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| 1 | ·1eaf3c6f26458a427c5b1d19458ff3db·11856272·debug·optional·coq-dbgsym_8.16.1+dfsg-1_armhf.deb | 1 | ·1eaf3c6f26458a427c5b1d19458ff3db·11856272·debug·optional·coq-dbgsym_8.16.1+dfsg-1_armhf.deb |
| 2 | · | 2 | ·d7ead5c484e997df28425b5617f22e9b·89869572·math·optional·coq_8.16.1+dfsg-1_armhf.deb |
| 3 | ·579f604be64da650161528cc3d81342a·653524·debug·optional·coqide-dbgsym_8.16.1+dfsg-1_armhf.deb | 3 | ·579f604be64da650161528cc3d81342a·653524·debug·optional·coqide-dbgsym_8.16.1+dfsg-1_armhf.deb |
| 4 | ·e682adb250352eafe15c2ef6802d5919·1835288·math·optional·coqide_8.16.1+dfsg-1_armhf.deb | 4 | ·e682adb250352eafe15c2ef6802d5919·1835288·math·optional·coqide_8.16.1+dfsg-1_armhf.deb |
| 5 | ·a0ea08501e72bf42861ebb1ec7878e08·1811244·debug·optional·libcoq-core-ocaml-dbgsym_8.16.1+dfsg-1_armhf.deb | 5 | ·a0ea08501e72bf42861ebb1ec7878e08·1811244·debug·optional·libcoq-core-ocaml-dbgsym_8.16.1+dfsg-1_armhf.deb |
| 6 | ·e | 6 | ·d3ec85ac8c209648cc0c1ddcb5fa03f2·43114532·ocaml·optional·libcoq-core-ocaml-dev_8.16.1+dfsg-1_armhf.deb |
| 7 | · | 7 | ·ea3db4c488646f8375ea2837da820bab·22260896·ocaml·optional·libcoq-core-ocaml_8.16.1+dfsg-1_armhf.deb |
| 8 | · | 8 | ·196f0c17f5b2ccd06b69b1a16c8ee74f·21526220·math·optional·libcoq-stdlib_8.16.1+dfsg-1_armhf.deb |
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| 1 | -rw-r--r--···0········0········0········4·2022-11-29·08:28:08.000000·debian-binary | 1 | -rw-r--r--···0········0········0········4·2022-11-29·08:28:08.000000·debian-binary |
| 2 | -rw-r--r--···0········0········0·····2328·2022-11-29·08:28:08.000000·control.tar.xz | 2 | -rw-r--r--···0········0········0·····2328·2022-11-29·08:28:08.000000·control.tar.xz |
| 3 | -rw-r--r--···0········0········0·8986 | 3 | -rw-r--r--···0········0········0·89867052·2022-11-29·08:28:08.000000·data.tar.xz |
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| 1 | Package:·coq | 1 | Package:·coq |
| 2 | Version:·8.16.1+dfsg-1 | 2 | Version:·8.16.1+dfsg-1 |
| 3 | Architecture:·armhf | 3 | Architecture:·armhf |
| 4 | Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org> | 4 | Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org> |
| 5 | Installed-Size:·28618 | 5 | Installed-Size:·286182 |
| 6 | Depends:·libcoq-stdlib·(=·8.16.1+dfsg-1),·libcoq-core-ocaml-4ew58,·libfindlib-ocaml-3yei3,·libzarith-ocaml-aip06,·ocaml-base-4.13.1,·python3:any,·libc6·(>=·2.35),·libgmp10·(>=·2:6.2.1+dfsg1),·ocaml-nox,·ocaml-findlib | 6 | Depends:·libcoq-stdlib·(=·8.16.1+dfsg-1),·libcoq-core-ocaml-4ew58,·libfindlib-ocaml-3yei3,·libzarith-ocaml-aip06,·ocaml-base-4.13.1,·python3:any,·libc6·(>=·2.35),·libgmp10·(>=·2:6.2.1+dfsg1),·ocaml-nox,·ocaml-findlib |
| 7 | Suggests:·coqide·|·proofgeneral,·ledit·|·readline-editor,·libcoq-core-ocaml-dev,·why·(>=·2.19),·coq-doc | 7 | Suggests:·coqide·|·proofgeneral,·ledit·|·readline-editor,·libcoq-core-ocaml-dev,·why·(>=·2.19),·coq-doc |
| 8 | Breaks:·coq-libs·(<<·8.2.pl1) | 8 | Breaks:·coq-libs·(<<·8.2.pl1) |
| 9 | Replaces:·coq-libs·(<<·8.2.pl1) | 9 | Replaces:·coq-libs·(<<·8.2.pl1) |
| 10 | Provides:·coq-8.16.1+4.13.1 | 10 | Provides:·coq-8.16.1+4.13.1 |
| 11 | Section:·math | 11 | Section:·math |
| 12 | Priority:·optional | 12 | Priority:·optional |
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| 3 | drwxr-xr-x···0·root·········(0)·root·········(0)········0·2022-11-29·08:28:08.000000·./usr/bin/ | 3 | drwxr-xr-x···0·root·········(0)·root·········(0)········0·2022-11-29·08:28:08.000000·./usr/bin/ |
| 4 | -rwxr-xr-x···0·root·········(0)·root·········(0)······891·2022-11-29·08:28:08.000000·./usr/bin/coq-makefile-make-both-single-timing-files | 4 | -rwxr-xr-x···0·root·········(0)·root·········(0)······891·2022-11-29·08:28:08.000000·./usr/bin/coq-makefile-make-both-single-timing-files |
| 5 | -rwxr-xr-x···0·root·········(0)·root·········(0)······931·2022-11-29·08:28:08.000000·./usr/bin/coq-makefile-make-both-time-files | 5 | -rwxr-xr-x···0·root·········(0)·root·········(0)······931·2022-11-29·08:28:08.000000·./usr/bin/coq-makefile-make-both-time-files |
| 6 | -rwxr-xr-x···0·root·········(0)·root·········(0)······697·2022-11-29·08:28:08.000000·./usr/bin/coq-makefile-make-one-time-file | 6 | -rwxr-xr-x···0·root·········(0)·root·········(0)······697·2022-11-29·08:28:08.000000·./usr/bin/coq-makefile-make-one-time-file |
| 7 | -rwxr-xr-x···0·root·········(0)·root·········(0)···714660·2022-11-29·08:28:08.000000·./usr/bin/coq-tex | 7 | -rwxr-xr-x···0·root·········(0)·root·········(0)···714660·2022-11-29·08:28:08.000000·./usr/bin/coq-tex |
| 8 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1433548·2022-11-29·08:28:08.000000·./usr/bin/coq_makefile | 8 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1433548·2022-11-29·08:28:08.000000·./usr/bin/coq_makefile |
| 9 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663372·2022-11-29·08:28:08.000000·./usr/bin/coqc | 9 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663372·2022-11-29·08:28:08.000000·./usr/bin/coqc |
| 10 | -rwxr-xr-x···0·root·········(0)·root·········(0)·400650 | 10 | -rwxr-xr-x···0·root·········(0)·root·········(0)·40065020·2022-11-29·08:28:08.000000·./usr/bin/coqc.byte |
| 11 | -rwxr-xr-x···0·root·········(0)·root·········(0)··5285088·2022-11-29·08:28:08.000000·./usr/bin/coqchk | 11 | -rwxr-xr-x···0·root·········(0)·root·········(0)··5285088·2022-11-29·08:28:08.000000·./usr/bin/coqchk |
| 12 | -rwxr-xr-x···0·root·········(0)·root·········(0)··2093316·2022-11-29·08:28:08.000000·./usr/bin/coqdep | 12 | -rwxr-xr-x···0·root·········(0)·root·········(0)··2093316·2022-11-29·08:28:08.000000·./usr/bin/coqdep |
| 13 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1823120·2022-11-29·08:28:08.000000·./usr/bin/coqdoc | 13 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1823120·2022-11-29·08:28:08.000000·./usr/bin/coqdoc |
| 14 | -rwxr-xr-x···0·root·········(0)·root·········(0)·406284 | 14 | -rwxr-xr-x···0·root·········(0)·root·········(0)·40628463·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.byte |
| 15 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14847864·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.opt | 15 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14847864·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.opt |
| 16 | -rwxr-xr-x···0·root·········(0)·root·········(0)··5428976·2022-11-29·08:28:08.000000·./usr/bin/coqnative | 16 | -rwxr-xr-x···0·root·········(0)·root·········(0)··5428976·2022-11-29·08:28:08.000000·./usr/bin/coqnative |
| 17 | -rwxr-xr-x···0·root·········(0)·root·········(0)···928428·2022-11-29·08:28:08.000000·./usr/bin/coqpp | 17 | -rwxr-xr-x···0·root·········(0)·root·········(0)···928428·2022-11-29·08:28:08.000000·./usr/bin/coqpp |
| 18 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663692·2022-11-29·08:28:08.000000·./usr/bin/coqproofworker.opt | 18 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663692·2022-11-29·08:28:08.000000·./usr/bin/coqproofworker.opt |
| 19 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663692·2022-11-29·08:28:08.000000·./usr/bin/coqqueryworker.opt | 19 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663692·2022-11-29·08:28:08.000000·./usr/bin/coqqueryworker.opt |
| 20 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663696·2022-11-29·08:28:08.000000·./usr/bin/coqtacticworker.opt | 20 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663696·2022-11-29·08:28:08.000000·./usr/bin/coqtacticworker.opt |
| 21 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663468·2022-11-29·08:28:08.000000·./usr/bin/coqtop | 21 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663468·2022-11-29·08:28:08.000000·./usr/bin/coqtop |
| 22 | -rwxr-xr-x···0·root·········(0)·root·········(0)·575884 | 22 | -rwxr-xr-x···0·root·········(0)·root·········(0)·57588432·2022-11-29·08:28:08.000000·./usr/bin/coqtop.byte |
| 23 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663468·2022-11-29·08:28:08.000000·./usr/bin/coqtop.opt | 23 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14663468·2022-11-29·08:28:08.000000·./usr/bin/coqtop.opt |
| 24 | -rwxr-xr-x···0·root·········(0)·root·········(0)···562828·2022-11-29·08:28:08.000000·./usr/bin/coqwc | 24 | -rwxr-xr-x···0·root·········(0)·root·········(0)···562828·2022-11-29·08:28:08.000000·./usr/bin/coqwc |
| 25 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14580776·2022-11-29·08:28:08.000000·./usr/bin/coqworkmgr | 25 | -rwxr-xr-x···0·root·········(0)·root·········(0)·14580776·2022-11-29·08:28:08.000000·./usr/bin/coqworkmgr |
| 26 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17068032·2022-11-29·08:28:08.000000·./usr/bin/csdpcert | 26 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17068032·2022-11-29·08:28:08.000000·./usr/bin/csdpcert |
| 27 | -rwxr-xr-x···0·root·········(0)·root·········(0)···866872·2022-11-29·08:28:08.000000·./usr/bin/ocamllibdep | 27 | -rwxr-xr-x···0·root·········(0)·root·········(0)···866872·2022-11-29·08:28:08.000000·./usr/bin/ocamllibdep |
| 28 | -rwxr-xr-x···0·root·········(0)·root·········(0)···968392·2022-11-29·08:28:08.000000·./usr/bin/votour | 28 | -rwxr-xr-x···0·root·········(0)·root·········(0)···968392·2022-11-29·08:28:08.000000·./usr/bin/votour |
| 29 | drwxr-xr-x···0·root·········(0)·root·········(0)········0·2022-11-29·08:28:08.000000·./usr/lib/ | 29 | drwxr-xr-x···0·root·········(0)·root·········(0)········0·2022-11-29·08:28:08.000000·./usr/lib/ |
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| 000002e0:· | 000002e0:·a324·536f·6d65·6490·a004·0e40·4004·6f40··.$Somed....@@.o@ | ||
| 000002f0:· | 000002f0:·9204·0440·4041·40a0·5940·a040·4040·4004··...@@A@.Y@.@@@@. | ||
| 00000300:· | 00000300:·7240·4040·9204·15a3·266c·617a·795f·744e··r@@@....&lazy_tN | ||
| 00000310:· | 00000310:·0800·0038·00a0·c004·3b02·05f5·e100·4055··...8....;.....@U | ||
| 00000320:· | 00000320:·4041·4041·40a0·5940·a040·4040·4004·7a40··@A@A@.Y@.@@@@.z@ | ||
| 00000330:·4040· | 00000330:·4040·9204·07a3·296e·6174·6976·6569·6e74··@@....)nativeint | ||
| 00000340:· | 00000340:·4b08·0000·3800·4040·4041·4040·4040·4004··K...8.@@@A@@@@@. | ||
| 00000350:· | 00000350:·7e40·4040·9204·03a3·2569·6e74·3332·4c08··~@@@....%int32L. | ||
| 00000360:·0 | 00000360:·0000·3800·4040·4041·4040·4040·4004·8240··..8.@@@A@@@@@..@ | ||
| 00000370:·4040· | 00000370:·4040·9204·03a3·2569·6e74·3634·4d08·0000··@@....%int64M... | ||
| 00000380:· | 00000380:·3800·4040·4041·4040·4040·4004·8640·4040··8.@@@A@@@@@..@@@ | ||
| 00000390:· | 00000390:·9204·03a3·3a55·6e64·6566·696e·6564·5f72··....:Undefined_r | ||
| 000003a0:· | 000003a0:·6563·7572·7369·7665·5f6d·6f64·756c·655d··ecursive_module] | ||
| 000003b0:· | 000003b0:·0800·0020·0090·0457·4090·a0c0·92a0·c0b3··...·...W@....... | ||
| 000003c0:· | 000003c0:·9004·8340·9040·0205·f5e1·0040·4aa0·c0b3··...@.@.....@J... | ||
| 000003d0:· | 000003d0:·9004·9040·9040·0205·f5e1·0040·40a0·0405··...@.@.....@@... | ||
| 000003e0:· | 000003e0:·4002·05f5·e100·4056·4040·4104·9aa0·b0a0··@.....@V@@A..... | ||
| 000003f0:· | 000003f0:·3d6f·6361·6d6c·2e77·6172·6e5f·6f6e·5f6c··=ocaml.warn_on_l | ||
| 00000400:·6 | 00000400:·6974·6572·616c·5f70·6174·7465·726e·049e··iteral_pattern.. | ||
| 00000410:· | 00000410:·0461·049e·4092·0417·a32e·4173·7365·7274··.a..@.....Assert | ||
| 00000420:· | 00000420:·5f66·6169·6c75·7265·5c08·0000·2000·0418··_failure\...·... | ||
| 00000430:· | 00000430:·4090·a0c0·92a0·0417·a004·13a0·0414·4002··@.............@. | ||
| 00000440:·0 | 00000440:·05f5·e100·4058·4040·4104·a9a0·b0a0·040f··....@X@@A....... | ||
| 00000450:·0 | 00000450:·04ac·046f·04ac·4092·040d·a330·4469·7669··...o..@....0Divi | ||
| 00000460:· | 00000460:·7369·6f6e·5f62·795f·7a65·726f·5908·0000··sion_by_zeroY... | ||
| 00000470:· | 00000470:·2000·0426·4090·4040·4104·b1a0·b0a0·0417···..&@.@@A....... | ||
| 00000480:· | 00000480:·04b4·0477·04b4·4092·0407·a32b·456e·645f··...w..@....+End_ | ||
| 00000490:· | 00000490:·6f66·5f66·696c·6558·0800·0020·0004·2e40··of_fileX...·...@ | ||
| 000004a0:· | 000004a0:·9040·4041·04b9·a0b0·a004·1f04·bc04·7f04··.@@A............ | ||
| 000004b0:· | 000004b0:·bc40·9204·07a3·2953·7973·5f65·7272·6f72··.@....)Sys_error | ||
| 000004c0:· | 000004c0:·5708·0000·2000·0436·4090·a004·3240·4041··W...·..6@...2@@A | ||
| 000004d0:· | 000004d0:·04c2·a0b0·a004·2804·c504·8804·c540·9204··......(......@.. | ||
| 000004e0:·0 | 000004e0:·08a3·2e53·7973·5f62·6c6f·636b·6564·5f69··...Sys_blocked_i | ||
| 000004f0:·6 | 000004f0:·6f5b·0800·0020·0004·3f40·9040·4041·04ca··o[...·..?@.@@A.. | ||
| 00000500:· | 00000500:·a0b0·a004·3004·cd04·9004·cd40·9204·07a3··....0......@.... | ||
| 00000510:· | 00000510:·294e·6f74·5f66·6f75·6e64·5608·0000·2000··)Not_foundV...·. | ||
| 00000520:·0 | 00000520:·0447·4090·4040·4104·d2a0·b0a0·0438·04d5··.G@.@@A......8.. | ||
| 00000530:· | 00000530:·0498·04d5·4092·0407·a327·4661·696c·7572··....@....'Failur | ||
| 00000540:· | 00000540:·6555·0800·0020·0004·4f40·90a0·044b·4040··eU...·..O@...K@@ | ||
| 00000550:·404 | 00000550:·4104·dba0·b0a0·0441·04de·04a1·04de·4092··A......A......@. | ||
| 00000560:· | 00000560:·0408·a330·496e·7661·6c69·645f·6172·6775··...0Invalid_argu | ||
| 00000570:· | 00000570:·6d65·6e74·5408·0000·2000·0458·4090·a004··mentT...·..X@... | ||
| 00000580:· | 00000580:·5440·4041·04e4·a0b0·a004·4a04·e704·aa04··T@@A......J..... | ||
| 00000590:· | 00000590:·e740·9204·08a3·2e53·7461·636b·5f6f·7665··.@.....Stack_ove | ||
| 000005a0:· | 000005a0:·7266·6c6f·775a·0800·0020·0004·6140·9040··rflowZ...·..a@.@ | ||
| 000005b0:·404 | 000005b0:·4041·04ec·a0b0·a004·5204·ef04·b204·ef40··@A......R......@ | ||
| 000005c0:· | 000005c0:·9204·07a3·2d4f·7574·5f6f·665f·6d65·6d6f··....-Out_of_memo | ||
| 000005d0:· | 000005d0:·7279·5308·0000·2000·0469·4090·4040·4104··ryS...·..i@.@@A. | ||
| 000005e0:· | 000005e0:·f4a0·b0a0·045a·04f7·04ba·04f7·4092·0407··.....Z......@... | ||
| 000005f0:· | 000005f0:·a32d·4d61·7463·685f·6661·696c·7572·6552··.-Match_failureR | ||
| 00000600:· | 00000600:·0800·0020·0004·7140·90a0·c092·a004·70a0··...·..q@......p. | ||
| 00000610:· | 00000610:·046c·a004·6d40·0205·f5e1·0040·6340·4041··.l..m@.....@c@@A | ||
| 00000620:· | 00000620:·0501·02a0·b0a0·0468·0501·0504·c805·0105··.......h........ | ||
| 00000630:· | 00000630:·4092·040d·a325·6279·7465·7343·0800·0038··@....%bytesC...8 | ||
| 00000640:· | 00000640:·0040·4040·4140·4040·4040·0501·0940·4040··.@@@A@@@@@...@@@ | ||
| 00000650:· | 00000650:·9204·0390·9226·5374·646c·6962·9226·4f70··.....&Stdlib.&Op | ||
| 00000660:· | 00000660:·7469·6f6e·0501·4308·0000·3800·a0c0·9040··tion..C...8....@ | ||
| 00000670:· | 00000670:·0205·f5e1·0040·0103·1540·4140·4140·a047··.....@...@A@A@.G | ||
| 00000680:·4 | 00000680:·40a0·4240·4040·b0c0·0501·467b·0109·7001··@.B@@@....F{..p. | ||
| 00000690:· | 00000690:·0970·c005·0147·7b01·0970·0109·8040·4040··.p...G{..p...@@@ | ||
| 000006a0:·010 | 000006a0:·40a1·0501·5540·4040·b0c0·0501·4a7b·0109··@...U@@@....J{.. | ||
| 000006b0:· | 000006b0:·7001·0975·c005·014b·7b01·0970·0109·7740··p..u...K{..p..w@ | ||
| 000006c0:· | 000006c0:·40a0·4241·4008·0000·3800·a005·0147·4041··@.BA@...8....G@A | ||
| 000006d0:· | 000006d0:·4141·40a0·4f40·a040·4040·4004·0c40·4040··AA@.O@.@@@@..@@@ | ||
| 000006e0:· | 000006e0:·0409·4041·4140·040c·4040·b004·0c04·0b40··..@AA@..@@.....@ | ||
| 000006f0:· | 000006f0:·0800·0030·0005·0147·0501·4705·0147·0501··...0...G..G..G.. | ||
| 00000700:·4 | 00000700:·4705·0147·0501·4705·0147·0501·4740·0501··G..G..G..G..G@.. | ||
| 00000710:· | 00000710:·4540·40a0·b0a3·41a0·0800·0028·00b1·2662··E@@...A....(..&b | ||
| 00000720:· | 00000720:·6f78·6b65·7900·5242·a004·02b0·c005·015c··oxkey.RB.......\ | ||
| 00000730:·0 | 00000730:·7e01·09ce·0109·d3c0·0501·5d7e·0109·ce01··~.........]~.... | ||
| 00000740:· | 00000740:·09d9·4040·0800·0038·0040·40a1·a0e0·a023··..@@...8.@@....# | ||
| 00000750:·4 | 00000750:·426f·7800·5390·a0c0·b390·0501·6da0·c096··Box.S.......m... | ||
| 00000760:·0 | 00000760:·c090·9021·6102·05f5·e100·4001·031c·4440··...!a.....@...D@ | ||
| 00000770:· | 00000770:·0103·1840·9040·0205·f5e1·0040·0103·1a40··...@.@.....@...@ | ||
| 00000780:· | 00000780:·90c0·b390·041d·4090·4002·05f5·e100·4001··......@.@.....@. | ||
| 00000790:·0 | 00000790:·031d·b0c0·0501·777e·0109·ce01·09dc·c005··......w~........ | ||
| 000007a0:· | 000007a0:·0178·7e01·09ce·0109·f240·40a1·0501·8642··.x~......@@....B | ||
| 000007b0:·40 | 000007b0:·4041·4140·4040·4040·b0c0·0501·7b7e·0109··@AA@@@@@....{~.. | ||
| 000007c0:· | 000007c0:·ce01·09ce·c005·017c·7e01·09ce·0109·fe40··.......|~......@ | ||
| 000007d0:· | 000007d0:·a0b0·a027·756e·626f·7865·64b0·c005·0182··...'unboxed..... | ||
| Max diff block lines reached; 29382/209425 bytes (14.03%) of diff not shown. | |||
| Offset 3612, 4575 lines modified | Offset 3612, 4575 lines modified | ||
| 0000e1b0:·a0a0·2443·5369·6790·303f·15ce·d987·9a20··..$CSig.0?.....· | 0000e1b0:·a0a0·2443·5369·6790·303f·15ce·d987·9a20··..$CSig.0?.....· | ||
| 0000e1c0:·03e4·319b·37b8·b6e4·63a0·a024·434d·6170··..1.7...c..$CMap | 0000e1c0:·03e4·319b·37b8·b6e4·63a0·a024·434d·6170··..1.7...c..$CMap | ||
| 0000e1d0:·9030·c521·9eea·9ff8·a030·c33b·5031·f3eb··.0.!.....0.;P1.. | 0000e1d0:·9030·c521·9eea·9ff8·a030·c33b·5031·f3eb··.0.!.....0.;P1.. | ||
| 0000e1e0:·5df6·a0a0·2543·4c69·7374·9030·30fa·13ed··]...%CList.00... | 0000e1e0:·5df6·a0a0·2543·4c69·7374·9030·30fa·13ed··]...%CList.00... | ||
| 0000e1f0:·1dbf·f1a5·4610·6f95·708d·799c·4084·95a6··....F.o.p.y.@... | 0000e1f0:·1dbf·f1a5·4610·6f95·708d·799c·4084·95a6··....F.o.p.y.@... | ||
| 0000e200:·be00·0000·0800·0000·0400·0000·0b00·0000··................ | 0000e200:·be00·0000·0800·0000·0400·0000·0b00·0000··................ | ||
| 0000e210:·0ba0·40a0·41a0·9040·4043·616d·6c31·3939··..@.A..@@Caml199 | 0000e210:·0ba0·40a0·41a0·9040·4043·616d·6c31·3939··..@.A..@@Caml199 | ||
| 0000e220:·3954·3033·3084·95a6·be00·011d·5 | 0000e220:·3954·3033·3084·95a6·be00·011d·5900·0033··9T030.......Y..3 | ||
| 0000e230:·2 | 0000e230:·2700·00cb·bc00·00c8·6e08·0000·3400·2743··'.......n...4.'C | ||
| 0000e240:·5374·7269·6e67·92b0·a0b0·98d0·b121·5301··String.......!S. | 0000e240:·5374·7269·6e67·92b0·a0b0·98d0·b121·5301··String.......!S. | ||
| 0000e250:·0098·41a0·0402·b0c0·3063·6c69·622f·6353··..A.....0clib/cS | 0000e250:·0098·41a0·0402·b0c0·3063·6c69·622f·6353··..A.....0clib/cS | ||
| 0000e260:·7472·696e·672e·6d6c·694c·0102·d901·02e5··tring.mliL...... | 0000e260:·7472·696e·672e·6d6c·694c·0102·d901·02e5··tring.mliL...... | ||
| 0000e270:·c004·024c·0102·d901·02e6·4090·d094·d0a0··...L......@..... | 0000e270:·c004·024c·0102·d901·02e6·4090·d094·d0a0··...L......@..... | ||
| 0000e280:·a190·9226·5374·646c·6962·2653·7472·696e··...&Stdlib&Strin | 0000e280:·a190·9226·5374·646c·6962·2653·7472·696e··...&Stdlib&Strin | ||
| 0000e290:·67a0·9004·03b0·c004·104c·0102·d901·02f8··g........L...... | 0000e290:·67a0·9004·03b0·c004·104c·0102·d901·02f8··g........L...... | ||
| 0000e2a0:·c004·114c·0102·d901·02fe·40b0·0403·0402··...L......@..... | 0000e2a0:·c004·114c·0102·d901·02fe·40b0·0403·0402··...L......@..... | ||
| 0000e2b0:·4093·9092·2e53·7464·6c69·625f·5f53·7472··@....Stdlib__Str | 0000e2b0:·4093·9092·2e53·7464·6c69·625f·5f53·7472··@....Stdlib__Str | Diff chunk too large, falling back to line-by-line diff (1331 lines added, 1331 lines removed) | |
| 0000e2c0:·696e·6708·0000·3000·a040·4004·0104·0104··ing...0..@@..... | 0000e2c0:·696e·6708·0000·3000·a040·4004·0104·0104··ing...0..@@..... | ||
| 0000e2d0:·0104·0104·0104·0104·0140·aba | 0000e2d0:·0104·0104·0104·0104·0140·aba7·b1b2·b2b2··.........@...... | ||
| 0000e2e0:·b2b2·b2b2·b2b2·b2b2·b2b | 0000e2e0:·b2b2·b2b2·b2b2·b2b2·b2b1·b1b1·b1b1·b1b1··................ | ||
| 0000e2f0:·b1b1·b1b1·b1b1·b1b1·b1 | 0000e2f0:·b1b1·b1b1·b1b1·b1b1·b140·a32a·666c·6f61··.........@.*floa | ||
| 0000e300:· | 0000e300:·7461·7272·6179·5108·0000·3800·4040·4041··tarrayQ...8.@@@A | ||
| 0000e310:·4 | 0000e310:·4040·4040·40b0·c026·5f6e·6f6e·655f·4040··@@@@@..&_none_@@ | ||
| 0000e320:· | 0000e320:·00ff·0402·4140·4040·9204·06a3·3565·7874··....A@@@....5ext | ||
| 0000e330:· | 0000e330:·656e·7369·6f6e·5f63·6f6e·7374·7275·6374··ension_construct | ||
| 0000e340:· | 0000e340:·6f72·5008·0000·3800·4040·4041·4040·4040··orP...8.@@@A@@@@ | ||
| 0000e350:·40 | 0000e350:·4004·0740·4040·9204·03a3·2369·6e74·4108··@..@@@....#intA. | ||
| 0000e360:·0 | 0000e360:·0000·3800·4040·4041·4040·4040·4004·0b40··..8.@@@A@@@@@..@ | ||
| 0000e370:·4 | 0000e370:·4140·9204·03a3·2463·6861·7242·0800·0038··A@....$charB...8 | ||
| 0000e380:· | 0000e380:·0040·4040·4140·4040·4040·040f·4041·4092··.@@@A@@@@@..@A@. | ||
| 0000e390:· | 0000e390:·0403·a326·7374·7269·6e67·4f08·0000·3800··...&stringO...8. | ||
| 0000e3a0:· | 0000e3a0:·4040·4041·4040·4040·4004·1340·4040·9204··@@@A@@@@@..@@@.. | ||
| 0000e3b0:·0 | 0000e3b0:·03a3·2566·6c6f·6174·4408·0000·3800·4040··..%floatD...8.@@ | ||
| 0000e3c0:·404 | 0000e3c0:·4041·4040·4040·4004·1740·4040·9204·03a3··@A@@@@@..@@@.... | ||
| 0000e3d0:· | 0000e3d0:·2462·6f6f·6c45·0800·0038·0040·40a1·a0e0··$boolE...8.@@... | ||
| 0000e3e0:· | 0000e3e0:·a325·6661·6c73·655e·9040·4004·2140·9204··.%false^.@@.!@.. | ||
| 0000e3f0:·0 | 0000e3f0:·03a0·e0a3·2474·7275·655f·9040·4004·2740··....$true_.@@.'@ | ||
| 0000e400:· | 0000e400:·9204·0340·4041·4040·4040·4004·2840·4140··...@@A@@@@@.(@A@ | ||
| 0000e410:· | 0000e410:·9204·10a3·2475·6e69·7446·0800·0038·0040··....$unitF...8.@ | ||
| 0000e420:·40 | 0000e420:·40a1·a0e0·a322·2829·6090·4040·0432·4092··@...."()`.@@.2@. | ||
| 0000e430:· | 0000e430:·0403·4040·4140·4040·4040·0433·4041·4092··..@@A@@@@@.3@A@. | ||
| 0000e440:· | 0000e440:·040a·a323·6578·6e47·0800·0038·0040·4041··...#exnG...8.@@A | ||
| 0000e450:·414 | 0000e450:·4140·4040·4040·0437·4040·4092·0403·a325··A@@@@@.7@@@....% | ||
| 0000e460:· | 0000e460:·6172·7261·7948·0800·0038·00a0·c090·4002··arrayH...8....@. | ||
| 0000e470:·0 | 0000e470:·05f5·e100·404f·4041·4041·40a0·007f·40a0··....@O@A@A@...@. | ||
| 0000e480:· | 0000e480:·4040·4040·0440·4040·4092·0408·a324·6c69··@@@@.@@@@....$li | ||
| 0000e490:· | 0000e490:·7374·4908·0000·3800·a0c0·0409·0205·f5e1··stI...8......... | ||
| 0000e4a0:· | 0000e4a0:·0040·5040·41a1·a0e0·a322·5b5d·6190·4040··.@P@A...."[]a.@@ | ||
| 0000e4b0:· | 0000e4b0:·044c·4092·0403·a0e0·a322·3a3a·6290·a004··.L@......"::b... | ||
| 0000e4c0:·0 | 0000e4c0:·0ea0·c0b3·9004·16a0·0413·4090·4002·05f5··..........@.@... | ||
| 0000e4d0:· | 0000e4d0:·e100·4051·4040·0459·4092·040a·4040·4140··..@Q@@.Y@...@@A@ | ||
| 0000e4e0:· | 0000e4e0:·a059·40a0·4040·4040·045c·4040·4092·041b··.Y@.@@@@.\@@@... | ||
| 0000e4f0:· | 0000e4f0:·a326·6f70·7469·6f6e·4a08·0000·3800·a0c0··.&optionJ...8... | ||
| 0000e500:· | 0000e500:·0425·0205·f5e1·0040·5340·41a1·a0e0·a324··.%.....@S@A....$ | ||
| 0000e510:· | 0000e510:·4e6f·6e65·6390·4040·0468·4092·0403·a0e0··Nonec.@@.h@..... | ||
| 0000e520:· | 0000e520:·a324·536f·6d65·6490·a004·0e40·4004·6f40··.$Somed....@@.o@ | ||
| 0000e530:· | 0000e530:·9204·0440·4041·40a0·5940·a040·4040·4004··...@@A@.Y@.@@@@. | ||
| 0000e540:· | 0000e540:·7240·4040·9204·15a3·266c·617a·795f·744e··r@@@....&lazy_tN | ||
| 0000e550:· | 0000e550:·0800·0038·00a0·c004·3b02·05f5·e100·4055··...8....;.....@U | ||
| 0000e560:· | 0000e560:·4041·4041·40a0·5940·a040·4040·4004·7a40··@A@A@.Y@.@@@@.z@ | ||
| 0000e570:·4040· | 0000e570:·4040·9204·07a3·296e·6174·6976·6569·6e74··@@....)nativeint | ||
| 0000e580:· | 0000e580:·4b08·0000·3800·4040·4041·4040·4040·4004··K...8.@@@A@@@@@. | ||
| 0000e590:· | 0000e590:·7e40·4040·9204·03a3·2569·6e74·3332·4c08··~@@@....%int32L. | ||
| 0000e5a0:·0 | 0000e5a0:·0000·3800·4040·4041·4040·4040·4004·8240··..8.@@@A@@@@@..@ | ||
| 0000e5b0:·4040· | 0000e5b0:·4040·9204·03a3·2569·6e74·3634·4d08·0000··@@....%int64M... | ||
| 0000e5c0:· | 0000e5c0:·3800·4040·4041·4040·4040·4004·8640·4040··8.@@@A@@@@@..@@@ | ||
| 0000e5d0:· | 0000e5d0:·9204·03a3·3a55·6e64·6566·696e·6564·5f72··....:Undefined_r | ||
| 0000e5e0:· | 0000e5e0:·6563·7572·7369·7665·5f6d·6f64·756c·655d··ecursive_module] | ||
| 0000e5f0:· | 0000e5f0:·0800·0020·0090·0457·4090·a0c0·92a0·c0b3··...·...W@....... | ||
| 0000e600:· | 0000e600:·9004·8340·9040·0205·f5e1·0040·4aa0·c0b3··...@.@.....@J... | ||
| 0000e610:· | 0000e610:·9004·9040·9040·0205·f5e1·0040·40a0·0405··...@.@.....@@... | ||
| 0000e620:· | 0000e620:·4002·05f5·e100·4056·4040·4104·9aa0·b0a0··@.....@V@@A..... | ||
| 0000e630:· | 0000e630:·3d6f·6361·6d6c·2e77·6172·6e5f·6f6e·5f6c··=ocaml.warn_on_l | ||
| 0000e640:·6 | 0000e640:·6974·6572·616c·5f70·6174·7465·726e·049e··iteral_pattern.. | ||
| 0000e650:· | 0000e650:·0461·049e·4092·0417·a32e·4173·7365·7274··.a..@.....Assert | ||
| 0000e660:· | 0000e660:·5f66·6169·6c75·7265·5c08·0000·2000·0418··_failure\...·... | ||
| 0000e670:· | 0000e670:·4090·a0c0·92a0·0417·a004·13a0·0414·4002··@.............@. | ||
| 0000e680:·0 | 0000e680:·05f5·e100·4058·4040·4104·a9a0·b0a0·040f··....@X@@A....... | ||
| 0000e690:·0 | 0000e690:·04ac·046f·04ac·4092·040d·a330·4469·7669··...o..@....0Divi | ||
| 0000e6a0:· | 0000e6a0:·7369·6f6e·5f62·795f·7a65·726f·5908·0000··sion_by_zeroY... | ||
| 0000e6b0:· | 0000e6b0:·2000·0426·4090·4040·4104·b1a0·b0a0·0417···..&@.@@A....... | ||
| 0000e6c0:· | 0000e6c0:·04b4·0477·04b4·4092·0407·a32b·456e·645f··...w..@....+End_ | ||
| 0000e6d0:· | 0000e6d0:·6f66·5f66·696c·6558·0800·0020·0004·2e40··of_fileX...·...@ | ||
| 0000e6e0:· | 0000e6e0:·9040·4041·04b9·a0b0·a004·1f04·bc04·7f04··.@@A............ | ||
| 0000e6f0:· | 0000e6f0:·bc40·9204·07a3·2953·7973·5f65·7272·6f72··.@....)Sys_error | ||
| 0000e700:· | 0000e700:·5708·0000·2000·0436·4090·a004·3240·4041··W...·..6@...2@@A | ||
| 0000e710:· | 0000e710:·04c2·a0b0·a004·2804·c504·8804·c540·9204··......(......@.. | ||
| 0000e720:·0 | 0000e720:·08a3·2e53·7973·5f62·6c6f·636b·6564·5f69··...Sys_blocked_i | ||
| 0000e730:·6 | 0000e730:·6f5b·0800·0020·0004·3f40·9040·4041·04ca··o[...·..?@.@@A.. | ||
| 0000e740:· | 0000e740:·a0b0·a004·3004·cd04·9004·cd40·9204·07a3··....0......@.... | ||
| 0000e750:· | 0000e750:·294e·6f74·5f66·6f75·6e64·5608·0000·2000··)Not_foundV...·. | ||
| 0000e760:·0 | 0000e760:·0447·4090·4040·4104·d2a0·b0a0·0438·04d5··.G@.@@A......8.. | ||
| 0000e770:· | 0000e770:·0498·04d5·4092·0407·a327·4661·696c·7572··....@....'Failur | ||
| 0000e780:· | 0000e780:·6555·0800·0020·0004·4f40·90a0·044b·4040··eU...·..O@...K@@ | ||
| 0000e790:·404 | 0000e790:·4104·dba0·b0a0·0441·04de·04a1·04de·4092··A......A......@. | ||
| 0000e7a0:· | 0000e7a0:·0408·a330·496e·7661·6c69·645f·6172·6775··...0Invalid_argu | ||
| 0000e7b0:· | 0000e7b0:·6d65·6e74·5408·0000·2000·0458·4090·a004··mentT...·..X@... | ||
| 0000e7c0:· | 0000e7c0:·5440·4041·04e4·a0b0·a004·4a04·e704·aa04··T@@A......J..... | ||
| 0000e7d0:· | 0000e7d0:·e740·9204·08a3·2e53·7461·636b·5f6f·7665··.@.....Stack_ove | ||
| 0000e7e0:· | 0000e7e0:·7266·6c6f·775a·0800·0020·0004·6140·9040··rflowZ...·..a@.@ | ||
| 0000e7f0:·404 | 0000e7f0:·4041·04ec·a0b0·a004·5204·ef04·b204·ef40··@A......R......@ | ||
| 0000e800:· | 0000e800:·9204·07a3·2d4f·7574·5f6f·665f·6d65·6d6f··....-Out_of_memo | ||
| 0000e810:· | 0000e810:·7279·5308·0000·2000·0469·4090·4040·4104··ryS...·..i@.@@A. | ||
| 0000e820:· | 0000e820:·f4a0·b0a0·045a·04f7·04ba·04f7·4092·0407··.....Z......@... | ||
| 0000e830:· | 0000e830:·a32d·4d61·7463·685f·6661·696c·7572·6552··.-Match_failureR | ||
| 0000e840:· | 0000e840:·0800·0020·0004·7140·90a0·c092·a004·70a0··...·..q@......p. | ||
| 0000e850:· | 0000e850:·046c·a004·6d40·0205·f5e1·0040·6340·4041··.l..m@.....@c@@A | ||
| 0000e860:· | 0000e860:·0501·02a0·b0a0·0468·0501·0504·c805·0105··.......h........ | ||
| 0000e870:· | 0000e870:·4092·040d·a325·6279·7465·7343·0800·0038··@....%bytesC...8 | ||
| 0000e880:· | 0000e880:·0040·4040·4140·4040·4040·0501·0940·4040··.@@@A@@@@@...@@@ | ||
| 0000e890:· | 0000e890:·9204·0305·013c·9226·4f70·7469·6f6e·4040··.....<.&Option@@ | ||
| 0000e8a0:· | 0000e8a0:·4091·a0c1·a021·7400·5108·0000·3800·4040··@....!t.Q...8.@@ | ||
| 0000e8b0:· | 0000e8b0:·4041·90c0·b390·a326·7374·7269·6e67·4f40··@A.....&stringO@ | ||
| 0000e8c0:· | 0000e8c0:·9040·0205·f5e1·0040·0104·9440·4040·40b0··.@.....@...@@@@. | ||
| 0000e8d0:· | 0000e8d0:·c02a·7374·7269·6e67·2e6d·6c69·0052·010e··.*string.mli.R.. | ||
| 0000e8e0:·6 | 0000e8e0:·8601·0e86·c004·0200·5201·0e86·010e·9540··........R......@ | ||
| 0000e8f0:· | 0000e8f0:·4040·40a1·2e53·7464·6c69·625f·5f53·7472··@@@..Stdlib__Str | ||
| 0000e900:·6 | 0000e900:·696e·6740·4140·a0b0·a024·6d61·6b65·0052··ing@A@...$make.R | ||
| 0000e910:· | 0000e910:·d0c0·c140·c0b3·90a3·2369·6e74·4140·9040··...@....#intA@.@ | ||
| 0000e920:· | 0000e920:·0205·f5e1·0040·0104·93c0·c140·c0b3·90a3··.....@.....@.... | ||
| 0000e930:· | 0000e930:·2463·6861·7242·4090·4002·05f5·e100·4001··$charB@.@.....@. | ||
| 0000e940:·0 | 0000e940:·0492·c0b3·9004·2140·9040·0205·f5e1·0040··......!@.@.....@ | ||
| 0000e950:·0205·f5e1·0040·0104·9 | 0000e950:·0104·9140·0205·f5e1·0040·0104·9040·0205··...@.....@...@.. | ||
| 0000e960:· | 0000e960:·f5e1·0040·0104·8f40·b0c0·041f·0055·010e··...@...@.....U.. | ||
| 0000e970:· | 0000e970:·b401·0eb4·c004·2000·5501·0eb4·010e·d440··......·.U......@ | ||
| 0000e980:· | 0000e980:·40a1·041e·4140·a0b0·a024·696e·6974·0053··@...A@...$init.S | ||
| Max diff block lines reached; 443993/629148 bytes (70.57%) of diff not shown. | |||
| Offset 52, 350 lines modified | Offset 52, 350 lines modified | ||
| 00000330:·7464·6c69·6290·302d·0826·66be·7fc2·ba91··tdlib.0-.&f..... | 00000330:·7464·6c69·6290·302d·0826·66be·7fc2·ba91··tdlib.0-.&f..... | ||
| 00000340:·6e72·3339·7491·dfa0·a038·4361·6d6c·696e··nr39t....8Camlin | 00000340:·6e72·3339·7491·dfa0·a038·4361·6d6c·696e··nr39t....8Camlin | ||
| 00000350:·7465·726e·616c·466f·726d·6174·4261·7369··ternalFormatBasi | 00000350:·7465·726e·616c·466f·726d·6174·4261·7369··ternalFormatBasi | ||
| 00000360:·6373·9030·c4b5·83a7·27ec·28f5·bc9b·a36a··cs.0....'.(....j | 00000360:·6373·9030·c4b5·83a7·27ec·28f5·bc9b·a36a··cs.0....'.(....j | ||
| 00000370:·dc64·cfc7·4084·95a6·be00·0000·0800·0000··.d..@........... | 00000370:·dc64·cfc7·4084·95a6·be00·0000·0800·0000··.d..@........... | ||
| 00000380:·0400·0000·0b00·0000·0ba0·40a0·41a0·9040··..........@.A..@ | 00000380:·0400·0000·0b00·0000·0ba0·40a0·41a0·9040··..........@.A..@ | ||
| 00000390:·4043·616d·6c31·3939·3954·3033·3084·95a6··@Caml1999T030... | 00000390:·4043·616d·6c31·3939·3954·3033·3084·95a6··@Caml1999T030... | ||
| 000003a0:·be00·0015·5 | 000003a0:·be00·0015·5600·0003·7800·000e·e400·000e··....V...x....... | ||
| 000003b0:· | 000003b0:·2a08·0000·3400·2b4f·7264·6572·6564·5479··*...4.+OrderedTy | ||
| 000003c0:·7065·92b0·a0b0·98d0·b121·5300·5342·a004··pe.......!S.SB.. | 000003c0:·7065·92b0·a0b0·98d0·b121·5300·5342·a004··pe.......!S.SB.. | ||
| 000003d0:·02b0·c034·636c·6962·2f6f·7264·6572·6564··...4clib/ordered | 000003d0:·02b0·c034·636c·6962·2f6f·7264·6572·6564··...4clib/ordered | ||
| 000003e0:·5479·7065·2e6d·6c69·4b01·02a4·0102·b0c0··Type.mliK....... | 000003e0:·5479·7065·2e6d·6c69·4b01·02a4·0102·b0c0··Type.mliK....... | ||
| 000003f0:·0402·4b01·02a4·0102·b140·90d0·91b0·a0b0··..K......@...... | 000003f0:·0402·4b01·02a4·0102·b140·90d0·91b0·a0b0··..K......@...... | ||
| 00000400:·a141·a008·0000·2800·b121·7400·5141·a004··.A....(..!t.QA.. | 00000400:·a141·a008·0000·2800·b121·7400·5141·a004··.A....(..!t.QA.. | ||
| 00000410:·02b0·c004·104d·0102·b801·02bf·c004·114d··.....M.........M | 00000410:·02b0·c004·104d·0102·b801·02bf·c004·114d··.....M.........M | ||
| 00000420:·0102·b801·02c0·4040·0800·0038·0040·4040··......@@...8.@@@ | 00000420:·0102·b801·02c0·4040·0800·0038·0040·4040··......@@...8.@@@ | ||
| 00000430:·4140·4040·4040·b0c0·0414·4d01·02b8·0102··A@@@@@....M..... | 00000430:·4140·4040·4040·b0c0·0414·4d01·02b8·0102··A@@@@@....M..... | ||
| 00000440:·ba04·0440·4040·40a1·0421·4040·4041·4004··...@@@@..!@@@A@. | 00000440:·ba04·0440·4040·40a1·0421·4040·4041·4004··...@@@@..!@@@A@. | ||
| 00000450:·0340·4008·0000·3000·a040·4004·0104·0104··.@@...0..@@..... | 00000450:·0340·4008·0000·3000·a040·4004·0104·0104··.@@...0..@@..... | ||
| 00000460:·0104·0104·0104·0104·0140·aba | 00000460:·0104·0104·0104·0104·0140·aba7·b1b2·b2b2··.........@...... | ||
| 00000470:·b2b2·b2b2·b2b2·b2b2·b2b | 00000470:·b2b2·b2b2·b2b2·b2b2·b2b1·b1b1·b1b1·b1b1··................ | ||
| 00000480:·b1b1·b1b1·b1b1·b1b1·b1b1·40a3·2a66·6c6f··..........@.*flo | |||
| 00000490:·6174·6172·7261·7951·0800·0038·0040·4040··atarrayQ...8.@@@ | |||
| 000004 | 00000480:·b1b1·b1b1·b1b1·b1b1·b140·a32a·666c·6f61··.........@.*floa | ||
| 00000490:·7461·7272·6179·5108·0000·3800·4040·4041··tarrayQ...8.@@@A | |||
| 000004a0:·4040·4040·40b0·c026·5f6e·6f6e·655f·4040··@@@@@..&_none_@@ | |||
| 000004b0:· | 000004b0:·00ff·0402·4140·4040·9204·06a3·3565·7874··....A@@@....5ext | ||
| 000004c0:· | 000004c0:·656e·7369·6f6e·5f63·6f6e·7374·7275·6374··ension_construct | ||
| 000004d0:· | 000004d0:·6f72·5008·0000·3800·4040·4041·4040·4040··orP...8.@@@A@@@@ | ||
| 000004e0:·40 | 000004e0:·4004·0740·4040·9204·03a3·2369·6e74·4108··@..@@@....#intA. | ||
| 000004f0:·0 | 000004f0:·0000·3800·4040·4041·4040·4040·4004·0b40··..8.@@@A@@@@@..@ | ||
| 00000500:·4140·9204·03a3·2463·6861·7242·0800·0038··A@....$charB...8 | |||
| 00000500:·4041·4092·0403·a324·6368·6172·4208·0000··@A@....$charB... | |||
| 00000510:·3800·4040·4041·4040·4040·4004·0f40·4140··8.@@@A@@@@@..@A@ | |||
| 00000520:·9204·03a3·2673·7472·696e·674f·0800·0038··....&stringO...8 | |||
| 000005 | 00000510:·0040·4040·4140·4040·4040·040f·4041·4092··.@@@A@@@@@..@A@. | ||
| 000005 | 00000520:·0403·a326·7374·7269·6e67·4f08·0000·3800··...&stringO...8. | ||
| 000005 | 00000530:·4040·4041·4040·4040·4004·1340·4040·9204··@@@A@@@@@..@@@.. | ||
| 000005 | 00000540:·03a3·2566·6c6f·6174·4408·0000·3800·4040··..%floatD...8.@@ | ||
| 00000550:·4041·4040·4040·4004·1740·4040·9204·03a3··@A@@@@@..@@@.... | |||
| 00000560:·2462·6f6f·6c45·0800·0038·0040·40a1·a0e0··$boolE...8.@@... | |||
| 00000570:· | 00000570:·a325·6661·6c73·655e·9040·4004·2140·9204··.%false^.@@.!@.. | ||
| 00000580:·03a0·e0a3·2474·7275·655f·9040·4004·2740··....$true_.@@.'@ | |||
| 00000580:·0403·a0e0·a324·7472·7565·5f90·4040·0427··.....$true_.@@.' | |||
| 00000590:·4092·0403·4040·4140·4040·4040·0428·4041··@...@@A@@@@@.(@A | |||
| 000005a0:·4092·0410·a324·756e·6974·4608·0000·3800··@....$unitF...8. | |||
| 000005b0:·4040·a1a0·e0a3·2228·2960·9040·4004·3240··@@...."()`.@@.2@ | |||
| 000005 | 00000590:·9204·0340·4041·4040·4040·4004·2840·4140··...@@A@@@@@.(@A@ | ||
| 000005 | 000005a0:·9204·10a3·2475·6e69·7446·0800·0038·0040··....$unitF...8.@ | ||
| 000005b0:·40a1·a0e0·a322·2829·6090·4040·0432·4092··@...."()`.@@.2@. | |||
| 000005c0:·0403·4040·4140·4040·4040·0433·4041·4092··..@@A@@@@@.3@A@. | |||
| 000005d0:·040a·a323·6578·6e47·0800·0038·0040·4041··...#exnG...8.@@A | |||
| 000005e0:·414 | 000005e0:·4140·4040·4040·0437·4040·4092·0403·a325··A@@@@@.7@@@....% | ||
| 000005f0:·2561·7272·6179·4808·0000·3800·a0c0·9040··%arrayH...8....@ | |||
| 00000600:·0205·f5e1·0040·4f40·4140·4140·a000·7f40··.....@O@A@A@...@ | |||
| 00000 | 000005f0:·6172·7261·7948·0800·0038·00a0·c090·4002··arrayH...8....@. | ||
| 00000600:·05f5·e100·404f·4041·4041·40a0·007f·40a0··....@O@A@A@...@. | |||
| 00000610:·4040·4040·0440·4040·4092·0408·a324·6c69··@@@@.@@@@....$li | |||
| 00000620:· | 00000620:·7374·4908·0000·3800·a0c0·0409·0205·f5e1··stI...8......... | ||
| 00000630:·e100·4050·4041·a1a0·e0a3·225b·5d61·9040··..@P@A...."[]a.@ | |||
| 00000640:·4004·4c40·9204·03a0·e0a3·223a·3a62·90a0··@.L@......"::b.. | |||
| 00000650:·040e·a0c0·b390·0416·a004·1340·9040·0205··...........@.@.. | |||
| 000006 | 00000630:·0040·5040·41a1·a0e0·a322·5b5d·6190·4040··.@P@A...."[]a.@@ | ||
| 00000640:·044c·4092·0403·a0e0·a322·3a3a·6290·a004··.L@......"::b... | |||
| 00000650:·0ea0·c0b3·9004·16a0·0413·4090·4002·05f5··..........@.@... | |||
| 00000660:·e100·4051·4040·0459·4092·040a·4040·4140··..@Q@@.Y@...@@A@ | |||
| 00000670:· | 00000670:·a059·40a0·4040·4040·045c·4040·4092·041b··.Y@.@@@@.\@@@... | ||
| 00000680:· | 00000680:·a326·6f70·7469·6f6e·4a08·0000·3800·a0c0··.&optionJ...8... | ||
| 00000690:·c004·2502·05f5·e100·4053·4041·a1a0·e0a3··..%.....@S@A.... | |||
| 000006a0:·244e·6f6e·6563·9040·4004·6840·9204·03a0··$Nonec.@@.h@.... | |||
| 000006 | 00000690:·0425·0205·f5e1·0040·5340·41a1·a0e0·a324··.%.....@S@A....$ | ||
| 000006a0:·4e6f·6e65·6390·4040·0468·4092·0403·a0e0··Nonec.@@.h@..... | |||
| 000006b0:·a324·536f·6d65·6490·a004·0e40·4004·6f40··.$Somed....@@.o@ | |||
| 000006c0:· | 000006c0:·9204·0440·4041·40a0·5940·a040·4040·4004··...@@A@.Y@.@@@@. | ||
| 000006d0:· | 000006d0:·7240·4040·9204·15a3·266c·617a·795f·744e··r@@@....&lazy_tN | ||
| 000006e0:· | 000006e0:·0800·0038·00a0·c004·3b02·05f5·e100·4055··...8....;.....@U | ||
| 000006f0:· | 000006f0:·4041·4041·40a0·5940·a040·4040·4004·7a40··@A@A@.Y@.@@@@.z@ | ||
| 00000700:·4040· | 00000700:·4040·9204·07a3·296e·6174·6976·6569·6e74··@@....)nativeint | ||
| 00000710:· | 00000710:·4b08·0000·3800·4040·4041·4040·4040·4004··K...8.@@@A@@@@@. | ||
| 00000720:· | 00000720:·7e40·4040·9204·03a3·2569·6e74·3332·4c08··~@@@....%int32L. | ||
| 00000730:·0 | 00000730:·0000·3800·4040·4041·4040·4040·4004·8240··..8.@@@A@@@@@..@ | ||
| 00000740:·4040· | 00000740:·4040·9204·03a3·2569·6e74·3634·4d08·0000··@@....%int64M... | ||
| 00000750:·0038·0040·4040·4140·4040·4040·0486·4040··.8.@@@A@@@@@..@@ | |||
| 00000760:·4092·0403·a33a·556e·6465·6669·6e65·645f··@....:Undefined_ | |||
| 00000770:·7265·6375·7273·6976·655f·6d6f·6475·6c65··recursive_module | |||
| 00000780:·5d08·0000·2000·9004·5740·90a0·c092·a0c0··]...·...W@...... | |||
| 00000790:·b390·0483·4090·4002·05f5·e100·404a·a0c0··....@.@.....@J.. | |||
| 000007a0:·b390·0490·4090·4002·05f5·e100·4040·a004··....@.@.....@@.. | |||
| 000007b0:·0540·0205·f5e1·0040·5640·4041·049a·a0b0··.@.....@V@@A.... | |||
| 000007c0:·a03d·6f63·616d·6c2e·7761·726e·5f6f·6e5f··.=ocaml.warn_on_ | |||
| 000007d0:·6c69·7465·7261·6c5f·7061·7474·6572·6e04··literal_pattern. | |||
| 000007e0:·9e04·6104·9e40·9204·17a3·2e41·7373·6572··..a..@.....Asser | |||
| 000007f0:·745f·6661·696c·7572·655c·0800·0020·0004··t_failure\...·.. | |||
| 00000800:·1840·90a0·c092·a004·17a0·0413·a004·1440··.@.............@ | |||
| 00000810:·0205·f5e1·0040·5840·4041·04a9·a0b0·a004··.....@X@@A...... | |||
| 00000820:·0f04·ac04·6f04·ac40·9204·0da3·3044·6976··....o..@....0Div | |||
| 00000830:·6973·696f·6e5f·6279·5f7a·6572·6f59·0800··ision_by_zeroY.. | |||
| 00000840:·0020·0004·2640·9040·4041·04b1·a0b0·a004··.·..&@.@@A...... | |||
| 00000850:·1704·b404·7704·b440·9204·07a3·2b45·6e64··....w..@....+End | |||
| 00000860:·5f6f·665f·6669·6c65·5808·0000·2000·042e··_of_fileX...·... | |||
| 00000870:·4090·4040·4104·b9a0·b0a0·041f·04bc·047f··@.@@A........... | |||
| 00000880:·04bc·4092·0407·a329·5379·735f·6572·726f··..@....)Sys_erro | |||
| 00000890:·7257·0800·0020·0004·3640·90a0·0432·4040··rW...·..6@...2@@ | |||
| 000008a0:·4104·c2a0·b0a0·0428·04c5·0488·04c5·4092··A......(......@. | |||
| 000008b0:·0408·a32e·5379·735f·626c·6f63·6b65·645f··....Sys_blocked_ | |||
| 000008c0:·696f·5b08·0000·2000·043f·4090·4040·4104··io[...·..?@.@@A. | |||
| 000008d0:·caa0·b0a0·0430·04cd·0490·04cd·4092·0407··.....0......@... | |||
| 000008e0:·a329·4e6f·745f·666f·756e·6456·0800·0020··.)Not_foundV...· | |||
| 000008f0:·0004·4740·9040·4041·04d2·a0b0·a004·3804··..G@.@@A......8. | |||
| 00000900:·d504·9804·d540·9204·07a3·2746·6169·6c75··.....@....'Failu | |||
| 00000910:·7265·5508·0000·2000·044f·4090·a004·4b40··reU...·..O@...K@ | |||
| 00000920:·4041·04db·a0b0·a004·4104·de04·a104·de40··@A......A......@ | |||
| 00000930:·9204·08a3·3049·6e76·616c·6964·5f61·7267··....0Invalid_arg | |||
| 00000940:·756d·656e·7454·0800·0020·0004·5840·90a0··umentT...·..X@.. | |||
| 00000950:·0454·4040·4104·e4a0·b0a0·044a·04e7·04aa··.T@@A......J.... | |||
| 00000960:·04e7·4092·0408·a32e·5374·6163·6b5f·6f76··..@.....Stack_ov | |||
| 00000970:·6572·666c·6f77·5a08·0000·2000·0461·4090··erflowZ...·..a@. | |||
| 00000980:·4040·4104·eca0·b0a0·0452·04ef·04b2·04ef··@@A......R...... | |||
| 00000990:·4092·0407·a32d·4f75·745f·6f66·5f6d·656d··@....-Out_of_mem | |||
| 000009a0:·6f72·7953·0800·0020·0004·6940·9040·4041··oryS...·..i@.@@A | |||
| 000009b0:·04f4·a0b0·a004·5a04·f704·ba04·f740·9204··......Z......@.. | |||
| 000009c0:·07a3·2d4d·6174·6368·5f66·6169·6c75·7265··..-Match_failure | |||
| 000009d0:·5208·0000·2000·0471·4090·a0c0·92a0·0470··R...·..q@......p | |||
| 000009e0:·a004·6ca0·046d·4002·05f5·e100·4063·4040··..l..m@.....@c@@ | |||
| 000009f0:·4105·0102·a0b0·a004·6805·0105·04c8·0501··A.......h....... | |||
| 00000a00:·0540·9204·0da3·2562·7974·6573·4308·0000··.@....%bytesC... | |||
| 00000 | 00000750:·3800·4040·4041·4040·4040·4004·8640·4040··8.@@@A@@@@@..@@@ | ||
| Max diff block lines reached; 11100/46588 bytes (23.83%) of diff not shown. | |||
| Offset 1, 12817 lines modified | Offset 1, 12816 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0003·20 | 00000010:·0003·20dc·0000·9e58·0002·7ac1·0002·759a··..·....X..z...u. | ||
| 00000020:·0800·0034·0027·556e·6963·6f64·6591·b0a0··...4.'Unicode... | 00000020:·0800·0034·0027·556e·6963·6f64·6591·b0a0··...4.'Unicode... | ||
| 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... | ||
| 00000080:·0102·a4c0·0402·4b01·02a4·0102·bc40·4004··......K......@@. | 00000080:·0102·a4c0·0402·4b01·02a4·0102·bc40·4004··......K......@@. | ||
| 00000090:·0440·4040·0404·4004·0404·0408·0000·3000··.@@@..@.......0. | 00000090:·0440·4040·0404·4004·0404·0408·0000·3000··.@@@..@.......0. | Diff chunk too large, falling back to line-by-line diff (12327 lines added, 12327 lines removed) | |
| 000000a0:·a040·4004·0104·0104·0104·0104·0104·0104··.@@............. | 000000a0:·a040·4004·0104·0104·0104·0104·0104·0104··.@@............. | ||
| 000000b0:·0140·aba | 000000b0:·0140·aba7·b1b2·b2b2·b2b2·b2b2·b2b2·b2b2··.@.............. | ||
| 000000c0:·b2b | 000000c0:·b2b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1··................ | ||
| 000000d0:·b1 | 000000d0:·b140·a32a·666c·6f61·7461·7272·6179·5108··.@.*floatarrayQ. | ||
| 000000e0:·0 | 000000e0:·0000·3800·4040·4041·4040·4040·4004·3340··..8.@@@A@@@@@.3@ | ||
| 000000f0:·4040· | 000000f0:·4040·9204·03a3·3565·7874·656e·7369·6f6e··@@....5extension | ||
| 00000100:· | 00000100:·5f63·6f6e·7374·7275·6374·6f72·5008·0000··_constructorP... | ||
| 00000110:· | 00000110:·3800·4040·4041·4040·4040·4004·3740·4040··8.@@@A@@@@@.7@@@ | ||
| 00000120:· | 00000120:·9204·03a3·2369·6e74·4108·0000·3800·4040··....#intA...8.@@ | ||
| 00000130:·404 | 00000130:·4041·4040·4040·4004·3b40·4140·9204·03a3··@A@@@@@.;@A@.... | ||
| 00000140:· | 00000140:·2463·6861·7242·0800·0038·0040·4040·4140··$charB...8.@@@A@ | ||
| 00000150:·4040·4040· | 00000150:·4040·4040·043f·4041·4092·0403·a326·7374··@@@@.?@A@....&st | ||
| 00000160:·7 | 00000160:·7269·6e67·4f08·0000·3800·4040·4041·4040··ringO...8.@@@A@@ | ||
| 00000170:·4040·40 | 00000170:·4040·4004·4340·4040·9204·03a3·2566·6c6f··@@@.C@@@....%flo | ||
| 00000180:·6 | 00000180:·6174·4408·0000·3800·4040·4041·4040·4040··atD...8.@@@A@@@@ | ||
| 00000190:·40 | 00000190:·4004·4740·4040·9204·03a3·2462·6f6f·6c45··@.G@@@....$boolE | ||
| 000001a0:· | 000001a0:·0800·0038·0040·40a1·a0e0·a325·6661·6c73··...8.@@....%fals | ||
| 000001b0:· | 000001b0:·655e·9040·4004·5140·9204·03a0·e0a3·2474··e^.@@.Q@......$t | ||
| 000001c0:·7 | 000001c0:·7275·655f·9040·4004·5740·9204·0340·4041··rue_.@@.W@...@@A | ||
| 000001d0:·4 | 000001d0:·4040·4040·4004·5840·4140·9204·10a3·2475··@@@@@.X@A@....$u | ||
| 000001e0:· | 000001e0:·6e69·7446·0800·0038·0040·40a1·a0e0·a322··nitF...8.@@...." | ||
| 000001f0:·2 | 000001f0:·2829·6090·4040·0462·4092·0403·4040·4140··()`.@@.b@...@@A@ | ||
| 00000200:·4040·4040· | 00000200:·4040·4040·0463·4041·4092·040a·a323·6578··@@@@.c@A@....#ex | ||
| 00000210:· | 00000210:·6e47·0800·0038·0040·4041·4140·4040·4040··nG...8.@@AA@@@@@ | ||
| 00000220:· | 00000220:·0467·4040·4092·0403·a325·6172·7261·7948··.g@@@....%arrayH | ||
| 00000230:· | 00000230:·0800·0038·00a0·c090·4002·05f5·e100·404f··...8....@.....@O | ||
| 00000240:·4 | 00000240:·4041·4041·40a0·007f·40a0·4040·4040·0470··@A@A@...@.@@@@.p | ||
| 00000250:· | 00000250:·4040·4092·0408·a324·6c69·7374·4908·0000··@@@....$listI... | ||
| 00000260:· | 00000260:·3800·a0c0·0409·0205·f5e1·0040·5040·41a1··8..........@P@A. | ||
| 00000270:·a | 00000270:·a0e0·a322·5b5d·6190·4040·047c·4092·0403··..."[]a.@@.|@... | ||
| 00000280:· | 00000280:·a0e0·a322·3a3a·6290·a004·0ea0·c0b3·9004··..."::b......... | ||
| 00000290:· | 00000290:·16a0·0413·4090·4002·05f5·e100·4051·4040··....@.@.....@Q@@ | ||
| 000002a0:· | 000002a0:·0489·4092·040a·4040·4140·a059·40a0·4040··..@...@@A@.Y@.@@ | ||
| 000002b0:·4040· | 000002b0:·4040·048c·4040·4092·041b·a326·6f70·7469··@@..@@@....&opti | ||
| 000002c0:·6 | 000002c0:·6f6e·4a08·0000·3800·a0c0·0425·0205·f5e1··onJ...8....%.... | ||
| 000002d0:· | 000002d0:·0040·5340·41a1·a0e0·a324·4e6f·6e65·6390··.@S@A....$Nonec. | ||
| 000002e0:· | 000002e0:·4040·0498·4092·0403·a0e0·a324·536f·6d65··@@..@......$Some | ||
| 000002f0:·6 | 000002f0:·6490·a004·0e40·4004·9f40·9204·0440·4041··d....@@..@...@@A | ||
| 00000300:·4 | 00000300:·40a0·5940·a040·4040·4004·a240·4040·9204··@.Y@.@@@@..@@@.. | ||
| 00000310:· | 00000310:·15a3·266c·617a·795f·744e·0800·0038·00a0··..&lazy_tN...8.. | ||
| 00000320:· | 00000320:·c004·3b02·05f5·e100·4055·4041·4041·40a0··..;.....@U@A@A@. | ||
| 00000330:· | 00000330:·5940·a040·4040·4004·aa40·4040·9204·07a3··Y@.@@@@..@@@.... | ||
| 00000340:· | 00000340:·296e·6174·6976·6569·6e74·4b08·0000·3800··)nativeintK...8. | ||
| 00000350:· | 00000350:·4040·4041·4040·4040·4004·ae40·4040·9204··@@@A@@@@@..@@@.. | ||
| 00000360:·0 | 00000360:·03a3·2569·6e74·3332·4c08·0000·3800·4040··..%int32L...8.@@ | ||
| 00000370:·404 | 00000370:·4041·4040·4040·4004·b240·4040·9204·03a3··@A@@@@@..@@@.... | ||
| 00000380:· | 00000380:·2569·6e74·3634·4d08·0000·3800·4040·4041··%int64M...8.@@@A | ||
| 00000390:·4 | 00000390:·4040·4040·4004·b640·4040·9204·03a3·3a55··@@@@@..@@@....:U | ||
| 000003a0:· | 000003a0:·6e64·6566·696e·6564·5f72·6563·7572·7369··ndefined_recursi | ||
| 000003b0:· | 000003b0:·7665·5f6d·6f64·756c·655d·0800·0020·0090··ve_module]...·.. | ||
| 000003c0:· | 000003c0:·0457·4090·a0c0·92a0·c0b3·9004·8340·9040··.W@..........@.@ | ||
| 000003d0:· | 000003d0:·0205·f5e1·0040·4aa0·c0b3·9004·9040·9040··.....@J......@.@ | ||
| 000003e0:· | 000003e0:·0205·f5e1·0040·40a0·0405·4002·05f5·e100··.....@@...@..... | ||
| 000003f0:· | 000003f0:·4056·4040·4104·caa0·b0a0·3d6f·6361·6d6c··@V@@A.....=ocaml | ||
| 00000400:· | 00000400:·2e77·6172·6e5f·6f6e·5f6c·6974·6572·616c··.warn_on_literal | ||
| 00000410:· | 00000410:·5f70·6174·7465·726e·04ce·0461·04ce·4092··_pattern...a..@. | ||
| 00000420:· | 00000420:·0417·a32e·4173·7365·7274·5f66·6169·6c75··....Assert_failu | ||
| 00000430:·7 | 00000430:·7265·5c08·0000·2000·0418·4090·a0c0·92a0··re\...·...@..... | ||
| 00000440:· | 00000440:·0417·a004·13a0·0414·4002·05f5·e100·4058··........@.....@X | ||
| 00000450:· | 00000450:·4040·4104·d9a0·b0a0·040f·04dc·046f·04dc··@@A..........o.. | ||
| 00000460:· | 00000460:·4092·040d·a330·4469·7669·7369·6f6e·5f62··@....0Division_b | ||
| 00000470:· | 00000470:·795f·7a65·726f·5908·0000·2000·0426·4090··y_zeroY...·..&@. | ||
| 00000480:· | 00000480:·4040·4104·e1a0·b0a0·0417·04e4·0477·04e4··@@A..........w.. | ||
| 00000490:· | 00000490:·4092·0407·a32b·456e·645f·6f66·5f66·696c··@....+End_of_fil | ||
| 000004a0:·6 | 000004a0:·6558·0800·0020·0004·2e40·9040·4041·04e9··eX...·...@.@@A.. | ||
| 000004b0:· | 000004b0:·a0b0·a004·1f04·ec04·7f04·ec40·9204·07a3··...........@.... | ||
| 000004c0:· | 000004c0:·2953·7973·5f65·7272·6f72·5708·0000·2000··)Sys_errorW...·. | ||
| 000004d0:·0 | 000004d0:·0436·4090·a004·3240·4041·04f2·a0b0·a004··.6@...2@@A...... | ||
| 000004e0:· | 000004e0:·2804·f504·8804·f540·9204·08a3·2e53·7973··(......@.....Sys | ||
| 000004f0:· | 000004f0:·5f62·6c6f·636b·6564·5f69·6f5b·0800·0020··_blocked_io[...· | ||
| 00000500:· | 00000500:·0004·3f40·9040·4041·04fa·a0b0·a004·3004··..?@.@@A......0. | ||
| 00000510:· | 00000510:·fd04·9004·fd40·9204·07a3·294e·6f74·5f66··.....@....)Not_f | ||
| 00000520:·6 | 00000520:·6f75·6e64·5608·0000·2000·0447·4090·4040··oundV...·..G@.@@ | ||
| 00000530:·4 | 00000530:·4105·0102·a0b0·a004·3805·0105·0498·0501··A.......8....... | ||
| 00000540:·0 | 00000540:·0540·9204·07a3·2746·6169·6c75·7265·5508··.@....'FailureU. | ||
| 00000550:·0 | 00000550:·0000·2000·044f·4090·a004·4b40·4041·0501··..·..O@...K@@A.. | ||
| 00000560:·0 | 00000560:·0ba0·b0a0·0441·0501·0e04·a105·010e·4092··.....A........@. | ||
| 00000570:· | 00000570:·0408·a330·496e·7661·6c69·645f·6172·6775··...0Invalid_argu | ||
| 00000580:· | 00000580:·6d65·6e74·5408·0000·2000·0458·4090·a004··mentT...·..X@... | ||
| 00000590:· | 00000590:·5440·4041·0501·14a0·b0a0·044a·0501·1704··T@@A.......J.... | ||
| 000005a0:· | 000005a0:·aa05·0117·4092·0408·a32e·5374·6163·6b5f··....@.....Stack_ | ||
| 000005b0:· | 000005b0:·6f76·6572·666c·6f77·5a08·0000·2000·0461··overflowZ...·..a | ||
| 000005c0:· | 000005c0:·4090·4040·4105·011c·a0b0·a004·5205·011f··@.@@A.......R... | ||
| 000005d0:· | 000005d0:·04b2·0501·1f40·9204·07a3·2d4f·7574·5f6f··.....@....-Out_o | ||
| 000005e0:·6 | 000005e0:·665f·6d65·6d6f·7279·5308·0000·2000·0469··f_memoryS...·..i | ||
| 000005f0:· | 000005f0:·4090·4040·4105·0124·a0b0·a004·5a05·0127··@.@@A..$....Z..' | ||
| 00000600:· | 00000600:·04ba·0501·2740·9204·07a3·2d4d·6174·6368··....'@....-Match | ||
| 00000610:· | 00000610:·5f66·6169·6c75·7265·5208·0000·2000·0471··_failureR...·..q | ||
| 00000620:· | 00000620:·4090·a0c0·92a0·0470·a004·6ca0·046d·4002··@......p..l..m@. | ||
| 00000630:·0 | 00000630:·05f5·e100·4063·4040·4105·0132·a0b0·a004··....@c@@A..2.... | ||
| 00000640:· | 00000640:·6805·0135·04c8·0501·3540·9204·0da3·2562··h..5....5@....%b | ||
| 00000650:· | 00000650:·7974·6573·4308·0000·3800·4040·4041·4040··ytesC...8.@@@A@@ | ||
| 00000660:·4040·40 | 00000660:·4040·4005·0139·4040·4092·0403·9092·2653··@@@..9@@@.....&S | ||
| 00000670:· | 00000670:·7464·6c69·6292·264f·7074·696f·6e40·40a0··tdlib.&Option@@. | ||
| 00000680:· | 00000680:·b0a3·41a0·0800·0028·00b1·2673·7461·7475··..A....(..&statu | ||
| 00000690:· | 00000690:·7300·5141·a004·02b0·c005·013c·4d01·02be··s.QA.......<M... | ||
| 000006a0:·013 | 000006a0:·0102·c3c0·0501·3d4d·0102·be01·02c9·4040··......=M......@@ | ||
| 000006b0:· | 000006b0:·0800·0038·0040·40a1·a0e0·a026·4c65·7474··...8.@@....&Lett | ||
| 000006c0:· | 000006c0:·6572·0052·9040·40b0·c005·0146·4d01·02be··er.R.@@....FM... | ||
| 000006d0:·01 | 000006d0:·0102·ccc0·0501·474d·0102·be01·02d2·4040··......GM......@@ | ||
| 000006e0:· | 000006e0:·a105·015e·41a0·e0a0·2949·6465·6e74·5061··...^A...)IdentPa | ||
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| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab942"></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="lab | 57 | <a·id="lab943"></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="lab | 59 | <a·id="lab944"></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/> |
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| 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="lab45 | 57 | <a·id="lab945"></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="lab991"></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="lab961"></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="lab962"></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="lab | 53 | <a·id="lab1013"></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="lab | 61 | <a·id="lab1014"></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/> |
| Offset 48, 39 lines modified | Offset 48, 39 lines modified | ||
| 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="lab | 53 | <a·id="lab996"></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="lab | 61 | <a·id="lab997"></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="lab | 70 | <a·id="lab998"></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/> |
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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="lab947"></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="lab | 49 | <a·id="lab995"></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="lab | 53 | <a·id="lab1006"></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="lab1007"></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="lab1008"></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="lab | 49 | <a·id="lab948"></a><h1·class="section">Properties·of·multiplication.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | <a·id="lab | 51 | <a·id="lab949"></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="lab | 61 | <a·id="lab950"></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="lab | 63 | <a·id="lab951"></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="lab | 73 | <a·id="lab952"></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="lab | 83 | <a·id="lab953"></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="lab | 92 | <a·id="lab954"></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/> |
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| 144 | <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/> | 144 | <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 | <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/> | 145 | <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 | <br/> | 146 | <br/> |
| 147 | </div> | 147 | </div> |
| 148 | <div·class="doc"> | 148 | <div·class="doc"> |
| 149 | <a·id="lab | 149 | <a·id="lab977"></a><h2·class="section">Remaining·constants·not·defined·in·Coq.Init.Nat</h2> |
| 150 | <div·class="paragraph">·</div> | 150 | <div·class="paragraph">·</div> |
| 151 | ·NB:·Aliasing·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·mandatory,·since·only·a·Definition·can·implement | 151 | ·NB:·Aliasing·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·mandatory,·since·only·a·Definition·can·implement |
| 152 | ····an·interface·Parameter...· | 152 | ····an·interface·Parameter...· |
| 153 | </div> | 153 | </div> |
| 154 | <div·class="code"> | 154 | <div·class="code"> |
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| 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="lab969"></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="lab970"></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/> |
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| 80 | |·<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/> | 80 | |·<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/> |
| 81 | <span·class="id"·title="keyword">end</span>.<br/> | 81 | <span·class="id"·title="keyword">end</span>.<br/> |
| 82 | <br/> | 82 | <br/> |
| 83 | </div> | 83 | </div> |
| 84 | <div·class="doc"> | 84 | <div·class="doc"> |
| 85 | <a·id="lab | 85 | <a·id="lab794"></a><h1·class="section">Decidability</h1> |
| 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="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/> | 89 | <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/> |
| 90 | <br/> | 90 | <br/> |
| 91 | </div> | 91 | </div> |
| 92 | <div·class="doc"> | 92 | <div·class="doc"> |
| 93 | <a·id="lab | 93 | <a·id="lab795"></a><h1·class="section">Discrimination</h1> |
| 94 | </div> | 94 | </div> |
| 95 | <div·class="code"> | 95 | <div·class="code"> |
| 96 | <br/> | 96 | <br/> |
| 97 | <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/> | 97 | <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/> |
| 98 | #[<span·class="id"·title="var">global</span>]<br/> | 98 | #[<span·class="id"·title="var">global</span>]<br/> |
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| 61 | <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>.<br/> | 61 | <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>.<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/> | 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 | <br/> | 63 | <br/> |
| 64 | </div> | 64 | </div> |
| 65 | <div·class="doc"> | 65 | <div·class="doc"> |
| 66 | <a·id="lab | 66 | <a·id="lab809"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1> |
| 67 | </div> | 67 | </div> |
| 68 | <div·class="code"> | 68 | <div·class="code"> |
| 69 | <br/> | 69 | <br/> |
| 70 | <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/> | 70 | <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 | · | 71 | · |
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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="lab267"></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 |
| 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="lab285"></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="lab269"></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="lab | 49 | <a·id="lab281"></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="lab287"></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="lab | 49 | <a·id="lab283"></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="lab284"></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="lab | 49 | <a·id="lab274"></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="lab268"></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="lab282"></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 | ||
| 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:·6c61·623 | 000006d0:·6c61·6232·3738·223e·3c2f·613e·3c68·3120··lab278"></a><h1· | ||
| 000006e0:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section"> | 000006e0:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section"> | ||
| 000006f0:·5479·7065·636c·6173·732d·6261·7365·6420··Typeclass-based· | 000006f0:·5479·7065·636c·6173·732d·6261·7365·6420··Typeclass-based· | ||
| 00000700:·7265·6c61·7469·6f6e·732c·2074·6163·7469··relations,·tacti | 00000700:·7265·6c61·7469·6f6e·732c·2074·6163·7469··relations,·tacti | ||
| 00000710:·6373·2061·6e64·2073·7461·6e64·6172·6420··cs·and·standard· | 00000710:·6373·2061·6e64·2073·7461·6e64·6172·6420··cs·and·standard· | ||
| 00000720:·696e·7374·616e·6365·733c·2f68·313e·0a0a··instances</h1>.. | 00000720:·696e·7374·616e·6365·733c·2f68·313e·0a0a··instances</h1>.. | ||
| 00000730:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par | 00000730:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par | ||
| 00000740:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>. | 00000740:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</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="lab272"></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="lab277"></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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| 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="lab273"></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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| 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="lab276"></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="lab1 | 51 | <a·id="lab1037"></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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| 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="lab9 | 49 | <a·id="lab1094"></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. |
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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="lab1135"></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> |
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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="lab1088"></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"> |
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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="lab1161"></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> |
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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="lab1159"></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="lab1148"></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"> |
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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="lab1181"></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="lab1 | 49 | <a·id="lab1112"></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, 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="lab1 | 49 | <a·id="lab1116"></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/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab11"></a><h1·class="section">From·new·Weak·Sets·to·old·ones</h1> | 58 | <a·id="lab1117"></a><h1·class="section">From·new·Weak·Sets·to·old·ones</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="Backport_WSets"·class="idref"·href="#Backport_WSets"><span·class="id"·title="module">Backport_WSets</span></a><br/> | 62 | <span·class="id"·title="keyword">Module</span>·<a·id="Backport_WSets"·class="idref"·href="#Backport_WSets"><span·class="id"·title="module">Backport_WSets</span></a><br/> |
| 63 | (<span·class="id"·title="var">E</span>:<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType.DecidableType</span></a>)<br/> | 63 | (<span·class="id"·title="var">E</span>:<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType.DecidableType</span></a>)<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="lab1021"></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="lab1020"></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="lab1084"></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="lab1 | 49 | <a·id="lab1173"></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="lab1115"></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="lab1121"></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="lab1082"></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="lab1083"></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="lab1"></a><h1·class="section">Finite·sets·library</h1> | 49 | <a·id="lab1160"></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="lab793"></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="lab792"></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="lab787"></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="lab776"></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="lab777"></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/> |
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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.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="lab788"></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="lab789"></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"> |
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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="lab525"></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="lab | 58 | <a·id="lab504"></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="lab | 49 | <a·id="lab526"></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="lab | 49 | <a·id="lab529"></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 158, 16 lines modified | Offset 158, 16 lines modified | ||
| 000009d0:·733d·2269·6422·2074·6974·6c65·3d22·7661··s="id"·title="va | 000009d0:·733d·2269·6422·2074·6974·6c65·3d22·7661··s="id"·title="va | ||
| 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·6120·6964·3d22·6c61·623 | 00000a40:·6f63·223e·0a3c·6120·6964·3d22·6c61·6234··oc">.<a·id="lab4 | ||
| 00000a50:·3 | 00000a50:·3938·223e·3c2f·613e·3c68·3120·636c·6173··98"></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 |
| Offset 50, 15 lines modified | Offset 50, 15 lines modified | ||
| 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="lab | 55 | <a·id="lab510"></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="lab | 49 | <a·id="lab502"></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/> |
| 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.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="lab | 56 | <a·id="lab527"></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">elimtype</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">elimtype</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span> |
| 60 | ····and·other·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.· | 60 | ····and·other·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.· |
| 61 | </div> | 61 | </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="lab503"></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="lab | 49 | <a·id="lab524"></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="lab543"></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"> |
| Offset 49, 15 lines modified | Offset 49, 15 lines modified | ||
| 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="lab541"></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> |
| 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.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="lab542"></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"> |
| Offset 46, 15 lines modified | Offset 46, 15 lines modified | ||
| 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="lab907"></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> |
| Offset 129, 15 lines modified | Offset 129, 15 lines modified | ||
| 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="lab839"></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="lab851"></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="lab936"></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="lab939"></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="lab843"></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="lab903"></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="lab841"></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="lab865"></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="lab846"></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="lab840"></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="lab842"></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="lab | 49 | <a·id="lab189"></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> |
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| 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="lab109"></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 |
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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="lab156"></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="lab144"></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="lab157"></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="lab | 49 | <a·id="lab149"></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="lab125"></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"> |
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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="lab130"></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="lab | 49 | <a·id="lab223"></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="lab | 49 | <a·id="lab265"></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="lab | 58 | <a·id="lab266"></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="lab184"></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="lab764"></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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| 54 | <br/> | 54 | <br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab766"></a><h1·class="section">Binary·natural·numbers,·definitions·of·operations</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>.<br/> | 62 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>.<br/> |
| 63 | <br/> | 63 | <br/> |
| 64 | <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/> | 64 | <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/> |
| 65 | <br/> | 65 | <br/> |
| 66 | </div> | 66 | </div> |
| 67 | <div·class="doc"> | 67 | <div·class="doc"> |
| 68 | <a·id="lab | 68 | <a·id="lab767"></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> |
| 69 | </div> | 69 | </div> |
| 70 | <div·class="code"> | 70 | <div·class="code"> |
| 71 | <br/> | 71 | <br/> |
| 72 | <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/> | 72 | <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/> |
| 73 | <br/> | 73 | <br/> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="doc"> | 75 | <div·class="doc"> |
| 76 | <a·id="lab | 76 | <a·id="lab768"></a><h2·class="section">Constants</h2> |
| 77 | </div> | 77 | </div> |
| 78 | <div·class="code"> | 78 | <div·class="code"> |
| 79 | <br/> | 79 | <br/> |
| 80 | <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/> | 80 | <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/> |
| 81 | <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/> | 81 | <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/> |
| 82 | <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/> | 82 | <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/> |
| 83 | <br/> | 83 | <br/> |
| 84 | </div> | 84 | </div> |
| 85 | <div·class="doc"> | 85 | <div·class="doc"> |
| 86 | <a·id="lab | 86 | <a·id="lab769"></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> |
| 87 | </div> | 87 | </div> |
| 88 | <div·class="code"> | 88 | <div·class="code"> |
| 89 | <br/> | 89 | <br/> |
| 90 | <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/> | 90 | <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/> |
| 91 | <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/> | 91 | <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="lab762"></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="lab341"></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="lab340"></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="lab300"></a><h1·class="section">Signature·and·specification·of·a·bounded·integer·structure</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·specifies·how·to·represent·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·when·<span·class="inlinecode"><span·class="id"·title="var">n</span>=2^<span·class="id"·title="var">d</span></span>, | 51 | ·This·file·specifies·how·to·represent·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span>·when·<span·class="inlinecode"><span·class="id"·title="var">n</span>=2^<span·class="id"·title="var">d</span></span>, |
| 52 | ····<span·class="inlinecode"><span·class="id"·title="var">d</span></span>·being·the·number·of·digits·of·these·bounded·integers.· | 52 | ····<span·class="inlinecode"><span·class="id"·title="var">d</span></span>·being·the·number·of·digits·of·these·bounded·integers.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <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="lab299"></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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| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | This·library·has·been·deprecated·since·Coq·version·8.10.· | 51 | This·library·has·been·deprecated·since·Coq·version·8.10.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab | 53 | <a·id="lab303"></a><h1·class="section">Int31·numbers·defines·indeed·a·cyclic·structure·:·Z/(2^31)Z</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | Author:·Arnaud·Spiwack·(+·Pierre·Letouzey) | 56 | 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="lab317"></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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| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | This·library·has·been·deprecated·since·Coq·version·8.10.· | 51 | This·library·has·been·deprecated·since·Coq·version·8.10.· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <a·id="lab | 53 | <a·id="lab316"></a><h1·class="section">Int31·numbers·defines·Z/(2^31)Z,·and·can·hence·be·equipped</h1> |
| 54 | ······with·a·ring·structure·and·a·ring·tactic· | 54 | ······with·a·ring·structure·and·a·ring·tactic· |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="code"> | 56 | <div·class="code"> |
| 57 | <br/> | 57 | <br/> |
| 58 | <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/> | 58 | <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="lab301"></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) |
| 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="lab302"></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="lab326"></a><h1·class="section">Type·<span·class="inlinecode"><span·class="id"·title="var">Z</span></span>·viewed·modulo·a·particular·constant·corresponds·to·<span·class="inlinecode"><span·class="id"·title="var">Z</span>/<span·class="id"·title="var">nZ</span></span></h1> |
| 50 | ······as·defined·abstractly·in·CyclicAxioms.· | 50 | ······as·defined·abstractly·in·CyclicAxioms.· |
| 51 | <div·class="paragraph">·</div> | 51 | <div·class="paragraph">·</div> |
| 52 | ·Even·if·the·construction·provided·here·is·not·reused·for·building | 52 | ·Even·if·the·construction·provided·here·is·not·reused·for·building |
| 53 | ··the·efficient·arbitrary·precision·numbers,·it·provides·a·simple | 53 | ··the·efficient·arbitrary·precision·numbers,·it·provides·a·simple |
| 54 | ··implementation·of·CyclicAxioms,·hence·ensuring·its·coherence.· | 54 | ··implementation·of·CyclicAxioms,·hence·ensuring·its·coherence.· |
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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="lab336"></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="lab298"></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> |
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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="lab335"></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> |
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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="lab332"></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> |
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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="lab288"></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> |
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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="lab331"></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="lab334"></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="lab327"></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="lab333"></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="lab328"></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="lab364"></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="lab329"></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="lab339"></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="lab338"></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="lab330"></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="lab337"></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> |
| Offset 467, 15 lines modified | Offset 467, 15 lines modified | ||
| 467 | <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/> | 467 | <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/> |
| 468 | <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/> | 468 | <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/> |
| 469 | <br/> | 469 | <br/> |
| 470 | </div> | 470 | </div> |
| 471 | <div·class="doc"> | 471 | <div·class="doc"> |
| 472 | <a·id="lab | 472 | <a·id="lab362"></a><h1·class="section">Properties·of·shifts</h1> |
| 473 | <div·class="paragraph">·</div> | 473 | <div·class="paragraph">·</div> |
| 474 | ·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> | 474 | ·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> |
| 475 | ···below·(combined·with·<span·class="inlinecode"><span·class="id"·title="var">testbit_neg_r</span></span>)·is·equivalent·to | 475 | ···below·(combined·with·<span·class="inlinecode"><span·class="id"·title="var">testbit_neg_r</span></span>)·is·equivalent·to |
| 476 | ···<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>.· | 476 | ···<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>.· |
| 477 | </div> | 477 | </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="lab352"></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> |
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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="lab348"></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: |
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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="lab358"></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: |
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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.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="lab56 | 53 | <a·id="lab356"></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="lab57 | 53 | <a·id="lab357"></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="lab363"></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/> |
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| 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="lab289"></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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| 172 | <br/> | 172 | <br/> |
| 173 | <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: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>,·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:37"><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: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>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:37"><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:38"><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:37"><span·class="id"·title="variable">a</span></a>.<br/> | 173 | <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: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>,·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:37"><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: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>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#a:37"><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:38"><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:37"><span·class="id"·title="variable">a</span></a>.<br/> |
| 174 | <br/> | 174 | <br/> |
| 175 | </div> | 175 | </div> |
| 176 | <div·class="doc"> | 176 | <div·class="doc"> |
| 177 | <a·id="lab | 177 | <a·id="lab291"></a><h1·class="section">Basic·values·of·divisions·and·modulo.</h1> |
| 178 | </div> | 178 | </div> |
| 179 | <div·class="code"> | 179 | <div·class="code"> |
| 180 | <br/> | 180 | <br/> |
| 181 | <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:39"·class="idref"·href="#a:39"><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:39"><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:39"><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/> | 181 | <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:39"·class="idref"·href="#a:39"><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:39"><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:39"><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/> |
| Offset 71, 15 lines modified | Offset 71, 15 lines modified | ||
| 71 | <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/> | 71 | <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">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/> | 72 | <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 | <br/> | 73 | <br/> |
| 74 | </div> | 74 | </div> |
| 75 | <div·class="doc"> | 75 | <div·class="doc"> |
| 76 | <a·id="lab | 76 | <a·id="lab295"></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> |
| 77 | <div·class="paragraph">·</div> | 77 | <div·class="paragraph">·</div> |
| 78 | ·For·any·two·points,·one·is·an·iterated·successor·of·the·other.· | 78 | ·For·any·two·points,·one·is·an·iterated·successor·of·the·other.· |
| 79 | </div> | 79 | </div> |
| 80 | <div·class="code"> | 80 | <div·class="code"> |
| Offset 414, 15 lines modified | Offset 414, 15 lines modified | ||
| 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="lab290"></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"> |
| Offset 343, 15 lines modified | Offset 343, 15 lines modified | ||
| 343 | (<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/> | 343 | (<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">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/> | 344 | (<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 | <br/> | 345 | <br/> |
| 346 | </div> | 346 | </div> |
| 347 | <div·class="doc"> | 347 | <div·class="doc"> |
| 348 | <a·id="lab | 348 | <a·id="lab294"></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> |
| 349 | </div> | 349 | </div> |
| 350 | <div·class="code"> | 350 | <div·class="code"> |
| 351 | <br/> | 351 | <br/> |
| 352 | <br/> | 352 | <br/> |
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| 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="lab343"></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="lab342"></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="lab346"></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/> |
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| 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="lab347"></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/> |
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| 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="lab1 | 58 | <a·id="lab1"></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> |
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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="lab59"></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> |
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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">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="lab56"></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="lab1019"></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="lab1215"></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="lab1233"></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="lab1236"></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="lab1237"></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="lab1213"></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/> |
| Offset 48, 64 lines modified | Offset 48, 64 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.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="lab1194"></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="lab1195"></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="lab1196"></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="lab1197"></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="lab1198"></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 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.Reals.PartSum.html#"><span·class="id"·title="library">PartSum</span></a>.<br/> | 53 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.PartSum.html#"><span·class="id"·title="library">PartSum</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="lab6 | 58 | <a·id="lab680"></a><h1·class="section">Formalization·of·alternated·series</h1> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <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/> | 61 | <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="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/> | 62 | <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 | <br/> | 63 | <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="lab615"></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/> |
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| 00013af0:·7952·6561·6c73·4d75·6c74·2e68·746d·6c23··yRealsMult.html# | 00013af0:·7952·6561·6c73·4d75·6c74·2e68·746d·6c23··yRealsMult.html# | ||
| 00013b00:·783a·3130·3522·3e3c·7370·616e·2063·6c61··x:105"><span·cla | 00013b00:·783a·3130·3522·3e3c·7370·616e·2063·6c61··x:105"><span·cla | ||
| 00013b10:·7373·3d22·6964·2220·7469·746c·653d·2276··ss="id"·title="v | 00013b10:·7373·3d22·6964·2220·7469·746c·653d·2276··ss="id"·title="v | ||
| 00013b20:·6172·6961·626c·6522·3e78·3c2f·7370·616e··ariable">x</span | 00013b20:·6172·6961·626c·6522·3e78·3c2f·7370·616e··ariable">x</span | ||
| 00013b30:·3e3c·2f61·3e2e·3c62·722f·3e0a·0a3c·6272··></a>.<br/>..<br | 00013b30:·3e3c·2f61·3e2e·3c62·722f·3e0a·0a3c·6272··></a>.<br/>..<br | ||
| 00013b40:·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469·7620··/>.</div>..<div· | 00013b40:·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469·7620··/>.</div>..<div· | ||
| 00013b50:·636c·6173·733d·2264·6f63·223e·0a3c·6120··class="doc">.<a· | 00013b50:·636c·6173·733d·2264·6f63·223e·0a3c·6120··class="doc">.<a· | ||
| 00013b60:·6964·3d22·6c61·623 | 00013b60:·6964·3d22·6c61·6236·3134·223e·3c2f·613e··id="lab614"></a> | ||
| 00013b70:·3c68·3120·636c·6173·733d·2273·6563·7469··<h1·class="secti | 00013b70:·3c68·3120·636c·6173·733d·2273·6563·7469··<h1·class="secti | ||
| 00013b80:·6f6e·223e·4669·656c·643c·2f68·313e·0a0a··on">Field</h1>.. | 00013b80:·6f6e·223e·4669·656c·643c·2f68·313e·0a0a··on">Field</h1>.. | ||
| 00013b90:·3c2f·6469·763e·0a3c·6469·7620·636c·6173··</div>.<div·clas | 00013b90:·3c2f·6469·763e·0a3c·6469·7620·636c·6173··</div>.<div·clas | ||
| 00013ba0:·733d·2263·6f64·6522·3e0a·0a3c·6272·2f3e··s="code">..<br/> | 00013ba0:·733d·2263·6f64·6522·3e0a·0a3c·6272·2f3e··s="code">..<br/> | ||
| 00013bb0:·0a3c·7370·616e·2063·6c61·7373·3d22·6964··.<span·class="id | 00013bb0:·0a3c·7370·616e·2063·6c61·7373·3d22·6964··.<span·class="id | ||
| 00013bc0:·2220·7469·746c·653d·226b·6579·776f·7264··"·title="keyword | 00013bc0:·2220·7469·746c·653d·226b·6579·776f·7264··"·title="keyword | ||
| 00013bd0:·223e·4c65·6d6d·613c·2f73·7061·6e3e·203c··">Lemma</span>·< | 00013bd0:·223e·4c65·6d6d·613c·2f73·7061·6e3e·203c··">Lemma</span>·< |
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| 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="lab616"></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="lab583"></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/> |
| Offset 102, 15 lines modified | Offset 102, 15 lines modified | ||
| 00000650:·6e74·223e·0a0a·3c68·3120·636c·6173·733d··nt">..<h1·class= | 00000650:·6e74·223e·0a0a·3c68·3120·636c·6173·733d··nt">..<h1·class= | ||
| 00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra | 00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra | ||
| 00000670:·7279·2043·6f71·2e52·6561·6c73·2e52·496e··ry·Coq.Reals.RIn | 00000670:·7279·2043·6f71·2e52·6561·6c73·2e52·496e··ry·Coq.Reals.RIn | ||
| 00000680:·6571·3c2f·6831·3e0a·0a3c·6469·7620·636c··eq</h1>..<div·cl | 00000680:·6571·3c2f·6831·3e0a·0a3c·6469·7620·636c··eq</h1>..<div·cl | ||
| 00000690:·6173·733d·2263·6f64·6522·3e0a·0a3c·6272··ass="code">..<br | 00000690:·6173·733d·2263·6f64·6522·3e0a·0a3c·6272··ass="code">..<br | ||
| 000006a0:·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469·7620··/>.</div>..<div· | 000006a0:·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469·7620··/>.</div>..<div· | ||
| 000006b0:·636c·6173·733d·2264·6f63·223e·0a3c·6120··class="doc">.<a· | 000006b0:·636c·6173·733d·2264·6f63·223e·0a3c·6120··class="doc">.<a· | ||
| 000006c0:·6964·3d22·6c61·623 | 000006c0:·6964·3d22·6c61·6236·3839·223e·3c2f·613e··id="lab689"></a> | ||
| 000006d0:·3c68·3120·636c·6173·733d·2273·6563·7469··<h1·class="secti | 000006d0:·3c68·3120·636c·6173·733d·2273·6563·7469··<h1·class="secti | ||
| 000006e0:·6f6e·223e·4261·7369·6320·6c65·6d6d·6173··on">Basic·lemmas | 000006e0:·6f6e·223e·4261·7369·6320·6c65·6d6d·6173··on">Basic·lemmas | ||
| 000006f0:·2066·6f72·2074·6865·2063·6c61·7373·6963···for·the·classic | 000006f0:·2066·6f72·2074·6865·2063·6c61·7373·6963···for·the·classic | ||
| 00000700:·616c·2072·6561·6c20·6e75·6d62·6572·733c··al·real·numbers< | 00000700:·616c·2072·6561·6c20·6e75·6d62·6572·733c··al·real·numbers< | ||
| 00000710:·2f68·313e·0a0a·3c2f·6469·763e·0a3c·6469··/h1>..</div>.<di | 00000710:·2f68·313e·0a0a·3c2f·6469·763e·0a3c·6469··/h1>..</div>.<di | ||
| 00000720:·7620·636c·6173·733d·2263·6f64·6522·3e0a··v·class="code">. | 00000720:·7620·636c·6173·733d·2263·6f64·6522·3e0a··v·class="code">. | ||
| 00000730:·0a3c·6272·2f3e·0a3c·7370·616e·2063·6c61··.<br/>.<span·cla | 00000730:·0a3c·6272·2f3e·0a3c·7370·616e·2063·6c61··.<br/>.<span·cla |
| 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">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="lab7 | 56 | <a·id="lab733"></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/> |
| Offset 58, 15 lines modified | Offset 58, 15 lines modified | ||
| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<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.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="lab661"></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/> |
| 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.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="lab651"></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/> |
| Offset 53, 15 lines modified | Offset 53, 15 lines modified | ||
| 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="lab665"></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 56, 15 lines modified | Offset 56, 15 lines modified | ||
| 56 | <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/> | 56 | <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/> |
| 57 | <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 | <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 | <br/> | 58 | <br/> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="doc"> | 60 | <div·class="doc"> |
| 61 | <a·id="lab6 | 61 | <a·id="lab683"></a><h1·class="section">Preliminaries·lemmas</h1> |
| 62 | </div> | 62 | </div> |
| 63 | <div·class="code"> | 63 | <div·class="code"> |
| 64 | <br/> | 64 | <br/> |
| 65 | <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/> | 65 | <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/> |
| 66 | <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/> | 66 | <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 290, 21 lines modified | Offset 290, 21 lines modified | ||
| 00001210:·2220·7469·746c·653d·226b·6579·776f·7264··"·title="keyword | 00001210:·2220·7469·746c·653d·226b·6579·776f·7264··"·title="keyword | ||
| 00001220:·223e·5363·6f70·653c·2f73·7061·6e3e·203c··">Scope</span>·< | 00001220:·223e·5363·6f70·653c·2f73·7061·6e3e·203c··">Scope</span>·< | ||
| 00001230:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"· | 00001230:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"· | ||
| 00001240:·7469·746c·653d·2276·6172·223e·525f·7363··title="var">R_sc | 00001240:·7469·746c·653d·2276·6172·223e·525f·7363··title="var">R_sc | ||
| 00001250:·6f70·653c·2f73·7061·6e3e·2e3c·6272·2f3e··ope</span>.<br/> | 00001250:·6f70·653c·2f73·7061·6e3e·2e3c·6272·2f3e··ope</span>.<br/> | ||
| 00001260:·0a0a·3c62·722f·3e0a·3c2f·6469·763e·0a0a··..<br/>.</div>.. | 00001260:·0a0a·3c62·722f·3e0a·3c2f·6469·763e·0a0a··..<br/>.</div>.. | ||
| 00001270:·3c64·6976·2063·6c61·7373·3d22·646f·6322··<div·class="doc" | 00001270:·3c64·6976·2063·6c61·7373·3d22·646f·6322··<div·class="doc" | ||
| 00001280:·3e0a·3c61·2069·643d·226c·6162·363 | 00001280:·3e0a·3c61·2069·643d·226c·6162·3632·3122··>.<a·id="lab621" | ||
| 00001290:·3e3c·2f61·3e3c·6831·2063·6c61·7373·3d22··></a><h1·class=" | 00001290:·3e3c·2f61·3e3c·6831·2063·6c61·7373·3d22··></a><h1·class=" | ||
| 000012a0:·7365·6374·696f·6e22·3e50·7265·6c69·6d69··section">Prelimi | 000012a0:·7365·6374·696f·6e22·3e50·7265·6c69·6d69··section">Prelimi | ||
| 000012b0:·6e61·7269·6573·3c2f·6831·3e0a·0a3c·6469··naries</h1>..<di | 000012b0:·6e61·7269·6573·3c2f·6831·3e0a·0a3c·6469··naries</h1>..<di | ||
| 000012c0:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra | 000012c0:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra | ||
| 000012d0:·7068·223e·203c·2f64·6976·3e0a·0a3c·6120··ph">·</div>..<a· | 000012d0:·7068·223e·203c·2f64·6976·3e0a·0a3c·6120··ph">·</div>..<a· | ||
| 000012e0:·6964·3d22·6c61·6236·3 | 000012e0:·6964·3d22·6c61·6236·3232·223e·3c2f·613e··id="lab622"></a> | ||
| 000012f0:·3c68·3220·636c·6173·733d·2273·6563·7469··<h2·class="secti | 000012f0:·3c68·3220·636c·6173·733d·2273·6563·7469··<h2·class="secti | ||
| 00001300:·6f6e·223e·5661·7269·6f75·7320·6765·6e65··on">Various·gene | 00001300:·6f6e·223e·5661·7269·6f75·7320·6765·6e65··on">Various·gene | ||
| 00001310:·7269·6320·6c65·6d6d·6173·2077·6869·6368··ric·lemmas·which | 00001310:·7269·6320·6c65·6d6d·6173·2077·6869·6368··ric·lemmas·which | ||
| 00001320:·2070·726f·6261·626c·7920·7368·6f75·6c64···probably·should | 00001320:·2070·726f·6261·626c·7920·7368·6f75·6c64···probably·should | ||
| 00001330:·2067·6f20·736f·6d65·7768·6572·6520·656c···go·somewhere·el | 00001330:·2067·6f20·736f·6d65·7768·6572·6520·656c···go·somewhere·el | ||
| 00001340:·7365·3c2f·6832·3e0a·0a3c·2f64·6976·3e0a··se</h2>..</div>. | 00001340:·7365·3c2f·6832·3e0a·0a3c·2f64·6976·3e0a··se</h2>..</div>. | ||
| 00001350:·3c64·6976·2063·6c61·7373·3d22·636f·6465··<div·class="code | 00001350:·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="lab605"></a><h1·class="section">Field·operations</h1> |
| 71 | <div·class="paragraph">·</div> | 71 | <div·class="paragraph">·</div> |
| 72 | <a·id="lab | 72 | <a·id="lab606"></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="lab658"></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="lab646"></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="lab726"></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="lab577"></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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| 57 | <br/> | 57 | <br/> |
| 58 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> | 58 | <span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/> |
| 59 | <br/> | 59 | <br/> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="doc"> | 61 | <div·class="doc"> |
| 62 | <a·id="lab7 | 62 | <a·id="lab736"></a><h1·class="section">Each·bounded·subset·of·N·has·a·maximal·element</h1> |
| 63 | </div> | 63 | </div> |
| 64 | <div·class="code"> | 64 | <div·class="code"> |
| 65 | <br/> | 65 | <br/> |
| 66 | <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/> | 66 | <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 | <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/> | 67 | <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="lab653"></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="lab7 | 76 | <a·id="lab744"></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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| 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.Reals.Rbase.html#"><span·class="id"·title="library">Rbase</span></a>·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#"><span·class="id"·title="library">Rbasic_fun</span></a>·<a·class="idref"·href="Coq.Reals.ROrderedType.html#"><span·class="id"·title="library">ROrderedType</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.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>·<a·class="idref"·href="Coq.Reals.Rbase.html#"><span·class="id"·title="library">Rbase</span></a>·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#"><span·class="id"·title="library">Rbasic_fun</span></a>·<a·class="idref"·href="Coq.Reals.ROrderedType.html#"><span·class="id"·title="library">ROrderedType</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="lab581"></a><h1·class="section">Maximum·and·Minimum·of·two·real·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">R_scope</span>.<br/> | 57 | <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="lab594"></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="lab663"></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="lab6 | 60 | <a·id="lab648"></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="lab7 | 165 | <a·id="lab741"></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/> |
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| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Rbase.html#"><span·class="id"·title="library">Rbase</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#"><span·class="id"·title="library">Rfunctions</span></a>·<a·class="idref"·href="Coq.Reals.SeqSeries.html#"><span·class="id"·title="library">SeqSeries</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_fun.html#"><span·class="id"·title="library">Rtrigo_fun</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Reals.Rbase.html#"><span·class="id"·title="library">Rbase</span></a>·<a·class="idref"·href="Coq.Reals.Rfunctions.html#"><span·class="id"·title="library">Rfunctions</span></a>·<a·class="idref"·href="Coq.Reals.SeqSeries.html#"><span·class="id"·title="library">SeqSeries</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo_fun.html#"><span·class="id"·title="library">Rtrigo_fun</span></a>.<br/> |
| 50 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 50 | <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 | <br/> | 51 | <br/> |
| 52 | </div> | 52 | </div> |
| 53 | <div·class="doc"> | 53 | <div·class="doc"> |
| 54 | <a·id="lab | 54 | <a·id="lab574"></a><h1·class="section">Definition·of·exponential</h1> |
| 55 | </div> | 55 | </div> |
| 56 | <div·class="code"> | 56 | <div·class="code"> |
| 57 | <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/> | 57 | <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 | <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/> | 58 | <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 | <br/> | 59 | <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="lab599"></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="lab105"></a><h1·class="section">Some·properties·of·the·operators·on·relations</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab106"></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/> |
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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="lab86"></a><h1·class="section">Some·operators·on·relations</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab87"></a><h1·class="section">Initial·authors:·Bruno·Barras,·Cristina·Cornes</h1> |
| 51 | <a·id="lab | 51 | <a·id="lab88"></a><h1·class="section">···································································</h1> |
| 52 | <a·id="lab | 52 | <a·id="lab89"></a><h1·class="section">Some·of·the·initial·definitions·were·taken·from·:</h1> |
| 53 | <a·id="lab | 53 | <a·id="lab90"></a><h1·class="section">Constructing·Recursion·Operators·in·Type·Theory</h1> |
| 54 | <a·id="lab | 54 | <a·id="lab91"></a><h1·class="section">L.·Paulson··JSC·(1986)·2,·325-355</h1> |
| 55 | <a·id="lab | 55 | <a·id="lab92"></a><h1·class="section">···································································</h1> |
| 56 | <a·id="lab | 56 | <a·id="lab93"></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="lab94"></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="lab758"></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="lab747"></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="lab748"></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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| 83 | <br/> | 83 | <br/> |
| 84 | <span·class="id"·title="keyword">Section</span>·<a·id="Permut"·class="idref"·href="#Permut"><span·class="id"·title="section">Permut</span></a>.<br/> | 84 | <span·class="id"·title="keyword">Section</span>·<a·id="Permut"·class="idref"·href="#Permut"><span·class="id"·title="section">Permut</span></a>.<br/> |
| 85 | <br/> | 85 | <br/> |
| 86 | </div> | 86 | </div> |
| 87 | <div·class="doc"> | 87 | <div·class="doc"> |
| 88 | <a·id="lab | 88 | <a·id="lab759"></a><h1·class="section">From·lists·to·multisets</h1> |
| 89 | </div> | 89 | </div> |
| 90 | <div·class="code"> | 90 | <div·class="code"> |
| 91 | <br/> | 91 | <br/> |
| 92 | <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/> | 92 | <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/> |
| 93 | <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/> | 93 | <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="lab757"></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="lab538"></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="lab530"></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"> |
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| 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="lab1252"></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="lab1253"></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="lab1254"></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="lab1238"></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="lab1255"></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="lab1256"></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="lab1264"></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="lab1273"></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="lab1274"></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/> |
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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">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="lab1282"></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="inductive">Compare</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="inductive">Compare</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="lab1272"></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> |
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| 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="lab1251"></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="lab1284"></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="lab1239"></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="lab1240"></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="lab1241"></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="lab1291"></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="lab1245"></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="lab1270"></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/> |
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| 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="lab1266"></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 |
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| 78 | </div> | 78 | </div> |
| 79 | <div·class="doc"> | 79 | <div·class="doc"> |
| 80 | 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 | 80 | 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 |
| 81 | is·true·for·the·one·that·use·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span>· | 81 | is·true·for·the·one·that·use·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span>· |
| 82 | <div·class="paragraph">·</div> | 82 | <div·class="paragraph">·</div> |
| 83 | <a·id="lab | 83 | <a·id="lab1182"></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> |
| 84 | </div> | 84 | </div> |
| 85 | <div·class="code"> | 85 | <div·class="code"> |
| 86 | <br/> | 86 | <br/> |
| 87 | <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:20"·class="idref"·href="#A:20"><span·class="id"·title="binder">A</span></a>·<a·id="n:21"·class="idref"·href="#n:21"><span·class="id"·title="binder">n</span></a>·(<a·id="v1:22"·class="idref"·href="#v1:22"><span·class="id"·title="binder">v1</span></a>·<a·id="v2:23"·class="idref"·href="#v2:23"><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:20"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#n:21"><span·class="id"·title="variable">n</span></a>):<br/> | 87 | <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:20"·class="idref"·href="#A:20"><span·class="id"·title="binder">A</span></a>·<a·id="n:21"·class="idref"·href="#n:21"><span·class="id"·title="binder">n</span></a>·(<a·id="v1:22"·class="idref"·href="#v1:22"><span·class="id"·title="binder">v1</span></a>·<a·id="v2:23"·class="idref"·href="#v2:23"><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:20"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#n:21"><span·class="id"·title="variable">n</span></a>):<br/> |
| 88 | <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="p1:24"·class="idref"·href="#p1:24"><span·class="id"·title="binder">p1</span></a>·<a·id="p2:25"·class="idref"·href="#p2:25"><span·class="id"·title="binder">p2</span></a>,·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p1:24"><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#p2:25"><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#v1:22"><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#p1:24"><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#v2:23"><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#p2:25"><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#v1:22"><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#v2:23"><span·class="id"·title="variable">v2</span></a>.<br/> | 88 | <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="p1:24"·class="idref"·href="#p1:24"><span·class="id"·title="binder">p1</span></a>·<a·id="p2:25"·class="idref"·href="#p2:25"><span·class="id"·title="binder">p2</span></a>,·<a·class="idref"·href="Coq.Vectors.VectorSpec.html#p1:24"><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#p2:25"><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#v1:22"><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#p1:24"><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#v2:23"><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#p2:25"><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#v1:22"><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#v2:23"><span·class="id"·title="variable">v2</span></a>.<br/> |
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| 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>.<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>.<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="lab375"></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> |
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| 54 | <br/> | 54 | <br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab443"></a><h1·class="section">Binary·Integers,·Definitions·of·Operations</h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·Initial·author:·Pierre·Crégut,·CNET,·Lannion,·France· | 60 | ·Initial·author:·Pierre·Crégut,·CNET,·Lannion,·France· |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <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="lab433"></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="lab421"></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="lab440"></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/> |
| 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="lab423"></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/> |
| Offset 64, 30 lines modified | Offset 64, 30 lines modified | ||
| 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="lab425"></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="lab426"></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="lab472"></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"> |
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| 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="lab489"></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. |
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| 169 | <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/> | 169 | <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/> |
| 170 | <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/> | 170 | <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/> |
| 171 | <br/> | 171 | <br/> |
| 172 | </div> | 172 | </div> |
| 173 | <div·class="doc"> | 173 | <div·class="doc"> |
| 174 | <a·id="lab | 174 | <a·id="lab471"></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> |
| 175 | ··and·<span·class="inlinecode"><span·class="id"·title="var">Zodd</span></span>· | 175 | ··and·<span·class="inlinecode"><span·class="id"·title="var">Zodd</span></span>· |
| 176 | <div·class="paragraph">·</div> | 176 | <div·class="paragraph">·</div> |
| 177 | ·Properties·of·<span·class="inlinecode"><span·class="id"·title="var">Z.div2</span></span>· | 177 | ·Properties·of·<span·class="inlinecode"><span·class="id"·title="var">Z.div2</span></span>· |
| 178 | </div> | 178 | </div> |
| 179 | <div·class="code"> | 179 | <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="lab488"></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 |
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| 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="lab418"></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="lab419"></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="lab437"></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 238, 15 lines modified | Offset 238, 15 lines modified | ||
| 238 | <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/> | 238 | <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/> |
| 239 | 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/> | 239 | 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/> |
| 240 | <br/> | 240 | <br/> |
| 241 | </div> | 241 | </div> |
| 242 | <div·class="doc"> | 242 | <div·class="doc"> |
| 243 | <a·id="lab | 243 | <a·id="lab493"></a><h1·class="section">Greatest·common·divisor·(gcd).</h1> |
| 244 | <div·class="paragraph">·</div> | 244 | <div·class="paragraph">·</div> |
| 245 | ·There·is·no·unicity·of·the·gcd;·hence·we·define·the·predicate | 245 | ·There·is·no·unicity·of·the·gcd;·hence·we·define·the·predicate |
| 246 | ····<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>. | 246 | ····<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>. |
| 247 | ····(We·show·later·that·the·<span·class="inlinecode"><span·class="id"·title="var">gcd</span></span>·is·actually·unique·if·we·discard·its·sign.)· | 247 | ····(We·show·later·that·the·<span·class="inlinecode"><span·class="id"·title="var">gcd</span></span>·is·actually·unique·if·we·discard·its·sign.)· |
| 248 | </div> | 248 | </div> |
| Offset 67, 30 lines modified | Offset 67, 30 lines modified | ||
| 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="lab365"></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="lab366"></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="lab492"></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="lab490"></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="lab415"></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: |
| Offset 149, 15 lines modified | Offset 149, 15 lines modified | ||
| 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="lab483"></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/> |
| Offset 64, 15 lines modified | Offset 64, 15 lines modified | ||
| 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="lab482"></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="lab | 145 | <a·id="lab837"></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"> |
| 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="lab822"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab823"></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="lab824"></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="lab828"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab829"></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="lab830"></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/> |
| Offset 43, 23 lines modified | Offset 43, 23 lines modified | ||
| 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="lab812"></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="lab813"></a><h1·class="section">into·equivalent·versions·in·Haskell's·Prelude·that·are·defined</h1> |
| 49 | <a·id="lab | 49 | <a·id="lab814"></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="lab815"></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="lab816"></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/> |
| Offset 43, 15 lines modified | Offset 43, 15 lines modified | ||
| 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="lab831"></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/> |
| 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.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="lab825"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab826"></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="lab827"></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/> |
| Offset 43, 23 lines modified | Offset 43, 23 lines modified | ||
| 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="lab817"></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="lab818"></a><h1·class="section">into·equivalent·versions·in·Haskell's·Prelude·that·are·defined</h1> |
| 49 | <a·id="lab | 49 | <a·id="lab819"></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="lab820"></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="lab821"></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/> |
| 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">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="lab775"></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 |
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| 000191d0:·2f3e·0a3c·6120·6872·6566·3d22·436f·712e··/>.<a·href="Coq. | 000191d0:·2f3e·0a3c·6120·6872·6566·3d22·436f·712e··/>.<a·href="Coq. | ||
| 000191e0:·496e·6974·2e42·7974·652e·6874·6d6c·2362··Init.Byte.html#b | 000191e0:·496e·6974·2e42·7974·652e·6874·6d6c·2362··Init.Byte.html#b | ||
| 000191f0:·7974·6522·3e62·7974·653c·2f61·3e20·5b69··yte">byte</a>·[i | 000191f0:·7974·6522·3e62·7974·653c·2f61·3e20·5b69··yte">byte</a>·[i | ||
| 00019200:·6e64·7563·7469·7665·2c20·696e·203c·6120··nductive,·in·<a· | 00019200:·6e64·7563·7469·7665·2c20·696e·203c·6120··nductive,·in·<a· | ||
| 00019210:·6872·6566·3d22·436f·712e·496e·6974·2e42··href="Coq.Init.B | 00019210:·6872·6566·3d22·436f·712e·496e·6974·2e42··href="Coq.Init.B | ||
| 00019220:·7974·652e·6874·6d6c·223e·436f·712e·496e··yte.html">Coq.In | 00019220:·7974·652e·6874·6d6c·223e·436f·712e·496e··yte.html">Coq.In | ||
| 00019230:·6974·2e42·7974·653c·2f61·3e5d·3c62·722f··it.Byte</a>]<br/ | 00019230:·6974·2e42·7974·653c·2f61·3e5d·3c62·722f··it.Byte</a>]<br/ | ||
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| 558 | <hr/><a·id="global_T"></a><h2>T·</h2> | 558 | <hr/><a·id="global_T"></a><h2>T·</h2> |
| 559 | <a·href="Coq.Numbers.Cyclic.ZModulo.ZModulo.html#t">t</a>·[definition,·in·<a·href="Coq.Numbers.Cyclic.ZModulo.ZModulo.html">Coq.Numbers.Cyclic.ZModulo.ZModulo</a>]<br/> | 559 | <a·href="Coq.Numbers.Cyclic.ZModulo.ZModulo.html#t">t</a>·[definition,·in·<a·href="Coq.Numbers.Cyclic.ZModulo.ZModulo.html">Coq.Numbers.Cyclic.ZModulo.ZModulo</a>]<br/> |
| 560 | <a·href="Coq.Init.Nat.html#t">t</a>·[definition,·in·<a·href="Coq.Init.Nat.html">Coq.Init.Nat</a>]<br/> | 560 | <a·href="Coq.Init.Nat.html#t">t</a>·[definition,·in·<a·href="Coq.Init.Nat.html">Coq.Init.Nat</a>]<br/> |
| 561 | <a·href="Coq.Vectors.Fin.html#t">t</a>·[inductive,·in·<a·href="Coq.Vectors.Fin.html">Coq.Vectors.Fin</a>]<br/> | 561 | <a·href="Coq.Vectors.Fin.html#t">t</a>·[inductive,·in·<a·href="Coq.Vectors.Fin.html">Coq.Vectors.Fin</a>]<br/> |
| 562 | <a·href="Coq.micromega.VarMap.html#t">t</a>·[abbreviation,·in·<a·href="Coq.micromega.VarMap.html">Coq.micromega.VarMap</a>]<br/> | 562 | <a·href="Coq.micromega.VarMap.html#t">t</a>·[abbreviation,·in·<a·href="Coq.micromega.VarMap.html">Coq.micromega.VarMap</a>]<br/> |
| 563 | <a·href="Coq.micromega.VarMap.html#t">t</a>·[inductive,·in·<a·href="Coq.micromega.VarMap.html">Coq.micromega.VarMap</a>]<br/> | 563 | <a·href="Coq.micromega.VarMap.html#t">t</a>·[inductive,·in·<a·href="Coq.micromega.VarMap.html">Coq.micromega.VarMap</a>]<br/> |
| 564 | <a·href="Coq.Vectors.VectorDef.html#t">t</a>·[inductive,·in·<a·href="Coq.Vectors.VectorDef.html">Coq.Vectors.VectorDef</a>]<br/> | 564 | <a·href="Coq.Vectors.VectorDef.html#t">t</a>·[inductive,·in·<a·href="Coq.Vectors.VectorDef.html">Coq.Vectors.VectorDef</a>]<br/> |
| 565 | <a·href="Coq.Init.Tactics.html">Tactics</a>·[library]<br/> | ||
| 566 | <a·href="Coq.Program.Tactics.html">Tactics</a>·[library]<br/> | 565 | <a·href="Coq.Program.Tactics.html">Tactics</a>·[library]<br/> |
| 566 | <a·href="Coq.Init.Tactics.html">Tactics</a>·[library]<br/> | ||
| 567 | <a·href="Coq.ssr.ssreflect.html#tactic_view">tactic_view</a>·[constructor,·in·<a·href="Coq.ssr.ssreflect.html">Coq.ssr.ssreflect</a>]<br/> | 567 | <a·href="Coq.ssr.ssreflect.html#tactic_view">tactic_view</a>·[constructor,·in·<a·href="Coq.ssr.ssreflect.html">Coq.ssr.ssreflect</a>]<br/> |
| 568 | <a·href="Coq.rtauto.Bintree.html#Tadd">Tadd</a>·[definition,·in·<a·href="Coq.rtauto.Bintree.html">Coq.rtauto.Bintree</a>]<br/> | 568 | <a·href="Coq.rtauto.Bintree.html#Tadd">Tadd</a>·[definition,·in·<a·href="Coq.rtauto.Bintree.html">Coq.rtauto.Bintree</a>]<br/> |
| 569 | <a·href="Coq.ssr.ssrfun.html#tag">tag</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> | 569 | <a·href="Coq.ssr.ssrfun.html#tag">tag</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> |
| 570 | <a·href="Coq.ssr.ssrfun.html#Tag">Tag</a>·[section,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> | 570 | <a·href="Coq.ssr.ssrfun.html#Tag">Tag</a>·[section,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> |
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| 572 | <a·href="Coq.ssr.ssrfun.html#tagged">tagged</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> | 572 | <a·href="Coq.ssr.ssrfun.html#tagged">tagged</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> |
| 573 | <a·href="Coq.ssr.ssrfun.html#Tagged2">Tagged2</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> | 573 | <a·href="Coq.ssr.ssrfun.html#Tagged2">Tagged2</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> |
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| 984 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties.s''">WEqProperties_fun.BasicProperties.s''</a>·[variable,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 984 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties.s''">WEqProperties_fun.BasicProperties.s''</a>·[variable,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 985 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties.s'">WEqProperties_fun.BasicProperties.s'</a>·[variable,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 985 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties.s'">WEqProperties_fun.BasicProperties.s'</a>·[variable,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 986 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties.s">WEqProperties_fun.BasicProperties.s</a>·[variable,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 986 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties.s">WEqProperties_fun.BasicProperties.s</a>·[variable,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 987 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties">WEqProperties_fun.BasicProperties</a>·[section,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 987 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.BasicProperties">WEqProperties_fun.BasicProperties</a>·[section,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 988 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.Add">WEqProperties_fun.Add</a>·[definition,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 988 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.Add">WEqProperties_fun.Add</a>·[definition,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 989 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.MP">WEqProperties_fun.MP</a>·[module,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 989 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun.MP">WEqProperties_fun.MP</a>·[module,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 990 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun">WEqProperties_fun</a>·[module,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> | 990 | <a·href="Coq.FSets.FSetEqProperties.html#WEqProperties_fun">WEqProperties_fun</a>·[module,·in·<a·href="Coq.FSets.FSetEqProperties.html">Coq.FSets.FSetEqProperties</a>]<br/> |
| 991 | <a·href="Coq.Init.Wf.html">Wf</a>·[library]<br/> | ||
| 992 | <a·href="Coq.Program.Wf.html">Wf</a>·[library]<br/> | 991 | <a·href="Coq.Program.Wf.html">Wf</a>·[library]<br/> |
| 992 | <a·href="Coq.Init.Wf.html">Wf</a>·[library]<br/> | ||
| 993 | <a·href="Coq.FSets.FMapFacts.html#WFacts">WFacts</a>·[module,·in·<a·href="Coq.FSets.FMapFacts.html">Coq.FSets.FMapFacts</a>]<br/> | 993 | <a·href="Coq.FSets.FMapFacts.html#WFacts">WFacts</a>·[module,·in·<a·href="Coq.FSets.FMapFacts.html">Coq.FSets.FMapFacts</a>]<br/> |
| 994 | <a·href="Coq.MSets.MSetFacts.html#WFacts">WFacts</a>·[module,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> | 994 | <a·href="Coq.MSets.MSetFacts.html#WFacts">WFacts</a>·[module,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> |
| 995 | <a·href="Coq.FSets.FSetFacts.html#WFacts">WFacts</a>·[module,·in·<a·href="Coq.FSets.FSetFacts.html">Coq.FSets.FSetFacts</a>]<br/> | 995 | <a·href="Coq.FSets.FSetFacts.html#WFacts">WFacts</a>·[module,·in·<a·href="Coq.FSets.FSetFacts.html">Coq.FSets.FSetFacts</a>]<br/> |
| 996 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn">WFactsOn</a>·[module,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> | 996 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn">WFactsOn</a>·[module,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> |
| 997 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn.add_s_m">WFactsOn.add_s_m</a>·[instance,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> | 997 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn.add_s_m">WFactsOn.add_s_m</a>·[instance,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> |
| 998 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn.add_m">WFactsOn.add_m</a>·[instance,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> | 998 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn.add_m">WFactsOn.add_m</a>·[instance,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> |
| 999 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn.add_neq_b">WFactsOn.add_neq_b</a>·[lemma,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> | 999 | <a·href="Coq.MSets.MSetFacts.html#WFactsOn.add_neq_b">WFactsOn.add_neq_b</a>·[lemma,·in·<a·href="Coq.MSets.MSetFacts.html">Coq.MSets.MSetFacts</a>]<br/> |
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| 573 | <a·href="Ltac2.Bool.html">Bool</a>·<br/> | 573 | <a·href="Ltac2.Bool.html">Bool</a>·<br/> |
| 574 | <a·href="Coq.Bool.Bool.html">Bool</a>·<br/> | 574 | <a·href="Coq.Bool.Bool.html">Bool</a>·<br/> |
| 575 | <a·href="Coq.Bool.BoolEq.html">BoolEq</a>·<br/> | 575 | <a·href="Coq.Bool.BoolEq.html">BoolEq</a>·<br/> |
| 576 | <a·href="Coq.Bool.BoolOrder.html">BoolOrder</a>·<br/> | 576 | <a·href="Coq.Bool.BoolOrder.html">BoolOrder</a>·<br/> |
| 577 | <a·href="Coq.Arith.Bool_nat.html">Bool_nat</a>·<br/> | 577 | <a·href="Coq.Arith.Bool_nat.html">Bool_nat</a>·<br/> |
| 578 | <a·href="Coq.btauto.Btauto.html">Btauto</a>·<br/> | 578 | <a·href="Coq.btauto.Btauto.html">Btauto</a>·<br/> |
| 579 | <a·href="Coq.Bool.Bvector.html">Bvector</a>·<br/> | 579 | <a·href="Coq.Bool.Bvector.html">Bvector</a>·<br/> |
| 580 | <a·href="Coq.Init.Byte.html">Byte</a>·<br/> | ||
| 581 | <a·href="Coq.Strings.Byte.html">Byte</a>·<br/> | 580 | <a·href="Coq.Strings.Byte.html">Byte</a>·<br/> |
| 581 | <a·href="Coq.Init.Byte.html">Byte</a>·<br/> | ||
| 582 | <a·href="Coq.Strings.ByteVector.html">ByteVector</a>·<br/> | 582 | <a·href="Coq.Strings.ByteVector.html">ByteVector</a>·<br/> |
| 583 | <br/><br/>····<div·id="sidebarWrapper"> | 583 | <br/><br/>····<div·id="sidebarWrapper"> |
| 584 | ······<div·id="sidebar"> | 584 | ······<div·id="sidebar"> |
| 585 | ········<div·class="block"> | 585 | ········<div·class="block"> |
| 586 | ··········<h2·class="title">Navigation</h2> | 586 | ··········<h2·class="title">Navigation</h2> |
| 587 | ··········<div·class="content"> | 587 | ··········<div·class="content"> |
| 588 | ············<ul·class="menu"> | 588 | ············<ul·class="menu"> |
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| 552 | <td><a·href="index_record_Z.html">Z</a></td> | 552 | <td><a·href="index_record_Z.html">Z</a></td> |
| 553 | <td>_</td> | 553 | <td>_</td> |
| 554 | <td>other</td> | 554 | <td>other</td> |
| 555 | <td>(166·entries)</td> | 555 | <td>(166·entries)</td> |
| 556 | </tr> | 556 | </tr> |
| 557 | </table> | 557 | </table> |
| 558 | <hr/><a·id="library_T"></a><h2>T·(library)</h2> | 558 | <hr/><a·id="library_T"></a><h2>T·(library)</h2> |
| 559 | <a·href="Coq.Init.Tactics.html">Tactics</a>·<br/> | ||
| 560 | <a·href="Coq.Program.Tactics.html">Tactics</a>·<br/> | 559 | <a·href="Coq.Program.Tactics.html">Tactics</a>·<br/> |
| 560 | <a·href="Coq.Init.Tactics.html">Tactics</a>·<br/> | ||
| 561 | <a·href="Coq.micromega.Tauto.html">Tauto</a>·<br/> | 561 | <a·href="Coq.micromega.Tauto.html">Tauto</a>·<br/> |
| 562 | <a·href="Coq.Init.Tauto.html">Tauto</a>·<br/> | 562 | <a·href="Coq.Init.Tauto.html">Tauto</a>·<br/> |
| 563 | <a·href="Coq.Wellfounded.Transitive_Closure.html">Transitive_Closure</a>·<br/> | 563 | <a·href="Coq.Wellfounded.Transitive_Closure.html">Transitive_Closure</a>·<br/> |
| 564 | <br/><br/>····<div·id="sidebarWrapper"> | 564 | <br/><br/>····<div·id="sidebarWrapper"> |
| 565 | ······<div·id="sidebar"> | 565 | ······<div·id="sidebar"> |
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| 567 | ··········<h2·class="title">Navigation</h2> | 567 | ··········<h2·class="title">Navigation</h2> |
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| 555 | <td>(166·entries)</td> | 555 | <td>(166·entries)</td> |
| 556 | </tr> | 556 | </tr> |
| 557 | </table> | 557 | </table> |
| 558 | <hr/><a·id="library_W"></a><h2>W·(library)</h2> | 558 | <hr/><a·id="library_W"></a><h2>W·(library)</h2> |
| 559 | <a·href="Coq.Logic.WeakFan.html">WeakFan</a>·<br/> | 559 | <a·href="Coq.Logic.WeakFan.html">WeakFan</a>·<br/> |
| 560 | <a·href="Coq.Wellfounded.Wellfounded.html">Wellfounded</a>·<br/> | 560 | <a·href="Coq.Wellfounded.Wellfounded.html">Wellfounded</a>·<br/> |
| 561 | <a·href="Coq.Wellfounded.Well_Ordering.html">Well_Ordering</a>·<br/> | 561 | <a·href="Coq.Wellfounded.Well_Ordering.html">Well_Ordering</a>·<br/> |
| 562 | <a·href="Coq.Init.Wf.html">Wf</a>·<br/> | ||
| 563 | <a·href="Coq.Program.Wf.html">Wf</a>·<br/> | 562 | <a·href="Coq.Program.Wf.html">Wf</a>·<br/> |
| 563 | <a·href="Coq.Init.Wf.html">Wf</a>·<br/> | ||
| 564 | <a·href="Coq.Arith.Wf_nat.html">Wf_nat</a>·<br/> | 564 | <a·href="Coq.Arith.Wf_nat.html">Wf_nat</a>·<br/> |
| 565 | <a·href="Coq.ZArith.Wf_Z.html">Wf_Z</a>·<br/> | 565 | <a·href="Coq.ZArith.Wf_Z.html">Wf_Z</a>·<br/> |
| 566 | <a·href="Coq.Logic.WKL.html">WKL</a>·<br/> | 566 | <a·href="Coq.Logic.WKL.html">WKL</a>·<br/> |
| 567 | <br/><br/>····<div·id="sidebarWrapper"> | 567 | <br/><br/>····<div·id="sidebarWrapper"> |
| 568 | ······<div·id="sidebar"> | 568 | ······<div·id="sidebar"> |
| 569 | ········<div·class="block"> | 569 | ········<div·class="block"> |
| 570 | ··········<h2·class="title">Navigation</h2> | 570 | ··········<h2·class="title">Navigation</h2> |