div> |
| 72 | <div·class="doc"> | 72 | <div·class="doc"> |
| 73 | <a·id="lab34 | 73 | <a·id="lab347"></a><h2·class="section">1·is·neutral</h2> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="code"> | 75 | <div·class="code"> |
| 76 | <br/> | 76 | <br/> |
| 77 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_1_l·instead.")]<br/> | 77 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_1_l·instead.")]<br/> |
| 78 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_1_l"·class="idref"·href="#mult_1_l"><span·class="id"·title="abbreviation">mult_1_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_1_l"><span·class="id"·title="lemma">Nat.mul_1_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).·#[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_1_r·instead.")]<br/> | 78 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_1_l"·class="idref"·href="#mult_1_l"><span·class="id"·title="abbreviation">mult_1_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_1_l"><span·class="id"·title="lemma">Nat.mul_1_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).·#[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_1_r·instead.")]<br/> |
| 79 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_1_r"·class="idref"·href="#mult_1_r"><span·class="id"·title="abbreviation">mult_1_r</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_1_r"><span·class="id"·title="lemma">Nat.mul_1_r</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· | 79 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_1_r"·class="idref"·href="#mult_1_r"><span·class="id"·title="abbreviation">mult_1_r</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_1_r"><span·class="id"·title="lemma">Nat.mul_1_r</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· |
| 80 | <br/> | 80 | <br/> |
| 81 | </div> | 81 | </div> |
| 82 | <div·class="doc"> | 82 | <div·class="doc"> |
| 83 | <a·id="lab34 | 83 | <a·id="lab348"></a><h2·class="section">Commutativity</h2> |
| 84 | </div> | 84 | </div> |
| 85 | <div·class="code"> | 85 | <div·class="code"> |
| 86 | <br/> | 86 | <br/> |
| 87 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_comm·instead.")]<br/> | 87 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_comm·instead.")]<br/> |
| 88 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_comm"·class="idref"·href="#mult_comm"><span·class="id"·title="abbreviation">mult_comm</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_comm"><span·class="id"·title="lemma">Nat.mul_comm</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· | 88 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_comm"·class="idref"·href="#mult_comm"><span·class="id"·title="abbreviation">mult_comm</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.mul_comm"><span·class="id"·title="lemma">Nat.mul_comm</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).· |
| 89 | <br/> | 89 | <br/> |
| 90 | </div> | 90 | </div> |
| 91 | <div·class="doc"> | 91 | <div·class="doc"> |
| 92 | <a·id="lab34 | 92 | <a·id="lab349"></a><h2·class="section">Distributivity</h2> |
| 93 | </div> | 93 | </div> |
| 94 | <div·class="code"> | 94 | <div·class="code"> |
| 95 | <br/> | 95 | <br/> |
| 96 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_add_distr_r·instead.")]<br/> | 96 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_add_distr_r·instead.")]<br/> |
| 97 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_plus_distr_r"·class="idref"·href="#mult_plus_distr_r"><span·class="id"·title="abbreviation">mult_plus_distr_r</span></a>·:=<br/> | 97 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_plus_distr_r"·class="idref"·href="#mult_plus_distr_r"><span·class="id"·title="abbreviation">mult_plus_distr_r</span></a>·:=<br/> |
| Offset 145, 15 lines modified | Offset 145, 15 lines modified | ||
| 145 | <a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#Proper"><span·class="id"·title="class">Proper</span></a>·(<a·class="idref"·href="Coq.Init.Logic.html#eq"><span·class="id"·title="inductive">eq</span></a><a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a><a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a><a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a><a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>)·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> | 145 | <a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#Proper"><span·class="id"·title="class">Proper</span></a>·(<a·class="idref"·href="Coq.Init.Logic.html#eq"><span·class="id"·title="inductive">eq</span></a><a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a><a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a><a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a><a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>)·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> |
| 146 | <span·class="id"·title="keyword">forall</span>·<a·id="n:14"·class="idref"·href="#n:14"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>))·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>)).<br/> | 146 | <span·class="id"·title="keyword">forall</span>·<a·id="n:14"·class="idref"·href="#n:14"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>))·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>)).<br/> |
| 147 | <br/> | 147 | <br/> |
| 148 | </div> | 148 | </div> |
| 149 | <div·class="doc"> | 149 | <div·class="doc"> |
| 150 | <a·id="lab | 150 | <a·id="lab311"></a><h2·class="section">Remaining·constants·not·defined·in·Coq.Init.Nat</h2> |
| 151 | <div·class="paragraph">·</div> | 151 | <div·class="paragraph">·</div> |
| 152 | ·NB:·Aliasing·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·mandatory,·since·only·a·Definition·can·implement | 152 | ·NB:·Aliasing·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·mandatory,·since·only·a·Definition·can·implement |
| 153 | ····an·interface·Parameter...· | 153 | ····an·interface·Parameter...· |
| 154 | </div> | 154 | </div> |
| 155 | <div·class="code"> | 155 | <div·class="code"> |
| Offset 48, 30 lines modified | Offset 48, 30 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | Properties·of·addition.· | 51 | Properties·of·addition.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab | 53 | <a·id="lab326"></a><h1·class="section">This·file·is·OBSOLETE,·see·<span·class="inlinecode"><span·class="id"·title="var">Arith_base</span></span>·instead.</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> | 57 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#"><span·class="id"·title="library">Arith_prebase</span></a>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> | 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab327"></a><h1·class="section">Neutrality·of·0,·commutativity,·associativity</h1> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="code"> | 65 | <div·class="code"> |
| 66 | <br/> | 66 | <br/> |
| 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Plus·file·is·obsolete.·Use·Nat.add_0_l·instead.")]<br/> | 67 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Plus·file·is·obsolete.·Use·Nat.add_0_l·instead.")]<br/> |
| 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="plus_0_l"·class="idref"·href="#plus_0_l"><span·class="id"·title="abbreviation">plus_0_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.add_0_l"><span·class="id"·title="lemma">Nat.add_0_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 68 | <span·class="id"·title="keyword">Notation</span>·<a·id="plus_0_l"·class="idref"·href="#plus_0_l"><span·class="id"·title="abbreviation">plus_0_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.add_0_l"><span·class="id"·title="lemma">Nat.add_0_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
| Offset 86, 27 lines modified | Offset 86, 27 lines modified | ||
| 86 | |·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·=>·<a·class="idref"·href="Coq.Init.Logic.html#False"><span·class="id"·title="inductive">False</span></a><br/> | 86 | |·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·=>·<a·class="idref"·href="Coq.Init.Logic.html#False"><span·class="id"·title="inductive">False</span></a><br/> |
| 87 | <span·class="id"·title="keyword">end</span>.<br/> | 87 | <span·class="id"·title="keyword">end</span>.<br/> |
| 88 | <br/> | 88 | <br/> |
| 89 | </div> | 89 | </div> |
| 90 | <div·class="doc"> | 90 | <div·class="doc"> |
| 91 | <a·id="lab | 91 | <a·id="lab228"></a><h1·class="section">Decidability</h1> |
| 92 | </div> | 92 | </div> |
| 93 | <div·class="code"> | 93 | <div·class="code"> |
| 94 | <br/> | 94 | <br/> |
| 95 | <span·class="id"·title="keyword">Lemma</span>·<a·id="bool_dec"·class="idref"·href="#bool_dec"><span·class="id"·title="lemma">bool_dec</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:3"·class="idref"·href="#b1:3"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:4"·class="idref"·href="#b2:4"><span·class="id"·title="binder">b2</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>,·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/> | 95 | <span·class="id"·title="keyword">Lemma</span>·<a·id="bool_dec"·class="idref"·href="#bool_dec"><span·class="id"·title="lemma">bool_dec</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:3"·class="idref"·href="#b1:3"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:4"·class="idref"·href="#b2:4"><span·class="id"·title="binder">b2</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>,·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/> |
| 96 | <br/> | 96 | <br/> |
| 97 | </div> | 97 | </div> |
| 98 | <div·class="doc"> | 98 | <div·class="doc"> |
| 99 | <a·id="lab | 99 | <a·id="lab229"></a><h1·class="section">Discrimination</h1> |
| 100 | </div> | 100 | </div> |
| 101 | <div·class="code"> | 101 | <div·class="code"> |
| 102 | <br/> | 102 | <br/> |
| 103 | <span·class="id"·title="keyword">Lemma</span>·<a·id="diff_true_false"·class="idref"·href="#diff_true_false"><span·class="id"·title="lemma">diff_true_false</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>.<br/> | 103 | <span·class="id"·title="keyword">Lemma</span>·<a·id="diff_true_false"·class="idref"·href="#diff_true_false"><span·class="id"·title="lemma">diff_true_false</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>.<br/> |
| 104 | #[<span·class="id"·title="var">global</span>]<br/> | 104 | #[<span·class="id"·title="var">global</span>]<br/> |
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| 62 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations"><span·class="id"·title="module">BoolNotations</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations"><span·class="id"·title="module">BoolNotations</span></a>.<br/> |
| 63 | <span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">typeclass_instances</span>·<span·class="id"·title="var">relations</span>.<br/> | 63 | <span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">typeclass_instances</span>·<span·class="id"·title="var">relations</span>.<br/> |
| 64 | <br/> | 64 | <br/> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="doc"> | 66 | <div·class="doc"> |
| 67 | <a·id="lab | 67 | <a·id="lab243"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="code"> | 69 | <div·class="code"> |
| 70 | <br/> | 70 | <br/> |
| 71 | <span·class="id"·title="keyword">Lemma</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="lemma">le_refl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:1"·class="idref"·href="#b:1"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>.<br/> | 71 | <span·class="id"·title="keyword">Lemma</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="lemma">le_refl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:1"·class="idref"·href="#b:1"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>.<br/> |
| 72 | · | 72 | · |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab21"></a><h1·class="section">Typeclass-based·setoids.·Definitions·on·<span·class="inlinecode"><span·class="id"·title="var">Equivalence</span></span>.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab5"></a><h1·class="section">Typeclass-based·morphism·definition·and·standard,·minimal·instances</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab2"></a><h1·class="section">Typeclass-based·relations,·tactics·and·standard·instances</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···This·is·the·basic·theory·needed·to·formalize·morphisms·and·setoids. | 51 | ···This·is·the·basic·theory·needed·to·formalize·morphisms·and·setoids. |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab14"></a><h1·class="section">A·typeclass·to·ease·the·handling·of·decidable·properties.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·A·proposition·is·decidable·whenever·it·is·reflected·by·a·boolean.· | 51 | ·A·proposition·is·decidable·whenever·it·is·reflected·by·a·boolean.· |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab20"></a><h1·class="section">Decidable·equivalences.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab10"></a><h1·class="section">Typeclass-based·setoids.·Definitions·on·<span·class="inlinecode"><span·class="id"·title="var">Equivalence</span></span>.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab15"></a><h1·class="section">Initialization·code·for·typeclasses,·setting·up·the·default·tactic</h1> |
| 50 | ···for·instance·search. | 50 | ···for·instance·search. |
| 51 | <div·class="paragraph">·</div> | 51 | <div·class="paragraph">·</div> |
| 52 | ···Author:·Matthieu·Sozeau | 52 | ···Author:·Matthieu·Sozeau |
| 53 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 53 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab11"></a><h1·class="section">Typeclass-based·morphism·definition·and·standard,·minimal·instances</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">Proper</span></span>·instances·for·propositional·connectives.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab9"></a><h1·class="section">Morphism·instances·for·relations.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra | 00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra | ||
| 00000670:·7279·2043·6f71·2e43·6c61·7373·6573·2e52··ry·Coq.Classes.R | 00000670:·7279·2043·6f71·2e43·6c61·7373·6573·2e52··ry·Coq.Classes.R | ||
| 00000680:·656c·6174·696f·6e43·6c61·7373·6573·3c2f··elationClasses</ | 00000680:·656c·6174·696f·6e43·6c61·7373·6573·3c2f··elationClasses</ | ||
| 00000690:·6831·3e0a·0a3c·6469·7620·636c·6173·733d··h1>..<div·class= | 00000690:·6831·3e0a·0a3c·6469·7620·636c·6173·733d··h1>..<div·class= | ||
| 000006a0:·2263·6f64·6522·3e0a·0a3c·6272·2f3e·0a3c··"code">..<br/>.< | 000006a0:·2263·6f64·6522·3e0a·0a3c·6272·2f3e·0a3c··"code">..<br/>.< | ||
| 000006b0:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas | 000006b0:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas | Diff chunk too large, falling back to line-by-line diff (7824 lines added, 7824 lines removed) | |
| 000006c0:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id=" | 000006c0:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id=" | ||
| 000006d0:·6c61·6231·3 | 000006d0:·6c61·6231·3722·3e3c·2f61·3e3c·6831·2063··lab17"></a><h1·c | ||
| 000006e0:· | 000006e0:·6c61·7373·3d22·7365·6374·696f·6e22·3e54··lass="section">T | ||
| 000006f0:· | 000006f0:·7970·6563·6c61·7373·2d62·6173·6564·2072··ypeclass-based·r | ||
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| 00000ce0:·6 | 00000ce0:·6c65·3d22·7661·7222·3e54·3c2f·7370·616e··le="var">T</span | ||
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| 00000d00:· | 00000d00:·6422·2074·6974·6c65·3d22·7661·7222·3e55··d"·title="var">U | ||
| 00000d10:·3 | 00000d10:·3c2f·7370·616e·3e20·3c73·7061·6e20·636c··</span>·<span·cl | ||
| 00000d20:·6 | 00000d20:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | ||
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| 00000e00:· | 00000e00:·6f63·223e·0a57·6520·616c·6c6f·7720·746f··oc">.We·allow·to | ||
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| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab13"></a><h1·class="section">Relations·over·pairs</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.SetoidList.html#"><span·class="id"·title="library">SetoidList</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.SetoidList.html#"><span·class="id"·title="library">SetoidList</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relations.html#"><span·class="id"·title="library">Relations</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relations.html#"><span·class="id"·title="library">Relations</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab8"></a><h1·class="section">Typeclass-based·setoids,·tactics·and·standard·instances.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab16"></a><h1·class="section">Decidable·setoid·equality·theory.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab7"></a><h1·class="section">Tactics·for·typeclass-based·setoids.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 47 | <br/> | 47 | <br/> |
| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | <a·id="lab | 51 | <a·id="lab631"></a><h1·class="section">FMapAVL</h1> |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | ·This·module·implements·maps·using·AVL·trees. | 53 | ·This·module·implements·maps·using·AVL·trees. |
| 54 | ····It·follows·the·implementation·from·Ocaml's·standard·library. | 54 | ····It·follows·the·implementation·from·Ocaml's·standard·library. |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
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| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab611"></a><h1·class="section">Finite·maps·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>.·These | 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>.·These |
| 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>·written·using | 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>·written·using |
| 53 | ··different·styles:·equivalence·and·boolean·equalities. | 53 | ··different·styles:·equivalence·and·boolean·equalities. |
| Offset 47, 15 lines modified | Offset 47, 15 lines modified | ||
| 47 | <br/> | 47 | <br/> |
| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | <a·id="lab | 51 | <a·id="lab715"></a><h1·class="section">FMapFullAVL</h1> |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | ···This·file·contains·some·complements·to·<span·class="inlinecode"><span·class="id"·title="var">FMapAVL</span></span>. | 53 | ···This·file·contains·some·complements·to·<span·class="inlinecode"><span·class="id"·title="var">FMapAVL</span></span>. |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab706"></a><h1·class="section">Finite·map·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·interfaces·for·finite·maps· | 51 | ·This·file·proposes·interfaces·for·finite·maps· |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab570"></a><h1·class="section">Finite·map·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface |
| 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>·using·lists·of·pairs·ordered·(increasing)·with·respect·to | 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>·using·lists·of·pairs·ordered·(increasing)·with·respect·to |
| 53 | ·left·projection.· | 53 | ·left·projection.· |
| 54 | </div> | 54 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab713"></a><h1·class="section">FMapPositive·:·an·implementation·of·FMapInterface·for·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·keys.</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>·<a·class="idref"·href="Coq.Structures.OrderedTypeEx.html#"><span·class="id"·title="library">OrderedTypeEx</span></a>·<a·class="idref"·href="Coq.FSets.FMapInterface.html#"><span·class="id"·title="library">FMapInterface</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>·<a·class="idref"·href="Coq.Structures.OrderedTypeEx.html#"><span·class="id"·title="library">OrderedTypeEx</span></a>·<a·class="idref"·href="Coq.FSets.FMapInterface.html#"><span·class="id"·title="library">FMapInterface</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab692"></a><h1·class="section">Finite·map·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface |
| 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.WS</span></span>·using·lists·of·pairs,·unordered·but·without·redundancy.· | 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.WS</span></span>·using·lists·of·pairs,·unordered·but·without·redundancy.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab609"></a><h1·class="section">FSetAVL·:·Implementation·of·FSetInterface·via·AVL·trees</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·implements·finite·sets·using·AVL·trees. | 51 | ·This·module·implements·finite·sets·using·AVL·trees. |
| 52 | ····It·follows·the·implementation·from·Ocaml's·standard·library, | 52 | ····It·follows·the·implementation·from·Ocaml's·standard·library, |
| 53 | <div·class="paragraph">·</div> | 53 | <div·class="paragraph">·</div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab703"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·implements·bridges·(as·functors)·from·dependent | 51 | ·This·module·implements·bridges·(as·functors)·from·dependent |
| 52 | ····to/from·non-dependent·set·signature.· | 52 | ····to/from·non-dependent·set·signature.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab596"></a><h1·class="section">Compatibility·functors·between·FSetInterface·and·MSetInterface.</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetInterface.html#"><span·class="id"·title="library">FSetInterface</span></a>·<a·class="idref"·href="Coq.FSets.FSetFacts.html#"><span·class="id"·title="library">FSetFacts</span></a>·<a·class="idref"·href="Coq.MSets.MSetInterface.html#"><span·class="id"·title="library">MSetInterface</span></a>·<a·class="idref"·href="Coq.MSets.MSetFacts.html#"><span·class="id"·title="library">MSetFacts</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetInterface.html#"><span·class="id"·title="library">FSetInterface</span></a>·<a·class="idref"·href="Coq.FSets.FSetFacts.html#"><span·class="id"·title="library">FSetFacts</span></a>·<a·class="idref"·href="Coq.MSets.MSetInterface.html#"><span·class="id"·title="library">MSetInterface</span></a>·<a·class="idref"·href="Coq.MSets.MSetFacts.html#"><span·class="id"·title="library">MSetFacts</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 54 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| Offset 71, 15 lines modified | Offset 71, 15 lines modified | ||
| 71 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun"·class="idref"·href="#WDecide_fun"><span·class="id"·title="module">WDecide_fun</span></a>·(<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>)(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">M</span>·:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>).<br/> | 71 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun"·class="idref"·href="#WDecide_fun"><span·class="id"·title="module">WDecide_fun</span></a>·(<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>)(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">M</span>·:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>).<br/> |
| 72 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.F"·class="idref"·href="#WDecide_fun.F"><span·class="id"·title="module">F</span></a>·:=·<a·class="idref"·href="Coq.FSets.FSetFacts.html#WFacts_fun"><span·class="id"·title="module">FSetFacts.WFacts_fun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#M"><span·class="id"·title="module">M</span></a>.<br/> | 72 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.F"·class="idref"·href="#WDecide_fun.F"><span·class="id"·title="module">F</span></a>·:=·<a·class="idref"·href="Coq.FSets.FSetFacts.html#WFacts_fun"><span·class="id"·title="module">FSetFacts.WFacts_fun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#M"><span·class="id"·title="module">M</span></a>.<br/> |
| 73 | <br/> | 73 | <br/> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="doc"> | 75 | <div·class="doc"> |
| 76 | <a·id="lab | 76 | <a·id="lab676"></a><h1·class="section">Overview</h1> |
| 77 | ····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will | 77 | ····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will |
| 78 | ····solve·any·valid·goal·of·the·form | 78 | ····solve·any·valid·goal·of·the·form |
| 79 | <pre> | 79 | <pre> |
| 80 | ····forall·s1·...·sn, | 80 | ····forall·s1·...·sn, |
| 81 | ····forall·x1·...·xm, | 81 | ····forall·x1·...·xm, |
| 82 | ····P1·->·...·->·Pk·->·P | 82 | ····P1·->·...·->·Pk·->·P |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab712"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·proves·many·properties·of·finite·sets·that | 51 | ·This·module·proves·many·properties·of·finite·sets·that |
| 52 | ····are·consequences·of·the·axiomatization·in·<span·class="inlinecode"><span·class="id"·title="var">FsetInterface</span></span> | 52 | ····are·consequences·of·the·axiomatization·in·<span·class="inlinecode"><span·class="id"·title="var">FsetInterface</span></span> |
| 53 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FsetProperties</span></span>·it·uses | 53 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FsetProperties</span></span>·it·uses |
| 54 | ····sets·operations·instead·of·predicates·over·sets,·i.e. | 54 | ····sets·operations·instead·of·predicates·over·sets,·i.e. |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab566"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>.·These | 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>.·These |
| 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>·written·using | 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>·written·using |
| 53 | ··different·styles:·equivalence·and·boolean·equalities. | 53 | ··different·styles:·equivalence·and·boolean·equalities. |
| 54 | ··Moreover,·we·prove·that·<span·class="inlinecode"><span·class="id"·title="var">E.Eq</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Equal</span></span>·are·setoid·equalities. | 54 | ··Moreover,·we·prove·that·<span·class="inlinecode"><span·class="id"·title="var">E.Eq</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Equal</span></span>·are·setoid·equalities. |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab601"></a><h1·class="section">Finite·set·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Set·interfaces,·inspired·by·the·one·of·Ocaml.·When·compared·with | 51 | ·Set·interfaces,·inspired·by·the·one·of·Ocaml.·When·compared·with |
| 52 | ····Ocaml,·the·main·differences·are: | 52 | ····Ocaml,·the·main·differences·are: |
| 53 | <ul·class="doclist"> | 53 | <ul·class="doclist"> |
| 54 | <li>·the·lack·of·<span·class="inlinecode"><span·class="id"·title="var">iter</span></span>·function,·useless·since·Coq·is·purely·functional | 54 | <li>·the·lack·of·<span·class="inlinecode"><span·class="id"·title="var">iter</span></span>·function,·useless·since·Coq·is·purely·functional |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab610"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent |
| 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>·using·strictly·ordered·list.· | 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>·using·strictly·ordered·list.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab582"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·functor·derives·additional·properties·from·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>. | 51 | ·This·functor·derives·additional·properties·from·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>. |
| 52 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FSetEqProperties</span></span>·it·uses | 52 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FSetEqProperties</span></span>·it·uses |
| 53 | ····predicates·over·sets·instead·of·sets·operations,·i.e. | 53 | ····predicates·over·sets·instead·of·sets·operations,·i.e. |
| 54 | ····<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span></span>·instead·of·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span>=<span·class="id"·title="var">true</span></span>, | 54 | ····<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span></span>·instead·of·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span>=<span·class="id"·title="var">true</span></span>, |
| Offset 45, 28 lines modified | Offset 45, 28 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab629"></a><h1·class="section">Finite·sets·library·:·conversion·to·old·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span></h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Sets.Ensembles.html#"><span·class="id"·title="library">Ensembles</span></a>·<a·class="idref"·href="Coq.Sets.Finite_sets.html#"><span·class="id"·title="library">Finite_sets</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Sets.Ensembles.html#"><span·class="id"·title="library">Ensembles</span></a>·<a·class="idref"·href="Coq.Sets.Finite_sets.html#"><span·class="id"·title="library">Finite_sets</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetInterface.html#"><span·class="id"·title="library">FSetInterface</span></a>·<a·class="idref"·href="Coq.FSets.FSetProperties.html#"><span·class="id"·title="library">FSetProperties</span></a>·<a·class="idref"·href="Coq.Structures.OrderedTypeEx.html#"><span·class="id"·title="library">OrderedTypeEx</span></a>·<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#"><span·class="id"·title="library">DecidableTypeEx</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetInterface.html#"><span·class="id"·title="library">FSetInterface</span></a>·<a·class="idref"·href="Coq.FSets.FSetProperties.html#"><span·class="id"·title="library">FSetProperties</span></a>·<a·class="idref"·href="Coq.Structures.OrderedTypeEx.html#"><span·class="id"·title="library">OrderedTypeEx</span></a>·<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#"><span·class="id"·title="library">DecidableTypeEx</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab630"></a><h1·class="section">Going·from·<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·with·usual·Leibniz·equality</h1> |
| 59 | ····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.· | 59 | ····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/> | 63 | <span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab714"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent |
| 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.WS</span></span>·using·lists·without·redundancy.· | 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.WS</span></span>·using·lists·without·redundancy.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab466"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Notation</span>·<a·id="valid_binary"·class="idref"·href="#valid_binary"><span·class="id"·title="abbreviation">valid_binary</span></a>·:=·(<a·class="idref"·href="Coq.Floats.SpecFloat.html#valid_binary"><span·class="id"·title="definition">valid_binary</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>).<br/> | 57 | <span·class="id"·title="keyword">Notation</span>·<a·id="valid_binary"·class="idref"·href="#valid_binary"><span·class="id"·title="abbreviation">valid_binary</span></a>·:=·(<a·class="idref"·href="Coq.Floats.SpecFloat.html#valid_binary"><span·class="id"·title="definition">valid_binary</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>).<br/> |
| Offset 49, 15 lines modified | Offset 49, 15 lines modified | ||
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>·<a·class="idref"·href="Coq.Floats.FloatAxioms.html#"><span·class="id"·title="library">FloatAxioms</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>·<a·class="idref"·href="Coq.Floats.FloatAxioms.html#"><span·class="id"·title="library">FloatAxioms</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Psatz.html#"><span·class="id"·title="library">Psatz</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Psatz.html#"><span·class="id"·title="library">Psatz</span></a>.<br/> |
| 51 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab467"></a><h1·class="section">Support·results·involving·frexp·and·ldexp</h1> |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="code"> | 56 | <div·class="code"> |
| 57 | <br/> | 57 | <br/> |
| 58 | <span·class="id"·title="keyword">Lemma</span>·<a·id="shift_value"·class="idref"·href="#shift_value"><span·class="id"·title="lemma">shift_value</span></a>·:·<a·class="idref"·href="Coq.Floats.FloatOps.html#shift"><span·class="id"·title="definition">shift</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·(2<a·class="idref"·href="Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>)%<span·class="id"·title="var">Z</span>.<br/> | 58 | <span·class="id"·title="keyword">Lemma</span>·<a·id="shift_value"·class="idref"·href="#shift_value"><span·class="id"·title="lemma">shift_value</span></a>·:·<a·class="idref"·href="Coq.Floats.FloatOps.html#shift"><span·class="id"·title="definition">shift</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·(2<a·class="idref"·href="Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>)%<span·class="id"·title="var">Z</span>.<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab468"></a><h1·class="section">Derived·operations·and·mapping·between·primitive·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>s·and·<span·class="inlinecode"><span·class="id"·title="var">spec_float</span></span>s</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Definition</span>·<a·id="prec"·class="idref"·href="#prec"><span·class="id"·title="definition">prec</span></a>·:=·53%<span·class="id"·title="var">Z</span>.<br/> | 57 | <span·class="id"·title="keyword">Definition</span>·<a·id="prec"·class="idref"·href="#prec"><span·class="id"·title="definition">prec</span></a>·:=·53%<span·class="id"·title="var">Z</span>.<br/> |
| 58 | <span·class="id"·title="keyword">Definition</span>·<a·id="emax"·class="idref"·href="#emax"><span·class="id"·title="definition">emax</span></a>·:=·1024%<span·class="id"·title="var">Z</span>.<br/> | 58 | <span·class="id"·title="keyword">Definition</span>·<a·id="emax"·class="idref"·href="#emax"><span·class="id"·title="definition">emax</span></a>·:=·1024%<span·class="id"·title="var">Z</span>.<br/> |
| Offset 48, 24 lines modified | Offset 48, 24 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.PrimInt63.html#"><span·class="id"·title="library">PrimInt63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.PrimInt63.html#"><span·class="id"·title="library">PrimInt63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab469"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | This·interface·provides·processor·operators·for·the·Binary64·format·of·the | 55 | This·interface·provides·processor·operators·for·the·Binary64·format·of·the |
| 56 | IEEE·754-2008·standard.· | 56 | IEEE·754-2008·standard.· |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | <a·id="lab | 58 | <a·id="lab470"></a><h2·class="section">Type·definition·for·the·co-domain·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h2> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <span·class="id"·title="keyword">Variant</span>·<a·id="float_comparison"·class="idref"·href="#float_comparison"><span·class="id"·title="inductive">float_comparison</span></a>·:·<span·class="id"·title="keyword">Set</span>·:=·<a·id="FEq"·class="idref"·href="#FEq"><span·class="id"·title="constructor">FEq</span></a>·|·<a·id="FLt"·class="idref"·href="#FLt"><span·class="id"·title="constructor">FLt</span></a>·|·<a·id="FGt"·class="idref"·href="#FGt"><span·class="id"·title="constructor">FGt</span></a>·|·<a·id="FNotComparable"·class="idref"·href="#FNotComparable"><span·class="id"·title="constructor">FNotComparable</span></a>.<br/> | 61 | <span·class="id"·title="keyword">Variant</span>·<a·id="float_comparison"·class="idref"·href="#float_comparison"><span·class="id"·title="inductive">float_comparison</span></a>·:·<span·class="id"·title="keyword">Set</span>·:=·<a·id="FEq"·class="idref"·href="#FEq"><span·class="id"·title="constructor">FEq</span></a>·|·<a·id="FLt"·class="idref"·href="#FLt"><span·class="id"·title="constructor">FLt</span></a>·|·<a·id="FGt"·class="idref"·href="#FGt"><span·class="id"·title="constructor">FGt</span></a>·|·<a·id="FNotComparable"·class="idref"·href="#FNotComparable"><span·class="id"·title="constructor">FNotComparable</span></a>.<br/> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_comparison</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_cmp</span>.<br/> | 63 | <span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_comparison</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_cmp</span>.<br/> |
| Offset 48, 24 lines modified | Offset 48, 24 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab480"></a><h1·class="section">Specification·of·floating-point·arithmetic</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | This·specification·is·mostly·borrowed·from·the·<span·class="inlinecode"><span·class="id"·title="var">IEEE754.Binary</span></span>·module | 55 | This·specification·is·mostly·borrowed·from·the·<span·class="inlinecode"><span·class="id"·title="var">IEEE754.Binary</span></span>·module |
| 56 | of·the·Flocq·library·(see·<a·href="http://flocq.gforge.inria.fr/">http://flocq.gforge.inria.fr/</a>)· | 56 | of·the·Flocq·library·(see·<a·href="http://flocq.gforge.inria.fr/">http://flocq.gforge.inria.fr/</a>)· |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | <a·id="lab | 58 | <a·id="lab481"></a><h2·class="section">Inductive·specification·of·floating-point·numbers</h2> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | Similar·to·<span·class="inlinecode"><span·class="id"·title="var">Flocq.IEEE754.Binary.full_float</span></span>,·but·with·no·NaN·payload.· | 60 | Similar·to·<span·class="inlinecode"><span·class="id"·title="var">Flocq.IEEE754.Binary.full_float</span></span>,·but·with·no·NaN·payload.· |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab2 | 49 | <a·id="lab266"></a><h1·class="section">Bytes</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Coq.Init.Ltac</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Coq.Init.Ltac</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Datatypes.html#"><span·class="id"·title="library">Coq.Init.Datatypes</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Datatypes.html#"><span·class="id"·title="library">Coq.Init.Datatypes</span></a>.<br/> |
| Offset 53, 15 lines modified | Offset 53, 15 lines modified | ||
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Ltac</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Ltac</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab2 | 58 | <a·id="lab246"></a><h1·class="section">Datatypes·with·zero·and·one·element</h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·<span·class="inlinecode"><span·class="id"·title="var">Empty_set</span></span>·is·a·datatype·with·no·inhabitant· | 60 | ·<span·class="inlinecode"><span·class="id"·title="var">Empty_set</span></span>·is·a·datatype·with·no·inhabitant· |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab2 | 49 | <a·id="lab271"></a><h1·class="section">Decimal·numbers</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·These·numbers·coded·in·base·10·will·be·used·for·parsing·and·printing | 51 | ·These·numbers·coded·in·base·10·will·be·used·for·parsing·and·printing |
| 52 | ····other·Coq·numeral·datatypes·in·an·human-readable·way. | 52 | ····other·Coq·numeral·datatypes·in·an·human-readable·way. |
| 53 | ····See·the·<span·class="inlinecode"><span·class="id"·title="var">Number</span></span>·<span·class="inlinecode"><span·class="id"·title="keyword">Notation</span></span>·command. | 53 | ····See·the·<span·class="inlinecode"><span·class="id"·title="var">Number</span></span>·<span·class="inlinecode"><span·class="id"·title="keyword">Notation</span></span>·command. |
| 54 | ····We·represent·numbers·in·base·10·as·lists·of·decimal·digits, | 54 | ····We·represent·numbers·in·base·10·as·lists·of·decimal·digits, |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab2 | 49 | <a·id="lab270"></a><h1·class="section">Hexadecimal·numbers</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·These·numbers·coded·in·base·16·will·be·used·for·parsing·and·printing | 51 | ·These·numbers·coded·in·base·16·will·be·used·for·parsing·and·printing |
| 52 | ····other·Coq·numeral·datatypes·in·an·human-readable·way. | 52 | ····other·Coq·numeral·datatypes·in·an·human-readable·way. |
| 53 | ····See·the·<span·class="inlinecode"><span·class="id"·title="var">Number</span></span>·<span·class="inlinecode"><span·class="id"·title="keyword">Notation</span></span>·command. | 53 | ····See·the·<span·class="inlinecode"><span·class="id"·title="var">Number</span></span>·<span·class="inlinecode"><span·class="id"·title="keyword">Notation</span></span>·command. |
| 54 | ····We·represent·numbers·in·base·16·as·lists·of·hexadecimal·digits, | 54 | ····We·represent·numbers·in·base·16·as·lists·of·hexadecimal·digits, |
| Offset 159, 15 lines modified | Offset 159, 15 lines modified | ||
| 000009e0:·7222·3e42·3c2f·7370·616e·3e29·203a·203c··r">B</span>)·:·< | 000009e0:·7222·3e42·3c2f·7370·616e·3e29·203a·203c··r">B</span>)·:·< | ||
| 000009f0:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"· | 000009f0:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"· | ||
| 00000a00:·7469·746c·653d·2276·6172·223e·7479·7065··title="var">type | 00000a00:·7469·746c·653d·2276·6172·223e·7479·7065··title="var">type | ||
| 00000a10:·5f73·636f·7065·3c2f·7370·616e·3e2e·3c62··_scope</span>.<b | 00000a10:·5f73·636f·7065·3c2f·7370·616e·3e2e·3c62··_scope</span>.<b | ||
| 00000a20:·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64·6976··r/>..<br/>.</div | 00000a20:·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64·6976··r/>..<br/>.</div | ||
| 00000a30:·3e0a·0a3c·6469·7620·636c·6173·733d·2264··>..<div·class="d | 00000a30:·3e0a·0a3c·6469·7620·636c·6173·733d·2264··>..<div·class="d | ||
| 00000a40:·6f63·223e·0a3c·6120·6964·3d22·6c61·6232··oc">.<a·id="lab2 | 00000a40:·6f63·223e·0a3c·6120·6964·3d22·6c61·6232··oc">.<a·id="lab2 | ||
| 00000a50:·3 | 00000a50:·3734·223e·3c2f·613e·3c68·3120·636c·6173··74"></a><h1·clas | ||
| 00000a60:·733d·2273·6563·7469·6f6e·223e·5072·6f70··s="section">Prop | 00000a60:·733d·2273·6563·7469·6f6e·223e·5072·6f70··s="section">Prop | ||
| 00000a70:·6f73·6974·696f·6e61·6c20·636f·6e6e·6563··ositional·connec | 00000a70:·6f73·6974·696f·6e61·6c20·636f·6e6e·6563··ositional·connec | ||
| 00000a80:·7469·7665·733c·2f68·313e·0a0a·3c64·6976··tives</h1>..<div | 00000a80:·7469·7665·733c·2f68·313e·0a0a·3c64·6976··tives</h1>..<div | ||
| 00000a90:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap | 00000a90:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap | ||
| 00000aa0:·6822·3e20·3c2f·6469·763e·0a0a·203c·7370··h">·</div>..·<sp | 00000aa0:·6822·3e20·3c2f·6469·763e·0a0a·203c·7370··h">·</div>..·<sp | ||
| 00000ab0:·616e·2063·6c61·7373·3d22·696e·6c69·6e65··an·class="inline | 00000ab0:·616e·2063·6c61·7373·3d22·696e·6c69·6e65··an·class="inline | ||
| 00000ac0:·636f·6465·223e·3c73·7061·6e20·636c·6173··code"><span·clas | 00000ac0:·636f·6465·223e·3c73·7061·6e20·636c·6173··code"><span·clas |
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| 50 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Init.Number.html#"><span·class="id"·title="library">Number</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Init.Number.html#"><span·class="id"·title="library">Number</span></a>.<br/> |
| 51 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> | 51 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> |
| 52 | <br/> | 52 | <br/> |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | <a·id="lab2 | 55 | <a·id="lab252"></a><h1·class="section">Peano·natural·numbers,·definitions·of·operations</h1> |
| 56 | <div·class="paragraph">·</div> | 56 | <div·class="paragraph">·</div> |
| 57 | ·This·file·is·meant·to·be·used·as·a·whole·module, | 57 | ·This·file·is·meant·to·be·used·as·a·whole·module, |
| 58 | ····without·importing·it,·leading·to·qualified·definitions | 58 | ····without·importing·it,·leading·to·qualified·definitions |
| 59 | ····(e.g.·Nat.pred)· | 59 | ····(e.g.·Nat.pred)· |
| 60 | </div> | 60 | </div> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab2 | 49 | <a·id="lab267"></a><h1·class="section">Decimal·or·Hexadecimal·numbers</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>.<br/> |
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| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Specif.html#"><span·class="id"·title="library">Specif</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Specif.html#"><span·class="id"·title="library">Specif</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab26 | 56 | <a·id="lab268"></a><h1·class="section">Useful·tactics</h1> |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal. | 58 | ·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal. |
| 59 | ····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.· | 59 | ····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab2 | 49 | <a·id="lab273"></a><h1·class="section">The·tauto·and·intuition·tactics</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Notations.html#"><span·class="id"·title="library">Notations</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Notations.html#"><span·class="id"·title="library">Notations</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Ltac</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Ltac</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab27 | 49 | <a·id="lab272"></a><h1·class="section">This·module·proves·the·validity·of</h1> |
| 50 | <ul·class="doclist"> | 50 | <ul·class="doclist"> |
| 51 | <li>·well-founded·recursion·(also·known·as·course·of·values) | 51 | <li>·well-founded·recursion·(also·known·as·course·of·values) |
| 52 | </li> | 52 | </li> |
| 53 | <li>·well-founded·induction | 53 | <li>·well-founded·induction |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab1091"></a><h1·class="section">Basics:·definition·of·polymorphic·lists·and·some·operations</h1> |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·The·definition·of·<span·class="inlinecode"><span·class="id"·title="var">list</span></span>·is·now·in·<span·class="inlinecode"><span·class="id"·title="var">Init</span>/<span·class="id"·title="var">Datatypes</span></span>, | 58 | ·The·definition·of·<span·class="inlinecode"><span·class="id"·title="var">list</span></span>·is·now·in·<span·class="inlinecode"><span·class="id"·title="var">Init</span>/<span·class="id"·title="var">Datatypes</span></span>, |
| 59 | ····as·well·as·the·definitions·of·<span·class="inlinecode"><span·class="id"·title="var">length</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">app</span></span>· | 59 | ····as·well·as·the·definitions·of·<span·class="inlinecode"><span·class="id"·title="var">length</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">app</span></span>· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
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| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Lists.List.html#"><span·class="id"·title="library">List</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Lists.List.html#"><span·class="id"·title="library">List</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Sorting.Sorted.html#"><span·class="id"·title="library">Sorted</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Sorting.Sorted.html#"><span·class="id"·title="library">Sorted</span></a>.<br/> |
| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Program.Basics.html#"><span·class="id"·title="library">Basics</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Program.Basics.html#"><span·class="id"·title="library">Basics</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | <a·id="lab | 55 | <a·id="lab1122"></a><h1·class="section">Logical·relations·over·lists·with·respect·to·a·setoid·equality</h1> |
| 56 | ······or·ordering.· | 56 | ······or·ordering.· |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·This·can·be·seen·as·a·complement·of·predicate·<span·class="inlinecode"><span·class="id"·title="var">lelistA</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">sort</span></span> | 58 | ·This·can·be·seen·as·a·complement·of·predicate·<span·class="inlinecode"><span·class="id"·title="var">lelistA</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">sort</span></span> |
| 59 | ····found·in·<span·class="inlinecode"><span·class="id"·title="var">Sorting</span></span>.· | 59 | ····found·in·<span·class="inlinecode"><span·class="id"·title="var">Sorting</span></span>.· |
| 60 | </div> | 60 | </div> |
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| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Logic.Eqdep_dec.html#"><span·class="id"·title="library">Eqdep_dec</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Logic.Eqdep_dec.html#"><span·class="id"·title="library">Eqdep_dec</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.Streams.html#"><span·class="id"·title="library">Streams</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.Streams.html#"><span·class="id"·title="library">Streams</span></a>.<br/> |
| 51 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab1090"></a><h1·class="section">Memoization</h1> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | ·Successive·outputs·of·a·given·function·<span·class="inlinecode"><span·class="id"·title="var">f</span></span>·are·stored·in | 56 | ·Successive·outputs·of·a·given·function·<span·class="inlinecode"><span·class="id"·title="var">f</span></span>·are·stored·in |
| 57 | ····a·stream·in·order·to·avoid·duplicated·computations.· | 57 | ····a·stream·in·order·to·avoid·duplicated·computations.· |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="code"> | 59 | <div·class="code"> |
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| 46 | <div·class="code"> | 46 | <div·class="code"> |
| 47 | <br/> | 47 | <br/> |
| 48 | </div> | 48 | </div> |
| 49 | <div·class="doc"> | 49 | <div·class="doc"> |
| 50 | Some·facts·and·definitions·concerning·choice·and·description·in | 50 | Some·facts·and·definitions·concerning·choice·and·description·in |
| 51 | ·······intuitionistic·logic.·<a·id="lab | 51 | ·······intuitionistic·logic.·<a·id="lab1017"></a><h1·class="section">References:</h1> |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory, | 53 | <span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory, |
| 54 | unpublished. | 54 | unpublished. |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
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| 129 | <br/> | 129 | <br/> |
| 130 | <span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/> | 130 | <span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/> |
| 131 | <br/> | 131 | <br/> |
| 132 | </div> | 132 | </div> |
| 133 | <div·class="doc"> | 133 | <div·class="doc"> |
| 134 | <a·id="lab | 134 | <a·id="lab1016"></a><h3·class="section">Weaker·lemmas·(compatibility·lemmas)</h3> |
| 135 | </div> | 135 | </div> |
| 136 | <div·class="code"> | 136 | <div·class="code"> |
| 137 | <br/> | 137 | <br/> |
| 138 | <span·class="id"·title="keyword">Theorem</span>·<a·id="choice"·class="idref"·href="#choice"><span·class="id"·title="lemma">choice</span></a>·:<br/> | 138 | <span·class="id"·title="keyword">Theorem</span>·<a·id="choice"·class="idref"·href="#choice"><span·class="id"·title="lemma">choice</span></a>·:<br/> |
| 139 | <span·class="id"·title="keyword">forall</span>·(<a·id="A:37"·class="idref"·href="#A:37"><span·class="id"·title="binder">A</span></a>·<a·id="B:38"·class="idref"·href="#B:38"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:39"·class="idref"·href="#R:39"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:37"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#B:38"><span·class="id"·title="variable">B</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>),<br/> | 139 | <span·class="id"·title="keyword">forall</span>·(<a·id="A:37"·class="idref"·href="#A:37"><span·class="id"·title="binder">A</span></a>·<a·id="B:38"·class="idref"·href="#B:38"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:39"·class="idref"·href="#R:39"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:37"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#B:38"><span·class="id"·title="variable">B</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>),<br/> |
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| 110 | <div·class="paragraph">·</div> | 110 | <div·class="paragraph">·</div> |
| 111 | 4.2·Classical·logic·=·choice·of·representatives·in·a·partition·of·bool | 111 | 4.2·Classical·logic·=·choice·of·representatives·in·a·partition·of·bool |
| 112 | <div·class="paragraph">·</div> | 112 | <div·class="paragraph">·</div> |
| 113 | <a·id="lab | 113 | <a·id="lab959"></a><h1·class="section">Prop·degeneracy·=·excluded-middle·+·prop·extensionality</h1> |
| 114 | <div·class="paragraph">·</div> | 114 | <div·class="paragraph">·</div> |
| 115 | ·i.e.········<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">True</span></span>·<span·class="inlinecode">\/</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">False</span>)</span> | 115 | ·i.e.········<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">True</span></span>·<span·class="inlinecode">\/</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">False</span>)</span> |
| 116 | ·························<span·class="inlinecode"><-></span> | 116 | ·························<span·class="inlinecode"><-></span> |
| 117 | ·······<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>\/~<span·class="id"·title="var">A</span>)</span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span>,</span>·<span·class="inlinecode">(<span·class="id"·title="var">A</span><-><span·class="id"·title="var">B</span>)</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">B</span>)</span> | 117 | ·······<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>\/~<span·class="id"·title="var">A</span>)</span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span>,</span>·<span·class="inlinecode">(<span·class="id"·title="var">A</span><-><span·class="id"·title="var">B</span>)</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">B</span>)</span> |
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| 104 | <div·class="code"> | 104 | <div·class="code"> |
| 105 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#"><span·class="id"·title="library">ClassicalFacts</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#"><span·class="id"·title="library">ChoiceFacts</span></a>.<br/> | 105 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#"><span·class="id"·title="library">ClassicalFacts</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#"><span·class="id"·title="library">ChoiceFacts</span></a>.<br/> |
| 106 | <br/> | 106 | <br/> |
| 107 | </div> | 107 | </div> |
| 108 | <div·class="doc"> | 108 | <div·class="doc"> |
| 109 | <a·id="lab | 109 | <a·id="lab973"></a><h1·class="section">Pred.·Ext.·+·Rel.·Axiom·of·Choice·->·Excluded-Middle</h1> |
| 110 | </div> | 110 | </div> |
| 111 | <div·class="code"> | 111 | <div·class="code"> |
| 112 | <br/> | 112 | <br/> |
| 113 | <span·class="id"·title="keyword">Section</span>·<a·id="PredExt_RelChoice_imp_EM"·class="idref"·href="#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/> | 113 | <span·class="id"·title="keyword">Section</span>·<a·id="PredExt_RelChoice_imp_EM"·class="idref"·href="#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/> |
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| 112 | ···equalities·assuming·axiom·eq_rect_eq | 112 | ···equalities·assuming·axiom·eq_rect_eq |
| 113 | <div·class="paragraph">·</div> | 113 | <div·class="paragraph">·</div> |
| 114 | <div·class="paragraph">·</div> | 114 | <div·class="paragraph">·</div> |
| 115 | <a·id="lab | 115 | <a·id="lab944"></a><h1·class="section">Definition·of·dependent·equality·and·equivalence·with·equality·of·dependent·pairs</h1> |
| 116 | </div> | 116 | </div> |
| 117 | <div·class="code"> | 117 | <div·class="code"> |
| 118 | <br/> | 118 | <br/> |
| 119 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#EqNotations"><span·class="id"·title="module">EqNotations</span></a>.<br/> | 119 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#EqNotations"><span·class="id"·title="module">EqNotations</span></a>.<br/> |
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| 84 | ·····from·a·proof·of·decidability·of·equality·for·a·set·in·Set | 84 | ·····from·a·proof·of·decidability·of·equality·for·a·set·in·Set |
| 85 | <div·class="paragraph">·</div> | 85 | <div·class="paragraph">·</div> |
| 86 | <div·class="paragraph">·</div> | 86 | <div·class="paragraph">·</div> |
| 87 | <a·id="lab | 87 | <a·id="lab952"></a><h1·class="section">Streicher's·K·and·injectivity·of·dependent·pair·hold·on·decidable·types</h1> |
| 88 | </div> | 88 | </div> |
| 89 | <div·class="code"> | 89 | <div·class="code"> |
| 90 | <br/> | 90 | <br/> |
| 91 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 91 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
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| 101 | <br/> | 101 | <br/> |
| 102 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 102 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 103 | <br/> | 103 | <br/> |
| 104 | </div> | 104 | </div> |
| 105 | <div·class="doc"> | 105 | <div·class="doc"> |
| 106 | <a·id="lab | 106 | <a·id="lab955"></a><h1·class="section">Definitions</h1> |
| 107 | <div·class="paragraph">·</div> | 107 | <div·class="paragraph">·</div> |
| 108 | ·Being·an·inverse· | 108 | ·Being·an·inverse· |
| 109 | </div> | 109 | </div> |
| 110 | <div·class="code"> | 110 | <div·class="code"> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1014"></a><h1·class="section">Functions·on·finite·domains</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Main·result·:·for·functions·<span·class="inlinecode"><span·class="id"·title="var">f</span>:<span·class="id"·title="var">A</span>-><span·class="id"·title="var">A</span></span>·with·finite·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>, | 51 | ·Main·result·:·for·functions·<span·class="inlinecode"><span·class="id"·title="var">f</span>:<span·class="id"·title="var">A</span>-><span·class="id"·title="var">A</span></span>·with·finite·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>, |
| 52 | ····f·injective·<->·f·bijective·<->·f·surjective.· | 52 | ····f·injective·<->·f·bijective·<->·f·surjective.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
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| 172 | <br/> | 172 | <br/> |
| 173 | <br/> | 173 | <br/> |
| 174 | </div> | 174 | </div> |
| 175 | <div·class="doc"> | 175 | <div·class="doc"> |
| 176 | <a·id="lab | 176 | <a·id="lab976"></a><h1·class="section">A·modular·proof·of·Hurkens's·paradox.</h1> |
| 177 | <div·class="paragraph">·</div> | 177 | <div·class="paragraph">·</div> |
| 178 | ·It·relies·on·an·axiomatisation·of·a·shallow·embedding·of·system·U- | 178 | ·It·relies·on·an·axiomatisation·of·a·shallow·embedding·of·system·U- |
| 179 | ····(i.e.··types·of·U-·are·interpreted·by·types·of·Coq).·The | 179 | ····(i.e.··types·of·U-·are·interpreted·by·types·of·Coq).·The |
| 180 | ····universes·are·encoded·in·a·style,·due·to·Martin-Löf,·where·they | 180 | ····universes·are·encoded·in·a·style,·due·to·Martin-Löf,·where·they |
| 181 | ····are·given·by·a·set·of·names·and·a·family·<span·class="inlinecode"><span·class="id"·title="var">El</span>:<span·class="id"·title="var">Name</span>-><span·class="id"·title="keyword">Type</span></span>·which | 181 | ····are·given·by·a·set·of·names·and·a·family·<span·class="inlinecode"><span·class="id"·title="var">El</span>:<span·class="id"·title="var">Name</span>-><span·class="id"·title="keyword">Type</span></span>·which |
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| 113 | <br/> | 113 | <br/> |
| 114 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 114 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 115 | <br/> | 115 | <br/> |
| 116 | </div> | 116 | </div> |
| 117 | <div·class="doc"> | 117 | <div·class="doc"> |
| 118 | <a·id="lab | 118 | <a·id="lab947"></a><h1·class="section">Definitions</h1> |
| 119 | <div·class="paragraph">·</div> | 119 | <div·class="paragraph">·</div> |
| 120 | ·Propositional·extensionality· | 120 | ·Propositional·extensionality· |
| 121 | </div> | 121 | </div> |
| 122 | <div·class="code"> | 122 | <div·class="code"> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab943"></a><h1·class="section">Basic·facts·about·Prop·as·a·type</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·An·intuitionistic·theorem·from·topos·theory·<span·class="inlinecode">[<span·class="id"·title="var">LambekScott</span>]</span> | 51 | ·An·intuitionistic·theorem·from·topos·theory·<span·class="inlinecode">[<span·class="id"·title="var">LambekScott</span>]</span> |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1015"></a><h1·class="section">The·Set·universe·seen·as·a·synonym·for·Type</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·After·loading·this·file,·Set·becomes·just·another·name·for·Type. | 51 | ·After·loading·this·file,·Set·becomes·just·another·name·for·Type. |
| 52 | ····This·allows·easily·performing·a·Set-to-Type·migration,·or·at·least | 52 | ····This·allows·easily·performing·a·Set-to-Type·migration,·or·at·least |
| 53 | ····test·whether·a·development·relies·or·not·on·specific·features·of | 53 | ····test·whether·a·development·relies·or·not·on·specific·features·of |
| 54 | ····Set:·simply·insert·some·Require·Export·of·this·file·at·starting | 54 | ····Set:·simply·insert·some·Require·Export·of·this·file·at·starting |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1"></a><h1·class="section">MSetAVL·:·Implementation·of·MSetInterface·via·AVL·trees</h1> | 49 | <a·id="lab1164"></a><h1·class="section">MSetAVL·:·Implementation·of·MSetInterface·via·AVL·trees</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·implements·finite·sets·using·AVL·trees. | 51 | ·This·module·implements·finite·sets·using·AVL·trees. |
| 52 | ····It·follows·the·implementation·from·Ocaml's·standard·library, | 52 | ····It·follows·the·implementation·from·Ocaml's·standard·library, |
| 53 | <div·class="paragraph">·</div> | 53 | <div·class="paragraph">·</div> |
| Offset 71, 15 lines modified | Offset 71, 15 lines modified | ||
| 71 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecideOn"·class="idref"·href="#WDecideOn"><span·class="id"·title="module">WDecideOn</span></a>·(<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>)(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">M</span>·:·<a·class="idref"·href="Coq.MSets.MSetInterface.html#WSetsOn"><span·class="id"·title="module">WSetsOn</span></a>·<a·class="idref"·href="Coq.MSets.MSetDecide.html#E"><span·class="id"·title="module">E</span></a>).<br/> | 71 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecideOn"·class="idref"·href="#WDecideOn"><span·class="id"·title="module">WDecideOn</span></a>·(<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>)(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">M</span>·:·<a·class="idref"·href="Coq.MSets.MSetInterface.html#WSetsOn"><span·class="id"·title="module">WSetsOn</span></a>·<a·class="idref"·href="Coq.MSets.MSetDecide.html#E"><span·class="id"·title="module">E</span></a>).<br/> |
| 72 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecideOn.F"·class="idref"·href="#WDecideOn.F"><span·class="id"·title="module">F</span></a>·:=·<a·class="idref"·href="Coq.MSets.MSetFacts.html#WFactsOn"><span·class="id"·title="module">MSetFacts.WFactsOn</span></a>·<a·class="idref"·href="Coq.MSets.MSetDecide.html#E"><span·class="id"·title="module">E</span></a>·<a·class="idref"·href="Coq.MSets.MSetDecide.html#M"><span·class="id"·title="module">M</span></a>.<br/> | 72 | <span·class="id"·title="keyword">Module</span>·<a·id="WDecideOn.F"·class="idref"·href="#WDecideOn.F"><span·class="id"·title="module">F</span></a>·:=·<a·class="idref"·href="Coq.MSets.MSetFacts.html#WFactsOn"><span·class="id"·title="module">MSetFacts.WFactsOn</span></a>·<a·class="idref"·href="Coq.MSets.MSetDecide.html#E"><span·class="id"·title="module">E</span></a>·<a·class="idref"·href="Coq.MSets.MSetDecide.html#M"><span·class="id"·title="module">M</span></a>.<br/> |
| 73 | <br/> | 73 | <br/> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="doc"> | 75 | <div·class="doc"> |
| 76 | <a·id="lab | 76 | <a·id="lab1265"></a><h1·class="section">Overview</h1> |
| 77 | ····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will | 77 | ····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will |
| 78 | ····solve·any·valid·goal·of·the·form | 78 | ····solve·any·valid·goal·of·the·form |
| 79 | <pre> | 79 | <pre> |
| 80 | ····forall·s1·...·sn, | 80 | ····forall·s1·...·sn, |
| 81 | ····forall·x1·...·xm, | 81 | ····forall·x1·...·xm, |
| 82 | ····P1·->·...·->·Pk·->·P | 82 | ····P1·->·...·->·Pk·->·P |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1257"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·proves·many·properties·of·finite·sets·that | 51 | ·This·module·proves·many·properties·of·finite·sets·that |
| 52 | ····are·consequences·of·the·axiomatization·in·<span·class="inlinecode"><span·class="id"·title="var">FsetInterface</span></span> | 52 | ····are·consequences·of·the·axiomatization·in·<span·class="inlinecode"><span·class="id"·title="var">FsetInterface</span></span> |
| 53 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FsetProperties</span></span>·it·uses | 53 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FsetProperties</span></span>·it·uses |
| 54 | ····sets·operations·instead·of·predicates·over·sets,·i.e. | 54 | ····sets·operations·instead·of·predicates·over·sets,·i.e. |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1247"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>.·These | 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>.·These |
| 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>·written·using | 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>·written·using |
| 53 | ··different·styles:·equivalence·and·boolean·equalities. | 53 | ··different·styles:·equivalence·and·boolean·equalities. |
| 54 | ··Moreover,·we·prove·that·<span·class="inlinecode"><span·class="id"·title="var">E.Eq</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Equal</span></span>·are·setoid·equalities. | 54 | ··Moreover,·we·prove·that·<span·class="inlinecode"><span·class="id"·title="var">E.Eq</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Equal</span></span>·are·setoid·equalities. |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1137"></a><h1·class="section">MSetGenTree·:·sets·via·generic·trees</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····This·module·factorizes·common·parts·in·implementations | 51 | ····This·module·factorizes·common·parts·in·implementations |
| 52 | ····of·finite·sets·as·AVL·trees·and·as·Red-Black·trees.·The·nodes | 52 | ····of·finite·sets·as·AVL·trees·and·as·Red-Black·trees.·The·nodes |
| 53 | ····of·the·trees·defined·here·include·an·generic·information | 53 | ····of·the·trees·defined·here·include·an·generic·information |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1198"></a><h1·class="section">Finite·set·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Set·interfaces,·inspired·by·the·one·of·Ocaml.·When·compared·with | 51 | ·Set·interfaces,·inspired·by·the·one·of·Ocaml.·When·compared·with |
| 52 | ····Ocaml,·the·main·differences·are: | 52 | ····Ocaml,·the·main·differences·are: |
| 53 | <ul·class="doclist"> | 53 | <ul·class="doclist"> |
| 54 | <li>·the·lack·of·<span·class="inlinecode"><span·class="id"·title="var">iter</span></span>·function,·useless·since·Coq·is·purely·functional | 54 | <li>·the·lack·of·<span·class="inlinecode"><span·class="id"·title="var">iter</span></span>·function,·useless·since·Coq·is·purely·functional |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1260"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent |
| 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>·using·strictly·ordered·list.· | 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>·using·strictly·ordered·list.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1123"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·functor·derives·additional·properties·from·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>. | 51 | ·This·functor·derives·additional·properties·from·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>. |
| 52 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">MSetEqProperties</span></span>·it·uses | 52 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">MSetEqProperties</span></span>·it·uses |
| 53 | ····predicates·over·sets·instead·of·sets·operations,·i.e. | 53 | ····predicates·over·sets·instead·of·sets·operations,·i.e. |
| 54 | ····<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span></span>·instead·of·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span>=<span·class="id"·title="var">true</span></span>, | 54 | ····<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span></span>·instead·of·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">s</span>=<span·class="id"·title="var">true</span></span>, |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab10 | 49 | <a·id="lab1205"></a><h1·class="section">MSetRBT·:·Implementation·of·MSetInterface·via·Red-Black·trees</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Initial·author:·Andrew·W.·Appel,·2011. | 51 | ·Initial·author:·Andrew·W.·Appel,·2011. |
| 52 | ····Extra·modifications·by:·Pierre·Letouzey | 52 | ····Extra·modifications·by:·Pierre·Letouzey |
| 53 | <div·class="paragraph">·</div> | 53 | <div·class="paragraph">·</div> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab15 | 49 | <a·id="lab1258"></a><h1·class="section">Finite·sets·library·:·conversion·to·old·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span></h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Sets.Ensembles.html#"><span·class="id"·title="library">Ensembles</span></a>·<a·class="idref"·href="Coq.Sets.Finite_sets.html#"><span·class="id"·title="library">Finite_sets</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Sets.Ensembles.html#"><span·class="id"·title="library">Ensembles</span></a>·<a·class="idref"·href="Coq.Sets.Finite_sets.html#"><span·class="id"·title="library">Finite_sets</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.MSets.MSetInterface.html#"><span·class="id"·title="library">MSetInterface</span></a>·<a·class="idref"·href="Coq.MSets.MSetProperties.html#"><span·class="id"·title="library">MSetProperties</span></a>·<a·class="idref"·href="Coq.Structures.OrdersEx.html#"><span·class="id"·title="library">OrdersEx</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.MSets.MSetInterface.html#"><span·class="id"·title="library">MSetInterface</span></a>·<a·class="idref"·href="Coq.MSets.MSetProperties.html#"><span·class="id"·title="library">MSetProperties</span></a>·<a·class="idref"·href="Coq.Structures.OrdersEx.html#"><span·class="id"·title="library">OrdersEx</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab15 | 58 | <a·id="lab1259"></a><h1·class="section">Going·from·<span·class="inlinecode"><span·class="id"·title="var">MSets</span></span>·with·usual·Leibniz·equality</h1> |
| 59 | ····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.· | 59 | ····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.Equalities.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.MSets.MSetInterface.html#WSetsOn"><span·class="id"·title="module">WSetsOn</span></a>·<a·class="idref"·href="Coq.MSets.MSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/> | 63 | <span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.Equalities.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.MSets.MSetInterface.html#WSetsOn"><span·class="id"·title="module">WSetsOn</span></a>·<a·class="idref"·href="Coq.MSets.MSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1252"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent |
| 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">MSetWeakInterface.S</span></span>·using·lists·without·redundancy.· | 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">MSetWeakInterface.S</span></span>·using·lists·without·redundancy.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
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| 51 | <a·class="idref"·href="Coq.Structures.Equalities.html#"><span·class="id"·title="library">Equalities</span></a>·<a·class="idref"·href="Coq.Structures.OrdersFacts.html#"><span·class="id"·title="library">OrdersFacts</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#"><span·class="id"·title="library">NAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NMaxMin.html#"><span·class="id"·title="library">NMaxMin</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NProperties.html#"><span·class="id"·title="library">NProperties</span></a>.<br/> | 51 | <a·class="idref"·href="Coq.Structures.Equalities.html#"><span·class="id"·title="library">Equalities</span></a>·<a·class="idref"·href="Coq.Structures.OrdersFacts.html#"><span·class="id"·title="library">OrdersFacts</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#"><span·class="id"·title="library">NAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NMaxMin.html#"><span·class="id"·title="library">NMaxMin</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NProperties.html#"><span·class="id"·title="library">NProperties</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.NArith.BinNatDef.html#"><span·class="id"·title="library">BinNatDef</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.NArith.BinNatDef.html#"><span·class="id"·title="library">BinNatDef</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab1295"></a><h1·class="section">Binary·natural·numbers,·operations·and·properties</h1> |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">N0</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Npos</span></span>·are·now | 58 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">N0</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Npos</span></span>·are·now |
| 59 | ····defined·in·<span·class="inlinecode"><span·class="id"·title="var">BinNums.v</span></span>· | 59 | ····defined·in·<span·class="inlinecode"><span·class="id"·title="var">BinNums.v</span></span>· |
| 60 | <div·class="paragraph">·</div> | 60 | <div·class="paragraph">·</div> |
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| 57 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Npos"><span·class="id"·title="constructor">Npos</span></a>·1).<br/> | 57 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Npos"><span·class="id"·title="constructor">Npos</span></a>·1).<br/> |
| 58 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'2'"·class="idref"·href="#:::'2'"><span·class="id"·title="notation">"</span></a>2"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Npos"><span·class="id"·title="constructor">Npos</span></a>·2).<br/> | 58 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'2'"·class="idref"·href="#:::'2'"><span·class="id"·title="notation">"</span></a>2"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Npos"><span·class="id"·title="constructor">Npos</span></a>·2).<br/> |
| 59 | <br/> | 59 | <br/> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="doc"> | 61 | <div·class="doc"> |
| 62 | <a·id="lab | 62 | <a·id="lab1297"></a><h1·class="section">Binary·natural·numbers,·definitions·of·operations</h1> |
| 63 | </div> | 63 | </div> |
| 64 | <div·class="code"> | 64 | <div·class="code"> |
| 65 | <br/> | 65 | <br/> |
| 66 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>.<br/> | 66 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>.<br/> |
| 67 | <br/> | 67 | <br/> |
| 68 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.t"·class="idref"·href="#N.t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.Numbers.BinNums.html#N"><span·class="id"·title="inductive">N</span></a>.<br/> | 68 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.t"·class="idref"·href="#N.t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.Numbers.BinNums.html#N"><span·class="id"·title="inductive">N</span></a>.<br/> |
| 69 | <br/> | 69 | <br/> |
| 70 | </div> | 70 | </div> |
| 71 | <div·class="doc"> | 71 | <div·class="doc"> |
| 72 | <a·id="lab | 72 | <a·id="lab1298"></a><h2·class="section">Nicer·name·<span·class="inlinecode"><span·class="id"·title="var">N.pos</span></span>·for·constructor·<span·class="inlinecode"><span·class="id"·title="var">Npos</span></span></h2> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="code"> | 74 | <div·class="code"> |
| 75 | <br/> | 75 | <br/> |
| 76 | <span·class="id"·title="keyword">Notation</span>·<a·id="N.pos"·class="idref"·href="#N.pos"><span·class="id"·title="abbreviation">pos</span></a>·:=·<a·class="idref"·href="Coq.Numbers.BinNums.html#Npos"><span·class="id"·title="constructor">Npos</span></a>.<br/> | 76 | <span·class="id"·title="keyword">Notation</span>·<a·id="N.pos"·class="idref"·href="#N.pos"><span·class="id"·title="abbreviation">pos</span></a>·:=·<a·class="idref"·href="Coq.Numbers.BinNums.html#Npos"><span·class="id"·title="constructor">Npos</span></a>.<br/> |
| 77 | <br/> | 77 | <br/> |
| 78 | </div> | 78 | </div> |
| 79 | <div·class="doc"> | 79 | <div·class="doc"> |
| 80 | <a·id="lab | 80 | <a·id="lab1299"></a><h2·class="section">Constants</h2> |
| 81 | </div> | 81 | </div> |
| 82 | <div·class="code"> | 82 | <div·class="code"> |
| 83 | <br/> | 83 | <br/> |
| 84 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.zero"·class="idref"·href="#N.zero"><span·class="id"·title="definition">zero</span></a>·:=·0.<br/> | 84 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.zero"·class="idref"·href="#N.zero"><span·class="id"·title="definition">zero</span></a>·:=·0.<br/> |
| 85 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.one"·class="idref"·href="#N.one"><span·class="id"·title="definition">one</span></a>·:=·1.<br/> | 85 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.one"·class="idref"·href="#N.one"><span·class="id"·title="definition">one</span></a>·:=·1.<br/> |
| 86 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.two"·class="idref"·href="#N.two"><span·class="id"·title="definition">two</span></a>·:=·2.<br/> | 86 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.two"·class="idref"·href="#N.two"><span·class="id"·title="definition">two</span></a>·:=·2.<br/> |
| 87 | <br/> | 87 | <br/> |
| 88 | </div> | 88 | </div> |
| 89 | <div·class="doc"> | 89 | <div·class="doc"> |
| 90 | <a·id="lab | 90 | <a·id="lab1300"></a><h2·class="section">Operation·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode">-></span>·<span·class="inlinecode">2*<span·class="id"·title="var">x</span>+1</span></h2> |
| 91 | </div> | 91 | </div> |
| 92 | <div·class="code"> | 92 | <div·class="code"> |
| 93 | <br/> | 93 | <br/> |
| 94 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.succ_double"·class="idref"·href="#N.succ_double"><span·class="id"·title="definition">succ_double</span></a>·<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·:=<br/> | 94 | <span·class="id"·title="keyword">Definition</span>·<a·id="N.succ_double"·class="idref"·href="#N.succ_double"><span·class="id"·title="definition">succ_double</span></a>·<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·:=<br/> |
| 95 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.NArith.BinNatDef.html#x:1"><span·class="id"·title="variable">x</span></a>·<span·class="id"·title="keyword">with</span><br/> | 95 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.NArith.BinNatDef.html#x:1"><span·class="id"·title="variable">x</span></a>·<span·class="id"·title="keyword">with</span><br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.PArith.BinPos.html#"><span·class="id"·title="library">BinPos</span></a>·<a·class="idref"·href="Coq.NArith.BinNat.html#"><span·class="id"·title="library">BinNat</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>·<a·class="idref"·href="Coq.PArith.Pnat.html#"><span·class="id"·title="library">Pnat</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.PArith.BinPos.html#"><span·class="id"·title="library">BinPos</span></a>·<a·class="idref"·href="Coq.NArith.BinNat.html#"><span·class="id"·title="library">BinNat</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>·<a·class="idref"·href="Coq.PArith.Pnat.html#"><span·class="id"·title="library">Pnat</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab1306"></a><h1·class="section">Conversions·from·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>·to·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span></h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Module</span>·<a·id="N2Nat"·class="idref"·href="#N2Nat"><span·class="id"·title="module">N2Nat</span></a>.<br/> | 57 | <span·class="id"·title="keyword">Module</span>·<a·id="N2Nat"·class="idref"·href="#N2Nat"><span·class="id"·title="module">N2Nat</span></a>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab729"></a><h1·class="section">Alternative·Binary·Number·Notations</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Faster·but·less·safe·parsers·and·printers·of·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>.· | 51 | ·Faster·but·less·safe·parsers·and·printers·of·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | ·By·default,·literals·in·types·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·are·parsed·and | 53 | ·By·default,·literals·in·types·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·are·parsed·and |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab731"></a><h1·class="section">Binary·Numerical·Datatypes</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 53 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab794"></a><h1·class="section">Signature·and·specification·of·bounded·integers</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·specifies·<span·class="inlinecode"><span·class="id"·title="var">d</span></span>-bit·integers·as·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·with·<span·class="inlinecode"><span·class="id"·title="var">n</span>=2^<span·class="id"·title="var">d</span></span>· | 51 | ·This·file·specifies·<span·class="inlinecode"><span·class="id"·title="var">d</span></span>-bit·integers·as·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·with·<span·class="inlinecode"><span·class="id"·title="var">n</span>=2^<span·class="id"·title="var">d</span></span>· |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
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| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Abstract.CyclicAxioms.html#"><span·class="id"·title="library">CyclicAxioms</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Abstract.CyclicAxioms.html#"><span·class="id"·title="library">CyclicAxioms</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab793"></a><h1·class="section">From·<span·class="inlinecode"><span·class="id"·title="var">CyclicType</span></span>·to·<span·class="inlinecode"><span·class="id"·title="var">NZAxiomsSig</span></span></h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·A·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·representation·given·by·a·module·type·<span·class="inlinecode"><span·class="id"·title="var">CyclicType</span></span> | 60 | ·A·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·representation·given·by·a·module·type·<span·class="inlinecode"><span·class="id"·title="var">CyclicType</span></span> |
| 61 | ····implements·<span·class="inlinecode"><span·class="id"·title="var">NZAxiomsSig</span></span>,·e.g.·the·common·properties·between | 61 | ····implements·<span·class="inlinecode"><span·class="id"·title="var">NZAxiomsSig</span></span>,·e.g.·the·common·properties·between |
| 62 | ····N·and·Z·with·no·ordering.·Notice·that·the·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>·in·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·is | 62 | ····N·and·Z·with·no·ordering.·Notice·that·the·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>·in·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·is |
| 63 | ····a·power·of·2. | 63 | ····a·power·of·2. |
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| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab770"></a><h1·class="section">Int31·numbers·defines·indeed·a·cyclic·structure·:·Z/(2^31)Z</h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | <div·class="paragraph">·</div> | 60 | <div·class="paragraph">·</div> |
| 61 | Author:·Arnaud·Spiwack·(+·Pierre·Letouzey) | 61 | Author:·Arnaud·Spiwack·(+·Pierre·Letouzey) |
| Offset 61, 15 lines modified | Offset 61, 15 lines modified | ||
| 61 | <br/> | 61 | <br/> |
| 62 | <br/> | 62 | <br/> |
| 63 | </div> | 63 | </div> |
| 64 | <div·class="doc"> | 64 | <div·class="doc"> |
| 65 | <a·id="lab | 65 | <a·id="lab784"></a><h1·class="section">31-bit·integers</h1> |
| 66 | <div·class="paragraph">·</div> | 66 | <div·class="paragraph">·</div> |
| 67 | ·This·file·contains·basic·definitions·of·a·31-bit·integer | 67 | ·This·file·contains·basic·definitions·of·a·31-bit·integer |
| 68 | ··arithmetic.·In·fact·it·is·more·general·than·that.·The·only·reason | 68 | ··arithmetic.·In·fact·it·is·more·general·than·that.·The·only·reason |
| 69 | ··for·this·use·of·31·is·the·underlying·mechanism·for·hardware-efficient | 69 | ··for·this·use·of·31·is·the·underlying·mechanism·for·hardware-efficient |
| 70 | ··computations·by·A.·Spiwack.·Apart·from·this,·a·switch·to,·say, | 70 | ··computations·by·A.·Spiwack.·Apart·from·this,·a·switch·to,·say, |
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| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab783"></a><h1·class="section">Int31·numbers·defines·Z/(2^31)Z,·and·can·hence·be·equipped</h1> |
| 59 | ······with·a·ring·structure·and·a·ring·tactic· | 59 | ······with·a·ring·structure·and·a·ring·tactic· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int31.Int31.html#"><span·class="id"·title="library">Int31</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int31.Cyclic31.html#"><span·class="id"·title="library">Cyclic31</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Abstract.CyclicAxioms.html#"><span·class="id"·title="library">CyclicAxioms</span></a>.<br/> | 63 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int31.Int31.html#"><span·class="id"·title="library">Int31</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int31.Cyclic31.html#"><span·class="id"·title="library">Cyclic31</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Abstract.CyclicAxioms.html#"><span·class="id"·title="library">CyclicAxioms</span></a>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab769"></a><h1·class="section">Uint63·numbers·defines·indeed·a·cyclic·structure·:·Z/(2^63)Z</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | <div·class="paragraph">·</div> | 51 | <div·class="paragraph">·</div> |
| 52 | Author:·Arnaud·Spiwack·(+·Pierre·Letouzey) | 52 | Author:·Arnaud·Spiwack·(+·Pierre·Letouzey) |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab768"></a><h1·class="section">Uint63·numbers·defines·Z/(2^63)Z,·and·can·hence·be·equipped</h1> |
| 50 | ······with·a·ring·structure·and·a·ring·tactic· | 50 | ······with·a·ring·structure·and·a·ring·tactic· |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="code"> | 52 | <div·class="code"> |
| 53 | <br/> | 53 | <br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Cyclic63.html#"><span·class="id"·title="library">Cyclic63</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Abstract.CyclicAxioms.html#"><span·class="id"·title="library">CyclicAxioms</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Cyclic63.html#"><span·class="id"·title="library">Cyclic63</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Abstract.CyclicAxioms.html#"><span·class="id"·title="library">CyclicAxioms</span></a>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab795"></a><h1·class="section">Type·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·viewed·modulo·<span·class="inlinecode">2^<span·class="id"·title="var">d</span></span>·implements·CyclicAxioms.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·library·has·been·deprecated·since·Coq·version·8.17.· | 51 | ·This·library·has·been·deprecated·since·Coq·version·8.17.· |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab747"></a><h1·class="section">DecimalFacts·:·some·facts·about·Decimal·numbers</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab741"></a><h1·class="section">DecimalN</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">N</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">N</span></span> |
| 52 | ····are·bijections· | 52 | ····are·bijections· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab730"></a><h1·class="section">DecimalNat</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab802"></a><h1·class="section">DecimalPos</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab728"></a><h1·class="section">DecimalQ</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab804"></a><h1·class="section">DecimalR</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">R</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">R</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab745"></a><h1·class="section">Conversion·between·decimal·numbers·and·Coq·strings</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ·Pretty·straightforward,·which·is·precisely·the·point·of·the | 55 | ·Pretty·straightforward,·which·is·precisely·the·point·of·the |
| 56 | ····<span·class="inlinecode"><span·class="id"·title="var">Decimal.int</span></span>·datatype.·The·only·catch·is·<span·class="inlinecode"><span·class="id"·title="var">Decimal.Nil</span></span>·:·we·could | 56 | ····<span·class="inlinecode"><span·class="id"·title="var">Decimal.int</span></span>·datatype.·The·only·catch·is·<span·class="inlinecode"><span·class="id"·title="var">Decimal.Nil</span></span>·:·we·could |
| 57 | ····choose·to·convert·it·as·<span·class="inlinecode">""</span>·or·as·<span·class="inlinecode">"0"</span>.·In·the·first·case,·it·is | 57 | ····choose·to·convert·it·as·<span·class="inlinecode">""</span>·or·as·<span·class="inlinecode">"0"</span>.·In·the·first·case,·it·is |
| 58 | ····awkward·to·consider·""·(or·"-")·as·a·number,·while·in·the·second·case | 58 | ····awkward·to·consider·""·(or·"-")·as·a·number,·while·in·the·second·case |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab740"></a><h1·class="section">DecimalZ</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab766"></a><h1·class="section">HexadecimalFacts·:·some·facts·about·Hexadecimal·numbers</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab803"></a><h1·class="section">HexadecimalN</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">N</span></span> | 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">N</span></span> |
| 52 | ····are·bijections· | 52 | ····are·bijections· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab767"></a><h1·class="section">HexadecimalNat</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span> | 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab748"></a><h1·class="section">HexadecimalPos</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span> | 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab765"></a><h1·class="section">HexadecimalQ</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span> | 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab742"></a><h1·class="section">HexadecimalR</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">R</span></span> | 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">R</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab746"></a><h1·class="section">Conversion·between·hexadecimal·numbers·and·Coq·strings</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ·Pretty·straightforward,·which·is·precisely·the·point·of·the | 55 | ·Pretty·straightforward,·which·is·precisely·the·point·of·the |
| 56 | ····<span·class="inlinecode"><span·class="id"·title="var">Hexadecimal.int</span></span>·datatype.·The·only·catch·is·<span·class="inlinecode"><span·class="id"·title="var">Hexadecimal.Nil</span></span>·:·we·could | 56 | ····<span·class="inlinecode"><span·class="id"·title="var">Hexadecimal.int</span></span>·datatype.·The·only·catch·is·<span·class="inlinecode"><span·class="id"·title="var">Hexadecimal.Nil</span></span>·:·we·could |
| 57 | ····choose·to·convert·it·as·<span·class="inlinecode">""</span>·or·as·<span·class="inlinecode">"0"</span>.·In·the·first·case,·it·is | 57 | ····choose·to·convert·it·as·<span·class="inlinecode">""</span>·or·as·<span·class="inlinecode">"0"</span>.·In·the·first·case,·it·is |
| 58 | ····awkward·to·consider·""·(or·"-")·as·a·number,·while·in·the·second·case | 58 | ····awkward·to·consider·""·(or·"-")·as·a·number,·while·in·the·second·case |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab744"></a><h1·class="section">HexadecimalZ</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span> | 51 | ····Proofs·that·conversions·between·hexadecimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
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| 469 | <a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">exists</span></a>·<a·id="n:114"·class="idref"·href="#n:114"><span·class="id"·title="binder">n</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">,</span></a>·<span·class="id"·title="keyword">forall</span>·<a·id="m:115"·class="idref"·href="#m:115"><span·class="id"·title="binder">m</span></a>,·0<a·class="idref"·href="Coq.Structures.Orders.html#6733f1c73babd16d47d3db78628c74d6"><span·class="id"·title="notation"><=</span></a><a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:115"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#f:113"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:115"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#n:114"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZBits.html#08f35bb78d3f8b8e58903b9564c9213c"><span·class="id"·title="notation">.[</span></a><a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:115"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZBits.html#08f35bb78d3f8b8e58903b9564c9213c"><span·class="id"·title="notation">]</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a><br/> | 469 | <a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">exists</span></a>·<a·id="n:114"·class="idref"·href="#n:114"><span·class="id"·title="binder">n</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">,</span></a>·<span·class="id"·title="keyword">forall</span>·<a·id="m:115"·class="idref"·href="#m:115"><span·class="id"·title="binder">m</span></a>,·0<a·class="idref"·href="Coq.Structures.Orders.html#6733f1c73babd16d47d3db78628c74d6"><span·class="id"·title="notation"><=</span></a><a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:115"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#f:113"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:115"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#n:114"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZBits.html#08f35bb78d3f8b8e58903b9564c9213c"><span·class="id"·title="notation">.[</span></a><a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:115"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZBits.html#08f35bb78d3f8b8e58903b9564c9213c"><span·class="id"·title="notation">]</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a><br/> |
| 470 | <a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">exists</span></a>·<a·id="k:116"·class="idref"·href="#k:116"><span·class="id"·title="binder">k</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">,</span></a>·<span·class="id"·title="keyword">forall</span>·<a·id="m:117"·class="idref"·href="#m:117"><span·class="id"·title="binder">m</span></a>,·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#k:116"><span·class="id"·title="variable">k</span></a><a·class="idref"·href="Coq.Structures.Orders.html#6733f1c73babd16d47d3db78628c74d6"><span·class="id"·title="notation"><=</span></a><a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:117"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#f:113"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:117"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#f:113"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#k:116"><span·class="id"·title="variable">k</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">)</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">)</span></a>.<br/> | 470 | <a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">exists</span></a>·<a·id="k:116"·class="idref"·href="#k:116"><span·class="id"·title="binder">k</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">,</span></a>·<span·class="id"·title="keyword">forall</span>·<a·id="m:117"·class="idref"·href="#m:117"><span·class="id"·title="binder">m</span></a>,·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#k:116"><span·class="id"·title="variable">k</span></a><a·class="idref"·href="Coq.Structures.Orders.html#6733f1c73babd16d47d3db78628c74d6"><span·class="id"·title="notation"><=</span></a><a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:117"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#f:113"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#m:117"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#f:113"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZBits.html#k:116"><span·class="id"·title="variable">k</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">)</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">)</span></a>.<br/> |
| 471 | <br/> | 471 | <br/> |
| 472 | </div> | 472 | </div> |
| 473 | <div·class="doc"> | 473 | <div·class="doc"> |
| 474 | <a·id="lab | 474 | <a·id="lab749"></a><h1·class="section">Properties·of·shifts</h1> |
| 475 | <div·class="paragraph">·</div> | 475 | <div·class="paragraph">·</div> |
| 476 | ·First,·a·unified·specification·for·<span·class="inlinecode"><span·class="id"·title="var">shiftl</span></span>·:·the·<span·class="inlinecode"><span·class="id"·title="var">shiftl_spec</span></span> | 476 | ·First,·a·unified·specification·for·<span·class="inlinecode"><span·class="id"·title="var">shiftl</span></span>·:·the·<span·class="inlinecode"><span·class="id"·title="var">shiftl_spec</span></span> |
| 477 | ···below·(combined·with·<span·class="inlinecode"><span·class="id"·title="var">testbit_neg_r</span></span>)·is·equivalent·to | 477 | ···below·(combined·with·<span·class="inlinecode"><span·class="id"·title="var">testbit_neg_r</span></span>)·is·equivalent·to |
| 478 | ···<span·class="inlinecode"><span·class="id"·title="var">shiftl_spec_low</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">shiftl_spec_high</span></span>.· | 478 | ···<span·class="inlinecode"><span·class="id"·title="var">shiftl_spec_low</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">shiftl_spec_high</span></span>.· |
| 479 | </div> | 479 | </div> |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab759"></a><h1·class="section">Euclidean·Division·for·integers,·Euclid·convention</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ····We·use·here·the·"usual"·formulation·of·the·Euclid·Theorem | 55 | ····We·use·here·the·"usual"·formulation·of·the·Euclid·Theorem |
| 56 | ····<span·class="inlinecode"><span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span><>0</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="tactic">exists</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><span·class="id"·title="var">q</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode">=</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>*<span·class="id"·title="var">q</span>+<span·class="id"·title="var">r</span></span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">0</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><</span>·<span·class="inlinecode">|<span·class="id"·title="var">b</span>|</span>·<span·class="inlinecode"></span> | 56 | ····<span·class="inlinecode"><span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span><>0</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="tactic">exists</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><span·class="id"·title="var">q</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode">=</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>*<span·class="id"·title="var">q</span>+<span·class="id"·title="var">r</span></span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">0</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><</span>·<span·class="inlinecode">|<span·class="id"·title="var">b</span>|</span>·<span·class="inlinecode"></span> |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab755"></a><h1·class="section">Euclidean·Division·for·integers·(Floor·convention)</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ····We·use·here·the·convention·known·as·Floor,·or·Round-Toward-Bottom, | 55 | ····We·use·here·the·convention·known·as·Floor,·or·Round-Toward-Bottom, |
| 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·closest·integer·below·the·exact·fraction. | 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·closest·integer·below·the·exact·fraction. |
| 57 | ····It·can·be·summarized·by: | 57 | ····It·can·be·summarized·by: |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab751"></a><h1·class="section">Euclidean·Division·for·integers·(Trunc·convention)</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ····We·use·here·the·convention·known·as·Trunc,·or·Round-Toward-Zero, | 55 | ····We·use·here·the·convention·known·as·Trunc,·or·Round-Toward-Zero, |
| 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·integer·with·the·largest·absolute·value·to | 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·integer·with·the·largest·absolute·value·to |
| 57 | ····be·between·zero·and·the·exact·fraction.·It·can·be·summarized·by: | 57 | ····be·between·zero·and·the·exact·fraction.·It·can·be·summarized·by: |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZGcd.html#"><span·class="id"·title="library">ZGcd</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivTrunc.html#"><span·class="id"·title="library">ZDivTrunc</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivFloor.html#"><span·class="id"·title="library">ZDivFloor</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZGcd.html#"><span·class="id"·title="library">ZGcd</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivTrunc.html#"><span·class="id"·title="library">ZDivTrunc</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivFloor.html#"><span·class="id"·title="library">ZDivFloor</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab763"></a><h1·class="section">Least·Common·Multiple</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ·Unlike·other·functions·around,·we·will·define·lcm·below·instead·of | 55 | ·Unlike·other·functions·around,·we·will·define·lcm·below·instead·of |
| 56 | ··axiomatizing·it.·Indeed,·there·is·no·"prior·art"·about·lcm·in·the | 56 | ··axiomatizing·it.·Indeed,·there·is·no·"prior·art"·about·lcm·in·the |
| 57 | ··standard·library·to·be·compliant·with,·and·the·generic·definition | 57 | ··standard·library·to·be·compliant·with,·and·the·generic·definition |
| 58 | ··of·lcm·via·gcd·is·quite·reasonable. | 58 | ··of·lcm·via·gcd·is·quite·reasonable. |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab750"></a><h1·class="section">Properties·of·minimum·and·maximum·specific·to·integer·numbers</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="ZMaxMinProp"·class="idref"·href="#ZMaxMinProp"><span·class="id"·title="module">ZMaxMinProp</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">Z</span>·:·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#ZAxiomsMiniSig'"><span·class="id"·title="module">ZAxiomsMiniSig'</span></a>).<br/> | 57 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="ZMaxMinProp"·class="idref"·href="#ZMaxMinProp"><span·class="id"·title="module">ZMaxMinProp</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">Z</span>·:·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#ZAxiomsMiniSig'"><span·class="id"·title="module">ZAxiomsMiniSig'</span></a>).<br/> |
| 58 | <span·class="id"·title="keyword">Include</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#ZMulOrderProp"><span·class="id"·title="module">ZMulOrderProp</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMaxMin.html#Z"><span·class="id"·title="module">Z</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Include</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#ZMulOrderProp"><span·class="id"·title="module">ZMulOrderProp</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMaxMin.html#Z"><span·class="id"·title="module">Z</span></a>.<br/> |
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| 65 | <:·<a·class="idref"·href="Coq.Structures.Equalities.html#UsualDecidableTypeFull"><span·class="id"·title="module">UsualDecidableTypeFull</span></a><br/> | 65 | <:·<a·class="idref"·href="Coq.Structures.Equalities.html#UsualDecidableTypeFull"><span·class="id"·title="module">UsualDecidableTypeFull</span></a><br/> |
| 66 | :=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z"><span·class="id"·title="module">BinInt.Z</span></a>.<br/> | 66 | :=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z"><span·class="id"·title="module">BinInt.Z</span></a>.<br/> |
| 67 | <br/> | 67 | <br/> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="doc"> | 69 | <div·class="doc"> |
| 70 | <a·id="lab | 70 | <a·id="lab764"></a><h1·class="section">An·<span·class="inlinecode"><span·class="id"·title="var">order</span></span>·tactic·for·integers</h1> |
| 71 | </div> | 71 | </div> |
| 72 | <div·class="code"> | 72 | <div·class="code"> |
| 73 | <br/> | 73 | <br/> |
| 74 | <span·class="id"·title="keyword">Ltac</span>·<span·class="id"·title="var">z_order</span>·:=·<span·class="id"·title="var">Z.order</span>.<br/> | 74 | <span·class="id"·title="keyword">Ltac</span>·<span·class="id"·title="var">z_order</span>·:=·<span·class="id"·title="var">Z.order</span>.<br/> |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.List.html#"><span·class="id"·title="library">List</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.List.html#"><span·class="id"·title="library">List</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab743"></a><h1·class="section">Generic·dependently-typed·operators·about·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>-ary·functions</h1> |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·The·type·of·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>-ary·function:·<span·class="inlinecode"><span·class="id"·title="var">nfun</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span></span>·is | 58 | ·The·type·of·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>-ary·function:·<span·class="inlinecode"><span·class="id"·title="var">nfun</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span></span>·is |
| 59 | ····<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode">-></span>·<span·class="inlinecode">...</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span></span>·with·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>·occurrences·of·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·in·this·type.· | 59 | ····<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode">-></span>·<span·class="inlinecode">...</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span></span>·with·<span·class="inlinecode"><span·class="id"·title="var">n</span></span>·occurrences·of·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·in·this·type.· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
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| 186 | <br/> | 186 | <br/> |
| 187 | <span·class="id"·title="keyword">Theorem</span>·<a·id="NZDivProp.mod_small"·class="idref"·href="#NZDivProp.mod_small"><span·class="id"·title="lemma">mod_small</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:41"·class="idref"·href="#a:41"><span·class="id"·title="binder">a</span></a>·<a·id="b:42"·class="idref"·href="#b:42"><span·class="id"·title="binder">b</span></a>,·0<a·class="idref"·href="Coq.Structures.Orders.html#62fc7208ddb2fa310e7ae9ec2d794c92"><span·class="id"·title="notation"><=</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:41"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.Structures.Orders.html#62fc7208ddb2fa310e7ae9ec2d794c92"><span·class="id"·title="notation"><</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#b:42"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:41"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#DivModNotation.:::x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#b:42"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:41"><span·class="id"·title="variable">a</span></a>.<br/> | 187 | <span·class="id"·title="keyword">Theorem</span>·<a·id="NZDivProp.mod_small"·class="idref"·href="#NZDivProp.mod_small"><span·class="id"·title="lemma">mod_small</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:41"·class="idref"·href="#a:41"><span·class="id"·title="binder">a</span></a>·<a·id="b:42"·class="idref"·href="#b:42"><span·class="id"·title="binder">b</span></a>,·0<a·class="idref"·href="Coq.Structures.Orders.html#62fc7208ddb2fa310e7ae9ec2d794c92"><span·class="id"·title="notation"><=</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:41"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.Structures.Orders.html#62fc7208ddb2fa310e7ae9ec2d794c92"><span·class="id"·title="notation"><</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#b:42"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:41"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#DivModNotation.:::x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#b:42"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:41"><span·class="id"·title="variable">a</span></a>.<br/> |
| 188 | <br/> | 188 | <br/> |
| 189 | </div> | 189 | </div> |
| 190 | <div·class="doc"> | 190 | <div·class="doc"> |
| 191 | <a·id="lab | 191 | <a·id="lab737"></a><h1·class="section">Basic·values·of·divisions·and·modulo.</h1> |
| 192 | </div> | 192 | </div> |
| 193 | <div·class="code"> | 193 | <div·class="code"> |
| 194 | <br/> | 194 | <br/> |
| 195 | <span·class="id"·title="keyword">Lemma</span>·<a·id="NZDivProp.div_0_l"·class="idref"·href="#NZDivProp.div_0_l"><span·class="id"·title="lemma">div_0_l</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:43"·class="idref"·href="#a:43"><span·class="id"·title="binder">a</span></a>,·0<a·class="idref"·href="Coq.Structures.Orders.html#LtNotation.:::x_'<'_x"><span·class="id"·title="notation"><</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:43"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·0<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#553b1e0c6dfeea0fb5d9a4a0a1c7f719"><span·class="id"·title="notation">/</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:43"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·0.<br/> | 195 | <span·class="id"·title="keyword">Lemma</span>·<a·id="NZDivProp.div_0_l"·class="idref"·href="#NZDivProp.div_0_l"><span·class="id"·title="lemma">div_0_l</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:43"·class="idref"·href="#a:43"><span·class="id"·title="binder">a</span></a>,·0<a·class="idref"·href="Coq.Structures.Orders.html#LtNotation.:::x_'<'_x"><span·class="id"·title="notation"><</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:43"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·0<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#553b1e0c6dfeea0fb5d9a4a0a1c7f719"><span·class="id"·title="notation">/</span></a><a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:43"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·0.<br/> |
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| 72 | <span·class="id"·title="keyword">Module</span>·<a·id="NZDomainProp"·class="idref"·href="#NZDomainProp"><span·class="id"·title="module">NZDomainProp</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">NZ</span>:<a·class="idref"·href="Coq.Numbers.NatInt.NZAxioms.html#NZDomainSig'"><span·class="id"·title="module">NZDomainSig'</span></a>).<br/> | 72 | <span·class="id"·title="keyword">Module</span>·<a·id="NZDomainProp"·class="idref"·href="#NZDomainProp"><span·class="id"·title="module">NZDomainProp</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">NZ</span>:<a·class="idref"·href="Coq.Numbers.NatInt.NZAxioms.html#NZDomainSig'"><span·class="id"·title="module">NZDomainSig'</span></a>).<br/> |
| 73 | <span·class="id"·title="keyword">Include</span>·<a·class="idref"·href="Coq.Numbers.NatInt.NZBase.html#NZBaseProp"><span·class="id"·title="module">NZBaseProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDomain.html#NZ"><span·class="id"·title="module">NZ</span></a>.<br/> | 73 | <span·class="id"·title="keyword">Include</span>·<a·class="idref"·href="Coq.Numbers.NatInt.NZBase.html#NZBaseProp"><span·class="id"·title="module">NZBaseProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDomain.html#NZ"><span·class="id"·title="module">NZ</span></a>.<br/> |
| 74 | <br/> | 74 | <br/> |
| 75 | </div> | 75 | </div> |
| 76 | <div·class="doc"> | 76 | <div·class="doc"> |
| 77 | <a·id="lab | 77 | <a·id="lab734"></a><h1·class="section">Relationship·between·points·thanks·to·<span·class="inlinecode"><span·class="id"·title="var">succ</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">pred</span></span>.</h1> |
| 78 | <div·class="paragraph">·</div> | 78 | <div·class="paragraph">·</div> |
| 79 | ·For·any·two·points,·one·is·an·iterated·successor·of·the·other.· | 79 | ·For·any·two·points,·one·is·an·iterated·successor·of·the·other.· |
| 80 | </div> | 80 | </div> |
| 81 | <div·class="code"> | 81 | <div·class="code"> |
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| 414 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZPow.html#NZPowProp"><span·class="id"·title="module">NZPowProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>)<br/> | 414 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZPow.html#NZPowProp"><span·class="id"·title="module">NZPowProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>)<br/> |
| 415 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">F</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#NZLog2Prop"><span·class="id"·title="module">NZLog2Prop</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#C"><span·class="id"·title="module">C</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#E"><span·class="id"·title="module">E</span></a>).<br/> | 415 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">F</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#NZLog2Prop"><span·class="id"·title="module">NZLog2Prop</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#C"><span·class="id"·title="module">C</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#E"><span·class="id"·title="module">E</span></a>).<br/> |
| 416 | <br/> | 416 | <br/> |
| 417 | </div> | 417 | </div> |
| 418 | <div·class="doc"> | 418 | <div·class="doc"> |
| 419 | <a·id="lab | 419 | <a·id="lab733"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">log2_up</span></span>·:·a·binary·logarithm·that·rounds·up·instead·of·down</h1> |
| 420 | <div·class="paragraph">·</div> | 420 | <div·class="paragraph">·</div> |
| 421 | ·For·once,·we·define·instead·of·axiomatizing,·thanks·to·log2· | 421 | ·For·once,·we·define·instead·of·axiomatizing,·thanks·to·log2· |
| 422 | </div> | 422 | </div> |
| 423 | <div·class="code"> | 423 | <div·class="code"> |
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| 344 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">C</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZMulOrder.html#NZMulOrderProp"><span·class="id"·title="module">NZMulOrderProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#A"><span·class="id"·title="module">A</span></a>)<br/> | 344 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">C</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZMulOrder.html#NZMulOrderProp"><span·class="id"·title="module">NZMulOrderProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#A"><span·class="id"·title="module">A</span></a>)<br/> |
| 345 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">D</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#NZSqrtProp"><span·class="id"·title="module">NZSqrtProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#C"><span·class="id"·title="module">C</span></a>).<br/> | 345 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">D</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#NZSqrtProp"><span·class="id"·title="module">NZSqrtProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZSqrt.html#C"><span·class="id"·title="module">C</span></a>).<br/> |
| 346 | <br/> | 346 | <br/> |
| 347 | </div> | 347 | </div> |
| 348 | <div·class="doc"> | 348 | <div·class="doc"> |
| 349 | <a·id="lab | 349 | <a·id="lab732"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">sqrt_up</span></span>·:·a·square·root·that·rounds·up·instead·of·down</h1> |
| 350 | </div> | 350 | </div> |
| 351 | <div·class="code"> | 351 | <div·class="code"> |
| 352 | <br/> | 352 | <br/> |
| 353 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id="81ce257586db1a446e3fc4dfd8f6874e"·class="idref"·href="#81ce257586db1a446e3fc4dfd8f6874e"><span·class="id"·title="notation">"</span></a>a·²"·:=·(<span·class="id"·title="var">a</span><a·class="idref"·href="Coq.Numbers.NatInt.NZAxioms.html#797430c1791311f6dff109b908d9a13c"><span·class="id"·title="notation">*</span></a><span·class="id"·title="var">a</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·5,·<span·class="id"·title="keyword">no</span>·<span·class="id"·title="keyword">associativity</span>,·<span·class="id"·title="var">format</span>·"a·²").<br/> | 353 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id="81ce257586db1a446e3fc4dfd8f6874e"·class="idref"·href="#81ce257586db1a446e3fc4dfd8f6874e"><span·class="id"·title="notation">"</span></a>a·²"·:=·(<span·class="id"·title="var">a</span><a·class="idref"·href="Coq.Numbers.NatInt.NZAxioms.html#797430c1791311f6dff109b908d9a13c"><span·class="id"·title="notation">*</span></a><span·class="id"·title="var">a</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·5,·<span·class="id"·title="keyword">no</span>·<span·class="id"·title="keyword">associativity</span>,·<span·class="id"·title="var">format</span>·"a·²").<br/> |
| Offset 154, 15 lines modified | Offset 154, 15 lines modified | ||
| 154 | <br/> | 154 | <br/> |
| 155 | <span·class="id"·title="keyword">Theorem</span>·<a·id="NDivProp.mod_small"·class="idref"·href="#NDivProp.mod_small"><span·class="id"·title="lemma">mod_small</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:25"·class="idref"·href="#a:25"><span·class="id"·title="binder">a</span></a>·<a·id="b:26"·class="idref"·href="#b:26"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:25"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.Structures.Orders.html#LtNotation.:::x_'<'_x"><span·class="id"·title="notation"><</span></a><a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#b:26"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:25"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#DivModNotation.:::x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#b:26"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:25"><span·class="id"·title="variable">a</span></a>.<br/> | 155 | <span·class="id"·title="keyword">Theorem</span>·<a·id="NDivProp.mod_small"·class="idref"·href="#NDivProp.mod_small"><span·class="id"·title="lemma">mod_small</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:25"·class="idref"·href="#a:25"><span·class="id"·title="binder">a</span></a>·<a·id="b:26"·class="idref"·href="#b:26"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:25"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.Structures.Orders.html#LtNotation.:::x_'<'_x"><span·class="id"·title="notation"><</span></a><a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#b:26"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:25"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#DivModNotation.:::x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#b:26"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:25"><span·class="id"·title="variable">a</span></a>.<br/> |
| 156 | · | 156 | · |
| 157 | <br/> | 157 | <br/> |
| 158 | </div> | 158 | </div> |
| 159 | <div·class="doc"> | 159 | <div·class="doc"> |
| 160 | <a·id="lab | 160 | <a·id="lab798"></a><h1·class="section">Basic·values·of·divisions·and·modulo.</h1> |
| 161 | </div> | 161 | </div> |
| 162 | <div·class="code"> | 162 | <div·class="code"> |
| 163 | <br/> | 163 | <br/> |
| 164 | <span·class="id"·title="keyword">Lemma</span>·<a·id="NDivProp.div_0_l"·class="idref"·href="#NDivProp.div_0_l"><span·class="id"·title="lemma">div_0_l</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:27"·class="idref"·href="#a:27"><span·class="id"·title="binder">a</span></a>,·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:27"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.Structures.Equalities.html#9cff58ff8ba6db8841aeeecf7f406093"><span·class="id"·title="notation">~=</span></a>0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·0<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#553b1e0c6dfeea0fb5d9a4a0a1c7f719"><span·class="id"·title="notation">/</span></a><a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:27"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·0.<br/> | 164 | <span·class="id"·title="keyword">Lemma</span>·<a·id="NDivProp.div_0_l"·class="idref"·href="#NDivProp.div_0_l"><span·class="id"·title="lemma">div_0_l</span></a>:·<span·class="id"·title="keyword">forall</span>·<a·id="a:27"·class="idref"·href="#a:27"><span·class="id"·title="binder">a</span></a>,·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:27"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.Structures.Equalities.html#9cff58ff8ba6db8841aeeecf7f406093"><span·class="id"·title="notation">~=</span></a>0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·0<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#553b1e0c6dfeea0fb5d9a4a0a1c7f719"><span·class="id"·title="notation">/</span></a><a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#a:27"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#a3a0575149632d194e420f6edb9b2362"><span·class="id"·title="notation">==</span></a>·0.<br/> |
| 165 | · | 165 | · |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#"><span·class="id"·title="library">NAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NSub.html#"><span·class="id"·title="library">NSub</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#"><span·class="id"·title="library">NDiv</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NGcd.html#"><span·class="id"·title="library">NGcd</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#"><span·class="id"·title="library">NAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NSub.html#"><span·class="id"·title="library">NSub</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NDiv.html#"><span·class="id"·title="library">NDiv</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NGcd.html#"><span·class="id"·title="library">NGcd</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab797"></a><h1·class="section">Least·Common·Multiple</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ·Unlike·other·functions·around,·we·will·define·lcm·below·instead·of | 55 | ·Unlike·other·functions·around,·we·will·define·lcm·below·instead·of |
| 56 | ··axiomatizing·it.·Indeed,·there·is·no·"prior·art"·about·lcm·in·the | 56 | ··axiomatizing·it.·Indeed,·there·is·no·"prior·art"·about·lcm·in·the |
| 57 | ··standard·library·to·be·compliant·with,·and·the·generic·definition | 57 | ··standard·library·to·be·compliant·with,·and·the·generic·definition |
| 58 | ··of·lcm·via·gcd·is·quite·reasonable. | 58 | ··of·lcm·via·gcd·is·quite·reasonable. |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#"><span·class="id"·title="library">NAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NSub.html#"><span·class="id"·title="library">NSub</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#"><span·class="id"·title="library">NAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NSub.html#"><span·class="id"·title="library">NSub</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab796"></a><h1·class="section">Properties·of·minimum·and·maximum·specific·to·natural·numbers</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="NMaxMinProp"·class="idref"·href="#NMaxMinProp"><span·class="id"·title="module">NMaxMinProp</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">N</span>·:·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#NAxiomsMiniSig'"><span·class="id"·title="module">NAxiomsMiniSig'</span></a>).<br/> | 57 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="NMaxMinProp"·class="idref"·href="#NMaxMinProp"><span·class="id"·title="module">NMaxMinProp</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">N</span>·:·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#NAxiomsMiniSig'"><span·class="id"·title="module">NAxiomsMiniSig'</span></a>).<br/> |
| 58 | <span·class="id"·title="keyword">Include</span>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NSub.html#NSubProp"><span·class="id"·title="module">NSubProp</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NMaxMin.html#N"><span·class="id"·title="module">N</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Include</span>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NSub.html#NSubProp"><span·class="id"·title="module">NSubProp</span></a>·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NMaxMin.html#N"><span·class="id"·title="module">N</span></a>.<br/> |
| Offset 53, 15 lines modified | Offset 53, 15 lines modified | ||
| 53 | <br/> | 53 | <br/> |
| 54 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">N_scope</span>.<br/> | 54 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">N_scope</span>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab801"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">BinNat.N</span></span>·already·implements·<span·class="inlinecode"><span·class="id"·title="var">NAxiomSig</span></span></h1> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <br/> | 61 | <br/> |
| 62 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>·<:·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#NAxiomsSig"><span·class="id"·title="module">NAxiomsSig</span></a>·:=·<a·class="idref"·href="Coq.NArith.BinNat.html#N"><span·class="id"·title="module">N</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>·<:·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#NAxiomsSig"><span·class="id"·title="module">NAxiomsSig</span></a>·:=·<a·class="idref"·href="Coq.NArith.BinNat.html#N"><span·class="id"·title="module">N</span></a>.<br/> |
| Offset 53, 15 lines modified | Offset 53, 15 lines modified | ||
| 53 | <br/> | 53 | <br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.PArith.BinPosDef.html#"><span·class="id"·title="library">BinPosDef</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.PArith.BinPosDef.html#"><span·class="id"·title="library">BinPosDef</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab386"></a><h1·class="section">Binary·positive·numbers,·operations·and·properties</h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· | 60 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· |
| 61 | <div·class="paragraph">·</div> | 61 | <div·class="paragraph">·</div> |
| 62 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> | 62 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab359"></a><h1·class="section">Binary·positive·numbers,·operations</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· | 51 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> | 53 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">positive_scope</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">positive_scope</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab356"></a><h1·class="section">DecidableType·structure·for·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·numbers</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Module</span>·<a·id="Positive_as_DT"·class="idref"·href="#Positive_as_DT"><span·class="id"·title="module">Positive_as_DT</span></a>·<:·<a·class="idref"·href="Coq.Structures.Equalities.html#UsualDecidableTypeFull"><span·class="id"·title="module">UsualDecidableTypeFull</span></a>·:=·<a·class="idref"·href="Coq.PArith.BinPos.html#Pos"><span·class="id"·title="module">Pos</span></a>.<br/> | 60 | <span·class="id"·title="keyword">Module</span>·<a·id="Positive_as_DT"·class="idref"·href="#Positive_as_DT"><span·class="id"·title="module">Positive_as_DT</span></a>·<:·<a·class="idref"·href="Coq.Structures.Equalities.html#UsualDecidableTypeFull"><span·class="id"·title="module">UsualDecidableTypeFull</span></a>·:=·<a·class="idref"·href="Coq.PArith.BinPos.html#Pos"><span·class="id"·title="module">Pos</span></a>.<br/> |
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| 43 | <h1·class="libtitle">Library·Coq.Program.Combinators</h1> | 43 | <h1·class="libtitle">Library·Coq.Program.Combinators</h1> |
| 44 | <div·class="code"> | 44 | <div·class="code"> |
| 45 | </div> | 45 | </div> |
| 46 | <div·class="doc"> | 46 | <div·class="doc"> |
| 47 | <a·id="lab | 47 | <a·id="lab805"></a><h1·class="section">Proofs·about·standard·combinators,·exports·functional·extensionality.</h1> |
| 48 | <div·class="paragraph">·</div> | 48 | <div·class="paragraph">·</div> |
| 49 | ···Author:·Matthieu·Sozeau | 49 | ···Author:·Matthieu·Sozeau |
| 50 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 50 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.ZArith.BinInt.html#"><span·class="id"·title="library">ZArith.BinInt</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.ZArith.BinInt.html#"><span·class="id"·title="library">ZArith.BinInt</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab1053"></a><h1·class="section">Definition·of·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span>·and·basic·properties</h1> |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·Rationals·are·pairs·of·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·numbers.· | 58 | ·Rationals·are·pairs·of·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·numbers.· |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
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| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Q_scope</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Q_scope</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab1050"></a><h1·class="section">DecidableType·structure·for·rational·numbers</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Module</span>·<a·id="Q_as_DT"·class="idref"·href="#Q_as_DT"><span·class="id"·title="module">Q_as_DT</span></a>·<:·<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableTypeFull"><span·class="id"·title="module">DecidableTypeFull</span></a>.<br/> | 60 | <span·class="id"·title="keyword">Module</span>·<a·id="Q_as_DT"·class="idref"·href="#Q_as_DT"><span·class="id"·title="module">Q_as_DT</span></a>·<:·<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableTypeFull"><span·class="id"·title="module">DecidableTypeFull</span></a>.<br/> |
| 61 | <span·class="id"·title="keyword">Definition</span>·<a·id="Q_as_DT.t"·class="idref"·href="#Q_as_DT.t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.QArith.QArith_base.html#Q"><span·class="id"·title="record">Q</span></a>.<br/> | 61 | <span·class="id"·title="keyword">Definition</span>·<a·id="Q_as_DT.t"·class="idref"·href="#Q_as_DT.t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.QArith.QArith_base.html#Q"><span·class="id"·title="record">Q</span></a>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1046"></a><h1·class="section">An·absolute·value·for·normalized·rational·numbers.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Contributed·by·Cédric·Auger· | 51 | ·Contributed·by·Cédric·Auger· |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
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| 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.QArith.QArith_base.html#"><span·class="id"·title="library">QArith_base</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.QArith.QArith_base.html#"><span·class="id"·title="library">QArith_base</span></a>.<br/> |
| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.setoid_ring.NArithRing.html#"><span·class="id"·title="library">NArithRing</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.setoid_ring.NArithRing.html#"><span·class="id"·title="library">NArithRing</span></a>.<br/> |
| 52 | <br/> | 52 | <br/> |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | <a·id="lab | 55 | <a·id="lab1049"></a><h1·class="section">field·and·ring·tactics·for·rational·numbers</h1> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="code"> | 57 | <div·class="code"> |
| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Definition</span>·<a·id="Qsrt"·class="idref"·href="#Qsrt"><span·class="id"·title="definition">Qsrt</span></a>·:·<a·class="idref"·href="Coq.setoid_ring.Ring_theory.html#ring_theory"><span·class="id"·title="record">ring_theory</span></a>·0·1·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qplus"><span·class="id"·title="definition">Qplus</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qmult"><span·class="id"·title="definition">Qmult</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qminus"><span·class="id"·title="definition">Qminus</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qopp"><span·class="id"·title="definition">Qopp</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qeq"><span·class="id"·title="definition">Qeq</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Definition</span>·<a·id="Qsrt"·class="idref"·href="#Qsrt"><span·class="id"·title="definition">Qsrt</span></a>·:·<a·class="idref"·href="Coq.setoid_ring.Ring_theory.html#ring_theory"><span·class="id"·title="record">ring_theory</span></a>·0·1·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qplus"><span·class="id"·title="definition">Qplus</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qmult"><span·class="id"·title="definition">Qmult</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qminus"><span·class="id"·title="definition">Qminus</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qopp"><span·class="id"·title="definition">Qopp</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qeq"><span·class="id"·title="definition">Qeq</span></a>.<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.QArith.QArith_base.html#"><span·class="id"·title="library">QArith_base</span></a>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>·<a·class="idref"·href="Coq.QArith.QOrderedType.html#"><span·class="id"·title="library">QOrderedType</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.QArith.QArith_base.html#"><span·class="id"·title="library">QArith_base</span></a>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>·<a·class="idref"·href="Coq.QArith.QOrderedType.html#"><span·class="id"·title="library">QOrderedType</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab1047"></a><h1·class="section">Maximum·and·Minimum·of·two·rational·numbers</h1> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="code"> | 55 | <div·class="code"> |
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Q_scope</span>.<br/> | 57 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Q_scope</span>.<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#"><span·class="id"·title="library">Zpow_facts</span></a>·<a·class="idref"·href="Coq.QArith.Qfield.html#"><span·class="id"·title="library">Qfield</span></a>·<a·class="idref"·href="Coq.QArith.Qreduction.html#"><span·class="id"·title="library">Qreduction</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#"><span·class="id"·title="library">Zpow_facts</span></a>·<a·class="idref"·href="Coq.QArith.Qfield.html#"><span·class="id"·title="library">Qfield</span></a>·<a·class="idref"·href="Coq.QArith.Qreduction.html#"><span·class="id"·title="library">Qreduction</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab1071"></a><h1·class="section">Properties·of·Qpower_positive</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | <a·id="lab | 55 | <a·id="lab1072"></a><h2·class="section">Values·of·Qpower_positive·for·specific·arguments</h2> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="code"> | 57 | <div·class="code"> |
| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_positive_1"·class="idref"·href="#Qpower_positive_1"><span·class="id"·title="lemma">Qpower_positive_1</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·1·<a·class="idref"·href="Coq.QArith.Qpower.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·1.<br/> | 59 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_positive_1"·class="idref"·href="#Qpower_positive_1"><span·class="id"·title="lemma">Qpower_positive_1</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·1·<a·class="idref"·href="Coq.QArith.Qpower.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·1.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_positive_0"·class="idref"·href="#Qpower_positive_0"><span·class="id"·title="lemma">Qpower_positive_0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·0·<a·class="idref"·href="Coq.QArith.Qpower.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·0.<br/> | 61 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_positive_0"·class="idref"·href="#Qpower_positive_0"><span·class="id"·title="lemma">Qpower_positive_0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·0·<a·class="idref"·href="Coq.QArith.Qpower.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·0.<br/> |
| 62 | <br/> | 62 | <br/> |
| 63 | </div> | 63 | </div> |
| 64 | <div·class="doc"> | 64 | <div·class="doc"> |
| 65 | <a·id="lab | 65 | <a·id="lab1073"></a><h2·class="section">Relation·of·Qpower_positive·to·zero</h2> |
| 66 | </div> | 66 | </div> |
| 67 | <div·class="code"> | 67 | <div·class="code"> |
| 68 | <br/> | 68 | <br/> |
| 69 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_not_0_positive"·class="idref"·href="#Qpower_not_0_positive"><span·class="id"·title="lemma">Qpower_not_0_positive</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="a:3"·class="idref"·href="#a:3"><span·class="id"·title="binder">a</span></a>·<a·id="n:4"·class="idref"·href="#n:4"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#a:3"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:3"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:4"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·0.<br/> | 69 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_not_0_positive"·class="idref"·href="#Qpower_not_0_positive"><span·class="id"·title="lemma">Qpower_not_0_positive</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="a:3"·class="idref"·href="#a:3"><span·class="id"·title="binder">a</span></a>·<a·id="n:4"·class="idref"·href="#n:4"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#a:3"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:3"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:4"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·0.<br/> |
| 70 | <br/> | 70 | <br/> |
| 71 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_pos_positive"·class="idref"·href="#Qpower_pos_positive"><span·class="id"·title="lemma">Qpower_pos_positive</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="p:5"·class="idref"·href="#p:5"><span·class="id"·title="binder">p</span></a>·<a·id="n:6"·class="idref"·href="#n:6"><span·class="id"·title="binder">n</span></a>,·0·<a·class="idref"·href="Coq.QArith.QArith_base.html#bb2ae1e3b1d37b46a15737373eedf343"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#p:5"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·0·<a·class="idref"·href="Coq.QArith.QArith_base.html#bb2ae1e3b1d37b46a15737373eedf343"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#p:5"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:6"><span·class="id"·title="variable">n</span></a>.<br/> | 71 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_pos_positive"·class="idref"·href="#Qpower_pos_positive"><span·class="id"·title="lemma">Qpower_pos_positive</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="p:5"·class="idref"·href="#p:5"><span·class="id"·title="binder">p</span></a>·<a·id="n:6"·class="idref"·href="#n:6"><span·class="id"·title="binder">n</span></a>,·0·<a·class="idref"·href="Coq.QArith.QArith_base.html#bb2ae1e3b1d37b46a15737373eedf343"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#p:5"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·0·<a·class="idref"·href="Coq.QArith.QArith_base.html#bb2ae1e3b1d37b46a15737373eedf343"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#p:5"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:6"><span·class="id"·title="variable">n</span></a>.<br/> |
| 72 | <br/> | 72 | <br/> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="doc"> | 74 | <div·class="doc"> |
| 75 | <a·id="lab | 75 | <a·id="lab1074"></a><h2·class="section">Qpower_positive·and·multiplication,·exponent·subtraction</h2> |
| 76 | </div> | 76 | </div> |
| 77 | <div·class="code"> | 77 | <div·class="code"> |
| 78 | <br/> | 78 | <br/> |
| 79 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qmult_power_positive"·class="idref"·href="#Qmult_power_positive"><span·class="id"·title="lemma">Qmult_power_positive</span></a>··:·<span·class="id"·title="keyword">forall</span>·<a·id="a:7"·class="idref"·href="#a:7"><span·class="id"·title="binder">a</span></a>·<a·id="b:8"·class="idref"·href="#b:8"><span·class="id"·title="binder">b</span></a>·<a·id="n:9"·class="idref"·href="#n:9"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·(<a·class="idref"·href="Coq.QArith.Qpower.html#a:7"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#b:8"><span·class="id"·title="variable">b</span></a>)·<a·class="idref"·href="Coq.QArith.Qpower.html#n:9"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:7"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:9"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)*(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#b:8"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:9"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)</span></a>.<br/> | 79 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qmult_power_positive"·class="idref"·href="#Qmult_power_positive"><span·class="id"·title="lemma">Qmult_power_positive</span></a>··:·<span·class="id"·title="keyword">forall</span>·<a·id="a:7"·class="idref"·href="#a:7"><span·class="id"·title="binder">a</span></a>·<a·id="b:8"·class="idref"·href="#b:8"><span·class="id"·title="binder">b</span></a>·<a·id="n:9"·class="idref"·href="#n:9"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·(<a·class="idref"·href="Coq.QArith.Qpower.html#a:7"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#b:8"><span·class="id"·title="variable">b</span></a>)·<a·class="idref"·href="Coq.QArith.Qpower.html#n:9"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:7"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:9"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)*(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#b:8"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:9"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)</span></a>.<br/> |
| 80 | <br/> | 80 | <br/> |
| 81 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_plus_positive"·class="idref"·href="#Qpower_plus_positive"><span·class="id"·title="lemma">Qpower_plus_positive</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="a:10"·class="idref"·href="#a:10"><span·class="id"·title="binder">a</span></a>·<a·id="n:11"·class="idref"·href="#n:11"><span·class="id"·title="binder">n</span></a>·<a·id="m:12"·class="idref"·href="#m:12"><span·class="id"·title="binder">m</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:10"><span·class="id"·title="variable">a</span></a>·(<a·class="idref"·href="Coq.QArith.Qpower.html#n:11"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span·class="id"·title="notation">+</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#m:12"><span·class="id"·title="variable">m</span></a>)·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:10"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:11"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)*(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:10"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#m:12"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)</span></a>.<br/> | 81 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qpower_plus_positive"·class="idref"·href="#Qpower_plus_positive"><span·class="id"·title="lemma">Qpower_plus_positive</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="a:10"·class="idref"·href="#a:10"><span·class="id"·title="binder">a</span></a>·<a·id="n:11"·class="idref"·href="#n:11"><span·class="id"·title="binder">n</span></a>·<a·id="m:12"·class="idref"·href="#m:12"><span·class="id"·title="binder">m</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:10"><span·class="id"·title="variable">a</span></a>·(<a·class="idref"·href="Coq.QArith.Qpower.html#n:11"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span·class="id"·title="notation">+</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#m:12"><span·class="id"·title="variable">m</span></a>)·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:10"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:11"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)*(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:10"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#m:12"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#a913d2882dcbf2c07ae6ccbc47eaad45"><span·class="id"·title="notation">)</span></a>.<br/> |
| 82 | <br/> | 82 | <br/> |
| 83 | </div> | 83 | </div> |
| 84 | <div·class="doc"> | 84 | <div·class="doc"> |
| 85 | <a·id="lab | 85 | <a·id="lab1075"></a><h2·class="section">Qpower_positive·and·inversion,·division,·exponent·subtraction</h2> |
| 86 | </div> | 86 | </div> |
| 87 | <div·class="code"> | 87 | <div·class="code"> |
| 88 | <br/> | 88 | <br/> |
| 89 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qinv_power_positive"·class="idref"·href="#Qinv_power_positive"><span·class="id"·title="lemma">Qinv_power_positive</span></a>··:·<span·class="id"·title="keyword">forall</span>·<a·id="a:13"·class="idref"·href="#a:13"><span·class="id"·title="binder">a</span></a>·<a·id="n:14"·class="idref"·href="#n:14"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·(<a·class="idref"·href="Coq.QArith.QArith_base.html#8bbb7e6c4363db1e6ff873634c45a65e"><span·class="id"·title="notation">/</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#a:13"><span·class="id"·title="variable">a</span></a>)·<a·class="idref"·href="Coq.QArith.Qpower.html#n:14"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#8bbb7e6c4363db1e6ff873634c45a65e"><span·class="id"·title="notation">/(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:13"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:14"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#8bbb7e6c4363db1e6ff873634c45a65e"><span·class="id"·title="notation">)</span></a>.<br/> | 89 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Qinv_power_positive"·class="idref"·href="#Qinv_power_positive"><span·class="id"·title="lemma">Qinv_power_positive</span></a>··:·<span·class="id"·title="keyword">forall</span>·<a·id="a:13"·class="idref"·href="#a:13"><span·class="id"·title="binder">a</span></a>·<a·id="n:14"·class="idref"·href="#n:14"><span·class="id"·title="binder">n</span></a>,·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·(<a·class="idref"·href="Coq.QArith.QArith_base.html#8bbb7e6c4363db1e6ff873634c45a65e"><span·class="id"·title="notation">/</span></a><a·class="idref"·href="Coq.QArith.Qpower.html#a:13"><span·class="id"·title="variable">a</span></a>)·<a·class="idref"·href="Coq.QArith.Qpower.html#n:14"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#0eaf04f08ecf51ebfc2703f9e1d22838"><span·class="id"·title="notation">==</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#8bbb7e6c4363db1e6ff873634c45a65e"><span·class="id"·title="notation">/(</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#Qpower_positive"><span·class="id"·title="definition">Qpower_positive</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#a:13"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.QArith.Qpower.html#n:14"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.QArith.QArith_base.html#8bbb7e6c4363db1e6ff873634c45a65e"><span·class="id"·title="notation">)</span></a>.<br/> |
| Offset 54, 15 lines modified | Offset 54, 15 lines modified | ||
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lra.html#"><span·class="id"·title="library">Lra</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lra.html#"><span·class="id"·title="library">Lra</span></a>.<br/> |
| 55 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 55 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 56 | <br/> | 56 | <br/> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="doc"> | 58 | <div·class="doc"> |
| 59 | <a·id="lab | 59 | <a·id="lab225"></a><h1·class="section">Formalization·of·alternated·series</h1> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <span·class="id"·title="keyword">Definition</span>·<a·id="tg_alt"·class="idref"·href="#tg_alt"><span·class="id"·title="definition">tg_alt</span></a>·(<a·id="Un:1"·class="idref"·href="#Un:1"><span·class="id"·title="binder">Un</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·(<a·id="i:2"·class="idref"·href="#i:2"><span·class="id"·title="binder">i</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">(</span></a>-1<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">^</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#i:2"><span·class="id"·title="variable">i</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#Un:1"><span·class="id"·title="variable">Un</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#i:2"><span·class="id"·title="variable">i</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Definition</span>·<a·id="tg_alt"·class="idref"·href="#tg_alt"><span·class="id"·title="definition">tg_alt</span></a>·(<a·id="Un:1"·class="idref"·href="#Un:1"><span·class="id"·title="binder">Un</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·(<a·id="i:2"·class="idref"·href="#i:2"><span·class="id"·title="binder">i</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">(</span></a>-1<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">^</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#i:2"><span·class="id"·title="variable">i</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#Un:1"><span·class="id"·title="variable">Un</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#i:2"><span·class="id"·title="variable">i</span></a>.<br/> |
| 63 | <span·class="id"·title="keyword">Definition</span>·<a·id="positivity_seq"·class="idref"·href="#positivity_seq"><span·class="id"·title="definition">positivity_seq</span></a>·(<a·id="Un:3"·class="idref"·href="#Un:3"><span·class="id"·title="binder">Un</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<span·class="id"·title="keyword">forall</span>·<a·id="n:4"·class="idref"·href="#n:4"><span·class="id"·title="binder">n</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·0·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#77d341fe206b2891763e25999ffddca2"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#Un:3"><span·class="id"·title="variable">Un</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#n:4"><span·class="id"·title="variable">n</span></a>.<br/> | 63 | <span·class="id"·title="keyword">Definition</span>·<a·id="positivity_seq"·class="idref"·href="#positivity_seq"><span·class="id"·title="definition">positivity_seq</span></a>·(<a·id="Un:3"·class="idref"·href="#Un:3"><span·class="id"·title="binder">Un</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<span·class="id"·title="keyword">forall</span>·<a·id="n:4"·class="idref"·href="#n:4"><span·class="id"·title="binder">n</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·0·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#77d341fe206b2891763e25999ffddca2"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#Un:3"><span·class="id"·title="variable">Un</span></a>·<a·class="idref"·href="Coq.Reals.AltSeries.html#n:4"><span·class="id"·title="variable">n</span></a>.<br/> |
| 64 | <br/> | 64 | <br/> |
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| 404 | :·<a·class="idref"·href="Coq.Classes.CMorphisms.html#Proper"><span·class="id"·title="class">CMorphisms.Proper</span></a><br/> | 404 | :·<a·class="idref"·href="Coq.Classes.CMorphisms.html#Proper"><span·class="id"·title="class">CMorphisms.Proper</span></a><br/> |
| 405 | (<a·class="idref"·href="Coq.Classes.CMorphisms.html#respectful"><span·class="id"·title="definition">CMorphisms.respectful</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qeq"><span·class="id"·title="definition">Qeq</span></a>·<a·class="idref"·href="Coq.Reals.Cauchy.ConstructiveCauchyReals.html#CRealEq"><span·class="id"·title="definition">CRealEq</span></a>)·<a·class="idref"·href="Coq.Reals.Cauchy.ConstructiveCauchyReals.html#inject_Q"><span·class="id"·title="definition">inject_Q</span></a>.<br/> | 405 | (<a·class="idref"·href="Coq.Classes.CMorphisms.html#respectful"><span·class="id"·title="definition">CMorphisms.respectful</span></a>·<a·class="idref"·href="Coq.QArith.QArith_base.html#Qeq"><span·class="id"·title="definition">Qeq</span></a>·<a·class="idref"·href="Coq.Reals.Cauchy.ConstructiveCauchyReals.html#CRealEq"><span·class="id"·title="definition">CRealEq</span></a>)·<a·class="idref"·href="Coq.Reals.Cauchy.ConstructiveCauchyReals.html#inject_Q"><span·class="id"·title="definition">inject_Q</span></a>.<br/> |
| 406 | <br/> | 406 | <br/> |
| 407 | </div> | 407 | </div> |
| 408 | <div·class="doc"> | 408 | <div·class="doc"> |
| 409 | <a·id="lab | 409 | <a·id="lab166"></a><h1·class="section">Algebraic·operations</h1> |
| 410 | <div·class="paragraph">·</div> | 410 | <div·class="paragraph">·</div> |
| 411 | ·We·reduce·the·rational·numbers·to·accelerate·calculations.· | 411 | ·We·reduce·the·rational·numbers·to·accelerate·calculations.· |
| 412 | </div> | 412 | </div> |
| 413 | <div·class="code"> | 413 | <div·class="code"> |
| 414 | <span·class="id"·title="keyword">Definition</span>·<a·id="CReal_plus_seq"·class="idref"·href="#CReal_plus_seq"><span·class="id"·title="definition">CReal_plus_seq</span></a>·(<a·id="x:112"·class="idref"·href="#x:112"><span·class="id"·title="binder">x</span></a>·<a·id="y:113"·class="idref"·href="#y:113"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Reals.Cauchy.ConstructiveCauchyReals.html#CReal"><span·class="id"·title="record">CReal</span></a>)·:=<br/> | 414 | <span·class="id"·title="keyword">Definition</span>·<a·id="CReal_plus_seq"·class="idref"·href="#CReal_plus_seq"><span·class="id"·title="definition">CReal_plus_seq</span></a>·(<a·id="x:112"·class="idref"·href="#x:112"><span·class="id"·title="binder">x</span></a>·<a·id="y:113"·class="idref"·href="#y:113"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Reals.Cauchy.ConstructiveCauchyReals.html#CReal"><span·class="id"·title="record">CReal</span></a>)·:=<br/> |
| Offset 5556, 16 lines modified | Offset 5556, 16 lines modified | ||
| 00015b30:·6976·6543·6175·6368·7952·6561·6c73·4d75··iveCauchyRealsMu | 00015b30:·6976·6543·6175·6368·7952·6561·6c73·4d75··iveCauchyRealsMu | ||
| 00015b40:·6c74·2e68·746d·6c23·783a·3130·3522·3e3c··lt.html#x:105">< | 00015b40:·6c74·2e68·746d·6c23·783a·3130·3522·3e3c··lt.html#x:105">< | ||
| 00015b50:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"· | 00015b50:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"· | ||
| 00015b60:·7469·746c·653d·2276·6172·6961·626c·6522··title="variable" | 00015b60:·7469·746c·653d·2276·6172·6961·626c·6522··title="variable" | ||
| 00015b70:·3e78·3c2f·7370·616e·3e3c·2f61·3e2e·3c62··>x</span></a>.<b | 00015b70:·3e78·3c2f·7370·616e·3e3c·2f61·3e2e·3c62··>x</span></a>.<b | ||
| 00015b80:·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64·6976··r/>..<br/>.</div | 00015b80:·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64·6976··r/>..<br/>.</div | ||
| 00015b90:·3e0a·0a3c·6469·7620·636c·6173·733d·2264··>..<div·class="d | 00015b90:·3e0a·0a3c·6469·7620·636c·6173·733d·2264··>..<div·class="d | ||
| 00015ba0:·6f63·223e·0a3c·6120·6964·3d22·6c61·623 | 00015ba0:·6f63·223e·0a3c·6120·6964·3d22·6c61·6231··oc">.<a·id="lab1 | ||
| 00015bb0:·3 | 00015bb0:·3732·223e·3c2f·613e·3c68·3120·636c·6173··72"></a><h1·clas | ||
| 00015bc0:·733d·2273·6563·7469·6f6e·223e·4669·656c··s="section">Fiel | 00015bc0:·733d·2273·6563·7469·6f6e·223e·4669·656c··s="section">Fiel | ||
| 00015bd0:·643c·2f68·313e·0a0a·3c2f·6469·763e·0a3c··d</h1>..</div>.< | 00015bd0:·643c·2f68·313e·0a0a·3c2f·6469·763e·0a3c··d</h1>..</div>.< | ||
| 00015be0:·6469·7620·636c·6173·733d·2263·6f64·6522··div·class="code" | 00015be0:·6469·7620·636c·6173·733d·2263·6f64·6522··div·class="code" | ||
| 00015bf0:·3e0a·0a3c·6272·2f3e·0a3c·7370·616e·2063··>..<br/>.<span·c | 00015bf0:·3e0a·0a3c·6272·2f3e·0a3c·7370·616e·2063··>..<br/>.<span·c | ||
| 00015c00:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title= | 00015c00:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title= | ||
| 00015c10:·226b·6579·776f·7264·223e·4c65·6d6d·613c··"keyword">Lemma< | 00015c10:·226b·6579·776f·7264·223e·4c65·6d6d·613c··"keyword">Lemma< | ||
| 00015c20:·2f73·7061·6e3e·203c·6120·6964·3d22·4352··/span>·<a·id="CR | 00015c20:·2f73·7061·6e3e·203c·6120·6964·3d22·4352··/span>·<a·id="CR |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lqa.html#"><span·class="id"·title="library">Lqa</span></a>.·<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Cauchy.PosExtra.html#"><span·class="id"·title="library">PosExtra</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lqa.html#"><span·class="id"·title="library">Lqa</span></a>.·<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Cauchy.PosExtra.html#"><span·class="id"·title="library">PosExtra</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab167"></a><h1·class="section">Power·of·2·open·and·closed·upper·and·lower·bounds·for·<span·class="inlinecode"><span·class="id"·title="var">q</span></span>·<span·class="inlinecode">:</span>·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span></h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Fixpoint</span>·<a·id="Pos_log2floor_plus1"·class="idref"·href="#Pos_log2floor_plus1"><span·class="id"·title="definition">Pos_log2floor_plus1</span></a>·(<a·id="p:1"·class="idref"·href="#p:1"><span·class="id"·title="binder">p</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>)·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>·:=<br/> | 60 | <span·class="id"·title="keyword">Fixpoint</span>·<a·id="Pos_log2floor_plus1"·class="idref"·href="#Pos_log2floor_plus1"><span·class="id"·title="definition">Pos_log2floor_plus1</span></a>·(<a·id="p:1"·class="idref"·href="#p:1"><span·class="id"·title="binder">p</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>)·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>·:=<br/> |
| 61 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Reals.Cauchy.QExtra.html#p:1"><span·class="id"·title="variable">p</span></a>·<span·class="id"·title="keyword">with</span><br/> | 61 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Reals.Cauchy.QExtra.html#p:1"><span·class="id"·title="variable">p</span></a>·<span·class="id"·title="keyword">with</span><br/> |
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| 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Cauchy.QExtra.html#"><span·class="id"·title="library">QExtra</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Cauchy.QExtra.html#"><span·class="id"·title="library">QExtra</span></a>.<br/> |
| 60 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Classes.CMorphisms.html#"><span·class="id"·title="library">CMorphisms</span></a>.<br/> | 60 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Classes.CMorphisms.html#"><span·class="id"·title="library">CMorphisms</span></a>.<br/> |
| 61 | <br/> | 61 | <br/> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="doc"> | 63 | <div·class="doc"> |
| 64 | <a·id="lab | 64 | <a·id="lab83"></a><h1·class="section">Q·Auxiliary·Lemmas</h1> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="code"> | 66 | <div·class="code"> |
| 67 | <br/> | 67 | <br/> |
| 68 | <br/> | 68 | <br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab112"></a><h1·class="section">Basic·lemmas·for·the·classical·real·numbers</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·provides·several·hundred·basic·lemmas·about·foundamental | 51 | ·This·file·provides·several·hundred·basic·lemmas·about·foundamental |
| 52 | ····operations·on·R: | 52 | ····operations·on·R: |
| 53 | <ul·class="doclist"> | 53 | <ul·class="doclist"> |
| 54 | <li>·addition·denoted·by·<span·class="inlinecode"><span·class="id"·title="var">Rplus</span></span>·(notation:·infix·<span·class="inlinecode">+</span>), | 54 | <li>·addition·denoted·by·<span·class="inlinecode"><span·class="id"·title="var">Rplus</span></span>·(notation:·infix·<span·class="inlinecode">+</span>), |
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| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab54"></a><h1·class="section">DecidableType·structure·for·real·numbers</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Req_dec"·class="idref"·href="#Req_dec"><span·class="id"·title="lemma">Req_dec</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="r1:1"·class="idref"·href="#r1:1"><span·class="id"·title="binder">r1</span></a>·<a·id="r2:2"·class="idref"·href="#r2:2"><span·class="id"·title="binder">r2</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Reals.ROrderedType.html#r1:1"><span·class="id"·title="variable">r1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.ROrderedType.html#r2:2"><span·class="id"·title="variable">r2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Reals.ROrderedType.html#r1:1"><span·class="id"·title="variable">r1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·<a·class="idref"·href="Coq.Reals.ROrderedType.html#r2:2"><span·class="id"·title="variable">r2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/> | 60 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Req_dec"·class="idref"·href="#Req_dec"><span·class="id"·title="lemma">Req_dec</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="r1:1"·class="idref"·href="#r1:1"><span·class="id"·title="binder">r1</span></a>·<a·id="r2:2"·class="idref"·href="#r2:2"><span·class="id"·title="binder">r2</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Reals.ROrderedType.html#r1:1"><span·class="id"·title="variable">r1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.ROrderedType.html#r2:2"><span·class="id"·title="variable">r2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Reals.ROrderedType.html#r1:1"><span·class="id"·title="variable">r1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·<a·class="idref"·href="Coq.Reals.ROrderedType.html#r2:2"><span·class="id"·title="variable">r2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/> |
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| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.micromega.Ztac.html#"><span·class="id"·title="library">Ztac</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.micromega.Ztac.html#"><span·class="id"·title="library">Ztac</span></a>.<br/> |
| 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 59 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab193"></a><h1·class="section">Fractional·part</h1> |
| 64 | <div·class="paragraph">·</div> | 64 | <div·class="paragraph">·</div> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="code"> | 66 | <div·class="code"> |
| 67 | <span·class="id"·title="keyword">Definition</span>·<a·id="Int_part"·class="idref"·href="#Int_part"><span·class="id"·title="definition">Int_part</span></a>·(<a·id="r:1"·class="idref"·href="#r:1"><span·class="id"·title="binder">r</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>·:=·(<a·class="idref"·href="Coq.Reals.Rdefinitions.html#up"><span·class="id"·title="definition">up</span></a>·<a·class="idref"·href="Coq.Reals.R_Ifp.html#r:1"><span·class="id"·title="variable">r</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·1)%<span·class="id"·title="var">Z</span>.<br/> | 67 | <span·class="id"·title="keyword">Definition</span>·<a·id="Int_part"·class="idref"·href="#Int_part"><span·class="id"·title="definition">Int_part</span></a>·(<a·id="r:1"·class="idref"·href="#r:1"><span·class="id"·title="binder">r</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>·:=·(<a·class="idref"·href="Coq.Reals.Rdefinitions.html#up"><span·class="id"·title="definition">up</span></a>·<a·class="idref"·href="Coq.Reals.R_Ifp.html#r:1"><span·class="id"·title="variable">r</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·1)%<span·class="id"·title="var">Z</span>.<br/> |
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| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Rsqrt_def.html#"><span·class="id"·title="library">Rsqrt_def</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Rsqrt_def.html#"><span·class="id"·title="library">Rsqrt_def</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab223"></a><h1·class="section">Continuous·extension·of·Rsqrt·on·R</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <span·class="id"·title="keyword">Definition</span>·<a·id="sqrt"·class="idref"·href="#sqrt"><span·class="id"·title="definition">sqrt</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=<br/> | 59 | <span·class="id"·title="keyword">Definition</span>·<a·id="sqrt"·class="idref"·href="#sqrt"><span·class="id"·title="definition">sqrt</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=<br/> |
| 60 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#Rcase_abs"><span·class="id"·title="lemma">Rcase_abs</span></a>·<a·class="idref"·href="Coq.Reals.R_sqrt.html#x:1"><span·class="id"·title="variable">x</span></a>·<span·class="id"·title="keyword">with</span><br/> | 60 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#Rcase_abs"><span·class="id"·title="lemma">Rcase_abs</span></a>·<a·class="idref"·href="Coq.Reals.R_sqrt.html#x:1"><span·class="id"·title="variable">x</span></a>·<span·class="id"·title="keyword">with</span><br/> |
| 61 | |·<a·class="idref"·href="Coq.Init.Specif.html#left"><span·class="id"·title="constructor">left</span></a>·<span·class="id"·title="var">_</span>·=>·0<br/> | 61 | |·<a·class="idref"·href="Coq.Init.Specif.html#left"><span·class="id"·title="constructor">left</span></a>·<span·class="id"·title="var">_</span>·=>·0<br/> |
| 62 | |·<a·class="idref"·href="Coq.Init.Specif.html#right"><span·class="id"·title="constructor">right</span></a>·<span·class="id"·title="var">a</span>·=>·<a·class="idref"·href="Coq.Reals.Rsqrt_def.html#Rsqrt"><span·class="id"·title="definition">Rsqrt</span></a>·(<a·class="idref"·href="Coq.Reals.RIneq.html#mknonnegreal"><span·class="id"·title="constructor">mknonnegreal</span></a>·<a·class="idref"·href="Coq.Reals.R_sqrt.html#x:1"><span·class="id"·title="variable">x</span></a>·(<a·class="idref"·href="Coq.Reals.RIneq.html#Rge_le"><span·class="id"·title="lemma">Rge_le</span></a>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">a</span>))<br/> | 62 | |·<a·class="idref"·href="Coq.Init.Specif.html#right"><span·class="id"·title="constructor">right</span></a>·<span·class="id"·title="var">a</span>·=>·<a·class="idref"·href="Coq.Reals.Rsqrt_def.html#Rsqrt"><span·class="id"·title="definition">Rsqrt</span></a>·(<a·class="idref"·href="Coq.Reals.RIneq.html#mknonnegreal"><span·class="id"·title="constructor">mknonnegreal</span></a>·<a·class="idref"·href="Coq.Reals.R_sqrt.html#x:1"><span·class="id"·title="variable">x</span></a>·(<a·class="idref"·href="Coq.Reals.RIneq.html#Rge_le"><span·class="id"·title="lemma">Rge_le</span></a>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">a</span>))<br/> |
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| 53 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 53 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 54 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Type</span>·<span·class="id"·title="var">f</span>·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Type</span>·<span·class="id"·title="var">f</span>·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab60"></a><h1·class="section">Basic·operations·on·functions</h1> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <span·class="id"·title="keyword">Definition</span>·<a·id="plus_fct"·class="idref"·href="#plus_fct"><span·class="id"·title="definition">plus_fct</span></a>·<a·id="f1:1"·class="idref"·href="#f1:1"><span·class="id"·title="binder">f1</span></a>·<a·id="f2:2"·class="idref"·href="#f2:2"><span·class="id"·title="binder">f2</span></a>·(<a·id="x:3"·class="idref"·href="#x:3"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f1:1"><span·class="id"·title="variable">f1</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:3"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f2:2"><span·class="id"·title="variable">f2</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:3"><span·class="id"·title="variable">x</span></a>.<br/> | 61 | <span·class="id"·title="keyword">Definition</span>·<a·id="plus_fct"·class="idref"·href="#plus_fct"><span·class="id"·title="definition">plus_fct</span></a>·<a·id="f1:1"·class="idref"·href="#f1:1"><span·class="id"·title="binder">f1</span></a>·<a·id="f2:2"·class="idref"·href="#f2:2"><span·class="id"·title="binder">f2</span></a>·(<a·id="x:3"·class="idref"·href="#x:3"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f1:1"><span·class="id"·title="variable">f1</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:3"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f2:2"><span·class="id"·title="variable">f2</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:3"><span·class="id"·title="variable">x</span></a>.<br/> |
| 62 | <span·class="id"·title="keyword">Definition</span>·<a·id="opp_fct"·class="idref"·href="#opp_fct"><span·class="id"·title="definition">opp_fct</span></a>·<a·id="f:4"·class="idref"·href="#f:4"><span·class="id"·title="binder">f</span></a>·(<a·id="x:5"·class="idref"·href="#x:5"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f:4"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:5"><span·class="id"·title="variable">x</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Definition</span>·<a·id="opp_fct"·class="idref"·href="#opp_fct"><span·class="id"·title="definition">opp_fct</span></a>·<a·id="f:4"·class="idref"·href="#f:4"><span·class="id"·title="binder">f</span></a>·(<a·id="x:5"·class="idref"·href="#x:5"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f:4"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:5"><span·class="id"·title="variable">x</span></a>.<br/> |
| 63 | <span·class="id"·title="keyword">Definition</span>·<a·id="mult_fct"·class="idref"·href="#mult_fct"><span·class="id"·title="definition">mult_fct</span></a>·<a·id="f1:6"·class="idref"·href="#f1:6"><span·class="id"·title="binder">f1</span></a>·<a·id="f2:7"·class="idref"·href="#f2:7"><span·class="id"·title="binder">f2</span></a>·(<a·id="x:8"·class="idref"·href="#x:8"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f1:6"><span·class="id"·title="variable">f1</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:8"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f2:7"><span·class="id"·title="variable">f2</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:8"><span·class="id"·title="variable">x</span></a>.<br/> | 63 | <span·class="id"·title="keyword">Definition</span>·<a·id="mult_fct"·class="idref"·href="#mult_fct"><span·class="id"·title="definition">mult_fct</span></a>·<a·id="f1:6"·class="idref"·href="#f1:6"><span·class="id"·title="binder">f1</span></a>·<a·id="f2:7"·class="idref"·href="#f2:7"><span·class="id"·title="binder">f2</span></a>·(<a·id="x:8"·class="idref"·href="#x:8"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f1:6"><span·class="id"·title="variable">f1</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:8"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f2:7"><span·class="id"·title="variable">f2</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:8"><span·class="id"·title="variable">x</span></a>.<br/> |
| 64 | <span·class="id"·title="keyword">Definition</span>·<a·id="mult_real_fct"·class="idref"·href="#mult_real_fct"><span·class="id"·title="definition">mult_real_fct</span></a>·(<a·id="a:9"·class="idref"·href="#a:9"><span·class="id"·title="binder">a</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·<a·id="f:10"·class="idref"·href="#f:10"><span·class="id"·title="binder">f</span></a>·(<a·id="x:11"·class="idref"·href="#x:11"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#a:9"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f:10"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:11"><span·class="id"·title="variable">x</span></a>.<br/> | 64 | <span·class="id"·title="keyword">Definition</span>·<a·id="mult_real_fct"·class="idref"·href="#mult_real_fct"><span·class="id"·title="definition">mult_real_fct</span></a>·(<a·id="a:9"·class="idref"·href="#a:9"><span·class="id"·title="binder">a</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·<a·id="f:10"·class="idref"·href="#f:10"><span·class="id"·title="binder">f</span></a>·(<a·id="x:11"·class="idref"·href="#x:11"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#a:9"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#f:10"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis1.html#x:11"><span·class="id"·title="variable">x</span></a>.<br/> |
| Offset 59, 15 lines modified | Offset 59, 15 lines modified | ||
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">rorders</span>·<span·class="id"·title="var">real</span>.<br/> | 60 | <span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">rorders</span>·<span·class="id"·title="var">real</span>.<br/> |
| 61 | <br/> | 61 | <br/> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="doc"> | 63 | <div·class="doc"> |
| 64 | <a·id="lab | 64 | <a·id="lab178"></a><h1·class="section">Preliminaries·lemmas</h1> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="code"> | 66 | <div·class="code"> |
| 67 | <br/> | 67 | <br/> |
| 68 | <span·class="id"·title="keyword">Lemma</span>·<a·id="f_incr_implies_g_incr_interv"·class="idref"·href="#f_incr_implies_g_incr_interv"><span·class="id"·title="lemma">f_incr_implies_g_incr_interv</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="f:1"·class="idref"·href="#f:1"><span·class="id"·title="binder">f</span></a>·<a·id="g:2"·class="idref"·href="#g:2"><span·class="id"·title="binder">g</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<span·class="id"·title="keyword">forall</span>·<a·id="lb:3"·class="idref"·href="#lb:3"><span·class="id"·title="binder">lb</span></a>·<a·id="ub:4"·class="idref"·href="#ub:4"><span·class="id"·title="binder">ub</span></a>,<br/> | 68 | <span·class="id"·title="keyword">Lemma</span>·<a·id="f_incr_implies_g_incr_interv"·class="idref"·href="#f_incr_implies_g_incr_interv"><span·class="id"·title="lemma">f_incr_implies_g_incr_interv</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="f:1"·class="idref"·href="#f:1"><span·class="id"·title="binder">f</span></a>·<a·id="g:2"·class="idref"·href="#g:2"><span·class="id"·title="binder">g</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<span·class="id"·title="keyword">forall</span>·<a·id="lb:3"·class="idref"·href="#lb:3"><span·class="id"·title="binder">lb</span></a>·<a·id="ub:4"·class="idref"·href="#ub:4"><span·class="id"·title="binder">ub</span></a>,<br/> |
| 69 | <a·class="idref"·href="Coq.Reals.Ranalysis5.html#lb:3"><span·class="id"·title="variable">lb</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis5.html#ub:4"><span·class="id"·title="variable">ub</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> | 69 | <a·class="idref"·href="Coq.Reals.Ranalysis5.html#lb:3"><span·class="id"·title="variable">lb</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Reals.Ranalysis5.html#ub:4"><span·class="id"·title="variable">ub</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> |
| Offset 311, 21 lines modified | Offset 311, 21 lines modified | ||
| 00001360:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | 00001360:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | ||
| 00001370:·7661·7222·3e72·6561·6c3c·2f73·7061·6e3e··var">real</span> | 00001370:·7661·7222·3e72·6561·6c3c·2f73·7061·6e3e··var">real</span> | ||
| 00001380:·203c·7370·616e·2063·6c61·7373·3d22·6964···<span·class="id | 00001380:·203c·7370·616e·2063·6c61·7373·3d22·6964···<span·class="id | ||
| 00001390:·2220·7469·746c·653d·2276·6172·223e·6172··"·title="var">ar | 00001390:·2220·7469·746c·653d·2276·6172·223e·6172··"·title="var">ar | ||
| 000013a0:·6974·683c·2f73·7061·6e3e·2e3c·6272·2f3e··ith</span>.<br/> | 000013a0:·6974·683c·2f73·7061·6e3e·2e3c·6272·2f3e··ith</span>.<br/> | ||
| 000013b0:·0a0a·3c62·722f·3e0a·3c2f·6469·763e·0a0a··..<br/>.</div>.. | 000013b0:·0a0a·3c62·722f·3e0a·3c2f·6469·763e·0a0a··..<br/>.</div>.. | ||
| 000013c0:·3c64·6976·2063·6c61·7373·3d22·646f·6322··<div·class="doc" | 000013c0:·3c64·6976·2063·6c61·7373·3d22·646f·6322··<div·class="doc" | ||
| 000013d0:·3e0a·3c61·2069·643d·226c·6162·3 | 000013d0:·3e0a·3c61·2069·643d·226c·6162·3139·3522··>.<a·id="lab195" | ||
| 000013e0:·3e3c·2f61·3e3c·6831·2063·6c61·7373·3d22··></a><h1·class=" | 000013e0:·3e3c·2f61·3e3c·6831·2063·6c61·7373·3d22··></a><h1·class=" | ||
| 000013f0:·7365·6374·696f·6e22·3e50·7265·6c69·6d69··section">Prelimi | 000013f0:·7365·6374·696f·6e22·3e50·7265·6c69·6d69··section">Prelimi | ||
| 00001400:·6e61·7269·6573·3c2f·6831·3e0a·0a3c·6469··naries</h1>..<di | 00001400:·6e61·7269·6573·3c2f·6831·3e0a·0a3c·6469··naries</h1>..<di | ||
| 00001410:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra | 00001410:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra | ||
| 00001420:·7068·223e·203c·2f64·6976·3e0a·0a3c·6120··ph">·</div>..<a· | 00001420:·7068·223e·203c·2f64·6976·3e0a·0a3c·6120··ph">·</div>..<a· | ||
| 00001430:·6964·3d22·6c61·623 | 00001430:·6964·3d22·6c61·6231·3936·223e·3c2f·613e··id="lab196"></a> | ||
| 00001440:·3c68·3220·636c·6173·733d·2273·6563·7469··<h2·class="secti | 00001440:·3c68·3220·636c·6173·733d·2273·6563·7469··<h2·class="secti | ||
| 00001450:·6f6e·223e·5661·7269·6f75·7320·6765·6e65··on">Various·gene | 00001450:·6f6e·223e·5661·7269·6f75·7320·6765·6e65··on">Various·gene | ||
| 00001460:·7269·6320·6c65·6d6d·6173·2077·6869·6368··ric·lemmas·which | 00001460:·7269·6320·6c65·6d6d·6173·2077·6869·6368··ric·lemmas·which | ||
| 00001470:·2070·726f·6261·626c·7920·7368·6f75·6c64···probably·should | 00001470:·2070·726f·6261·626c·7920·7368·6f75·6c64···probably·should | ||
| 00001480:·2067·6f20·736f·6d65·7768·6572·6520·656c···go·somewhere·el | 00001480:·2067·6f20·736f·6d65·7768·6572·6520·656c···go·somewhere·el | ||
| 00001490:·7365·3c2f·6832·3e0a·0a3c·2f64·6976·3e0a··se</h2>..</div>. | 00001490:·7365·3c2f·6832·3e0a·0a3c·2f64·6976·3e0a··se</h2>..</div>. | ||
| 000014a0:·3c64·6976·2063·6c61·7373·3d22·636f·6465··<div·class="code | 000014a0:·3c64·6976·2063·6c61·7373·3d22·636f·6465··<div·class="code |
| Offset 65, 19 lines modified | Offset 65, 19 lines modified | ||
| 65 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#"><span·class="id"·title="library">Rdefinitions</span></a>.<br/> | 65 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#"><span·class="id"·title="library">Rdefinitions</span></a>.<br/> |
| 66 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 66 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 67 | <br/> | 67 | <br/> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="doc"> | 69 | <div·class="doc"> |
| 70 | <a·id="lab | 70 | <a·id="lab45"></a><h1·class="section">Field·operations</h1> |
| 71 | <div·class="paragraph">·</div> | 71 | <div·class="paragraph">·</div> |
| 72 | <a·id="lab | 72 | <a·id="lab46"></a><h2·class="section">Addition</h2> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="code"> | 74 | <div·class="code"> |
| 75 | <br/> | 75 | <br/> |
| 76 | <span·class="id"·title="keyword">Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">CReal_scope</span>.<br/> | 76 | <span·class="id"·title="keyword">Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">CReal_scope</span>.<br/> |
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| 61 | <br/> | 61 | <br/> |
| 62 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Type</span>·<span·class="id"·title="var">r</span>·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Type</span>·<span·class="id"·title="var">r</span>·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>.<br/> |
| 63 | <br/> | 63 | <br/> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="doc"> | 65 | <div·class="doc"> |
| 66 | <a·id="lab | 66 | <a·id="lab107"></a><h1·class="section">Rmin</h1> |
| 67 | <div·class="paragraph">·</div> | 67 | <div·class="paragraph">·</div> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="code"> | 69 | <div·class="code"> |
| 70 | <span·class="id"·title="keyword">Definition</span>·<a·id="Rmin"·class="idref"·href="#Rmin"><span·class="id"·title="definition">Rmin</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·<a·id="y:2"·class="idref"·href="#y:2"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=<br/> | 70 | <span·class="id"·title="keyword">Definition</span>·<a·id="Rmin"·class="idref"·href="#Rmin"><span·class="id"·title="definition">Rmin</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·<a·id="y:2"·class="idref"·href="#y:2"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=<br/> |
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| 174 | <span·class="id"·title="keyword">Notation</span>·<a·id="::R_scope:x_'<'_x_'<'_x"·class="idref"·href="#::R_scope:x_'<'_x_'<'_x"><span·class="id"·title="notation">"</span></a>x·<·y·<·z"···:=·(<span·class="id"·title="var">x</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>··<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>··<span·class="id"·title="var">z</span>)·:·<span·class="id"·title="var">R_scope</span>.<br/> | 174 | <span·class="id"·title="keyword">Notation</span>·<a·id="::R_scope:x_'<'_x_'<'_x"·class="idref"·href="#::R_scope:x_'<'_x_'<'_x"><span·class="id"·title="notation">"</span></a>x·<·y·<·z"···:=·(<span·class="id"·title="var">x</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>··<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>··<span·class="id"·title="var">z</span>)·:·<span·class="id"·title="var">R_scope</span>.<br/> |
| 175 | <span·class="id"·title="keyword">Notation</span>·<a·id="655439258f20e6256090528bc533ae29"·class="idref"·href="#655439258f20e6256090528bc533ae29"><span·class="id"·title="notation">"</span></a>x·<·y·<=·z"··:=·(<span·class="id"·title="var">x</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>··<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#77d341fe206b2891763e25999ffddca2"><span·class="id"·title="notation"><=</span></a>·<span·class="id"·title="var">z</span>)·:·<span·class="id"·title="var">R_scope</span>.<br/> | 175 | <span·class="id"·title="keyword">Notation</span>·<a·id="655439258f20e6256090528bc533ae29"·class="idref"·href="#655439258f20e6256090528bc533ae29"><span·class="id"·title="notation">"</span></a>x·<·y·<=·z"··:=·(<span·class="id"·title="var">x</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>··<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<span·class="id"·title="var">y</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#77d341fe206b2891763e25999ffddca2"><span·class="id"·title="notation"><=</span></a>·<span·class="id"·title="var">z</span>)·:·<span·class="id"·title="var">R_scope</span>.<br/> |
| 176 | <br/> | 176 | <br/> |
| 177 | </div> | 177 | </div> |
| 178 | <div·class="doc"> | 178 | <div·class="doc"> |
| 179 | <a·id="lab | 179 | <a·id="lab110"></a><h1·class="section">Injection·from·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·to·<span·class="inlinecode"><span·class="id"·title="var">R</span></span></h1> |
| 180 | </div> | 180 | </div> |
| 181 | <div·class="code"> | 181 | <div·class="code"> |
| 182 | <br/> | 182 | <br/> |
| 183 | <span·class="id"·title="keyword">Fixpoint</span>·<a·id="IPR_2"·class="idref"·href="#IPR_2"><span·class="id"·title="definition">IPR_2</span></a>·(<a·id="p:50"·class="idref"·href="#p:50"><span·class="id"·title="binder">p</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#R"><span·class="id"·title="abbreviation">R</span></a>·:=<br/> | 183 | <span·class="id"·title="keyword">Fixpoint</span>·<a·id="IPR_2"·class="idref"·href="#IPR_2"><span·class="id"·title="definition">IPR_2</span></a>·(<a·id="p:50"·class="idref"·href="#p:50"><span·class="id"·title="binder">p</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#R"><span·class="id"·title="abbreviation">R</span></a>·:=<br/> |
| 184 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#p:50"><span·class="id"·title="variable">p</span></a>·<span·class="id"·title="keyword">with</span><br/> | 184 | <span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#p:50"><span·class="id"·title="variable">p</span></a>·<span·class="id"·title="keyword">with</span><br/> |
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| 70 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> | 70 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/> |
| 71 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 71 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 72 | <br/> | 72 | <br/> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="doc"> | 74 | <div·class="doc"> |
| 75 | <a·id="lab | 75 | <a·id="lab159"></a><h1·class="section">Lemmas·about·factorial</h1> |
| 76 | </div> | 76 | </div> |
| 77 | <div·class="code"> | 77 | <div·class="code"> |
| 78 | <span·class="id"·title="keyword">Lemma</span>·<a·id="INR_fact_neq_0"·class="idref"·href="#INR_fact_neq_0"><span·class="id"·title="lemma">INR_fact_neq_0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Reals.Raxioms.html#INR"><span·class="id"·title="definition">INR</span></a>·(<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:1"><span·class="id"·title="variable">n</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0.<br/> | 78 | <span·class="id"·title="keyword">Lemma</span>·<a·id="INR_fact_neq_0"·class="idref"·href="#INR_fact_neq_0"><span·class="id"·title="lemma">INR_fact_neq_0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Reals.Raxioms.html#INR"><span·class="id"·title="definition">INR</span></a>·(<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:1"><span·class="id"·title="variable">n</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0.<br/> |
| 79 | <br/> | 79 | <br/> |
| 80 | <span·class="id"·title="keyword">Lemma</span>·<a·id="fact_simpl"·class="idref"·href="#fact_simpl"><span·class="id"·title="lemma">fact_simpl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:2"><span·class="id"·title="variable">n</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:2"><span·class="id"·title="variable">n</span></a>)%<span·class="id"·title="var">nat</span>.<br/> | 80 | <span·class="id"·title="keyword">Lemma</span>·<a·id="fact_simpl"·class="idref"·href="#fact_simpl"><span·class="id"·title="lemma">fact_simpl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:2"><span·class="id"·title="variable">n</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#n:2"><span·class="id"·title="variable">n</span></a>)%<span·class="id"·title="var">nat</span>.<br/> |
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| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.R_sqrt.html#"><span·class="id"·title="library">R_sqrt</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.R_sqrt.html#"><span·class="id"·title="library">R_sqrt</span></a>.<br/> |
| 54 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 54 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab189"></a><h1·class="section">Distance</h1> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <br/> | 61 | <br/> |
| 62 | <span·class="id"·title="keyword">Definition</span>·<a·id="dist_euc"·class="idref"·href="#dist_euc"><span·class="id"·title="definition">dist_euc</span></a>·(<a·id="x0:1"·class="idref"·href="#x0:1"><span·class="id"·title="binder">x0</span></a>·<a·id="y0:2"·class="idref"·href="#y0:2"><span·class="id"·title="binder">y0</span></a>·<a·id="x1:3"·class="idref"·href="#x1:3"><span·class="id"·title="binder">x1</span></a>·<a·id="y1:4"·class="idref"·href="#y1:4"><span·class="id"·title="binder">y1</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=<br/> | 62 | <span·class="id"·title="keyword">Definition</span>·<a·id="dist_euc"·class="idref"·href="#dist_euc"><span·class="id"·title="definition">dist_euc</span></a>·(<a·id="x0:1"·class="idref"·href="#x0:1"><span·class="id"·title="binder">x0</span></a>·<a·id="y0:2"·class="idref"·href="#y0:2"><span·class="id"·title="binder">y0</span></a>·<a·id="x1:3"·class="idref"·href="#x1:3"><span·class="id"·title="binder">x1</span></a>·<a·id="y1:4"·class="idref"·href="#y1:4"><span·class="id"·title="binder">y1</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·:=<br/> |
| 63 | <a·class="idref"·href="Coq.Reals.R_sqrt.html#sqrt"><span·class="id"·title="definition">sqrt</span></a>·(<a·class="idref"·href="Coq.Reals.RIneq.html#Rsqr"><span·class="id"·title="definition">Rsqr</span></a>·(<a·class="idref"·href="Coq.Reals.Rgeom.html#x0:1"><span·class="id"·title="variable">x0</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rgeom.html#x1:3"><span·class="id"·title="variable">x1</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.RIneq.html#Rsqr"><span·class="id"·title="definition">Rsqr</span></a>·(<a·class="idref"·href="Coq.Reals.Rgeom.html#y0:2"><span·class="id"·title="variable">y0</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rgeom.html#y1:4"><span·class="id"·title="variable">y1</span></a>)).<br/> | 63 | <a·class="idref"·href="Coq.Reals.R_sqrt.html#sqrt"><span·class="id"·title="definition">sqrt</span></a>·(<a·class="idref"·href="Coq.Reals.RIneq.html#Rsqr"><span·class="id"·title="definition">Rsqr</span></a>·(<a·class="idref"·href="Coq.Reals.Rgeom.html#x0:1"><span·class="id"·title="variable">x0</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rgeom.html#x1:3"><span·class="id"·title="variable">x1</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.RIneq.html#Rsqr"><span·class="id"·title="definition">Rsqr</span></a>·(<a·class="idref"·href="Coq.Reals.Rgeom.html#y0:2"><span·class="id"·title="variable">y0</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rgeom.html#y1:4"><span·class="id"·title="variable">y1</span></a>)).<br/> |
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| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 59 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab184"></a><h1·class="section">Each·bounded·subset·of·N·has·a·maximal·element</h1> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="code"> | 65 | <div·class="code"> |
| 66 | <br/> | 66 | <br/> |
| 67 | <span·class="id"·title="keyword">Definition</span>·<a·id="Nbound"·class="idref"·href="#Nbound"><span·class="id"·title="definition">Nbound</span></a>·(<a·id="I:1"·class="idref"·href="#I:1"><span·class="id"·title="binder">I</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·:·<span·class="id"·title="keyword">Prop</span>·:=<br/> | 67 | <span·class="id"·title="keyword">Definition</span>·<a·id="Nbound"·class="idref"·href="#Nbound"><span·class="id"·title="definition">Nbound</span></a>·(<a·id="I:1"·class="idref"·href="#I:1"><span·class="id"·title="binder">I</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·:·<span·class="id"·title="keyword">Prop</span>·:=<br/> |
| 68 | <a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">exists</span></a>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">,</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">(</span></a><span·class="id"·title="keyword">forall</span>·<a·id="i:3"·class="idref"·href="#i:3"><span·class="id"·title="binder">i</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#I:1"><span·class="id"·title="variable">I</span></a>·<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#i:3"><span·class="id"·title="variable">i</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·(<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#i:3"><span·class="id"·title="variable">i</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#n:2"><span·class="id"·title="variable">n</span></a>)%<span·class="id"·title="var">nat</span><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">)</span></a>.<br/> | 68 | <a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">exists</span></a>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">,</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">(</span></a><span·class="id"·title="keyword">forall</span>·<a·id="i:3"·class="idref"·href="#i:3"><span·class="id"·title="binder">i</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#I:1"><span·class="id"·title="variable">I</span></a>·<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#i:3"><span·class="id"·title="variable">i</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·(<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#i:3"><span·class="id"·title="variable">i</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Reals.RiemannInt_SF.html#n:2"><span·class="id"·title="variable">n</span></a>)%<span·class="id"·title="var">nat</span><a·class="idref"·href="Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span·class="id"·title="notation">)</span></a>.<br/> |
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| 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lra.html#"><span·class="id"·title="library">Lra</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lra.html#"><span·class="id"·title="library">Lra</span></a>.<br/> |
| 60 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 60 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 61 | <br/> | 61 | <br/> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="doc"> | 63 | <div·class="doc"> |
| 64 | <a·id="lab | 64 | <a·id="lab94"></a><h1·class="section">Calculus</h1> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="code"> | 66 | <div·class="code"> |
| 67 | <span·class="id"·title="keyword">Lemma</span>·<a·id="eps2_Rgt_R0"·class="idref"·href="#eps2_Rgt_R0"><span·class="id"·title="lemma">eps2_Rgt_R0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="eps:1"·class="idref"·href="#eps:1"><span·class="id"·title="binder">eps</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:1"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:1"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·0.<br/> | 67 | <span·class="id"·title="keyword">Lemma</span>·<a·id="eps2_Rgt_R0"·class="idref"·href="#eps2_Rgt_R0"><span·class="id"·title="lemma">eps2_Rgt_R0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="eps:1"·class="idref"·href="#eps:1"><span·class="id"·title="binder">eps</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:1"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:1"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·0.<br/> |
| 68 | <br/> | 68 | <br/> |
| 69 | <span·class="id"·title="keyword">Lemma</span>·<a·id="eps2"·class="idref"·href="#eps2"><span·class="id"·title="lemma">eps2</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="eps:2"·class="idref"·href="#eps:2"><span·class="id"·title="binder">eps</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:2"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:2"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·2·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:2"><span·class="id"·title="variable">eps</span></a>.<br/> | 69 | <span·class="id"·title="keyword">Lemma</span>·<a·id="eps2"·class="idref"·href="#eps2"><span·class="id"·title="lemma">eps2</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="eps:2"·class="idref"·href="#eps:2"><span·class="id"·title="binder">eps</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:2"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:2"><span·class="id"·title="variable">eps</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·2·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.Rlimit.html#eps:2"><span·class="id"·title="variable">eps</span></a>.<br/> |
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| 71 | <br/> | 71 | <br/> |
| 72 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.RIneq.html#"><span·class="id"·title="library">RIneq</span></a>.<br/> | 72 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.RIneq.html#"><span·class="id"·title="library">RIneq</span></a>.<br/> |
| 73 | <br/> | 73 | <br/> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="doc"> | 75 | <div·class="doc"> |
| 76 | <a·id="lab | 76 | <a·id="lab57"></a><h1·class="section">Decidability·of·arithmetical·statements</h1> |
| 77 | <div·class="paragraph">·</div> | 77 | <div·class="paragraph">·</div> |
| 78 | ·One·can·iterate·this·lemma·and·use·classical·logic·to·decide·any | 78 | ·One·can·iterate·this·lemma·and·use·classical·logic·to·decide·any |
| 79 | statement·in·the·arithmetical·hierarchy.· | 79 | statement·in·the·arithmetical·hierarchy.· |
| 80 | </div> | 80 | </div> |
| 81 | <div·class="code"> | 81 | <div·class="code"> |
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| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">real</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">real</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab173"></a><h1·class="section">Maximum·and·Minimum·of·two·real·numbers</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 60 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
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| 72 | <br/> | 72 | <br/> |
| 73 | <span·class="id"·title="keyword">Lemma</span>·<a·id="exp_le_3"·class="idref"·href="#exp_le_3"><span·class="id"·title="lemma">exp_le_3</span></a>·:·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#exp"><span·class="id"·title="definition">exp</span></a>·1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#77d341fe206b2891763e25999ffddca2"><span·class="id"·title="notation"><=</span></a>·3.<br/> | 73 | <span·class="id"·title="keyword">Lemma</span>·<a·id="exp_le_3"·class="idref"·href="#exp_le_3"><span·class="id"·title="lemma">exp_le_3</span></a>·:·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#exp"><span·class="id"·title="definition">exp</span></a>·1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#77d341fe206b2891763e25999ffddca2"><span·class="id"·title="notation"><=</span></a>·3.<br/> |
| 74 | <br/> | 74 | <br/> |
| 75 | </div> | 75 | </div> |
| 76 | <div·class="doc"> | 76 | <div·class="doc"> |
| 77 | <a·id="lab | 77 | <a·id="lab99"></a><h1·class="section">Properties·of··Exp</h1> |
| 78 | </div> | 78 | </div> |
| 79 | <div·class="code"> | 79 | <div·class="code"> |
| 80 | <br/> | 80 | <br/> |
| 81 | <span·class="id"·title="keyword">Lemma</span>·<a·id="exp_neq_0"·class="idref"·href="#exp_neq_0"><span·class="id"·title="lemma">exp_neq_0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="x:14"·class="idref"·href="#x:14"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#exp"><span·class="id"·title="definition">exp</span></a>·<a·class="idref"·href="Coq.Reals.Rpower.html#x:14"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0.<br/> | 81 | <span·class="id"·title="keyword">Lemma</span>·<a·id="exp_neq_0"·class="idref"·href="#exp_neq_0"><span·class="id"·title="lemma">exp_neq_0</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="x:14"·class="idref"·href="#x:14"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#exp"><span·class="id"·title="definition">exp</span></a>·<a·class="idref"·href="Coq.Reals.Rpower.html#x:14"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0.<br/> |
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| 54 | <br/> | 54 | <br/> |
| 55 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Type</span>·<span·class="id"·title="var">r</span>·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>.<br/> | 55 | <span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Type</span>·<span·class="id"·title="var">r</span>·:·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>.<br/> |
| 56 | <br/> | 56 | <br/> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="doc"> | 58 | <div·class="doc"> |
| 59 | <a·id="lab | 59 | <a·id="lab75"></a><h1·class="section">Definition·of·sequence·and·properties</h1> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Section</span>·<a·id="sequence"·class="idref"·href="#sequence"><span·class="id"·title="section">sequence</span></a>.<br/> | 63 | <span·class="id"·title="keyword">Section</span>·<a·id="sequence"·class="idref"·href="#sequence"><span·class="id"·title="section">sequence</span></a>.<br/> |
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| 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Logic.Classical_Pred_Type.html#"><span·class="id"·title="library">Classical_Pred_Type</span></a>.<br/> | 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Logic.Classical_Pred_Type.html#"><span·class="id"·title="library">Classical_Pred_Type</span></a>.<br/> |
| 56 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 56 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 57 | <br/> | 57 | <br/> |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="doc"> | 59 | <div·class="doc"> |
| 60 | <a·id="lab | 60 | <a·id="lab220"></a><h1·class="section">General·definitions·and·propositions</h1> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <div·class="code"> |
| 63 | <br/> | 63 | <br/> |
| 64 | <span·class="id"·title="keyword">Definition</span>·<a·id="included"·class="idref"·href="#included"><span·class="id"·title="definition">included</span></a>·(<a·id="D1:1"·class="idref"·href="#D1:1"><span·class="id"·title="binder">D1</span></a>·<a·id="D2:2"·class="idref"·href="#D2:2"><span·class="id"·title="binder">D2</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<span·class="id"·title="keyword">forall</span>·<a·id="x:3"·class="idref"·href="#x:3"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtopology.html#D1:1"><span·class="id"·title="variable">D1</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#D2:2"><span·class="id"·title="variable">D2</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>.<br/> | 64 | <span·class="id"·title="keyword">Definition</span>·<a·id="included"·class="idref"·href="#included"><span·class="id"·title="definition">included</span></a>·(<a·id="D1:1"·class="idref"·href="#D1:1"><span·class="id"·title="binder">D1</span></a>·<a·id="D2:2"·class="idref"·href="#D2:2"><span·class="id"·title="binder">D2</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<span·class="id"·title="keyword">forall</span>·<a·id="x:3"·class="idref"·href="#x:3"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtopology.html#D1:1"><span·class="id"·title="variable">D1</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#D2:2"><span·class="id"·title="variable">D2</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>.<br/> |
| 65 | <span·class="id"·title="keyword">Definition</span>·<a·id="disc"·class="idref"·href="#disc"><span·class="id"·title="definition">disc</span></a>·(<a·id="x:4"·class="idref"·href="#x:4"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·(<a·id="delta:5"·class="idref"·href="#delta:5"><span·class="id"·title="binder">delta</span></a>:<a·class="idref"·href="Coq.Reals.RIneq.html#posreal"><span·class="id"·title="record">posreal</span></a>)·(<a·id="y:6"·class="idref"·href="#y:6"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#Rabs"><span·class="id"·title="definition">Rabs</span></a>·(<a·class="idref"·href="Coq.Reals.Rtopology.html#y:6"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:4"><span·class="id"·title="variable">x</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#delta:5"><span·class="id"·title="variable">delta</span></a>.<br/> | 65 | <span·class="id"·title="keyword">Definition</span>·<a·id="disc"·class="idref"·href="#disc"><span·class="id"·title="definition">disc</span></a>·(<a·id="x:4"·class="idref"·href="#x:4"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·(<a·id="delta:5"·class="idref"·href="#delta:5"><span·class="id"·title="binder">delta</span></a>:<a·class="idref"·href="Coq.Reals.RIneq.html#posreal"><span·class="id"·title="record">posreal</span></a>)·(<a·id="y:6"·class="idref"·href="#y:6"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#Rabs"><span·class="id"·title="definition">Rabs</span></a>·(<a·class="idref"·href="Coq.Reals.Rtopology.html#y:6"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:4"><span·class="id"·title="variable">x</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#delta:5"><span·class="id"·title="variable">delta</span></a>.<br/> |
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| 160 | 1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> | 160 | 1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> |
| 161 | <a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·(<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">/</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a>1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>.<br/> | 161 | <a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·(<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">/</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a>1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>.<br/> |
| 162 | <br/> | 162 | <br/> |
| 163 | </div> | 163 | </div> |
| 164 | <div·class="doc"> | 164 | <div·class="doc"> |
| 165 | <a·id="lab | 165 | <a·id="lab104"></a><h1·class="section">Some·properties·of·cos,·sin·and·tan</h1> |
| 166 | </div> | 166 | </div> |
| 167 | <div·class="code"> | 167 | <div·class="code"> |
| 168 | <br/> | 168 | <br/> |
| 169 | <span·class="id"·title="keyword">Lemma</span>·<a·id="sin_2a"·class="idref"·href="#sin_2a"><span·class="id"·title="lemma">sin_2a</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="x:54"·class="idref"·href="#x:54"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#sin"><span·class="id"·title="definition">sin</span></a>·(2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:54"><span·class="id"·title="variable">x</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#sin"><span·class="id"·title="definition">sin</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:54"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#cos"><span·class="id"·title="definition">cos</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:54"><span·class="id"·title="variable">x</span></a>.<br/> | 169 | <span·class="id"·title="keyword">Lemma</span>·<a·id="sin_2a"·class="idref"·href="#sin_2a"><span·class="id"·title="lemma">sin_2a</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="x:54"·class="idref"·href="#x:54"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#sin"><span·class="id"·title="definition">sin</span></a>·(2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:54"><span·class="id"·title="variable">x</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·2·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#sin"><span·class="id"·title="definition">sin</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:54"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#cos"><span·class="id"·title="definition">cos</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:54"><span·class="id"·title="variable">x</span></a>.<br/> |
| Offset 50, 15 lines modified | Offset 50, 15 lines modified | ||
| 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lra.html#"><span·class="id"·title="library">Lra</span></a>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Lra.html#"><span·class="id"·title="library">Lra</span></a>·<a·class="idref"·href="Coq.micromega.Lia.html#"><span·class="id"·title="library">Lia</span></a>.<br/> |
| 51 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 51 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 52 | <br/> | 52 | <br/> |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | <a·id="lab | 55 | <a·id="lab175"></a><h1·class="section">Definition·of·exponential</h1> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="code"> | 57 | <div·class="code"> |
| 58 | <span·class="id"·title="keyword">Definition</span>·<a·id="exp_in"·class="idref"·href="#exp_in"><span·class="id"·title="definition">exp_in</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·<a·id="l:2"·class="idref"·href="#l:2"><span·class="id"·title="binder">l</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=<br/> | 58 | <span·class="id"·title="keyword">Definition</span>·<a·id="exp_in"·class="idref"·href="#exp_in"><span·class="id"·title="definition">exp_in</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·<a·id="l:2"·class="idref"·href="#l:2"><span·class="id"·title="binder">l</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=<br/> |
| 59 | <a·class="idref"·href="Coq.Reals.Rfunctions.html#infinite_sum"><span·class="id"·title="definition">infinite_sum</span></a>·(<span·class="id"·title="keyword">fun</span>·<a·id="i:3"·class="idref"·href="#i:3"><span·class="id"·title="binder">i</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·=>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·<a·class="idref"·href="Coq.Reals.Raxioms.html#INR"><span·class="id"·title="definition">INR</span></a>·(<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#i:3"><span·class="id"·title="variable">i</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#x:1"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">^</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#i:3"><span·class="id"·title="variable">i</span></a>)·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#l:2"><span·class="id"·title="variable">l</span></a>.<br/> | 59 | <a·class="idref"·href="Coq.Reals.Rfunctions.html#infinite_sum"><span·class="id"·title="definition">infinite_sum</span></a>·(<span·class="id"·title="keyword">fun</span>·<a·id="i:3"·class="idref"·href="#i:3"><span·class="id"·title="binder">i</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·=>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#2299a6ef56270a66cc1b95489d0f6e3a"><span·class="id"·title="notation">/</span></a>·<a·class="idref"·href="Coq.Reals.Raxioms.html#INR"><span·class="id"·title="definition">INR</span></a>·(<a·class="idref"·href="Coq.Arith.Factorial.html#fact"><span·class="id"·title="definition">fact</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#i:3"><span·class="id"·title="variable">i</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#x:1"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#1fc859ef9f60fc2e3a52c48371d07707"><span·class="id"·title="notation">^</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#i:3"><span·class="id"·title="variable">i</span></a>)·<a·class="idref"·href="Coq.Reals.Rtrigo_def.html#l:2"><span·class="id"·title="variable">l</span></a>.<br/> |
| 60 | <br/> | 60 | <br/> |
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| 57 | <br/> | 57 | <br/> |
| 58 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 58 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 59 | <br/> | 59 | <br/> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="doc"> | 61 | <div·class="doc"> |
| 62 | <a·id="lab | 62 | <a·id="lab77"></a><h1·class="section">Bounds·of·expressions·with·trigonometric·functions</h1> |
| 63 | </div> | 63 | </div> |
| 64 | <div·class="code"> | 64 | <div·class="code"> |
| 65 | <br/> | 65 | <br/> |
| 66 | <span·class="id"·title="keyword">Lemma</span>·<a·id="sin2_bound"·class="idref"·href="#sin2_bound"><span·class="id"·title="lemma">sin2_bound</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>,<br/> | 66 | <span·class="id"·title="keyword">Lemma</span>·<a·id="sin2_bound"·class="idref"·href="#sin2_bound"><span·class="id"·title="lemma">sin2_bound</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>,<br/> |
| 67 | 0·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#fbdebc8c1a732d40499de973c18a9a13"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Reals.RIneq.html#2774338496db27b8a3a3c51e5609a509"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Reals.Rtrigo_def.html#sin"><span·class="id"·title="definition">sin</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_facts.html#x:1"><span·class="id"·title="variable">x</span></a><a·class="idref"·href="Coq.Reals.RIneq.html#2774338496db27b8a3a3c51e5609a509"><span·class="id"·title="notation">)²</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#fbdebc8c1a732d40499de973c18a9a13"><span·class="id"·title="notation"><=</span></a>·1.<br/> | 67 | 0·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#fbdebc8c1a732d40499de973c18a9a13"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Reals.RIneq.html#2774338496db27b8a3a3c51e5609a509"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Reals.Rtrigo_def.html#sin"><span·class="id"·title="definition">sin</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_facts.html#x:1"><span·class="id"·title="variable">x</span></a><a·class="idref"·href="Coq.Reals.RIneq.html#2774338496db27b8a3a3c51e5609a509"><span·class="id"·title="notation">)²</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#fbdebc8c1a732d40499de973c18a9a13"><span·class="id"·title="notation"><=</span></a>·1.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab22"></a><h1·class="section">Some·properties·of·the·operators·on·relations</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab23"></a><h1·class="section">Initial·version·by·Bruno·Barras</h1> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="code"> | 52 | <div·class="code"> |
| 53 | <br/> | 53 | <br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#"><span·class="id"·title="library">Relation_Definitions</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#"><span·class="id"·title="library">Relation_Definitions</span></a>.<br/> |
| 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relation_Operators.html#"><span·class="id"·title="library">Relation_Operators</span></a>.<br/> | 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relation_Operators.html#"><span·class="id"·title="library">Relation_Operators</span></a>.<br/> |
| Offset 45, 34 lines modified | Offset 45, 34 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab26"></a><h1·class="section">Some·operators·on·relations</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab27"></a><h1·class="section">Initial·authors:·Bruno·Barras,·Cristina·Cornes</h1> |
| 51 | <a·id="lab | 51 | <a·id="lab28"></a><h1·class="section">···································································</h1> |
| 52 | <a·id="lab | 52 | <a·id="lab29"></a><h1·class="section">Some·of·the·initial·definitions·were·taken·from·:</h1> |
| 53 | <a·id="lab | 53 | <a·id="lab30"></a><h1·class="section">Constructing·Recursion·Operators·in·Type·Theory</h1> |
| 54 | <a·id="lab | 54 | <a·id="lab31"></a><h1·class="section">L.·Paulson··JSC·(1986)·2,·325-355</h1> |
| 55 | <a·id="lab | 55 | <a·id="lab32"></a><h1·class="section">···································································</h1> |
| 56 | <a·id="lab | 56 | <a·id="lab33"></a><h1·class="section">Further·extensions·by·Pierre·Castéran</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#"><span·class="id"·title="library">Relation_Definitions</span></a>.<br/> | 60 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#"><span·class="id"·title="library">Relation_Definitions</span></a>.<br/> |
| 61 | <br/> | 61 | <br/> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="doc"> | 63 | <div·class="doc"> |
| 64 | <a·id="lab | 64 | <a·id="lab34"></a><h2·class="section">Transitive·closure</h2> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="code"> | 66 | <div·class="code"> |
| 67 | <br/> | 67 | <br/> |
| 68 | <span·class="id"·title="keyword">Section</span>·<a·id="Transitive_Closure"·class="idref"·href="#Transitive_Closure"><span·class="id"·title="section">Transitive_Closure</span></a>.<br/> | 68 | <span·class="id"·title="keyword">Section</span>·<a·id="Transitive_Closure"·class="idref"·href="#Transitive_Closure"><span·class="id"·title="section">Transitive_Closure</span></a>.<br/> |
| 69 | <span·class="id"·title="keyword">Variable</span>·<a·id="Transitive_Closure.A"·class="idref"·href="#Transitive_Closure.A"><span·class="id"·title="variable">A</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> | 69 | <span·class="id"·title="keyword">Variable</span>·<a·id="Transitive_Closure.A"·class="idref"·href="#Transitive_Closure.A"><span·class="id"·title="variable">A</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab484"></a><h1·class="section">Circular·Shifts·(aka·Cyclic·Permutations)</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·The·main·inductive·<span·class="inlinecode"><span·class="id"·title="var">CPermutation</span></span>·relates·lists·up·to·circular·shifts·of·their·elements. | 51 | ·The·main·inductive·<span·class="inlinecode"><span·class="id"·title="var">CPermutation</span></span>·relates·lists·up·to·circular·shifts·of·their·elements. |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
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| 66 | <br/> | 66 | <br/> |
| 67 | <span·class="id"·title="keyword">Section</span>·<a·id="defs"·class="idref"·href="#defs"><span·class="id"·title="section">defs</span></a>.<br/> | 67 | <span·class="id"·title="keyword">Section</span>·<a·id="defs"·class="idref"·href="#defs"><span·class="id"·title="section">defs</span></a>.<br/> |
| 68 | <br/> | 68 | <br/> |
| 69 | </div> | 69 | </div> |
| 70 | <div·class="doc"> | 70 | <div·class="doc"> |
| 71 | <a·id="lab | 71 | <a·id="lab489"></a><h1·class="section">Trees·and·heap·trees</h1> |
| 72 | <div·class="paragraph">·</div> | 72 | <div·class="paragraph">·</div> |
| 73 | <a·id="lab | 73 | <a·id="lab490"></a><h2·class="section">Definition·of·trees·over·an·ordered·set</h2> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="code"> | 75 | <div·class="code"> |
| 76 | <br/> | 76 | <br/> |
| 77 | <span·class="id"·title="keyword">Variable</span>·<a·id="defs.A"·class="idref"·href="#defs.A"><span·class="id"·title="variable">A</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> | 77 | <span·class="id"·title="keyword">Variable</span>·<a·id="defs.A"·class="idref"·href="#defs.A"><span·class="id"·title="variable">A</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> |
| 78 | <span·class="id"·title="keyword">Variable</span>·<a·id="defs.leA"·class="idref"·href="#defs.leA"><span·class="id"·title="variable">leA</span></a>·:·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#relation"><span·class="id"·title="definition">relation</span></a>·<a·class="idref"·href="Coq.Sorting.Heap.html#defs.A"><span·class="id"·title="variable">A</span></a>.<br/> | 78 | <span·class="id"·title="keyword">Variable</span>·<a·id="defs.leA"·class="idref"·href="#defs.leA"><span·class="id"·title="variable">leA</span></a>·:·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#relation"><span·class="id"·title="definition">relation</span></a>·<a·class="idref"·href="Coq.Sorting.Heap.html#defs.A"><span·class="id"·title="variable">A</span></a>.<br/> |
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| 85 | <br/> | 85 | <br/> |
| 86 | <span·class="id"·title="keyword">Section</span>·<a·id="Permut"·class="idref"·href="#Permut"><span·class="id"·title="section">Permut</span></a>.<br/> | 86 | <span·class="id"·title="keyword">Section</span>·<a·id="Permut"·class="idref"·href="#Permut"><span·class="id"·title="section">Permut</span></a>.<br/> |
| 87 | <br/> | 87 | <br/> |
| 88 | </div> | 88 | </div> |
| 89 | <div·class="doc"> | 89 | <div·class="doc"> |
| 90 | <a·id="lab | 90 | <a·id="lab486"></a><h1·class="section">From·lists·to·multisets</h1> |
| 91 | </div> | 91 | </div> |
| 92 | <div·class="code"> | 92 | <div·class="code"> |
| 93 | <br/> | 93 | <br/> |
| 94 | <span·class="id"·title="keyword">Variable</span>·<a·id="Permut.A"·class="idref"·href="#Permut.A"><span·class="id"·title="variable">A</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> | 94 | <span·class="id"·title="keyword">Variable</span>·<a·id="Permut.A"·class="idref"·href="#Permut.A"><span·class="id"·title="variable">A</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> |
| 95 | <span·class="id"·title="keyword">Variable</span>·<a·id="Permut.eqA"·class="idref"·href="#Permut.eqA"><span·class="id"·title="variable">eqA</span></a>·:·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#relation"><span·class="id"·title="definition">relation</span></a>·<a·class="idref"·href="Coq.Sorting.PermutSetoid.html#Permut.A"><span·class="id"·title="variable">A</span></a>.<br/> | 95 | <span·class="id"·title="keyword">Variable</span>·<a·id="Permut.eqA"·class="idref"·href="#Permut.eqA"><span·class="id"·title="variable">eqA</span></a>·:·<a·class="idref"·href="Coq.Relations.Relation_Definitions.html#relation"><span·class="id"·title="definition">relation</span></a>·<a·class="idref"·href="Coq.Sorting.PermutSetoid.html#Permut.A"><span·class="id"·title="variable">A</span></a>.<br/> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab485"></a><h1·class="section">List·permutations·as·a·composition·of·adjacent·transpositions</h1> |
| 50 | </div> | 50 | </div> |
| 51 | <div·class="code"> | 51 | <div·class="code"> |
| 52 | <br/> | 52 | <br/> |
| 53 | <br/> | 53 | <br/> |
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| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>·<a·class="idref"·href="Coq.PArith.BinPos.html#"><span·class="id"·title="library">BinPos</span></a>·<a·class="idref"·href="Coq.NArith.BinNat.html#"><span·class="id"·title="library">BinNat</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>·<a·class="idref"·href="Coq.NArith.Nnat.html#"><span·class="id"·title="library">Nnat</span></a>·<a·class="idref"·href="Coq.Strings.Byte.html#"><span·class="id"·title="library">Coq.Strings.Byte</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#"><span·class="id"·title="library">Bool</span></a>·<a·class="idref"·href="Coq.PArith.BinPos.html#"><span·class="id"·title="library">BinPos</span></a>·<a·class="idref"·href="Coq.NArith.BinNat.html#"><span·class="id"·title="library">BinNat</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>·<a·class="idref"·href="Coq.NArith.Nnat.html#"><span·class="id"·title="library">Nnat</span></a>·<a·class="idref"·href="Coq.Strings.Byte.html#"><span·class="id"·title="library">Coq.Strings.Byte</span></a>.<br/> |
| 59 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Notations.html#IfNotations"><span·class="id"·title="module">IfNotations</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Notations.html#IfNotations"><span·class="id"·title="module">IfNotations</span></a>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab553"></a><h1·class="section">Definition·of·ascii·characters</h1> |
| 64 | <div·class="paragraph">·</div> | 64 | <div·class="paragraph">·</div> |
| 65 | ·Definition·of·ascii·character·as·a·8·bits·constructor· | 65 | ·Definition·of·ascii·character·as·a·8·bits·constructor· |
| 66 | </div> | 66 | </div> |
| 67 | <div·class="code"> | 67 | <div·class="code"> |
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| 61 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Strings.Byte.html#"><span·class="id"·title="library">Coq.Strings.Byte</span></a>.<br/> | 61 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Strings.Byte.html#"><span·class="id"·title="library">Coq.Strings.Byte</span></a>.<br/> |
| 62 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Notations.html#IfNotations"><span·class="id"·title="module">IfNotations</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Notations.html#IfNotations"><span·class="id"·title="module">IfNotations</span></a>.<br/> |
| 63 | <br/> | 63 | <br/> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="doc"> | 65 | <div·class="doc"> |
| 66 | <a·id="lab | 66 | <a·id="lab556"></a><h3·class="section">Definition·of·strings</h3> |
| 67 | <div·class="paragraph">·</div> | 67 | <div·class="paragraph">·</div> |
| 68 | ·Implementation·of·string·as·list·of·ascii·characters· | 68 | ·Implementation·of·string·as·list·of·ascii·characters· |
| 69 | </div> | 69 | </div> |
| 70 | <div·class="code"> | 70 | <div·class="code"> |
| Offset 56, 39 lines modified | Offset 56, 39 lines modified | ||
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | NB:·This·file·is·here·only·for·compatibility·with·earlier·version·of | 58 | NB:·This·file·is·here·only·for·compatibility·with·earlier·version·of |
| 59 | ····<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">FMap</span></span>.·Please·use·<span·class="inlinecode"><span·class="id"·title="var">Structures</span>/<span·class="id"·title="var">Equalities.v</span></span>·directly·now.· | 59 | ····<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">FMap</span></span>.·Please·use·<span·class="inlinecode"><span·class="id"·title="var">Structures</span>/<span·class="id"·title="var">Equalities.v</span></span>·directly·now.· |
| 60 | <div·class="paragraph">·</div> | 60 | <div·class="paragraph">·</div> |
| 61 | <a·id="lab | 61 | <a·id="lab539"></a><h1·class="section">Types·with·Equalities,·and·nothing·more·(for·subtyping·purpose)</h1> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="code"> | 63 | <div·class="code"> |
| 64 | <br/> | 64 | <br/> |
| 65 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="EqualityType"·class="idref"·href="#EqualityType"><span·class="id"·title="module">EqualityType</span></a>·:=·<a·class="idref"·href="Coq.Structures.Equalities.html#EqualityTypeOrig"><span·class="id"·title="module">Equalities.EqualityTypeOrig</span></a>.<br/> | 65 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="EqualityType"·class="idref"·href="#EqualityType"><span·class="id"·title="module">EqualityType</span></a>·:=·<a·class="idref"·href="Coq.Structures.Equalities.html#EqualityTypeOrig"><span·class="id"·title="module">Equalities.EqualityTypeOrig</span></a>.<br/> |
| 66 | <br/> | 66 | <br/> |
| 67 | </div> | 67 | </div> |
| 68 | <div·class="doc"> | 68 | <div·class="doc"> |
| 69 | <a·id="lab | 69 | <a·id="lab540"></a><h1·class="section">Types·with·decidable·Equalities·(but·no·ordering)</h1> |
| 70 | </div> | 70 | </div> |
| 71 | <div·class="code"> | 71 | <div·class="code"> |
| 72 | <br/> | 72 | <br/> |
| 73 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="DecidableType"·class="idref"·href="#DecidableType"><span·class="id"·title="module">DecidableType</span></a>·:=·<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableTypeOrig"><span·class="id"·title="module">Equalities.DecidableTypeOrig</span></a>.<br/> | 73 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="DecidableType"·class="idref"·href="#DecidableType"><span·class="id"·title="module">DecidableType</span></a>·:=·<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableTypeOrig"><span·class="id"·title="module">Equalities.DecidableTypeOrig</span></a>.<br/> |
| 74 | <br/> | 74 | <br/> |
| 75 | </div> | 75 | </div> |
| 76 | <div·class="doc"> | 76 | <div·class="doc"> |
| 77 | <a·id="lab | 77 | <a·id="lab541"></a><h1·class="section">Additional·notions·about·keys·and·datas·used·in·FMap</h1> |
| 78 | </div> | 78 | </div> |
| 79 | <div·class="code"> | 79 | <div·class="code"> |
| 80 | <br/> | 80 | <br/> |
| 81 | <span·class="id"·title="keyword">Module</span>·<a·id="KeyDecidableType"·class="idref"·href="#KeyDecidableType"><span·class="id"·title="module">KeyDecidableType</span></a>(<span·class="id"·title="var">D</span>:<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>).<br/> | 81 | <span·class="id"·title="keyword">Module</span>·<a·id="KeyDecidableType"·class="idref"·href="#KeyDecidableType"><span·class="id"·title="module">KeyDecidableType</span></a>(<span·class="id"·title="var">D</span>:<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>).<br/> |
| 82 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.DecidableType.html#D"><span·class="id"·title="module">D</span></a>.<br/> | 82 | <span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.DecidableType.html#D"><span·class="id"·title="module">D</span></a>.<br/> |
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| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | NB:·This·file·is·here·only·for·compatibility·with·earlier·version·of | 55 | NB:·This·file·is·here·only·for·compatibility·with·earlier·version·of |
| 56 | ····<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">FMap</span></span>.·Please·use·<span·class="inlinecode"><span·class="id"·title="var">Structures</span>/<span·class="id"·title="var">Equalities.v</span></span>·directly·now.· | 56 | ····<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">FMap</span></span>.·Please·use·<span·class="inlinecode"><span·class="id"·title="var">Structures</span>/<span·class="id"·title="var">Equalities.v</span></span>·directly·now.· |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | <a·id="lab | 58 | <a·id="lab532"></a><h1·class="section">Examples·of·Decidable·Type·structures.</h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·A·particular·case·of·<span·class="inlinecode"><span·class="id"·title="var">DecidableType</span></span>·where | 60 | ·A·particular·case·of·<span·class="inlinecode"><span·class="id"·title="var">DecidableType</span></span>·where |
| 61 | ····the·equality·is·the·usual·one·of·Coq.· | 61 | ····the·equality·is·the·usual·one·of·Coq.· |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="code"> | 63 | <div·class="code"> |
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| 65 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="Nop"·class="idref"·href="#Nop"><span·class="id"·title="module">Nop</span></a>.<br/> | 65 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="Nop"·class="idref"·href="#Nop"><span·class="id"·title="module">Nop</span></a>.<br/> |
| 66 | <span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Structures.Equalities.html#Nop"><span·class="id"·title="module">Nop</span></a>.<br/> | 66 | <span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Structures.Equalities.html#Nop"><span·class="id"·title="module">Nop</span></a>.<br/> |
| 67 | <br/> | 67 | <br/> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="doc"> | 69 | <div·class="doc"> |
| 70 | <a·id="lab | 70 | <a·id="lab522"></a><h1·class="section">Structure·with·just·a·base·type·<span·class="inlinecode"><span·class="id"·title="var">t</span></span></h1> |
| 71 | </div> | 71 | </div> |
| 72 | <div·class="code"> | 72 | <div·class="code"> |
| 73 | <br/> | 73 | <br/> |
| 74 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="Typ"·class="idref"·href="#Typ"><span·class="id"·title="module">Typ</span></a>.<br/> | 74 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="Typ"·class="idref"·href="#Typ"><span·class="id"·title="module">Typ</span></a>.<br/> |
| 75 | <span·class="id"·title="keyword">Parameter</span>·<span·class="id"·title="keyword">Inline</span>(10)·<a·id="Typ.t"·class="idref"·href="#Typ.t"><span·class="id"·title="axiom">t</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> | 75 | <span·class="id"·title="keyword">Parameter</span>·<span·class="id"·title="keyword">Inline</span>(10)·<a·id="Typ.t"·class="idref"·href="#Typ.t"><span·class="id"·title="axiom">t</span></a>·:·<span·class="id"·title="keyword">Type</span>.<br/> |
| 76 | <span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Structures.Equalities.html#Typ"><span·class="id"·title="module">Typ</span></a>.<br/> | 76 | <span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Structures.Equalities.html#Typ"><span·class="id"·title="module">Typ</span></a>.<br/> |
| 77 | <br/> | 77 | <br/> |
| 78 | </div> | 78 | </div> |
| 79 | <div·class="doc"> | 79 | <div·class="doc"> |
| 80 | <a·id="lab | 80 | <a·id="lab523"></a><h1·class="section">Structure·with·an·equality·relation·<span·class="inlinecode"><span·class="id"·title="var">eq</span></span></h1> |
| 81 | </div> | 81 | </div> |
| 82 | <div·class="code"> | 82 | <div·class="code"> |
| 83 | <br/> | 83 | <br/> |
| 84 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="HasEq"·class="idref"·href="#HasEq"><span·class="id"·title="module">HasEq</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">T</span>:<a·class="idref"·href="Coq.Structures.Equalities.html#Typ"><span·class="id"·title="module">Typ</span></a>).<br/> | 84 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="HasEq"·class="idref"·href="#HasEq"><span·class="id"·title="module">HasEq</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">T</span>:<a·class="idref"·href="Coq.Structures.Equalities.html#Typ"><span·class="id"·title="module">Typ</span></a>).<br/> |
| 85 | <span·class="id"·title="keyword">Parameter</span>·<span·class="id"·title="keyword">Inline</span>(30)·<a·id="HasEq.eq"·class="idref"·href="#HasEq.eq"><span·class="id"·title="axiom">eq</span></a>·:·<a·class="idref"·href="Coq.Structures.Equalities.html#T.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#T.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>.<br/> | 85 | <span·class="id"·title="keyword">Parameter</span>·<span·class="id"·title="keyword">Inline</span>(30)·<a·id="HasEq.eq"·class="idref"·href="#HasEq.eq"><span·class="id"·title="axiom">eq</span></a>·:·<a·class="idref"·href="Coq.Structures.Equalities.html#T.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#T.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>.<br/> |
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| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab542"></a><h1·class="section">Keys·and·datas·used·in·the·future·MMaps</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Module</span>·<a·id="KeyDecidableType"·class="idref"·href="#KeyDecidableType"><span·class="id"·title="module">KeyDecidableType</span></a>(<span·class="id"·title="var">D</span>:<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>).<br/> | 60 | <span·class="id"·title="keyword">Module</span>·<a·id="KeyDecidableType"·class="idref"·href="#KeyDecidableType"><span·class="id"·title="module">KeyDecidableType</span></a>(<span·class="id"·title="var">D</span>:<a·class="idref"·href="Coq.Structures.Equalities.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>).<br/> |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>·<a·class="idref"·href="Coq.Structures.OrdersTac.html#"><span·class="id"·title="library">OrdersTac</span></a>·<a·class="idref"·href="Coq.Structures.OrdersFacts.html#"><span·class="id"·title="library">OrdersFacts</span></a>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Program.Basics.html#"><span·class="id"·title="library">Basics</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>·<a·class="idref"·href="Coq.Structures.OrdersTac.html#"><span·class="id"·title="library">OrdersTac</span></a>·<a·class="idref"·href="Coq.Structures.OrdersFacts.html#"><span·class="id"·title="library">OrdersFacts</span></a>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Program.Basics.html#"><span·class="id"·title="library">Basics</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab544"></a><h1·class="section">A·Generic·construction·of·min·and·max</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | <a·id="lab | 55 | <a·id="lab545"></a><h2·class="section">First,·an·interface·for·types·with·<span·class="inlinecode"><span·class="id"·title="var">max</span></span>·and/or·<span·class="inlinecode"><span·class="id"·title="var">min</span></span></h2> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="code"> | 57 | <div·class="code"> |
| 58 | <br/> | 58 | <br/> |
| 59 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="HasMax"·class="idref"·href="#HasMax"><span·class="id"·title="module">HasMax</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">E</span>:<a·class="idref"·href="Coq.Structures.Orders.html#EqLe'"><span·class="id"·title="module">EqLe'</span></a>).<br/> | 59 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="HasMax"·class="idref"·href="#HasMax"><span·class="id"·title="module">HasMax</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">E</span>:<a·class="idref"·href="Coq.Structures.Orders.html#EqLe'"><span·class="id"·title="module">EqLe'</span></a>).<br/> |
| 60 | <span·class="id"·title="keyword">Parameter</span>·<span·class="id"·title="keyword">Inline</span>·<a·id="HasMax.max"·class="idref"·href="#HasMax.max"><span·class="id"·title="axiom">max</span></a>·:·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#E.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#E.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#E.t"><span·class="id"·title="axiom">t</span></a>.<br/> | 60 | <span·class="id"·title="keyword">Parameter</span>·<span·class="id"·title="keyword">Inline</span>·<a·id="HasMax.max"·class="idref"·href="#HasMax.max"><span·class="id"·title="axiom">max</span></a>·:·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#E.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#E.t"><span·class="id"·title="axiom">t</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#E.t"><span·class="id"·title="axiom">t</span></a>.<br/> |
| Offset 53, 15 lines modified | Offset 53, 15 lines modified | ||
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | NB:·This·file·is·here·only·for·compatibility·with·earlier·version·of | 55 | NB:·This·file·is·here·only·for·compatibility·with·earlier·version·of |
| 56 | ·····<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">FMap</span></span>.·Please·use·<span·class="inlinecode"><span·class="id"·title="var">Structures</span>/<span·class="id"·title="var">Orders.v</span></span>·directly·now.· | 56 | ·····<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">FMap</span></span>.·Please·use·<span·class="inlinecode"><span·class="id"·title="var">Structures</span>/<span·class="id"·title="var">Orders.v</span></span>·directly·now.· |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | <a·id="lab | 58 | <a·id="lab520"></a><h1·class="section">Ordered·types</h1> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <br/> | 61 | <br/> |
| 62 | <span·class="id"·title="keyword">Inductive</span>·<a·id="Compare"·class="idref"·href="#Compare"><span·class="id"·title="definition,·inductive"><span·id="Compare_rect"·class="id"><span·id="Compare_ind"·class="id"><span·id="Compare_rec"·class="id"><span·id="Compare_sind"·class="id">Compare</span></span></span></span></span></a>·(<a·id="X:1"·class="idref"·href="#X:1"><span·class="id"·title="binder">X</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="lt:2"·class="idref"·href="#lt:2"><span·class="id"·title="binder">lt</span></a>·<a·id="eq:3"·class="idref"·href="#eq:3"><span·class="id"·title="binder">eq</span></a>·:·<a·class="idref"·href="Coq.Structures.OrderedType.html#X:1"><span·class="id"·title="variable">X</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#X:1"><span·class="id"·title="variable">X</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·(<a·id="x:4"·class="idref"·href="#x:4"><span·class="id"·title="binder">x</span></a>·<a·id="y:5"·class="idref"·href="#y:5"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Structures.OrderedType.html#X:1"><span·class="id"·title="variable">X</span></a>)·:·<span·class="id"·title="keyword">Type</span>·:=<br/> | 62 | <span·class="id"·title="keyword">Inductive</span>·<a·id="Compare"·class="idref"·href="#Compare"><span·class="id"·title="definition,·inductive"><span·id="Compare_rect"·class="id"><span·id="Compare_ind"·class="id"><span·id="Compare_rec"·class="id"><span·id="Compare_sind"·class="id">Compare</span></span></span></span></span></a>·(<a·id="X:1"·class="idref"·href="#X:1"><span·class="id"·title="binder">X</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="lt:2"·class="idref"·href="#lt:2"><span·class="id"·title="binder">lt</span></a>·<a·id="eq:3"·class="idref"·href="#eq:3"><span·class="id"·title="binder">eq</span></a>·:·<a·class="idref"·href="Coq.Structures.OrderedType.html#X:1"><span·class="id"·title="variable">X</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#X:1"><span·class="id"·title="variable">X</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·(<a·id="x:4"·class="idref"·href="#x:4"><span·class="id"·title="binder">x</span></a>·<a·id="y:5"·class="idref"·href="#y:5"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Structures.OrderedType.html#X:1"><span·class="id"·title="variable">X</span></a>)·:·<span·class="id"·title="keyword">Type</span>·:=<br/> |
| 63 | |·<a·id="LT"·class="idref"·href="#LT"><span·class="id"·title="constructor">LT</span></a>·:·<a·class="idref"·href="Coq.Structures.OrderedType.html#lt:2"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#x:4"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#y:5"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#Compare:6"><span·class="id"·title="inductive">Compare</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#lt:2"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#eq:3"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#x:4"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#y:5"><span·class="id"·title="variable">y</span></a><br/> | 63 | |·<a·id="LT"·class="idref"·href="#LT"><span·class="id"·title="constructor">LT</span></a>·:·<a·class="idref"·href="Coq.Structures.OrderedType.html#lt:2"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#x:4"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#y:5"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#Compare:6"><span·class="id"·title="inductive">Compare</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#lt:2"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#eq:3"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#x:4"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Structures.OrderedType.html#y:5"><span·class="id"·title="variable">y</span></a><br/> |
| Offset 46, 15 lines modified | Offset 46, 15 lines modified | ||
| 46 | <div·class="code"> | 46 | <div·class="code"> |
| 47 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>.<br/> | 47 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>.<br/> |
| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | <a·id="lab | 51 | <a·id="lab504"></a><h1·class="section">An·alternative·(but·equivalent)·presentation·for·an·Ordered·Type</h1> |
| 52 | ···inferface.· | 52 | ···inferface.· |
| 53 | <div·class="paragraph">·</div> | 53 | <div·class="paragraph">·</div> |
| 54 | ·NB:·<span·class="inlinecode"><span·class="id"·title="var">comparison</span></span>,·defined·in·<span·class="inlinecode"><span·class="id"·title="var">Datatypes.v</span></span>·is·<span·class="inlinecode"><span·class="id"·title="var">Eq</span>|<span·class="id"·title="var">Lt</span>|<span·class="id"·title="var">Gt</span></span> | 54 | ·NB:·<span·class="inlinecode"><span·class="id"·title="var">comparison</span></span>,·defined·in·<span·class="inlinecode"><span·class="id"·title="var">Datatypes.v</span></span>·is·<span·class="inlinecode"><span·class="id"·title="var">Eq</span>|<span·class="id"·title="var">Lt</span>|<span·class="id"·title="var">Gt</span></span> |
| 55 | whereas·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span>,·defined·in·<span·class="inlinecode"><span·class="id"·title="var">OrderedType.v</span></span>·is·<span·class="inlinecode"><span·class="id"·title="var">EQ</span></span>·<span·class="inlinecode"><span·class="id"·title="var">_</span></span>·<span·class="inlinecode">|</span>·<span·class="inlinecode"><span·class="id"·title="var">LT</span></span>·<span·class="inlinecode"><span·class="id"·title="var">_</span></span>·<span·class="inlinecode">|</span>·<span·class="inlinecode"><span·class="id"·title="var">GT</span></span>·<span·class="inlinecode"><span·class="id"·title="var">_</span></span>·<span·class="inlinecode"></span> | 55 | whereas·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span>,·defined·in·<span·class="inlinecode"><span·class="id"·title="var">OrderedType.v</span></span>·is·<span·class="inlinecode"><span·class="id"·title="var">EQ</span></span>·<span·class="inlinecode"><span·class="id"·title="var">_</span></span>·<span·class="inlinecode">|</span>·<span·class="inlinecode"><span·class="id"·title="var">LT</span></span>·<span·class="inlinecode"><span·class="id"·title="var">_</span></span>·<span·class="inlinecode">|</span>·<span·class="inlinecode"><span·class="id"·title="var">GT</span></span>·<span·class="inlinecode"><span·class="id"·title="var">_</span></span>·<span·class="inlinecode"></span> |
| Offset 52, 15 lines modified | Offset 52, 15 lines modified | ||
| 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.NArith.NArith.html#"><span·class="id"·title="library">NArith</span></a>·<a·class="idref"·href="Coq.NArith.Ndec.html#"><span·class="id"·title="library">Ndec</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.NArith.NArith.html#"><span·class="id"·title="library">NArith</span></a>·<a·class="idref"·href="Coq.NArith.Ndec.html#"><span·class="id"·title="library">Ndec</span></a>.<br/> |
| 54 | <br/> | 54 | <br/> |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="doc"> | 56 | <div·class="doc"> |
| 57 | <a·id="lab | 57 | <a·id="lab531"></a><h1·class="section">Examples·of·Ordered·Type·structures.</h1> |
| 58 | <div·class="paragraph">·</div> | 58 | <div·class="paragraph">·</div> |
| 59 | ·First,·a·particular·case·of·<span·class="inlinecode"><span·class="id"·title="var">OrderedType</span></span>·where | 59 | ·First,·a·particular·case·of·<span·class="inlinecode"><span·class="id"·title="var">OrderedType</span></span>·where |
| 60 | ····the·equality·is·the·usual·one·of·Coq.· | 60 | ····the·equality·is·the·usual·one·of·Coq.· |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <div·class="code"> |
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| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Relations.Relations.html#"><span·class="id"·title="library">Relations</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#"><span·class="id"·title="library">Equalities</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Relations.Relations.html#"><span·class="id"·title="library">Relations</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#"><span·class="id"·title="library">Equalities</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 50 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 51 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab505"></a><h1·class="section">Ordered·types</h1> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | ·First,·signatures·with·only·the·order·relations· | 56 | ·First,·signatures·with·only·the·order·relations· |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
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| 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>.<br/> | 51 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.OrderedType.html#"><span·class="id"·title="library">OrderedType</span></a>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab514"></a><h1·class="section">Some·alternative·(but·equivalent)·presentations·for·an·Ordered·Type</h1> |
| 57 | ···inferface.· | 57 | ···inferface.· |
| 58 | <div·class="paragraph">·</div> | 58 | <div·class="paragraph">·</div> |
| 59 | <a·id="lab | 59 | <a·id="lab515"></a><h2·class="section">The·original·interface</h2> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="OrderedTypeOrig"·class="idref"·href="#OrderedTypeOrig"><span·class="id"·title="module">OrderedTypeOrig</span></a>·:=·<a·class="idref"·href="Coq.Structures.OrderedType.html#OrderedType"><span·class="id"·title="module">OrderedType.OrderedType</span></a>.<br/> | 63 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="OrderedTypeOrig"·class="idref"·href="#OrderedTypeOrig"><span·class="id"·title="module">OrderedTypeOrig</span></a>·:=·<a·class="idref"·href="Coq.Structures.OrderedType.html#OrderedType"><span·class="id"·title="module">OrderedType.OrderedType</span></a>.<br/> |
| 64 | <br/> | 64 | <br/> |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="doc"> | 66 | <div·class="doc"> |
| 67 | <a·id="lab | 67 | <a·id="lab516"></a><h2·class="section">An·interface·based·on·compare</h2> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="code"> | 69 | <div·class="code"> |
| 70 | <br/> | 70 | <br/> |
| 71 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="OrderedTypeAlt"·class="idref"·href="#OrderedTypeAlt"><span·class="id"·title="module">OrderedTypeAlt</span></a>.<br/> | 71 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="OrderedTypeAlt"·class="idref"·href="#OrderedTypeAlt"><span·class="id"·title="module">OrderedTypeAlt</span></a>.<br/> |
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| 52 | <a·class="idref"·href="Coq.Classes.RelationPairs.html#"><span·class="id"·title="library">RelationPairs</span></a>·<a·class="idref"·href="Coq.Structures.EqualitiesFacts.html#"><span·class="id"·title="library">EqualitiesFacts</span></a>.<br/> | 52 | <a·class="idref"·href="Coq.Classes.RelationPairs.html#"><span·class="id"·title="library">RelationPairs</span></a>·<a·class="idref"·href="Coq.Structures.EqualitiesFacts.html#"><span·class="id"·title="library">EqualitiesFacts</span></a>.<br/> |
| 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Strings.Ascii.html#"><span·class="id"·title="library">Ascii</span></a>·<a·class="idref"·href="Coq.Strings.String.html#"><span·class="id"·title="library">String</span></a>.<br/> |
| 54 | <br/> | 54 | <br/> |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="doc"> | 56 | <div·class="doc"> |
| 57 | <a·id="lab | 57 | <a·id="lab503"></a><h1·class="section">Examples·of·Ordered·Type·structures.</h1> |
| 58 | <div·class="paragraph">·</div> | 58 | <div·class="paragraph">·</div> |
| 59 | ·Ordered·Type·for·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Positive</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">ascii</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">string</span></span>·with·the·usual·or·lexicographic·order.· | 59 | ·Ordered·Type·for·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Positive</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">N</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">ascii</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">string</span></span>·with·the·usual·or·lexicographic·order.· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
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| 52 | <br/> | 52 | <br/> |
| 53 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 53 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 54 | <br/> | 54 | <br/> |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="doc"> | 56 | <div·class="doc"> |
| 57 | <a·id="lab | 57 | <a·id="lab533"></a><h1·class="section">Properties·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h1> |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="code"> | 59 | <div·class="code"> |
| 60 | <br/> | 60 | <br/> |
| 61 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="CompareFacts"·class="idref"·href="#CompareFacts"><span·class="id"·title="module">CompareFacts</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">O</span>:<a·class="idref"·href="Coq.Structures.Orders.html#DecStrOrder'"><span·class="id"·title="module">DecStrOrder'</span></a>).<br/> | 61 | <span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="CompareFacts"·class="idref"·href="#CompareFacts"><span·class="id"·title="module">CompareFacts</span></a>·(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">O</span>:<a·class="idref"·href="Coq.Structures.Orders.html#DecStrOrder'"><span·class="id"·title="module">DecStrOrder'</span></a>).<br/> |
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| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 52 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab512"></a><h1·class="section">Specialization·of·results·about·lists·modulo.</h1> |
| 57 | </div> | 57 | </div> |
| 58 | <div·class="code"> | 58 | <div·class="code"> |
| 59 | <br/> | 59 | <br/> |
| 60 | <span·class="id"·title="keyword">Module</span>·<a·id="OrderedTypeLists"·class="idref"·href="#OrderedTypeLists"><span·class="id"·title="module">OrderedTypeLists</span></a>·(<span·class="id"·title="var">O</span>:<a·class="idref"·href="Coq.Structures.Orders.html#OrderedType"><span·class="id"·title="module">OrderedType</span></a>).<br/> | 60 | <span·class="id"·title="keyword">Module</span>·<a·id="OrderedTypeLists"·class="idref"·href="#OrderedTypeLists"><span·class="id"·title="module">OrderedTypeLists</span></a>·(<span·class="id"·title="var">O</span>:<a·class="idref"·href="Coq.Structures.Orders.html#OrderedType"><span·class="id"·title="module">OrderedType</span></a>).<br/> |
| Offset 49, 15 lines modified | Offset 49, 15 lines modified | ||
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Program.Basics.html#"><span·class="id"·title="library">Basics</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#"><span·class="id"·title="library">Equalities</span></a>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="library">Setoid</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.Program.Basics.html#"><span·class="id"·title="library">Basics</span></a>·<a·class="idref"·href="Coq.Structures.Equalities.html#"><span·class="id"·title="library">Equalities</span></a>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 50 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 51 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab499"></a><h1·class="section">The·order·tactic</h1> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | ·This·tactic·is·designed·to·solve·systems·of·(in)equations | 56 | ·This·tactic·is·designed·to·solve·systems·of·(in)equations |
| 57 | ····involving·<span·class="inlinecode"><span·class="id"·title="var">eq</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·and·<span·class="inlinecode">~<span·class="id"·title="var">eq</span></span>·on·some·type.·This·tactic·is | 57 | ····involving·<span·class="inlinecode"><span·class="id"·title="var">eq</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·and·<span·class="inlinecode">~<span·class="id"·title="var">eq</span></span>·on·some·type.·This·tactic·is |
| 58 | ····domain-agnostic;·it·will·only·use·equivalence+order·axioms,·and | 58 | ····domain-agnostic;·it·will·only·use·equivalence+order·axioms,·and |
| 59 | ····not·analyze·elements·of·the·domain.·Hypothesis·or·goal·of·the·form | 59 | ····not·analyze·elements·of·the·domain.·Hypothesis·or·goal·of·the·form |
| Offset 5852, 15 lines modified | Offset 5852, 15 lines modified | ||
| 00016db0:·2068·7265·663d·2223·4954·4552·4154·4f52···href="#ITERATOR | 00016db0:·2068·7265·663d·2223·4954·4552·4154·4f52···href="#ITERATOR | ||
| 00016dc0:·5322·3e3c·7370·616e·2063·6c61·7373·3d22··S"><span·class=" | 00016dc0:·5322·3e3c·7370·616e·2063·6c61·7373·3d22··S"><span·class=" | ||
| 00016dd0:·6964·2220·7469·746c·653d·2273·6563·7469··id"·title="secti | 00016dd0:·6964·2220·7469·746c·653d·2273·6563·7469··id"·title="secti | ||
| 00016de0:·6f6e·223e·4954·4552·4154·4f52·533c·2f73··on">ITERATORS</s | 00016de0:·6f6e·223e·4954·4552·4154·4f52·533c·2f73··on">ITERATORS</s | ||
| 00016df0:·7061·6e3e·3c2f·613e·2e3c·6272·2f3e·0a3c··pan></a>.<br/>.< | 00016df0:·7061·6e3e·3c2f·613e·2e3c·6272·2f3e·0a3c··pan></a>.<br/>.< | ||
| 00016e00:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas | 00016e00:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas | ||
| 00016e10:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id=" | 00016e10:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id=" | ||
| 00016e20:·6c61·6231·323 | 00016e20:·6c61·6231·3238·3122·3e3c·2f61·3e3c·6831··lab1281"></a><h1 | ||
| 00016e30:·2063·6c61·7373·3d22·7365·6374·696f·6e22···class="section" | 00016e30:·2063·6c61·7373·3d22·7365·6374·696f·6e22···class="section" | ||
| 00016e40:·3e48·6572·6520·6172·6520·7370·6563·6961··>Here·are·specia | 00016e40:·3e48·6572·6520·6172·6520·7370·6563·6961··>Here·are·specia | ||
| 00016e50:·6c20·6e6f·6e20·6465·7065·6e64·656e·7420··l·non·dependent· | 00016e50:·6c20·6e6f·6e20·6465·7065·6e64·656e·7420··l·non·dependent· | ||
| 00016e60:·7573·6566·756c·2069·6e73·7461·6e74·6961··useful·instantia | 00016e60:·7573·6566·756c·2069·6e73·7461·6e74·6961··useful·instantia | ||
| 00016e70:·7469·6f6e·206f·6620·696e·6475·6374·696f··tion·of·inductio | 00016e70:·7469·6f6e·206f·6620·696e·6475·6374·696f··tion·of·inductio | ||
| 00016e80:·6e20·7363·6865·6d65·733c·2f68·313e·0a0a··n·schemes</h1>.. | 00016e80:·6e20·7363·6865·6d65·733c·2f68·313e·0a0a··n·schemes</h1>.. | ||
| 00016e90:·3c64·6976·2063·6c61·7373·3d22·7061·7261··<div·class="para | 00016e90:·3c64·6976·2063·6c61·7373·3d22·7061·7261··<div·class="para |
| Offset 81, 15 lines modified | Offset 81, 15 lines modified | ||
| 81 | </div> | 81 | </div> |
| 82 | <div·class="doc"> | 82 | <div·class="doc"> |
| 83 | Lemmas·are·done·for·functions·that·use·<span·class="inlinecode"><span·class="id"·title="var">Fin.t</span></span>·but·thanks·to·<span·class="inlinecode"><span·class="id"·title="var">Peano_dec.le_unique</span></span>,·all | 83 | Lemmas·are·done·for·functions·that·use·<span·class="inlinecode"><span·class="id"·title="var">Fin.t</span></span>·but·thanks·to·<span·class="inlinecode"><span·class="id"·title="var">Peano_dec.le_unique</span></span>,·all |
| 84 | is·true·for·the·one·that·use·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span>· | 84 | is·true·for·the·one·that·use·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span>· |
| 85 | <div·class="paragraph">·</div> | 85 | <div·class="paragraph">·</div> |
| 86 | <a·id="lab12 | 86 | <a·id="lab1283"></a><h2·class="section">Properties·of·<span·class="inlinecode"><span·class="id"·title="var">nth</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">nth_order</span></span></h2> |
| 87 | </div> | 87 | </div> |
| 88 | <div·class="code"> | 88 | <div·class="code"> |
| 89 | <br/> | 89 | <br/> |
| 90 | <span·class="id"·title="keyword">Lemma</span>·<a·id="eq_nth_iff"·class="idref"·href="#eq_nth_iff"><span·class="id"·title="lemma">eq_nth_iff</span></a>·<a·id="A:26"·class="idref"·href="#A:26"><span·class="id"·title="binder">A</span></a>·<a·id="n:27"·class="idref"·href="#n:27"><span·class="id"·title="binder">n</span></a>·(<a·id="v1:28"·class="idref"·href="#v1:28"><span·class="id"·title="binder">v1</span></a>·<a·id="v2:29"·class="idref"·href="#v2:29"><span·class="id"·title="binder">v2</span></a>:·<a·class="idref"·href="Coq.Vectors.VectorDef.html#t"><span·class="id"·title="inductive">t</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#A:26"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#n:27"><span·class="id"·title="variable">n</span></a>):<br/> | 90 | <span·class="id"·title="keyword">Lemma</span>·<a·id="eq_nth_iff"·class="idref"·href="#eq_nth_iff"><span·class="id"·title="lemma">eq_nth_iff</span></a>·<a·id="A:26"·class="idref"·href="#A:26"><span·class="id"·title="binder">A</span></a>·<a·id="n:27"·class="idref"·href="#n:27"><span·class="id"·title="binder">n</span></a>·(<a·id="v1:28"·class="idref"·href="#v1:28"><span·class="id"·title="binder">v1</span></a>·<a·id="v2:29"·class="idref"·href="#v2:29"><span·class="id"·title="binder">v2</span></a>:·<a·class="idref"·href="Coq.Vectors.VectorDef.html#t"><span·class="id"·title="inductive">t</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#A:26"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#n:27"><span·class="id"·title="variable">n</span></a>):<br/> |
| 91 | <a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">(</span></a><span·class="id"·title="keyword">forall</span>·<a·id="p1:30"·class="idref"·href="#p1:30"><span·class="id"·title="binder">p1</span></a>·<a·id="p2:31"·class="idref"·href="#p2:31"><span·class="id"·title="binder">p2</span></a>,·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p1:30"><span·class="id"·title="variable">p1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p2:31"><span·class="id"·title="variable">p2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v1:28"><span·class="id"·title="variable">v1</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">[@</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p1:30"><span·class="id"·title="variable">p1</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">]</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v2:29"><span·class="id"·title="variable">v2</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">[@</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p2:31"><span·class="id"·title="variable">p2</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">]</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v1:28"><span·class="id"·title="variable">v1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v2:29"><span·class="id"·title="variable">v2</span></a>.<br/> | 91 | <a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">(</span></a><span·class="id"·title="keyword">forall</span>·<a·id="p1:30"·class="idref"·href="#p1:30"><span·class="id"·title="binder">p1</span></a>·<a·id="p2:31"·class="idref"·href="#p2:31"><span·class="id"·title="binder">p2</span></a>,·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p1:30"><span·class="id"·title="variable">p1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p2:31"><span·class="id"·title="variable">p2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v1:28"><span·class="id"·title="variable">v1</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">[@</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p1:30"><span·class="id"·title="variable">p1</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">]</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v2:29"><span·class="id"·title="variable">v2</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">[@</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p2:31"><span·class="id"·title="variable">p2</span></a>·<a·class="idref"·href="Coq.Vectors.VectorDef.html#3e2ab5a54c52b47134e5ddb66aa31ebd"><span·class="id"·title="notation">]</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v1:28"><span·class="id"·title="variable">v1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#v2:29"><span·class="id"·title="variable">v2</span></a>.<br/> |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <a·class="idref"·href="Coq.Structures.OrdersFacts.html#"><span·class="id"·title="library">OrdersFacts</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZProperties.html#"><span·class="id"·title="library">ZProperties</span></a>·<a·class="idref"·href="Coq.Classes.DecidableClass.html#"><span·class="id"·title="library">DecidableClass</span></a>.<br/> | 51 | <a·class="idref"·href="Coq.Structures.OrdersFacts.html#"><span·class="id"·title="library">OrdersFacts</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZProperties.html#"><span·class="id"·title="library">ZProperties</span></a>·<a·class="idref"·href="Coq.Classes.DecidableClass.html#"><span·class="id"·title="library">DecidableClass</span></a>.<br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.ZArith.BinIntDef.html#"><span·class="id"·title="library">BinIntDef</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.ZArith.BinIntDef.html#"><span·class="id"·title="library">BinIntDef</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab811"></a><h1·class="section">Binary·Integers</h1> |
| 57 | <div·class="paragraph">·</div> | 57 | <div·class="paragraph">·</div> |
| 58 | ·Initial·author:·Pierre·Crégut,·CNET,·Lannion,·France· | 58 | ·Initial·author:·Pierre·Crégut,·CNET,·Lannion,·France· |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">Z0</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Zpos</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Zneg</span></span> | 60 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">Z0</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Zpos</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Zneg</span></span> |
| Offset 57, 15 lines modified | Offset 57, 15 lines modified | ||
| 57 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Zpos"><span·class="id"·title="constructor">Zpos</span></a>·1).<br/> | 57 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Zpos"><span·class="id"·title="constructor">Zpos</span></a>·1).<br/> |
| 58 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'2'"·class="idref"·href="#:::'2'"><span·class="id"·title="notation">"</span></a>2"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Zpos"><span·class="id"·title="constructor">Zpos</span></a>·2).<br/> | 58 | <span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'2'"·class="idref"·href="#:::'2'"><span·class="id"·title="notation">"</span></a>2"·:=·(<a·class="idref"·href="Coq.Numbers.BinNums.html#Zpos"><span·class="id"·title="constructor">Zpos</span></a>·2).<br/> |
| 59 | <br/> | 59 | <br/> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="doc"> | 61 | <div·class="doc"> |
| 62 | <a·id="lab | 62 | <a·id="lab893"></a><h1·class="section">Binary·Integers,·Definitions·of·Operations</h1> |
| 63 | <div·class="paragraph">·</div> | 63 | <div·class="paragraph">·</div> |
| 64 | ·Initial·author:·Pierre·Crégut,·CNET,·Lannion,·France· | 64 | ·Initial·author:·Pierre·Crégut,·CNET,·Lannion,·France· |
| 65 | </div> | 65 | </div> |
| 66 | <div·class="code"> | 66 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab856"></a><h1·class="section">An·light·axiomatization·of·integers·(used·in·MSetAVL).</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·We·define·a·signature·for·an·integer·datatype·based·on·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>. | 51 | ·We·define·a·signature·for·an·integer·datatype·based·on·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>. |
| 52 | ····The·goal·is·to·allow·a·switch·after·extraction·to·ocaml's | 52 | ····The·goal·is·to·allow·a·switch·after·extraction·to·ocaml's |
| 53 | ····<span·class="inlinecode"><span·class="id"·title="var">big_int</span></span>·or·even·<span·class="inlinecode"><span·class="id"·title="var">int</span></span>·when·finiteness·isn't·a·problem | 53 | ····<span·class="inlinecode"><span·class="id"·title="var">big_int</span></span>·or·even·<span·class="inlinecode"><span·class="id"·title="var">int</span></span>·when·finiteness·isn't·a·problem |
| 54 | ····(typically·:·when·measuring·the·height·of·an·AVL·tree). | 54 | ····(typically·:·when·measuring·the·height·of·an·AVL·tree). |
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| 70 | <br/> | 70 | <br/> |
| 71 | <span·class="id"·title="keyword">Variables</span>·<a·id="decidability.x"·class="idref"·href="#decidability.x"><span·class="id"·title="variable">x</span></a>·<a·id="decidability.y"·class="idref"·href="#decidability.y"><span·class="id"·title="variable">y</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>.<br/> | 71 | <span·class="id"·title="keyword">Variables</span>·<a·id="decidability.x"·class="idref"·href="#decidability.x"><span·class="id"·title="variable">x</span></a>·<a·id="decidability.y"·class="idref"·href="#decidability.y"><span·class="id"·title="variable">y</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>.<br/> |
| 72 | <br/> | 72 | <br/> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="doc"> | 74 | <div·class="doc"> |
| 75 | <a·id="lab | 75 | <a·id="lab808"></a><h1·class="section">Decidability·of·order·on·binary·integers</h1> |
| 76 | </div> | 76 | </div> |
| 77 | <div·class="code"> | 77 | <div·class="code"> |
| 78 | <br/> | 78 | <br/> |
| 79 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z_lt_dec"·class="idref"·href="#Z_lt_dec"><span·class="id"·title="definition">Z_lt_dec</span></a>·:·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.x"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.y"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.x"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.y"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/> | 79 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z_lt_dec"·class="idref"·href="#Z_lt_dec"><span·class="id"·title="definition">Z_lt_dec</span></a>·:·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.x"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.y"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.x"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#decidability.y"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/> |
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| 70 | <br/> | 70 | <br/> |
| 71 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 71 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 72 | <br/> | 72 | <br/> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="doc"> | 74 | <div·class="doc"> |
| 75 | <a·id="lab | 75 | <a·id="lab870"></a><h1·class="section">Properties·of·absolute·value</h1> |
| 76 | </div> | 76 | </div> |
| 77 | <div·class="code"> | 77 | <div·class="code"> |
| 78 | <br/> | 78 | <br/> |
| 79 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zabs_non_eq"·class="idref"·href="#Zabs_non_eq"><span·class="id"·title="abbreviation">Zabs_non_eq</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.abs_neq"><span·class="id"·title="lemma">Z.abs_neq</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 79 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zabs_non_eq"·class="idref"·href="#Zabs_non_eq"><span·class="id"·title="abbreviation">Zabs_non_eq</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.abs_neq"><span·class="id"·title="lemma">Z.abs_neq</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
| 80 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zabs_Zopp"·class="idref"·href="#Zabs_Zopp"><span·class="id"·title="abbreviation">Zabs_Zopp</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.abs_opp"><span·class="id"·title="lemma">Z.abs_opp</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 80 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zabs_Zopp"·class="idref"·href="#Zabs_Zopp"><span·class="id"·title="abbreviation">Zabs_Zopp</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.abs_opp"><span·class="id"·title="lemma">Z.abs_opp</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
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| 148 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Z.lnot_sub"·class="idref"·href="#Z.lnot_sub"><span·class="id"·title="lemma">lnot_sub</span></a>·<a·id="x:18"·class="idref"·href="#x:18"><span·class="id"·title="binder">x</span></a>·<a·id="y:19"·class="idref"·href="#y:19"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.lnot"><span·class="id"·title="definition">Z.lnot</span></a>·(<a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:18"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:19"><span·class="id"·title="variable">y</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.lnot"><span·class="id"·title="definition">Z.lnot</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:18"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:19"><span·class="id"·title="variable">y</span></a>.<br/> | 148 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Z.lnot_sub"·class="idref"·href="#Z.lnot_sub"><span·class="id"·title="lemma">lnot_sub</span></a>·<a·id="x:18"·class="idref"·href="#x:18"><span·class="id"·title="binder">x</span></a>·<a·id="y:19"·class="idref"·href="#y:19"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.lnot"><span·class="id"·title="definition">Z.lnot</span></a>·(<a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:18"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:19"><span·class="id"·title="variable">y</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.lnot"><span·class="id"·title="definition">Z.lnot</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:18"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:19"><span·class="id"·title="variable">y</span></a>.<br/> |
| 149 | · | 149 | · |
| 150 | <br/> | 150 | <br/> |
| 151 | </div> | 151 | </div> |
| 152 | <div·class="doc"> | 152 | <div·class="doc"> |
| 153 | Explicit·formulas·for·carry·bits·during·addition.·Conceptually,·the·theory | 153 | Explicit·formulas·for·carry·bits·during·addition.·Conceptually,·the·theory |
| 154 | <a·id="lab | 154 | <a·id="lab851"></a><h1·class="section">here·matches·the·bitblasting·rules·for·integers.·However,·the·vector·of</h1> |
| 155 | <a·id="lab | 155 | <a·id="lab852"></a><h1·class="section">carry·bits·is·represented·as·a·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·so·it·can·be·used·in·bitwise·operations.</h1> |
| 156 | <a·id="lab | 156 | <a·id="lab853"></a><h1·class="section">The·last·three·lemmas·about·<span·class="inlinecode"><span·class="id"·title="var">addcarries</span></span>·are·the·main·interface,·but·the</h1> |
| 157 | <a·id="lab | 157 | <a·id="lab854"></a><h1·class="section">generalization·<span·class="inlinecode"><span·class="id"·title="var">adccarries</span></span>·is·provided·as·the·same·theory·applies.</h1> |
| 158 | </div> | 158 | </div> |
| 159 | <div·class="code"> | 159 | <div·class="code"> |
| 160 | <br/> | 160 | <br/> |
| 161 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z.adccarries"·class="idref"·href="#Z.adccarries"><span·class="id"·title="definition">adccarries</span></a>·(<a·id="x:20"·class="idref"·href="#x:20"><span·class="id"·title="binder">x</span></a>·<a·id="y:21"·class="idref"·href="#y:21"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>)·(<a·id="c:22"·class="idref"·href="#c:22"><span·class="id"·title="binder">c</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>)·:=·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:20"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:21"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.b2z"><span·class="id"·title="definition">Z.b2z</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#c:22"><span·class="id"·title="variable">c</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:20"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:21"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>.<br/> | 161 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z.adccarries"·class="idref"·href="#Z.adccarries"><span·class="id"·title="definition">adccarries</span></a>·(<a·id="x:20"·class="idref"·href="#x:20"><span·class="id"·title="binder">x</span></a>·<a·id="y:21"·class="idref"·href="#y:21"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>)·(<a·id="c:22"·class="idref"·href="#c:22"><span·class="id"·title="binder">c</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>)·:=·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:20"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:21"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.b2z"><span·class="id"·title="definition">Z.b2z</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#c:22"><span·class="id"·title="variable">c</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:20"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:21"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>.<br/> |
| 162 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z.addcarries"·class="idref"·href="#Z.addcarries"><span·class="id"·title="definition">addcarries</span></a>·(<a·id="x:23"·class="idref"·href="#x:23"><span·class="id"·title="binder">x</span></a>·<a·id="y:24"·class="idref"·href="#y:24"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>)············:=·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:23"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:24"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>···········<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:23"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:24"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>.<br/> | 162 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z.addcarries"·class="idref"·href="#Z.addcarries"><span·class="id"·title="definition">addcarries</span></a>·(<a·id="x:23"·class="idref"·href="#x:23"><span·class="id"·title="binder">x</span></a>·<a·id="y:24"·class="idref"·href="#y:24"><span·class="id"·title="binder">y</span></a>·:·<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>)············:=·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:23"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:24"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>···········<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#x:23"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">.^</span></a>·<a·class="idref"·href="Coq.ZArith.Zbitwise.html#y:24"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.ZArith.Zbitwise.html#ec72b1098f7cf4b330263cf8ab5afa36"><span·class="id"·title="notation">)</span></a>.<br/> |
| Offset 56, 15 lines modified | Offset 56, 15 lines modified | ||
| 56 | <br/> | 56 | <br/> |
| 57 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 57 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 58 | <br/> | 58 | <br/> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="doc"> | 60 | <div·class="doc"> |
| 61 | <a·id="lab | 61 | <a·id="lab891"></a><h1·class="section">Boolean·operations·from·decidability·of·order</h1> |
| 62 | ·The·decidability·of·equality·and·order·relations·over | 62 | ·The·decidability·of·equality·and·order·relations·over |
| 63 | ····type·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·gives·some·boolean·functions·with·the·adequate·specification.· | 63 | ····type·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·gives·some·boolean·functions·with·the·adequate·specification.· |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="code"> | 65 | <div·class="code"> |
| 66 | <br/> | 66 | <br/> |
| 67 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z_lt_ge_bool"·class="idref"·href="#Z_lt_ge_bool"><span·class="id"·title="definition">Z_lt_ge_bool</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·<a·id="y:2"·class="idref"·href="#y:2"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>)·:=·<a·class="idref"·href="Coq.Bool.Sumbool.html#bool_of_sumbool"><span·class="id"·title="definition">bool_of_sumbool</span></a>·(<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#Z_lt_ge_dec"><span·class="id"·title="definition">Z_lt_ge_dec</span></a>·<a·class="idref"·href="Coq.ZArith.Zbool.html#x:1"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.Zbool.html#y:2"><span·class="id"·title="variable">y</span></a>).<br/> | 67 | <span·class="id"·title="keyword">Definition</span>·<a·id="Z_lt_ge_bool"·class="idref"·href="#Z_lt_ge_bool"><span·class="id"·title="definition">Z_lt_ge_bool</span></a>·(<a·id="x:1"·class="idref"·href="#x:1"><span·class="id"·title="binder">x</span></a>·<a·id="y:2"·class="idref"·href="#y:2"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>)·:=·<a·class="idref"·href="Coq.Bool.Sumbool.html#bool_of_sumbool"><span·class="id"·title="definition">bool_of_sumbool</span></a>·(<a·class="idref"·href="Coq.ZArith.ZArith_dec.html#Z_lt_ge_dec"><span·class="id"·title="definition">Z_lt_ge_dec</span></a>·<a·class="idref"·href="Coq.ZArith.Zbool.html#x:1"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.ZArith.Zbool.html#y:2"><span·class="id"·title="variable">y</span></a>).<br/> |
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| 64 | <br/> | 64 | <br/> |
| 65 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 65 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 66 | <br/> | 66 | <br/> |
| 67 | </div> | 67 | </div> |
| 68 | <div·class="doc"> | 68 | <div·class="doc"> |
| 69 | <a·id="lab | 69 | <a·id="lab883"></a><h1·class="section">Comparison·on·integers</h1> |
| 70 | </div> | 70 | </div> |
| 71 | <div·class="code"> | 71 | <div·class="code"> |
| 72 | <br/> | 72 | <br/> |
| 73 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zcompare_Gt_Lt_antisym"·class="idref"·href="#Zcompare_Gt_Lt_antisym"><span·class="id"·title="lemma">Zcompare_Gt_Lt_antisym</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Gt"><span·class="id"·title="constructor">Gt</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#m:2"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#n:1"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>.<br/> | 73 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zcompare_Gt_Lt_antisym"·class="idref"·href="#Zcompare_Gt_Lt_antisym"><span·class="id"·title="lemma">Zcompare_Gt_Lt_antisym</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Gt"><span·class="id"·title="constructor">Gt</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#m:2"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#n:1"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>.<br/> |
| 74 | <br/> | 74 | <br/> |
| 75 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zcompare_antisym"·class="idref"·href="#Zcompare_antisym"><span·class="id"·title="lemma">Zcompare_antisym</span></a>·<a·id="n:3"·class="idref"·href="#n:3"><span·class="id"·title="binder">n</span></a>·<a·id="m:4"·class="idref"·href="#m:4"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#CompOpp"><span·class="id"·title="definition">CompOpp</span></a>·(<a·class="idref"·href="Coq.ZArith.Zcompare.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#m:4"><span·class="id"·title="variable">m</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#m:4"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#n:3"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>.<br/> | 75 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zcompare_antisym"·class="idref"·href="#Zcompare_antisym"><span·class="id"·title="lemma">Zcompare_antisym</span></a>·<a·id="n:3"·class="idref"·href="#n:3"><span·class="id"·title="binder">n</span></a>·<a·id="m:4"·class="idref"·href="#m:4"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#CompOpp"><span·class="id"·title="definition">CompOpp</span></a>·(<a·class="idref"·href="Coq.ZArith.Zcompare.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#m:4"><span·class="id"·title="variable">m</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#m:4"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#n:3"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>.<br/> |
| 76 | <br/> | 76 | <br/> |
| 77 | </div> | 77 | </div> |
| 78 | <div·class="doc"> | 78 | <div·class="doc"> |
| 79 | <a·id="lab | 79 | <a·id="lab884"></a><h1·class="section">Transitivity·of·comparison</h1> |
| 80 | </div> | 80 | </div> |
| 81 | <div·class="code"> | 81 | <div·class="code"> |
| 82 | <br/> | 82 | <br/> |
| 83 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zcompare_Lt_trans"·class="idref"·href="#Zcompare_Lt_trans"><span·class="id"·title="lemma">Zcompare_Lt_trans</span></a>·:<br/> | 83 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zcompare_Lt_trans"·class="idref"·href="#Zcompare_Lt_trans"><span·class="id"·title="lemma">Zcompare_Lt_trans</span></a>·:<br/> |
| 84 | <span·class="id"·title="keyword">forall</span>·<a·id="n:5"·class="idref"·href="#n:5"><span·class="id"·title="binder">n</span></a>·<a·id="m:6"·class="idref"·href="#m:6"><span·class="id"·title="binder">m</span></a>·<a·id="p:7"·class="idref"·href="#p:7"><span·class="id"·title="binder">p</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#n:5"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#m:6"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#m:6"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#p:7"><span·class="id"·title="variable">p</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#n:5"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#p:7"><span·class="id"·title="variable">p</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>.<br/> | 84 | <span·class="id"·title="keyword">forall</span>·<a·id="n:5"·class="idref"·href="#n:5"><span·class="id"·title="binder">n</span></a>·<a·id="m:6"·class="idref"·href="#m:6"><span·class="id"·title="binder">m</span></a>·<a·id="p:7"·class="idref"·href="#p:7"><span·class="id"·title="binder">p</span></a>:<a·class="idref"·href="Coq.Numbers.BinNums.html#Z"><span·class="id"·title="inductive">Z</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#n:5"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#m:6"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#m:6"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#p:7"><span·class="id"·title="variable">p</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zcompare.html#n:5"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span·class="id"·title="notation">?=</span></a>·<a·class="idref"·href="Coq.ZArith.Zcompare.html#p:7"><span·class="id"·title="variable">p</span></a><a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#Lt"><span·class="id"·title="constructor">Lt</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab873"></a><h1·class="section">Euclidean·Division</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Initial·Contribution·by·Claude·Marché·and·Xavier·Urbain· | 51 | ·Initial·Contribution·by·Claude·Marché·and·Xavier·Urbain· |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="code"> | 53 | <div·class="code"> |
| Offset 49, 15 lines modified | Offset 49, 15 lines modified | ||
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#"><span·class="id"·title="library">BinInt</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivEucl.html#"><span·class="id"·title="library">ZDivEucl</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Classes.Morphisms.html#"><span·class="id"·title="library">Morphisms</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#"><span·class="id"·title="library">BinInt</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivEucl.html#"><span·class="id"·title="library">ZDivEucl</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 50 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 51 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab855"></a><h1·class="section">Definitions·of·division·for·binary·integers,·Euclid·convention.</h1> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | ·In·this·convention,·the·remainder·is·always·positive. | 56 | ·In·this·convention,·the·remainder·is·always·positive. |
| 57 | ····For·other·conventions,·see·<span·class="inlinecode"><span·class="id"·title="var">Z.div</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.quot</span></span>·in·file·<span·class="inlinecode"><span·class="id"·title="var">BinIntDef</span></span>. | 57 | ····For·other·conventions,·see·<span·class="inlinecode"><span·class="id"·title="var">Z.div</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.quot</span></span>·in·file·<span·class="inlinecode"><span·class="id"·title="var">BinIntDef</span></span>. |
| 58 | ····To·avoid·collision·with·the·other·divisions,·we·place·this·one | 58 | ····To·avoid·collision·with·the·other·divisions,·we·place·this·one |
| 59 | ····under·a·module. | 59 | ····under·a·module. |
| Offset 172, 15 lines modified | Offset 172, 15 lines modified | ||
| 172 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zodd_bool_succ"·class="idref"·href="#Zodd_bool_succ"><span·class="id"·title="abbreviation">Zodd_bool_succ</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.odd_succ"><span·class="id"·title="lemma">Z.odd_succ</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 172 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zodd_bool_succ"·class="idref"·href="#Zodd_bool_succ"><span·class="id"·title="abbreviation">Zodd_bool_succ</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.odd_succ"><span·class="id"·title="lemma">Z.odd_succ</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
| 173 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zodd_bool_pred"·class="idref"·href="#Zodd_bool_pred"><span·class="id"·title="abbreviation">Zodd_bool_pred</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.odd_pred"><span·class="id"·title="lemma">Z.odd_pred</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 173 | <span·class="id"·title="keyword">Notation</span>·<a·id="Zodd_bool_pred"·class="idref"·href="#Zodd_bool_pred"><span·class="id"·title="abbreviation">Zodd_bool_pred</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.odd_pred"><span·class="id"·title="lemma">Z.odd_pred</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
| 174 | <br/> | 174 | <br/> |
| 175 | </div> | 175 | </div> |
| 176 | <div·class="doc"> | 176 | <div·class="doc"> |
| 177 | <a·id="lab | 177 | <a·id="lab860"></a><h1·class="section">Definition·of·<span·class="inlinecode"><span·class="id"·title="var">Z.quot2</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z.div2</span></span>·and·properties·wrt·<span·class="inlinecode"><span·class="id"·title="var">Zeven</span></span></h1> |
| 178 | ··and·<span·class="inlinecode"><span·class="id"·title="var">Zodd</span></span>· | 178 | ··and·<span·class="inlinecode"><span·class="id"·title="var">Zodd</span></span>· |
| 179 | <div·class="paragraph">·</div> | 179 | <div·class="paragraph">·</div> |
| 180 | ·Properties·of·<span·class="inlinecode"><span·class="id"·title="var">Z.div2</span></span>· | 180 | ·Properties·of·<span·class="inlinecode"><span·class="id"·title="var">Z.div2</span></span>· |
| 181 | </div> | 181 | </div> |
| 182 | <div·class="code"> | 182 | <div·class="code"> |
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| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab931"></a><h1·class="section">Zgcd_alt·:·an·alternate·version·of·Z.gcd,·based·on·Euclid's·algorithm</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | <div·class="paragraph">·</div> | 51 | <div·class="paragraph">·</div> |
| 52 | Author:·Pierre·Letouzey | 52 | Author:·Pierre·Letouzey |
| Offset 98, 29 lines modified | Offset 98, 29 lines modified | ||
| 98 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.Zmisc.html#"><span·class="id"·title="library">Zmisc</span></a>.<br/> | 98 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.Zmisc.html#"><span·class="id"·title="library">Zmisc</span></a>.<br/> |
| 99 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.Wf_Z.html#"><span·class="id"·title="library">Wf_Z</span></a>.<br/> | 99 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.Wf_Z.html#"><span·class="id"·title="library">Wf_Z</span></a>.<br/> |
| 100 | <br/> | 100 | <br/> |
| 101 | </div> | 101 | </div> |
| 102 | <div·class="doc"> | 102 | <div·class="doc"> |
| 103 | <a·id="lab | 103 | <a·id="lab861"></a><h1·class="section">Simplification·lemmas</h1> |
| 104 | <div·class="paragraph">·</div> | 104 | <div·class="paragraph">·</div> |
| 105 | ·No·subgoal·or·smaller·subgoals····································· | 105 | ·No·subgoal·or·smaller·subgoals····································· |
| 106 | </div> | 106 | </div> |
| 107 | <div·class="code"> | 107 | <div·class="code"> |
| 108 | <br/> | 108 | <br/> |
| 109 | #[<span·class="id"·title="var">global</span>]<br/> | 109 | #[<span·class="id"·title="var">global</span>]<br/> |
| 110 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span><br/> | 110 | <span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span><br/> |
| 111 | </div> | 111 | </div> |
| 112 | <div·class="doc"> | 112 | <div·class="doc"> |
| 113 | <a·id="lab | 113 | <a·id="lab862"></a><h2·class="section">Reversible·simplification·lemmas·(no·loss·of·information)</h2> |
| 114 | ·Should·clearly·be·declared·as·hints······························· | 114 | ·Should·clearly·be·declared·as·hints······························· |
| 115 | <div·class="paragraph">·</div> | 115 | <div·class="paragraph">·</div> |
| 116 | ·Lemmas·ending·by·eq· | 116 | ·Lemmas·ending·by·eq· |
| 117 | </div> | 117 | </div> |
| 118 | <div·class="code"> | 118 | <div·class="code"> |
| 119 | <span·class="id"·title="var">Zsucc_eq_compat</span>·<br/> | 119 | <span·class="id"·title="var">Zsucc_eq_compat</span>·<br/> |
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| 91 | <li>·module·Zabs2Nat·:·from·Z·to·nat·(via·the·absolute·value) | 91 | <li>·module·Zabs2Nat·:·from·Z·to·nat·(via·the·absolute·value) |
| 92 | </li> | 92 | </li> |
| 93 | </ul> | 93 | </ul> |
| 94 | <div·class="paragraph">·</div> | 94 | <div·class="paragraph">·</div> |
| 95 | <a·id="lab | 95 | <a·id="lab864"></a><h1·class="section">Chains·of·conversions</h1> |
| 96 | <div·class="paragraph">·</div> | 96 | <div·class="paragraph">·</div> |
| 97 | ·When·combining·successive·conversions,·we·have·the·following | 97 | ·When·combining·successive·conversions,·we·have·the·following |
| 98 | ····commutative·diagram: | 98 | ····commutative·diagram: |
| 99 | <pre> | 99 | <pre> |
| 100 | ······--->·Nat·---- | 100 | ······--->·Nat·---- |
| Offset 241, 15 lines modified | Offset 241, 15 lines modified | ||
| 241 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zdivide_mod_minus"·class="idref"·href="#Zdivide_mod_minus"><span·class="id"·title="lemma">Zdivide_mod_minus</span></a>·<a·id="a:44"·class="idref"·href="#a:44"><span·class="id"·title="binder">a</span></a>·<a·id="b:45"·class="idref"·href="#b:45"><span·class="id"·title="binder">b</span></a>·<a·id="c:46"·class="idref"·href="#c:46"><span·class="id"·title="binder">c</span></a>:<br/> | 241 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zdivide_mod_minus"·class="idref"·href="#Zdivide_mod_minus"><span·class="id"·title="lemma">Zdivide_mod_minus</span></a>·<a·id="a:44"·class="idref"·href="#a:44"><span·class="id"·title="binder">a</span></a>·<a·id="b:45"·class="idref"·href="#b:45"><span·class="id"·title="binder">b</span></a>·<a·id="c:46"·class="idref"·href="#c:46"><span·class="id"·title="binder">c</span></a>:<br/> |
| 242 | 0·<a·class="idref"·href="Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#c:46"><span·class="id"·title="variable">c</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#b:45"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#5cd21895b4d5f127eb2c0acb80e8d6fc"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Znumtheory.html#b:45"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#5cd21895b4d5f127eb2c0acb80e8d6fc"><span·class="id"·title="notation">|</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#a:44"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#c:46"><span·class="id"·title="variable">c</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#5cd21895b4d5f127eb2c0acb80e8d6fc"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#a:44"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#b:45"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#c:46"><span·class="id"·title="variable">c</span></a>.<br/> | 242 | 0·<a·class="idref"·href="Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#c:46"><span·class="id"·title="variable">c</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#b:45"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#5cd21895b4d5f127eb2c0acb80e8d6fc"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Znumtheory.html#b:45"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#5cd21895b4d5f127eb2c0acb80e8d6fc"><span·class="id"·title="notation">|</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#a:44"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#c:46"><span·class="id"·title="variable">c</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#5cd21895b4d5f127eb2c0acb80e8d6fc"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#a:44"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#b:45"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.Znumtheory.html#c:46"><span·class="id"·title="variable">c</span></a>.<br/> |
| 243 | <br/> | 243 | <br/> |
| 244 | </div> | 244 | </div> |
| 245 | <div·class="doc"> | 245 | <div·class="doc"> |
| 246 | <a·id="lab | 246 | <a·id="lab932"></a><h1·class="section">Greatest·common·divisor·(gcd).</h1> |
| 247 | <div·class="paragraph">·</div> | 247 | <div·class="paragraph">·</div> |
| 248 | ·There·is·no·unicity·of·the·gcd;·hence·we·define·the·predicate | 248 | ·There·is·no·unicity·of·the·gcd;·hence·we·define·the·predicate |
| 249 | ····<span·class="inlinecode"><span·class="id"·title="var">Zis_gcd</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">b</span></span>·<span·class="inlinecode"><span·class="id"·title="var">g</span></span>·expressing·that·<span·class="inlinecode"><span·class="id"·title="var">g</span></span>·is·a·gcd·of·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">b</span></span>. | 249 | ····<span·class="inlinecode"><span·class="id"·title="var">Zis_gcd</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">b</span></span>·<span·class="inlinecode"><span·class="id"·title="var">g</span></span>·expressing·that·<span·class="inlinecode"><span·class="id"·title="var">g</span></span>·is·a·gcd·of·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">b</span></span>. |
| 250 | ····(We·show·later·that·the·<span·class="inlinecode"><span·class="id"·title="var">gcd</span></span>·is·actually·unique·if·we·discard·its·sign.)· | 250 | ····(We·show·later·that·the·<span·class="inlinecode"><span·class="id"·title="var">gcd</span></span>·is·actually·unique·if·we·discard·its·sign.)· |
| 251 | </div> | 251 | </div> |
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| 67 | <br/> | 67 | <br/> |
| 68 | </div> | 68 | </div> |
| 69 | <div·class="doc"> | 69 | <div·class="doc"> |
| 70 | Properties·of·the·order·relations·on·binary·integers· | 70 | Properties·of·the·order·relations·on·binary·integers· |
| 71 | <div·class="paragraph">·</div> | 71 | <div·class="paragraph">·</div> |
| 72 | <a·id="lab | 72 | <a·id="lab921"></a><h1·class="section">Trichotomy</h1> |
| 73 | </div> | 73 | </div> |
| 74 | <div·class="code"> | 74 | <div·class="code"> |
| 75 | <br/> | 75 | <br/> |
| 76 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Ztrichotomy_inf"·class="idref"·href="#Ztrichotomy_inf"><span·class="id"·title="lemma">Ztrichotomy_inf</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.Zorder.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.Zorder.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.Zorder.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span·class="id"·title="notation">}</span></a>.<br/> | 76 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Ztrichotomy_inf"·class="idref"·href="#Ztrichotomy_inf"><span·class="id"·title="lemma">Ztrichotomy_inf</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.Zorder.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.Zorder.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.ZArith.Zorder.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:2"><span·class="id"·title="variable">m</span></a><a·class="idref"·href="Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span·class="id"·title="notation">}</span></a>.<br/> |
| 77 | <br/> | 77 | <br/> |
| 78 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Ztrichotomy"·class="idref"·href="#Ztrichotomy"><span·class="id"·title="lemma">Ztrichotomy</span></a>·<a·id="n:3"·class="idref"·href="#n:3"><span·class="id"·title="binder">n</span></a>·<a·id="m:4"·class="idref"·href="#m:4"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.ZArith.Zorder.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:4"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:4"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:4"><span·class="id"·title="variable">m</span></a>.<br/> | 78 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Ztrichotomy"·class="idref"·href="#Ztrichotomy"><span·class="id"·title="lemma">Ztrichotomy</span></a>·<a·id="n:3"·class="idref"·href="#n:3"><span·class="id"·title="binder">n</span></a>·<a·id="m:4"·class="idref"·href="#m:4"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.ZArith.Zorder.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:4"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:4"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#n:3"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·<a·class="idref"·href="Coq.ZArith.Zorder.html#m:4"><span·class="id"·title="variable">m</span></a>.<br/> |
| 79 | <br/> | 79 | <br/> |
| 80 | </div> | 80 | </div> |
| 81 | <div·class="doc"> | 81 | <div·class="doc"> |
| 82 | <a·id="lab | 82 | <a·id="lab922"></a><h1·class="section">Decidability·of·equality·and·order·on·Z</h1> |
| 83 | </div> | 83 | </div> |
| 84 | <div·class="code"> | 84 | <div·class="code"> |
| 85 | <br/> | 85 | <br/> |
| 86 | <span·class="id"·title="keyword">Notation</span>·<a·id="dec_eq"·class="idref"·href="#dec_eq"><span·class="id"·title="abbreviation">dec_eq</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.eq_decidable"><span·class="id"·title="lemma">Z.eq_decidable</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 86 | <span·class="id"·title="keyword">Notation</span>·<a·id="dec_eq"·class="idref"·href="#dec_eq"><span·class="id"·title="abbreviation">dec_eq</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.eq_decidable"><span·class="id"·title="lemma">Z.eq_decidable</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
| 87 | <span·class="id"·title="keyword">Notation</span>·<a·id="dec_Zle"·class="idref"·href="#dec_Zle"><span·class="id"·title="abbreviation">dec_Zle</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.le_decidable"><span·class="id"·title="lemma">Z.le_decidable</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> | 87 | <span·class="id"·title="keyword">Notation</span>·<a·id="dec_Zle"·class="idref"·href="#dec_Zle"><span·class="id"·title="abbreviation">dec_Zle</span></a>·:=·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.le_decidable"><span·class="id"·title="lemma">Z.le_decidable</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/> |
| Offset 49, 15 lines modified | Offset 49, 15 lines modified | ||
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.BinInt.html#"><span·class="id"·title="library">BinInt</span></a>·<a·class="idref"·href="Coq.setoid_ring.Ring_theory.html#"><span·class="id"·title="library">Ring_theory</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.BinInt.html#"><span·class="id"·title="library">BinInt</span></a>·<a·class="idref"·href="Coq.setoid_ring.Ring_theory.html#"><span·class="id"·title="library">Ring_theory</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 50 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 51 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab810"></a><h1·class="section">Power·functions·over·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span></h1> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | ·Nota·:·this·file·is·mostly·deprecated.·The·definition·of·<span·class="inlinecode"><span·class="id"·title="var">Z.pow</span></span> | 56 | ·Nota·:·this·file·is·mostly·deprecated.·The·definition·of·<span·class="inlinecode"><span·class="id"·title="var">Z.pow</span></span> |
| 57 | ····and·its·usual·properties·are·now·provided·by·module·<span·class="inlinecode"><span·class="id"·title="var">BinInt.Z</span></span>.· | 57 | ····and·its·usual·properties·are·now·provided·by·module·<span·class="inlinecode"><span·class="id"·title="var">BinInt.Z</span></span>.· |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="code"> | 59 | <div·class="code"> |
| Offset 128, 15 lines modified | Offset 128, 15 lines modified | ||
| 128 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zpower2_Psize"·class="idref"·href="#Zpower2_Psize"><span·class="id"·title="lemma">Zpower2_Psize</span></a>·<a·id="n:25"·class="idref"·href="#n:25"><span·class="id"·title="binder">n</span></a>·<a·id="p:26"·class="idref"·href="#p:26"><span·class="id"·title="binder">p</span></a>·:<br/> | 128 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Zpower2_Psize"·class="idref"·href="#Zpower2_Psize"><span·class="id"·title="lemma">Zpower2_Psize</span></a>·<a·id="n:25"·class="idref"·href="#n:25"><span·class="id"·title="binder">n</span></a>·<a·id="p:26"·class="idref"·href="#p:26"><span·class="id"·title="binder">p</span></a>·:<br/> |
| 129 | <a·class="idref"·href="Coq.Numbers.BinNums.html#Zpos"><span·class="id"·title="constructor">Zpos</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:26"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·2<a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">^(</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#Z.of_nat"><span·class="id"·title="definition">Z.of_nat</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:25"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·(<a·class="idref"·href="Coq.PArith.BinPos.html#Pos.size_nat"><span·class="id"·title="definition">Pos.size_nat</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:26"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:25"><span·class="id"·title="variable">n</span></a>)%<span·class="id"·title="var">nat</span>.<br/> | 129 | <a·class="idref"·href="Coq.Numbers.BinNums.html#Zpos"><span·class="id"·title="constructor">Zpos</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:26"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·2<a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">^(</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#Z.of_nat"><span·class="id"·title="definition">Z.of_nat</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:25"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·(<a·class="idref"·href="Coq.PArith.BinPos.html#Pos.size_nat"><span·class="id"·title="definition">Pos.size_nat</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:26"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:25"><span·class="id"·title="variable">n</span></a>)%<span·class="id"·title="var">nat</span>.<br/> |
| 130 | <br/> | 130 | <br/> |
| 131 | </div> | 131 | </div> |
| 132 | <div·class="doc"> | 132 | <div·class="doc"> |
| 133 | <a·id="lab | 133 | <a·id="lab806"></a><h1·class="section">Z.pow·and·modulo</h1> |
| 134 | </div> | 134 | </div> |
| 135 | <div·class="code"> | 135 | <div·class="code"> |
| 136 | <br/> | 136 | <br/> |
| 137 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Zpower_mod"·class="idref"·href="#Zpower_mod"><span·class="id"·title="lemma">Zpower_mod</span></a>·<a·id="p:27"·class="idref"·href="#p:27"><span·class="id"·title="binder">p</span></a>·<a·id="q:28"·class="idref"·href="#q:28"><span·class="id"·title="binder">q</span></a>·<a·id="n:29"·class="idref"·href="#n:29"><span·class="id"·title="binder">n</span></a>·:<br/> | 137 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Zpower_mod"·class="idref"·href="#Zpower_mod"><span·class="id"·title="lemma">Zpower_mod</span></a>·<a·id="p:27"·class="idref"·href="#p:27"><span·class="id"·title="binder">p</span></a>·<a·id="q:28"·class="idref"·href="#q:28"><span·class="id"·title="binder">q</span></a>·<a·id="n:29"·class="idref"·href="#n:29"><span·class="id"·title="binder">n</span></a>·:<br/> |
| 138 | 0·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:27"><span·class="id"·title="variable">p</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">^</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#q:28"><span·class="id"·title="variable">q</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:27"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">)^</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#q:28"><span·class="id"·title="variable">q</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a>.<br/> | 138 | 0·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:27"><span·class="id"·title="variable">p</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">^</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#q:28"><span·class="id"·title="variable">q</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#p:27"><span·class="id"·title="variable">p</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span·class="id"·title="notation">)^</span></a><a·class="idref"·href="Coq.ZArith.Zpow_facts.html#q:28"><span·class="id"·title="variable">q</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:x_'mod'_x"><span·class="id"·title="notation">mod</span></a>·<a·class="idref"·href="Coq.ZArith.Zpow_facts.html#n:29"><span·class="id"·title="variable">n</span></a>.<br/> |
| Offset 50, 15 lines modified | Offset 50, 15 lines modified | ||
| 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.ZArith.Zpow_def.html#"><span·class="id"·title="library">Zpow_def</span></a>.<br/> | 50 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.ZArith.Zpow_def.html#"><span·class="id"·title="library">Zpow_def</span></a>.<br/> |
| 51 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 51 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 52 | <br/> | 52 | <br/> |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="doc"> | 54 | <div·class="doc"> |
| 55 | <a·id="lab | 55 | <a·id="lab867"></a><h1·class="section">Power·functions·over·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span></h1> |
| 56 | <div·class="paragraph">·</div> | 56 | <div·class="paragraph">·</div> |
| 57 | ·Nota·:·this·file·is·mostly·deprecated.·The·definition·of·<span·class="inlinecode"><span·class="id"·title="var">Z.pow</span></span> | 57 | ·Nota·:·this·file·is·mostly·deprecated.·The·definition·of·<span·class="inlinecode"><span·class="id"·title="var">Z.pow</span></span> |
| 58 | ····and·its·usual·properties·are·now·provided·by·module·<span·class="inlinecode"><span·class="id"·title="var">BinInt.Z</span></span>. | 58 | ····and·its·usual·properties·are·now·provided·by·module·<span·class="inlinecode"><span·class="id"·title="var">BinInt.Z</span></span>. |
| 59 | ····Powers·of·2·are·also·available·there·(see·<span·class="inlinecode"><span·class="id"·title="var">Z.shiftl</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.shiftr</span></span>). | 59 | ····Powers·of·2·are·also·available·there·(see·<span·class="inlinecode"><span·class="id"·title="var">Z.shiftl</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.shiftr</span></span>). |
| 60 | ····Only·remain·here: | 60 | ····Only·remain·here: |
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| 149 | <br/> | 149 | <br/> |
| 150 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Z_rem_mult"·class="idref"·href="#Z_rem_mult"><span·class="id"·title="lemma">Z_rem_mult</span></a>·<a·id="a:6"·class="idref"·href="#a:6"><span·class="id"·title="binder">a</span></a>·<a·id="b:7"·class="idref"·href="#b:7"><span·class="id"·title="binder">b</span></a>·:·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.rem"><span·class="id"·title="definition">Z.rem</span></a>·(<a·class="idref"·href="Coq.ZArith.Zquot.html#a:6"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.ZArith.Zquot.html#b:7"><span·class="id"·title="variable">b</span></a>)·<a·class="idref"·href="Coq.ZArith.Zquot.html#b:7"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·0.<br/> | 150 | <span·class="id"·title="keyword">Lemma</span>·<a·id="Z_rem_mult"·class="idref"·href="#Z_rem_mult"><span·class="id"·title="lemma">Z_rem_mult</span></a>·<a·id="a:6"·class="idref"·href="#a:6"><span·class="id"·title="binder">a</span></a>·<a·id="b:7"·class="idref"·href="#b:7"><span·class="id"·title="binder">b</span></a>·:·<a·class="idref"·href="Coq.ZArith.BinInt.html#Z.rem"><span·class="id"·title="definition">Z.rem</span></a>·(<a·class="idref"·href="Coq.ZArith.Zquot.html#a:6"><span·class="id"·title="variable">a</span></a><a·class="idref"·href="Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.ZArith.Zquot.html#b:7"><span·class="id"·title="variable">b</span></a>)·<a·class="idref"·href="Coq.ZArith.Zquot.html#b:7"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·0.<br/> |
| 151 | · | 151 | · |
| 152 | <br/> | 152 | <br/> |
| 153 | </div> | 153 | </div> |
| 154 | <div·class="doc"> | 154 | <div·class="doc"> |
| 155 | <a·id="lab | 155 | <a·id="lab937"></a><h1·class="section">Division·and·Opposite</h1> |
| 156 | </div> | 156 | </div> |
| 157 | <div·class="code"> | 157 | <div·class="code"> |
| 158 | <br/> | 158 | <br/> |
| 159 | <br/> | 159 | <br/> |
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| 64 | <br/> | 64 | <br/> |
| 65 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 65 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 66 | <br/> | 66 | <br/> |
| 67 | </div> | 67 | </div> |
| 68 | <div·class="doc"> | 68 | <div·class="doc"> |
| 69 | <a·id="lab | 69 | <a·id="lab882"></a><h1·class="section">Moving·terms·from·one·side·to·the·other·of·an·inequality</h1> |
| 70 | </div> | 70 | </div> |
| 71 | <div·class="code"> | 71 | <div·class="code"> |
| 72 | <br/> | 72 | <br/> |
| 73 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Zne_left"·class="idref"·href="#Zne_left"><span·class="id"·title="lemma">Zne_left</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.ZArith.BinInt.html#Zne"><span·class="id"·title="definition">Zne</span></a>·<a·class="idref"·href="Coq.ZArith.auxiliary.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.auxiliary.html#m:2"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#Zne"><span·class="id"·title="definition">Zne</span></a>·(<a·class="idref"·href="Coq.ZArith.auxiliary.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.ZArith.auxiliary.html#m:2"><span·class="id"·title="variable">m</span></a>)·0.<br/> | 73 | <span·class="id"·title="keyword">Theorem</span>·<a·id="Zne_left"·class="idref"·href="#Zne_left"><span·class="id"·title="lemma">Zne_left</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.ZArith.BinInt.html#Zne"><span·class="id"·title="definition">Zne</span></a>·<a·class="idref"·href="Coq.ZArith.auxiliary.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.auxiliary.html#m:2"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#Zne"><span·class="id"·title="definition">Zne</span></a>·(<a·class="idref"·href="Coq.ZArith.auxiliary.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#::Z_scope:'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.ZArith.auxiliary.html#m:2"><span·class="id"·title="variable">m</span></a>)·0.<br/> |
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| 140 | <br/> | 140 | <br/> |
| 141 | <span·class="id"·title="keyword">Section</span>·<a·id="Definitions"·class="idref"·href="#Definitions"><span·class="id"·title="section">Definitions</span></a>.<br/> | 141 | <span·class="id"·title="keyword">Section</span>·<a·id="Definitions"·class="idref"·href="#Definitions"><span·class="id"·title="section">Definitions</span></a>.<br/> |
| 142 | <br/> | 142 | <br/> |
| 143 | </div> | 143 | </div> |
| 144 | <div·class="doc"> | 144 | <div·class="doc"> |
| 145 | <a·id="lab5 | 145 | <a·id="lab564"></a><h1·class="section">Global,·inductive·definitions.</h1> |
| 146 | <div·class="paragraph">·</div> | 146 | <div·class="paragraph">·</div> |
| 147 | ·A·Horner·polynomial·is·either·a·constant,·or·a·product·P·×·(i·+·Q),·where·i· | 147 | ·A·Horner·polynomial·is·either·a·constant,·or·a·product·P·×·(i·+·Q),·where·i· |
| 148 | ··is·a·variable.· | 148 | ··is·a·variable.· |
| 149 | </div> | 149 | </div> |
| 150 | <div·class="code"> | 150 | <div·class="code"> |
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| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>.<br/> |
| 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellNatNum.html#"><span·class="id"·title="library">ExtrHaskellNatNum</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellNatNum.html#"><span·class="id"·title="library">ExtrHaskellNatNum</span></a>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab447"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab448"></a><h1·class="section">by·extracting·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·into·<span·class="inlinecode"><span·class="id"·title="var">Int</span></span>·is·definitively·*not*·a·good·idea.</h1> |
| 65 | <a·id="lab | 65 | <a·id="lab449"></a><h1·class="section">See·comments·in·<span·class="inlinecode"><span·class="id"·title="var">ExtrOcamlNatInt.v</span></span>.</h1> |
| 66 | · | 66 | · |
| 67 | </div> | 67 | </div> |
| 68 | <div·class="code"> | 68 | <div·class="code"> |
| 69 | <br/> | 69 | <br/> |
| 70 | <span·class="id"·title="keyword">Extract</span>·<span·class="id"·title="keyword">Inductive</span>·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·=>·"Prelude.Int"·[·"0"·"Prelude.succ"·]<br/> | 70 | <span·class="id"·title="keyword">Extract</span>·<span·class="id"·title="keyword">Inductive</span>·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·=>·"Prelude.Int"·[·"0"·"Prelude.succ"·]<br/> |
| Offset 58, 19 lines modified | Offset 58, 19 lines modified | ||
| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.Arith.html#"><span·class="id"·title="library">Arith</span></a>.<br/> |
| 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellNatNum.html#"><span·class="id"·title="library">ExtrHaskellNatNum</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellNatNum.html#"><span·class="id"·title="library">ExtrHaskellNatNum</span></a>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab444"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab445"></a><h1·class="section">by·extracting·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·into·<span·class="inlinecode"><span·class="id"·title="var">Integer</span></span>·isn't·necessarily·a·good·idea.</h1> |
| 65 | <a·id="lab | 65 | <a·id="lab446"></a><h1·class="section">See·comments·in·<span·class="inlinecode"><span·class="id"·title="var">ExtrOcamlNatInt.v</span></span>.</h1> |
| 66 | </div> | 66 | </div> |
| 67 | <div·class="code"> | 67 | <div·class="code"> |
| 68 | <br/> | 68 | <br/> |
| 69 | <span·class="id"·title="keyword">Extract</span>·<span·class="id"·title="keyword">Inductive</span>·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·=>·"Prelude.Integer"·[·"0"·"Prelude.succ"·]<br/> | 69 | <span·class="id"·title="keyword">Extract</span>·<span·class="id"·title="keyword">Inductive</span>·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>·=>·"Prelude.Integer"·[·"0"·"Prelude.succ"·]<br/> |
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| 43 | <h1·class="libtitle">Library·Coq.extraction.ExtrHaskellNatNum</h1> | 43 | <h1·class="libtitle">Library·Coq.extraction.ExtrHaskellNatNum</h1> |
| 44 | <div·class="code"> | 44 | <div·class="code"> |
| 45 | </div> | 45 | </div> |
| 46 | <div·class="doc"> | 46 | <div·class="doc"> |
| 47 | <a·id="lab | 47 | <a·id="lab450"></a><h1·class="section">Efficient·(but·uncertified)·extraction·of·usual·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·functions</h1> |
| 48 | <a·id="lab | 48 | <a·id="lab451"></a><h1·class="section">into·equivalent·versions·in·Haskell's·Prelude·that·are·defined</h1> |
| 49 | <a·id="lab | 49 | <a·id="lab452"></a><h1·class="section">for·any·<span·class="inlinecode"><span·class="id"·title="var">Num</span></span>·typeclass·instances.··Useful·in·combination·with</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab453"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="keyword">Extract</span></span>·<span·class="inlinecode"><span·class="id"·title="keyword">Inductive</span></span>·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·that·maps·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·onto·a·Haskell·type·that</h1> |
| 51 | <a·id="lab | 51 | <a·id="lab454"></a><h1·class="section">implements·<span·class="inlinecode"><span·class="id"·title="var">Num</span></span>.</h1> |
| 52 | · | 52 | · |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| 55 | <br/> | 55 | <br/> |
| 56 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> | 56 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> |
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| 43 | <h1·class="libtitle">Library·Coq.extraction.ExtrHaskellString</h1> | 43 | <h1·class="libtitle">Library·Coq.extraction.ExtrHaskellString</h1> |
| 44 | <div·class="code"> | 44 | <div·class="code"> |
| 45 | </div> | 45 | </div> |
| 46 | <div·class="doc"> | 46 | <div·class="doc"> |
| 47 | <a·id="lab | 47 | <a·id="lab460"></a><h1·class="section">Special·handling·of·ascii·and·strings·for·extraction·to·Haskell.</h1> |
| 48 | · | 48 | · |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="code"> | 50 | <div·class="code"> |
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> |
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| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> |
| 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellZNum.html#"><span·class="id"·title="library">ExtrHaskellZNum</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellZNum.html#"><span·class="id"·title="library">ExtrHaskellZNum</span></a>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab441"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab442"></a><h1·class="section">by·extracting·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·into·<span·class="inlinecode"><span·class="id"·title="var">Int</span></span>·is·definitively·*not*·a·good·idea.</h1> |
| 65 | <a·id="lab | 65 | <a·id="lab443"></a><h1·class="section">See·comments·in·<span·class="inlinecode"><span·class="id"·title="var">ExtrOcamlNatInt.v</span></span>.</h1> |
| 66 | · | 66 | · |
| 67 | </div> | 67 | </div> |
| 68 | <div·class="code"> | 68 | <div·class="code"> |
| 69 | <br/> | 69 | <br/> |
| 70 | <span·class="id"·title="keyword">Extract</span>·<span·class="id"·title="keyword">Inductive</span>·<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>·=>·"Prelude.Int"·[<br/> | 70 | <span·class="id"·title="keyword">Extract</span>·<span·class="id"·title="keyword">Inductive</span>·<a·class="idref"·href="Coq.Numbers.BinNums.html#positive"><span·class="id"·title="inductive">positive</span></a>·=>·"Prelude.Int"·[<br/> |
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| 43 | <h1·class="libtitle">Library·Coq.extraction.ExtrHaskellZNum</h1> | 43 | <h1·class="libtitle">Library·Coq.extraction.ExtrHaskellZNum</h1> |
| 44 | <div·class="code"> | 44 | <div·class="code"> |
| 45 | </div> | 45 | </div> |
| 46 | <div·class="doc"> | 46 | <div·class="doc"> |
| 47 | <a·id="lab | 47 | <a·id="lab455"></a><h1·class="section">Efficient·(but·uncertified)·extraction·of·usual·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·functions</h1> |
| 48 | <a·id="lab | 48 | <a·id="lab456"></a><h1·class="section">into·equivalent·versions·in·Haskell's·Prelude·that·are·defined</h1> |
| 49 | <a·id="lab | 49 | <a·id="lab457"></a><h1·class="section">for·any·<span·class="inlinecode"><span·class="id"·title="var">Num</span></span>·typeclass·instances.··Useful·in·combination·with</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab458"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="keyword">Extract</span></span>·<span·class="inlinecode"><span·class="id"·title="keyword">Inductive</span></span>·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·that·maps·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·onto·a·Haskell·type·that</h1> |
| 51 | <a·id="lab | 51 | <a·id="lab459"></a><h1·class="section">implements·<span·class="inlinecode"><span·class="id"·title="var">Num</span></span>.</h1> |
| 52 | · | 52 | · |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| 55 | <br/> | 55 | <br/> |
| 56 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> | 56 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> |
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| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">Z_scope</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab942"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">Z.div_mod_to_equations</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z.quot_rem_to_equations</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z.to_euclidean_division_equations</span></span>:</h1> |
| 57 | ·····the·tactics·for·preprocessing·<span·class="inlinecode"><span·class="id"·title="var">Z.div</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.modulo</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z.quot</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.rem</span></span>· | 57 | ·····the·tactics·for·preprocessing·<span·class="inlinecode"><span·class="id"·title="var">Z.div</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.modulo</span></span>,·<span·class="inlinecode"><span·class="id"·title="var">Z.quot</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.rem</span></span>· |
| 58 | <div·class="paragraph">·</div> | 58 | <div·class="paragraph">·</div> |
| 59 | ·These·tactics·use·the·complete·specification·of·<span·class="inlinecode"><span·class="id"·title="var">Z.div</span></span>·and | 59 | ·These·tactics·use·the·complete·specification·of·<span·class="inlinecode"><span·class="id"·title="var">Z.div</span></span>·and |
| 60 | ····<span·class="inlinecode"><span·class="id"·title="var">Z.modulo</span></span>·(<span·class="inlinecode"><span·class="id"·title="var">Z.quot</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.rem</span></span>,·respectively)·to·remove·these | 60 | ····<span·class="inlinecode"><span·class="id"·title="var">Z.modulo</span></span>·(<span·class="inlinecode"><span·class="id"·title="var">Z.quot</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Z.rem</span></span>,·respectively)·to·remove·these |
| 61 | ····functions·from·the·goal·without·losing·information.··The | 61 | ····functions·from·the·goal·without·losing·information.··The |