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| 1 | ·768cbb51a9f31d343284853419be757d·14760844·debug·optional·coq-dbgsym_8.16.1+dfsg-1_i386.deb | 1 | ·768cbb51a9f31d343284853419be757d·14760844·debug·optional·coq-dbgsym_8.16.1+dfsg-1_i386.deb |
| 2 | · | 2 | ·e70be6d56e1f808b88b9018562ed370d·87499280·math·optional·coq_8.16.1+dfsg-1_i386.deb |
| 3 | ·9ae026b33b639db1f9c7a42694628b45·696488·debug·optional·coqide-dbgsym_8.16.1+dfsg-1_i386.deb | 3 | ·9ae026b33b639db1f9c7a42694628b45·696488·debug·optional·coqide-dbgsym_8.16.1+dfsg-1_i386.deb |
| 4 | ·768423573eb0a08ace5001af93b90c0e·1807228·math·optional·coqide_8.16.1+dfsg-1_i386.deb | 4 | ·768423573eb0a08ace5001af93b90c0e·1807228·math·optional·coqide_8.16.1+dfsg-1_i386.deb |
| 5 | ·66efd96ed3610617941fdfd0fbcc9398·2105500·debug·optional·libcoq-core-ocaml-dbgsym_8.16.1+dfsg-1_i386.deb | 5 | ·66efd96ed3610617941fdfd0fbcc9398·2105500·debug·optional·libcoq-core-ocaml-dbgsym_8.16.1+dfsg-1_i386.deb |
| 6 | ·cb | 6 | ·fcb1d563d8059176a9a21044d1709c46·42448608·ocaml·optional·libcoq-core-ocaml-dev_8.16.1+dfsg-1_i386.deb |
| 7 | ·a60ee628322b58f1e8ce0d1e55f73096·21742816·ocaml·optional·libcoq-core-ocaml_8.16.1+dfsg-1_i386.deb | ||
| 8 | · | 7 | ·6e76ecbdb22e1a71051982a06f524e82·21743912·ocaml·optional·libcoq-core-ocaml_8.16.1+dfsg-1_i386.deb |
| 8 | ·4bd6c19a5b5754a0e7943012f8529d7d·21527100·math·optional·libcoq-stdlib_8.16.1+dfsg-1_i386.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········0·8749 | 3 | -rw-r--r--···0········0········0·87496760·2022-11-29·08:28:08.000000·data.tar.xz |
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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)···931448·2022-11-29·08:28:08.000000·./usr/bin/coq-tex | 7 | -rwxr-xr-x···0·root·········(0)·root·········(0)···931448·2022-11-29·08:28:08.000000·./usr/bin/coq-tex |
| 8 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1795528·2022-11-29·08:28:08.000000·./usr/bin/coq_makefile | 8 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1795528·2022-11-29·08:28:08.000000·./usr/bin/coq_makefile |
| 9 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164264·2022-11-29·08:28:08.000000·./usr/bin/coqc | 9 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164264·2022-11-29·08:28:08.000000·./usr/bin/coqc |
| 10 | -rwxr-xr-x···0·root·········(0)·root·········(0)·400651 | 10 | -rwxr-xr-x···0·root·········(0)·root·········(0)·40065126·2022-11-29·08:28:08.000000·./usr/bin/coqc.byte |
| 11 | -rwxr-xr-x···0·root·········(0)·root·········(0)··6244264·2022-11-29·08:28:08.000000·./usr/bin/coqchk | 11 | -rwxr-xr-x···0·root·········(0)·root·········(0)··6244264·2022-11-29·08:28:08.000000·./usr/bin/coqchk |
| 12 | -rwxr-xr-x···0·root·········(0)·root·········(0)··2500840·2022-11-29·08:28:08.000000·./usr/bin/coqdep | 12 | -rwxr-xr-x···0·root·········(0)·root·········(0)··2500840·2022-11-29·08:28:08.000000·./usr/bin/coqdep |
| 13 | -rwxr-xr-x···0·root·········(0)·root·········(0)··2169480·2022-11-29·08:28:08.000000·./usr/bin/coqdoc | 13 | -rwxr-xr-x···0·root·········(0)·root·········(0)··2169480·2022-11-29·08:28:08.000000·./usr/bin/coqdoc |
| 14 | -rwxr-xr-x···0·root·········(0)·root·········(0)·40628555·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.byte | 14 | -rwxr-xr-x···0·root·········(0)·root·········(0)·40628555·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.byte |
| 15 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17182760·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.opt | 15 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17182760·2022-11-29·08:28:08.000000·./usr/bin/coqidetop.opt |
| 16 | -rwxr-xr-x···0·root·········(0)·root·········(0)··6425544·2022-11-29·08:28:08.000000·./usr/bin/coqnative | 16 | -rwxr-xr-x···0·root·········(0)·root·········(0)··6425544·2022-11-29·08:28:08.000000·./usr/bin/coqnative |
| 17 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1157864·2022-11-29·08:28:08.000000·./usr/bin/coqpp | 17 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1157864·2022-11-29·08:28:08.000000·./usr/bin/coqpp |
| 18 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164584·2022-11-29·08:28:08.000000·./usr/bin/coqproofworker.opt | 18 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164584·2022-11-29·08:28:08.000000·./usr/bin/coqproofworker.opt |
| 19 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164584·2022-11-29·08:28:08.000000·./usr/bin/coqqueryworker.opt | 19 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164584·2022-11-29·08:28:08.000000·./usr/bin/coqqueryworker.opt |
| 20 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164616·2022-11-29·08:28:08.000000·./usr/bin/coqtacticworker.opt | 20 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164616·2022-11-29·08:28:08.000000·./usr/bin/coqtacticworker.opt |
| 21 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164360·2022-11-29·08:28:08.000000·./usr/bin/coqtop | 21 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164360·2022-11-29·08:28:08.000000·./usr/bin/coqtop |
| 22 | -rwxr-xr-x···0·root·········(0)·root·········(0)·575886 | 22 | -rwxr-xr-x···0·root·········(0)·root·········(0)·57588641·2022-11-29·08:28:08.000000·./usr/bin/coqtop.byte |
| 23 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164360·2022-11-29·08:28:08.000000·./usr/bin/coqtop.opt | 23 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17164360·2022-11-29·08:28:08.000000·./usr/bin/coqtop.opt |
| 24 | -rwxr-xr-x···0·root·········(0)·root·········(0)···750824·2022-11-29·08:28:08.000000·./usr/bin/coqwc | 24 | -rwxr-xr-x···0·root·········(0)·root·········(0)···750824·2022-11-29·08:28:08.000000·./usr/bin/coqwc |
| 25 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17077512·2022-11-29·08:28:08.000000·./usr/bin/coqworkmgr | 25 | -rwxr-xr-x···0·root·········(0)·root·········(0)·17077512·2022-11-29·08:28:08.000000·./usr/bin/coqworkmgr |
| 26 | -rwxr-xr-x···0·root·········(0)·root·········(0)·19908424·2022-11-29·08:28:08.000000·./usr/bin/csdpcert | 26 | -rwxr-xr-x···0·root·········(0)·root·········(0)·19908424·2022-11-29·08:28:08.000000·./usr/bin/csdpcert |
| 27 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1120552·2022-11-29·08:28:08.000000·./usr/bin/ocamllibdep | 27 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1120552·2022-11-29·08:28:08.000000·./usr/bin/ocamllibdep |
| 28 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1247048·2022-11-29·08:28:08.000000·./usr/bin/votour | 28 | -rwxr-xr-x···0·root·········(0)·root·········(0)··1247048·2022-11-29·08:28:08.000000·./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/ |
| Offset 2270214, 233860 lines modified | Offset 2270214, 233858 lines modified | ||
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| 159 | -rw-r--r--···0·root·········(0)·root·········(0)·····5358·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.cmi | 159 | -rw-r--r--···0·root·········(0)·root·········(0)·····5358·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.cmi |
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| 162 | -rw-r--r--···0·root·········(0)·root·········(0)······205·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.cmx | 162 | -rw-r--r--···0·root·········(0)·root·········(0)······205·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.cmx |
| 163 | -rw-r--r--···0·root·········(0)·root·········(0)·····5735·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.ml | 163 | -rw-r--r--···0·root·········(0)·root·········(0)·····5735·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.ml |
| 164 | -rw-r--r--···0·root·········(0)·root·········(0)·····5823·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.mli | 164 | -rw-r--r--···0·root·········(0)·root·········(0)·····5823·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/option.mli |
| 165 | -rw-r--r--···0·root·········(0)·root·········(0)······913·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmi | 165 | -rw-r--r--···0·root·········(0)·root·········(0)······913·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmi |
| 166 | -rw-r--r--···0·root·········(0)·root·········(0)····12026·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmt | 166 | -rw-r--r--···0·root·········(0)·root·········(0)····12026·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmt |
| 167 | -rw-r--r--···0·root·········(0)·root·········(0)·····64 | 167 | -rw-r--r--···0·root·········(0)·root·········(0)·····6416·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmti |
| 168 | -rw-r--r--···0·root·········(0)·root·········(0)······347·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmx | 168 | -rw-r--r--···0·root·········(0)·root·········(0)······347·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.cmx |
| 169 | -rw-r--r--···0·root·········(0)·root·········(0)·····1160·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.ml | 169 | -rw-r--r--···0·root·········(0)·root·········(0)·····1160·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.ml |
| 170 | -rw-r--r--···0·root·········(0)·root·········(0)······848·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.mli | 170 | -rw-r--r--···0·root·········(0)·root·········(0)······848·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/orderedType.mli |
| 171 | -rw-r--r--···0·root·········(0)·root·········(0)·····3581·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmi | 171 | -rw-r--r--···0·root·········(0)·root·········(0)·····3581·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmi |
| 172 | -rw-r--r--···0·root·········(0)·root·········(0)····484 | 172 | -rw-r--r--···0·root·········(0)·root·········(0)····48447·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmt |
| 173 | -rw-r--r--···0·root·········(0)·root·········(0)····19068·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmti | 173 | -rw-r--r--···0·root·········(0)·root·········(0)····19068·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmti |
| 174 | -rw-r--r--···0·root·········(0)·root·········(0)······272·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmx | 174 | -rw-r--r--···0·root·········(0)·root·········(0)······272·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.cmx |
| 175 | -rw-r--r--···0·root·········(0)·root·········(0)·····3830·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.ml | 175 | -rw-r--r--···0·root·········(0)·root·········(0)·····3830·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.ml |
| 176 | -rw-r--r--···0·root·········(0)·root·········(0)·····2744·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.mli | 176 | -rw-r--r--···0·root·········(0)·root·········(0)·····2744·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/predicate.mli |
| 177 | -rw-r--r--···0·root·········(0)·root·········(0)·····1534·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/range.cmi | 177 | -rw-r--r--···0·root·········(0)·root·········(0)·····1534·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/range.cmi |
| 178 | -rw-r--r--···0·root·········(0)·root·········(0)····35912·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/range.cmt | 178 | -rw-r--r--···0·root·········(0)·root·········(0)····35912·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/range.cmt |
| 179 | -rw-r--r--···0·root·········(0)·root·········(0)····10005·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/range.cmti | 179 | -rw-r--r--···0·root·········(0)·root·········(0)····10005·2022-11-29·08:28:08.000000·./usr/lib/ocaml/coq-core/clib/range.cmti |
| Offset 1, 2799 lines modified | Offset 1, 2798 lines modified | ||
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| 00001020:·0 | 00001020:·0145·4d01·02c1·0102·c1c0·0501·464d·0102··.EM.........FM.. | ||
| 00001030:· | 00001030:·c101·02dc·4040·4040·a105·015d·4040·4041··....@@@@...]@@@A | ||
| 00001040:· | 00001040:·90d0·b304·09a0·9026·7374·7269·6e67·b0c0··.......&string.. | ||
| 00001050:· | 00001050:·0501·4f4d·0102·c101·02d6·040a·4040·0410··..OM........@@.. | ||
| 00001060:·0 | 00001060:·0800·0030·0005·014d·0501·4d05·014d·0501··...0...M..M..M.. | ||
| 00001070:· | 00001070:·4d05·014d·0501·4d05·014d·0501·4d40·b105··M..M..M..M..M@.. | ||
| 00001080:·0 | 00001080:·014d·041a·0800·0038·0040·4040·4190·c096··.M.....8.@@@A... | ||
| 00001090:· | 00001090:·c0b3·9004·2140·9040·4240·0103·1742·4001··....!@.@B@...B@. | ||
| 000010a0:·404 | 000010a0:·0315·4040·4040·0416·4040·4004·1340·41b0··..@@@@..@@@..@A. | ||
| 000010b0:· | 000010b0:·040c·0415·4040·0417·4040·0501·58b0·0417··....@@..@@..X... | ||
| Max diff block lines reached; 23171/106409 bytes (21.78%) of diff not shown. | |||
| Offset 1, 3883 lines modified | Offset 1, 3882 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0000·f28 | 00000010:·0000·f280·0000·2c76·0000·b5f6·0000·b423··......,v.......# | ||
| 00000020:·0800·0034·0025·4469·6666·3291·b0a0·b0a3··...4.%Diff2..... | 00000020:·0800·0034·0025·4469·6666·3291·b0a0·b0a3··...4.%Diff2..... | ||
| 00000030:·41a0·0800·0028·00b1·2663·6f6d·6d6f·6e00··A....(..&common. | 00000030:·41a0·0800·0028·00b1·2663·6f6d·6d6f·6e00··A....(..&common. | ||
| 00000040:·5141·a004·02b0·c02d·636c·6962·2f64·6966··QA.....-clib/dif | 00000040:·5141·a004·02b0·c02d·636c·6962·2f64·6966··QA.....-clib/dif | ||
| 00000050:·6632·2e6d·6c58·0104·b301·04bb·c004·0258··f2.mlX.........X | 00000050:·6632·2e6d·6c58·0104·b301·04bb·c004·0258··f2.mlX.........X | ||
| 00000060:·0104·b301·04c1·40a0·a0d0·9021·61c0·9090··......@....!a... | 00000060:·0104·b301·04c1·40a0·a0d0·9021·61c0·9090··......@....!a... | ||
| 00000070:·0404·0205·f5e1·0040·0103·1708·0000·3000··.......@......0. | 00000070:·0404·0205·f5e1·0040·0103·1708·0000·3000··.......@......0. | Diff chunk too large, falling back to line-by-line diff (1310 lines added, 1310 lines removed) | |
| 00000080:·a040·4004·0104·0104·0104·0104·0104·0104··.@@............. | 00000080:·a040·4004·0104·0104·0104·0104·0104·0104··.@@............. | ||
| 00000090:·0140·b1ab·a | 00000090:·0140·b1ab·a7b1·b2b2·b2b2·b2b2·b2b2·b2b2··.@.............. | ||
| 000000a0:·b2b2·b | 000000a0:·b2b2·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1··................ | ||
| 000000b0:·b1b1· | 000000b0:·b1b1·40a3·2a66·6c6f·6174·6172·7261·7951··..@.*floatarrayQ | ||
| 000000c0:· | 000000c0:·0800·0038·0040·4040·4140·4040·4040·b0c0··...8.@@@A@@@@@.. | ||
| 000000d0:· | 000000d0:·265f·6e6f·6e65·5f40·4000·ff04·0241·4040··&_none_@@....A@@ | ||
| 000000e0:·40 | 000000e0:·4092·0406·a335·6578·7465·6e73·696f·6e5f··@....5extension_ | ||
| 000000f0:· | 000000f0:·636f·6e73·7472·7563·746f·7250·0800·0038··constructorP...8 | ||
| 00000100:· | 00000100:·0040·4040·4140·4040·4040·0407·4040·4092··.@@@A@@@@@..@@@. | ||
| 00000110:· | 00000110:·0403·a323·696e·7441·0800·0038·0040·4040··...#intA...8.@@@ | ||
| 00000120:·4 | 00000120:·4140·4040·4040·040b·4041·4092·0403·a324··A@@@@@..@A@....$ | ||
| 00000130:· | 00000130:·6368·6172·4208·0000·3800·4040·4041·4040··charB...8.@@@A@@ | ||
| 00000140:·4040·40 | 00000140:·4040·4004·0f40·4140·9204·03a3·2673·7472··@@@..@A@....&str | ||
| 00000150:· | 00000150:·696e·674f·0800·0038·0040·4040·4140·4040··ingO...8.@@@A@@@ | ||
| 00000160:·4040· | 00000160:·4040·0413·4040·4092·0403·a325·666c·6f61··@@..@@@....%floa | ||
| 00000170:· | 00000170:·7444·0800·0038·0040·4040·4140·4040·4040··tD...8.@@@A@@@@@ | ||
| 00000180:· | 00000180:·0417·4040·4092·0403·a324·626f·6f6c·4508··..@@@....$boolE. | ||
| 00000190:·0 | 00000190:·0000·3800·4040·a1a0·e0a3·2566·616c·7365··..8.@@....%false | ||
| 000001a0:· | 000001a0:·5e90·4040·0421·4092·0403·a0e0·a324·7472··^.@@.!@......$tr | ||
| 000001b0:·7 | 000001b0:·7565·5f90·4040·0427·4092·0403·4040·4140··ue_.@@.'@...@@A@ | ||
| 000001c0:·4040·4040· | 000001c0:·4040·4040·0428·4041·4092·0410·a324·756e··@@@@.(@A@....$un | ||
| 000001d0:·6 | 000001d0:·6974·4608·0000·3800·4040·a1a0·e0a3·2228··itF...8.@@...."( | ||
| 000001e0:·2 | 000001e0:·2960·9040·4004·3240·9204·0340·4041·4040··)`.@@.2@...@@A@@ | ||
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| 00000200:· | 00000200:·4708·0000·3800·4040·4141·4040·4040·4004··G...8.@@AA@@@@@. | ||
| 00000210:· | 00000210:·3740·4040·9204·03a3·2561·7272·6179·4808··7@@@....%arrayH. | ||
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| 00000560:· | 00000560:·616c·6964·5f61·7267·756d·656e·7454·0800··alid_argumentT.. | ||
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| 00000590:· | 00000590:·5374·6163·6b5f·6f76·6572·666c·6f77·5a08··Stack_overflowZ. | ||
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| 000006b0:· | 000006b0:·c401·04e3·4040·4040·a105·015f·4040·40b0··....@@@@..._@@@. | ||
| 000006c0:· | 000006c0:·c005·0154·5801·04b3·0104·b8c0·0501·5558··...TX.........UX | ||
| 000006d0:· | 000006d0:·0104·b301·04ba·4040·a042·4140·0800·0038··......@@.BA@...8 | ||
| 000006e0:· | 000006e0:·00a0·0501·5140·4140·4190·c098·e0a0·a026··....Q@A@A......& | ||
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| 00000730:· | 00000730:·0040·0103·1b40·c040·0205·f5e1·0040·0103··.@...@.@.....@.. | ||
| 00000740:· | 00000740:·1d40·4140·4002·05f5·e100·4001·031e·a059··.@A@@.....@....Y | ||
| 00000750:·40 | 00000750:·40a0·4040·4040·0425·4040·4004·2240·4041··@.@@@@.%@@@."@@A | ||
| 00000760:· | 00000760:·90d0·b7a0·b0b0·a004·1cb0·c005·017c·5901··.............|Y. | ||
| 00000770:·c00 | 00000770:·04c4·0104·c8c0·0501·7d59·0104·c401·04cf··........}Y...... | ||
| 00000780:· | 00000780:·4040·a0d0·92a0·d0b3·041d·a090·2369·6e74··@@..........#int | ||
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| 000007a0:· | 000007a0:·5901·04c4·0104·d640·4004·2505·017f·b004··Y......@@.%..... | ||
| 000007b0:· | 000007b0:·0304·0240·40a0·d0b3·0422·a090·2369·6e74··...@@...."..#int | ||
| 000007c0:· | 000007c0:·b0c0·0501·9259·0104·c401·04d9·c005·0193··.....Y.......... | ||
| 000007d0:· | 000007d0:·5901·04c4·0104·dc40·4004·2a05·0189·b004··Y......@@.*..... | ||
| Max diff block lines reached; 353615/535176 bytes (66.07%) of diff not shown. | |||
| Offset 210, 16 lines modified | Offset 210, 16 lines modified | ||
| 00000d10:·3a0e·a0a0·2653·7464·6c69·6290·302d·0826··:...&Stdlib.0-.& | 00000d10:·3a0e·a0a0·2653·7464·6c69·6290·302d·0826··:...&Stdlib.0-.& | ||
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| 00000d60:·0000·0800·0000·0400·0000·0b00·0000·0ba0··................ | 00000d60:·0000·0800·0000·0400·0000·0b00·0000·0ba0··................ | ||
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| 00000d80:·3033·3084·95a6·be00·003e·7 | 00000d80:·3033·3084·95a6·be00·003e·7300·0008·c400··030......>s..... | ||
| 00000d90:·0027· | 00000d90:·0027·4b00·0024·a208·0000·3400·2544·6966··.'K..$....4.%Dif | ||
| 00000da0:·6632·92b0·a0b0·9eb0·a02a·6f63·616d·6c2e··f2.......*ocaml. | 00000da0:·6632·92b0·a0b0·9eb0·a02a·6f63·616d·6c2e··f2.......*ocaml. | ||
| 00000db0:·7465·7874·b0c0·265f·6e6f·6e65·5f40·4000··text..&_none_@@. | 00000db0:·7465·7874·b0c0·265f·6e6f·6e65·5f40·4000··text..&_none_@@. | ||
| 00000dc0:·ff04·0241·90a0·a0a0·c091·b20a·0000·0140··...A...........@ | 00000dc0:·ff04·0241·90a0·a0a0·c091·b20a·0000·0140··...A...........@ | ||
| 00000dd0:·0a20·2020·416e·2069·6d70·6c65·6d65·6e74··.···An·implement | 00000dd0:·0a20·2020·416e·2069·6d70·6c65·6d65·6e74··.···An·implement | ||
| 00000de0:·6174·696f·6e20·6f66·2045·7567·656e·6520··ation·of·Eugene· | 00000de0:·6174·696f·6e20·6f66·2045·7567·656e·6520··ation·of·Eugene· | ||
| 00000df0:·4d79·6572·7327·204f·284e·4429·2044·6966··Myers'·O(ND)·Dif | 00000df0:·4d79·6572·7327·204f·284e·4429·2044·6966··Myers'·O(ND)·Dif | ||
| 00000e00:·6665·7265·6e63·6520·416c·676f·7269·7468··ference·Algorith | 00000e00:·6665·7265·6e63·6520·416c·676f·7269·7468··ference·Algorith |
| Offset 94, 666 lines modified | Offset 94, 666 lines modified | ||
| 000005d0:·646c·6962·9030·2d08·2666·be7f·c2ba·916e··dlib.0-.&f.....n | 000005d0:·646c·6962·9030·2d08·2666·be7f·c2ba·916e··dlib.0-.&f.....n | ||
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| 000005f0:·6572·6e61·6c46·6f72·6d61·7442·6173·6963··ernalFormatBasic | 000005f0:·6572·6e61·6c46·6f72·6d61·7442·6173·6963··ernalFormatBasic | ||
| 00000600:·7390·30c4·b583·a727·ec28·f5bc·9ba3·6adc··s.0....'.(....j. | 00000600:·7390·30c4·b583·a727·ec28·f5bc·9ba3·6adc··s.0....'.(....j. | ||
| 00000610:·64cf·c740·8495·a6be·0000·0008·0000·0004··d..@............ | 00000610:·64cf·c740·8495·a6be·0000·0008·0000·0004··d..@............ | ||
| 00000620:·0000·000b·0000·000b·a040·a041·a090·4040··.........@.A..@@ | 00000620:·0000·000b·0000·000b·a040·a041·a090·4040··.........@.A..@@ | ||
| 00000630:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000630:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000640:·0000·291 | 00000640:·0000·2912·0000·0589·0000·1957·0000·1755··..)........W...U | ||
| 00000650:·0800·0034·0027·4578·6e69·6e66·6f92·b0a0··...4.'Exninfo... | 00000650:·0800·0034·0027·4578·6e69·6e66·6f92·b0a0··...4.'Exninfo... | ||
| 00000660:·b09e·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | 00000660:·b09e·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | ||
| 00000670:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | 00000670:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | ||
| 00000680:·a0a0·a0c0·91b2·092c·2041·6464·6974·696f··.......,·Additio | 00000680:·a0a0·a0c0·91b2·092c·2041·6464·6974·696f··.......,·Additio | ||
| 00000690:·6e61·6c20·696e·666f·726d·6174·696f·6e20··nal·information· | 00000690:·6e61·6c20·696e·666f·726d·6174·696f·6e20··nal·information· | ||
| 000006a0:·776f·726e·2062·7920·6578·6365·7074·696f··worn·by·exceptio | 000006a0:·776f·726e·2062·7920·6578·6365·7074·696f··worn·by·exceptio | ||
| 000006b0:·6e73·2e20·b0c0·3063·6c69·622f·6578·6e69··ns.·..0clib/exni | 000006b0:·6e73·2e20·b0c0·3063·6c69·622f·6578·6e69··ns.·..0clib/exni | ||
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| 000006d0:·024b·0102·a401·02d5·4040·0404·4040·4004··.K......@@..@@@. | 000006d0:·024b·0102·a401·02d5·4040·0404·4040·4004··.K......@@..@@@. | Diff chunk too large, falling back to line-by-line diff (483 lines added, 483 lines removed) | |
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| 00000740:· | 00000740:·6578·7465·6e73·696f·6e5f·636f·6e73·7472··extension_constr | ||
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| 000009e0:· | 000009e0:·4040·4092·0403·a33a·556e·6465·6669·6e65··@@@....:Undefine | ||
| 000009f0:·6 | 000009f0:·645f·7265·6375·7273·6976·655f·6d6f·6475··d_recursive_modu | ||
| 00000a00:· | 00000a00:·6c65·5d08·0000·2000·9004·5740·90a0·c092··le]...·...W@.... | ||
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| 00000a20:· | 00000a20:·a0c0·b390·0490·4090·4002·05f5·e100·4040··......@.@.....@@ | ||
| 00000a30:· | 00000a30:·a004·0540·0205·f5e1·0040·5640·4041·04ca··...@.....@V@@A.. | ||
| 00000a40:· | 00000a40:·a0b0·a03d·6f63·616d·6c2e·7761·726e·5f6f··...=ocaml.warn_o | ||
| 00000a50:·6 | 00000a50:·6e5f·6c69·7465·7261·6c5f·7061·7474·6572··n_literal_patter | ||
| 00000a60:· | 00000a60:·6e04·ce04·6104·ce40·9204·17a3·2e41·7373··n...a..@.....Ass | ||
| 00000a70:· | 00000a70:·6572·745f·6661·696c·7572·655c·0800·0020··ert_failure\...· | ||
| 00000a80:· | 00000a80:·0004·1840·90a0·c092·a004·17a0·0413·a004··...@............ | ||
| 00000a90:· | 00000a90:·1440·0205·f5e1·0040·5840·4041·04d9·a0b0··.@.....@X@@A.... | ||
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| 00000ad0:· | 00000ad0:·a004·1704·e404·7704·e440·9204·07a3·2b45··......w..@....+E | ||
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| 00000b60:· | 00000b60:·0407·a329·4e6f·745f·666f·756e·6456·0800··...)Not_foundV.. | ||
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| 00000b80:· | 00000b80:·0438·0501·0504·9805·0105·4092·0407·a327··.8........@....' | ||
| 00000b90:· | 00000b90:·4661·696c·7572·6555·0800·0020·0004·4f40··FailureU...·..O@ | ||
| 00000ba0:· | 00000ba0:·90a0·044b·4040·4105·010b·a0b0·a004·4105··...K@@A.......A. | ||
| 00000bb0:·0 | 00000bb0:·010e·04a1·0501·0e40·9204·08a3·3049·6e76··.......@....0Inv | ||
| 00000bc0:· | 00000bc0:·616c·6964·5f61·7267·756d·656e·7454·0800··alid_argumentT.. | ||
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| 00000c00:· | 00000c00:·775a·0800·0020·0004·6140·9040·4041·0501··wZ...·..a@.@@A.. | ||
| 00000c10:· | 00000c10:·1ca0·b0a0·0452·0501·1f04·b205·011f·4092··.....R........@. | ||
| 00000c20:· | 00000c20:·0407·a32d·4f75·745f·6f66·5f6d·656d·6f72··...-Out_of_memor | ||
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| 00000c50:· | 00000c50:·0407·a32d·4d61·7463·685f·6661·696c·7572··...-Match_failur | ||
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| 00000d40:· | 00000d40:·4140·a047·40a0·4240·4040·b0c0·0501·4f4d··A@.G@.B@@@....OM | ||
| 00000d50:· | 00000d50:·0102·d701·02d7·0413·40a0·b0a0·296f·6361··........@...)oca | ||
| 00000d60:· | 00000d60:·6d6c·2e64·6f63·0501·6090·a0a0·a0c0·91b2··ml.doc..`....... | ||
| 00000d70:·0 | 00000d70:·0926·2049·6e66·6f72·6d61·7469·6f6e·2063··.&·Information·c | ||
| 00000d80:· | 00000d80:·6f6e·7461·696e·696e·6720·6120·6769·7665··ontaining·a·give | ||
| 00000d90:·6e | 00000d90:·6e20·7479·7065·2e20·b0c0·0501·5d4e·0102··n·type.·....]N.. | ||
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| Max diff block lines reached; 22199/90255 bytes (24.60%) of diff not shown. | |||
| Offset 186, 1008 lines modified | Offset 186, 1007 lines modified | ||
| 00000b90:·6873·6574·9030·9463·0633·0b75·4f24·e774··hset.0.c.3.uO$.t | 00000b90:·6873·6574·9030·9463·0633·0b75·4f24·e774··hset.0.c.3.uO$.t | ||
| 00000ba0:·b52c·7336·3fa2·a0a0·3843·616d·6c69·6e74··.,s6?...8Camlint | 00000ba0:·b52c·7336·3fa2·a0a0·3843·616d·6c69·6e74··.,s6?...8Camlint | ||
| 00000bb0:·6572·6e61·6c46·6f72·6d61·7442·6173·6963··ernalFormatBasic | 00000bb0:·6572·6e61·6c46·6f72·6d61·7442·6173·6963··ernalFormatBasic | ||
| 00000bc0:·7390·30c4·b583·a727·ec28·f5bc·9ba3·6adc··s.0....'.(....j. | 00000bc0:·7390·30c4·b583·a727·ec28·f5bc·9ba3·6adc··s.0....'.(....j. | ||
| 00000bd0:·64cf·c740·8495·a6be·0000·0008·0000·0004··d..@............ | 00000bd0:·64cf·c740·8495·a6be·0000·0008·0000·0004··d..@............ | ||
| 00000be0:·0000·000b·0000·000b·a040·a041·a090·4040··.........@.A..@@ | 00000be0:·0000·000b·0000·000b·a040·a041·a090·4040··.........@.A..@@ | ||
| 00000bf0:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000bf0:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000c00:·0000·3e7 | 00000c00:·0000·3e70·0000·084e·0000·263e·0000·2376··..>p...N..&>..#v | ||
| 00000c10:·0800·0034·0028·4861·7368·636f·6e73·92b0··...4.(Hashcons.. | 00000c10:·0800·0034·0028·4861·7368·636f·6e73·92b0··...4.(Hashcons.. | ||
| 00000c20:·a0b0·9eb0·a02a·6f63·616d·6c2e·7465·7874··.....*ocaml.text | 00000c20:·a0b0·9eb0·a02a·6f63·616d·6c2e·7465·7874··.....*ocaml.text | ||
| 00000c30:·b0c0·265f·6e6f·6e65·5f40·4000·ff04·0241··..&_none_@@....A | 00000c30:·b0c0·265f·6e6f·6e65·5f40·4000·ff04·0241··..&_none_@@....A | ||
| 00000c40:·90a0·a0a0·c091·b237·2047·656e·6572·6963··.......7·Generic | 00000c40:·90a0·a0a0·c091·b237·2047·656e·6572·6963··.......7·Generic | ||
| 00000c50:·2068·6173·682d·636f·6e73·696e·672e·20b0···hash-consing.·. | 00000c50:·2068·6173·682d·636f·6e73·696e·672e·20b0···hash-consing.·. | ||
| 00000c60:·c031·636c·6962·2f68·6173·6863·6f6e·732e··.1clib/hashcons. | 00000c60:·c031·636c·6962·2f68·6173·6863·6f6e·732e··.1clib/hashcons. | ||
| 00000c70:·6d6c·694b·0102·a401·02a4·c004·024b·0102··mliK.........K.. | 00000c70:·6d6c·694b·0102·a401·02a4·c004·024b·0102··mliK.........K.. | ||
| 00000c80:·a401·02c0·4040·0404·4040·4004·0440·0404··....@@..@@@..@.. | 00000c80:·a401·02c0·4040·0404·4040·4004·0440·0404··....@@..@@@..@.. | Diff chunk too large, falling back to line-by-line diff (771 lines added, 771 lines removed) | |
| 00000c90:·0800·0030·00a0·4040·0401·0401·0401·0401··...0..@@........ | 00000c90:·0800·0030·00a0·4040·0401·0401·0401·0401··...0..@@........ | ||
| 00000ca0:·0401·0401·0401·40ab·a | 00000ca0:·0401·0401·0401·40ab·a7b1·b2b2·b2b2·b2b2··......@......... | ||
| 00000cb0:·b2b2·b2b2·b2b2·b | 00000cb0:·b2b2·b2b2·b2b2·b1b1·b1b1·b1b1·b1b1·b1b1··................ | ||
| 00000cc0:·b1b1·b1b1·b1b1· | 00000cc0:·b1b1·b1b1·b1b1·40a3·2a66·6c6f·6174·6172··......@.*floatar | ||
| 00000cd0:·72 | 00000cd0:·7261·7951·0800·0038·0040·4040·4140·4040··rayQ...8.@@@A@@@ | ||
| 00000ce0:·4040· | 00000ce0:·4040·0433·4040·4092·0403·a335·6578·7465··@@.3@@@....5exte | ||
| 00000cf0:·6 | 00000cf0:·6e73·696f·6e5f·636f·6e73·7472·7563·746f··nsion_constructo | ||
| 00000d00:· | 00000d00:·7250·0800·0038·0040·4040·4140·4040·4040··rP...8.@@@A@@@@@ | ||
| 00000d10:· | 00000d10:·0437·4040·4092·0403·a323·696e·7441·0800··.7@@@....#intA.. | ||
| 00000d20:·00 | 00000d20:·0038·0040·4040·4140·4040·4040·043b·4041··.8.@@@A@@@@@.;@A | ||
| 00000d30:·4 | 00000d30:·4092·0403·a324·6368·6172·4208·0000·3800··@....$charB...8. | ||
| 00000d40:· | 00000d40:·4040·4041·4040·4040·4004·3f40·4140·9204··@@@A@@@@@.?@A@.. | ||
| 00000d50:·0 | 00000d50:·03a3·2673·7472·696e·674f·0800·0038·0040··..&stringO...8.@ | ||
| 00000d60:·4040·4 | 00000d60:·4040·4140·4040·4040·0443·4040·4092·0403··@@A@@@@@.C@@@... | ||
| 00000d70:· | 00000d70:·a325·666c·6f61·7444·0800·0038·0040·4040··.%floatD...8.@@@ | ||
| 00000d80:·4 | 00000d80:·4140·4040·4040·0447·4040·4092·0403·a324··A@@@@@.G@@@....$ | ||
| 00000d90:· | 00000d90:·626f·6f6c·4508·0000·3800·4040·a1a0·e0a3··boolE...8.@@.... | ||
| 00000da0:· | 00000da0:·2566·616c·7365·5e90·4040·0451·4092·0403··%false^.@@.Q@... | ||
| 00000db0:· | 00000db0:·a0e0·a324·7472·7565·5f90·4040·0457·4092··...$true_.@@.W@. | ||
| 00000dc0:· | 00000dc0:·0403·4040·4140·4040·4040·0458·4041·4092··..@@A@@@@@.X@A@. | ||
| 00000dd0:· | 00000dd0:·0410·a324·756e·6974·4608·0000·3800·4040··...$unitF...8.@@ | ||
| 00000de0:· | 00000de0:·a1a0·e0a3·2228·2960·9040·4004·6240·9204··...."()`.@@.b@.. | ||
| 00000df0:·040 | 00000df0:·0340·4041·4040·4040·4004·6340·4140·9204··.@@A@@@@@.c@A@.. | ||
| 00000e00:·0 | 00000e00:·0aa3·2365·786e·4708·0000·3800·4040·4141··..#exnG...8.@@AA | ||
| 00000e10:·4 | 00000e10:·4040·4040·4004·6740·4040·9204·03a3·2561··@@@@@.g@@@....%a | ||
| 00000e20:· | 00000e20:·7272·6179·4808·0000·3800·a0c0·9040·0205··rrayH...8....@.. | ||
| 00000e30:· | 00000e30:·f5e1·0040·4f40·4140·4140·a000·7f40·a040··...@O@A@A@...@.@ | ||
| 00000e40:·4040·404 | 00000e40:·4040·4004·7040·4040·9204·08a3·246c·6973··@@@.p@@@....$lis | ||
| 00000e50:·7 | 00000e50:·7449·0800·0038·00a0·c004·0902·05f5·e100··tI...8.......... | ||
| 00000e60:· | 00000e60:·4050·4041·a1a0·e0a3·225b·5d61·9040·4004··@P@A...."[]a.@@. | ||
| 00000e70:· | 00000e70:·7c40·9204·03a0·e0a3·223a·3a62·90a0·040e··|@......"::b.... | ||
| 00000e80:· | 00000e80:·a0c0·b390·0416·a004·1340·9040·0205·f5e1··.........@.@.... | ||
| 00000e90:· | 00000e90:·0040·5140·4004·8940·9204·0a40·4041·40a0··.@Q@@..@...@@A@. | ||
| 00000ea0:· | 00000ea0:·5940·a040·4040·4004·8c40·4040·9204·1ba3··Y@.@@@@..@@@.... | ||
| 00000eb0:· | 00000eb0:·266f·7074·696f·6e4a·0800·0038·00a0·c004··&optionJ...8.... | ||
| 00000ec0:· | 00000ec0:·2502·05f5·e100·4053·4041·a1a0·e0a3·244e··%.....@S@A....$N | ||
| 00000ed0:· | 00000ed0:·6f6e·6563·9040·4004·9840·9204·03a0·e0a3··onec.@@..@...... | ||
| 00000ee0:· | 00000ee0:·2453·6f6d·6564·90a0·040e·4040·049f·4092··$Somed....@@..@. | ||
| 00000ef0:· | 00000ef0:·0404·4040·4140·a059·40a0·4040·4040·04a2··..@@A@.Y@.@@@@.. | ||
| 00000f00:· | 00000f00:·4040·4092·0415·a326·6c61·7a79·5f74·4e08··@@@....&lazy_tN. | ||
| 00000f10:·0 | 00000f10:·0000·3800·a0c0·043b·0205·f5e1·0040·5540··..8....;.....@U@ | ||
| 00000f20:·4 | 00000f20:·4140·4140·a059·40a0·4040·4040·04aa·4040··A@A@.Y@.@@@@..@@ | ||
| 00000f30:·40 | 00000f30:·4092·0407·a329·6e61·7469·7665·696e·744b··@....)nativeintK | ||
| 00000f40:· | 00000f40:·0800·0038·0040·4040·4140·4040·4040·04ae··...8.@@@A@@@@@.. | ||
| 00000f50:· | 00000f50:·4040·4092·0403·a325·696e·7433·324c·0800··@@@....%int32L.. | ||
| 00000f60:·00 | 00000f60:·0038·0040·4040·4140·4040·4040·04b2·4040··.8.@@@A@@@@@..@@ | ||
| 00000f70:·40 | 00000f70:·4092·0403·a325·696e·7436·344d·0800·0038··@....%int64M...8 | ||
| 00000f80:· | 00000f80:·0040·4040·4140·4040·4040·04b6·4040·4092··.@@@A@@@@@..@@@. | ||
| 00000f90:· | 00000f90:·0403·a33a·556e·6465·6669·6e65·645f·7265··...:Undefined_re | ||
| 00000fa0:·6 | 00000fa0:·6375·7273·6976·655f·6d6f·6475·6c65·5d08··cursive_module]. | ||
| 00000fb0:·0 | 00000fb0:·0000·2000·9004·5740·90a0·c092·a0c0·b390··..·...W@........ | ||
| 00000fc0:· | 00000fc0:·0483·4090·4002·05f5·e100·404a·a0c0·b390··..@.@.....@J.... | ||
| 00000fd0:· | 00000fd0:·0490·4090·4002·05f5·e100·4040·a004·0540··..@.@.....@@...@ | ||
| 00000fe0:· | 00000fe0:·0205·f5e1·0040·5640·4041·04ca·a0b0·a03d··.....@V@@A.....= | ||
| 00000ff0:· | 00000ff0:·6f63·616d·6c2e·7761·726e·5f6f·6e5f·6c69··ocaml.warn_on_li | ||
| 00001000:· | 00001000:·7465·7261·6c5f·7061·7474·6572·6e04·ce04··teral_pattern... | ||
| 00001010:· | 00001010:·6104·ce40·9204·17a3·2e41·7373·6572·745f··a..@.....Assert_ | ||
| 00001020:· | 00001020:·6661·696c·7572·655c·0800·0020·0004·1840··failure\...·...@ | ||
| 00001030:· | 00001030:·90a0·c092·a004·17a0·0413·a004·1440·0205··.............@.. | ||
| 00001040:· | 00001040:·f5e1·0040·5840·4041·04d9·a0b0·a004·0f04··...@X@@A........ | ||
| 00001050:· | 00001050:·dc04·6f04·dc40·9204·0da3·3044·6976·6973··..o..@....0Divis | ||
| 00001060:· | 00001060:·696f·6e5f·6279·5f7a·6572·6f59·0800·0020··ion_by_zeroY...· | ||
| 00001070:· | 00001070:·0004·2640·9040·4041·04e1·a0b0·a004·1704··..&@.@@A........ | ||
| 00001080:· | 00001080:·e404·7704·e440·9204·07a3·2b45·6e64·5f6f··..w..@....+End_o | ||
| 00001090:·6 | 00001090:·665f·6669·6c65·5808·0000·2000·042e·4090··f_fileX...·...@. | ||
| 000010a0:· | 000010a0:·4040·4104·e9a0·b0a0·041f·04ec·047f·04ec··@@A............. | ||
| 000010b0:· | 000010b0:·4092·0407·a329·5379·735f·6572·726f·7257··@....)Sys_errorW | ||
| 000010c0:· | 000010c0:·0800·0020·0004·3640·90a0·0432·4040·4104··...·..6@...2@@A. | ||
| 000010d0:· | 000010d0:·f2a0·b0a0·0428·04f5·0488·04f5·4092·0408··.....(......@... | ||
| 000010e0:· | 000010e0:·a32e·5379·735f·626c·6f63·6b65·645f·696f··..Sys_blocked_io | ||
| 000010f0:· | 000010f0:·5b08·0000·2000·043f·4090·4040·4104·faa0··[...·..?@.@@A... | ||
| 00001100:· | 00001100:·b0a0·0430·04fd·0490·04fd·4092·0407·a329··...0......@....) | ||
| 00001110:· | 00001110:·4e6f·745f·666f·756e·6456·0800·0020·0004··Not_foundV...·.. | ||
| 00001120:· | 00001120:·4740·9040·4041·0501·02a0·b0a0·0438·0501··G@.@@A.......8.. | ||
| 00001130:·0 | 00001130:·0504·9805·0105·4092·0407·a327·4661·696c··......@....'Fail | ||
| 00001140:· | 00001140:·7572·6555·0800·0020·0004·4f40·90a0·044b··ureU...·..O@...K | ||
| 00001150:·4 | 00001150:·4040·4105·010b·a0b0·a004·4105·010e·04a1··@@A.......A..... | ||
| 00001160:· | 00001160:·0501·0e40·9204·08a3·3049·6e76·616c·6964··...@....0Invalid | ||
| 00001170:· | 00001170:·5f61·7267·756d·656e·7454·0800·0020·0004··_argumentT...·.. | ||
| 00001180:· | 00001180:·5840·90a0·0454·4040·4105·0114·a0b0·a004··X@...T@@A....... | ||
| 00001190:· | 00001190:·4a05·0117·04aa·0501·1740·9204·08a3·2e53··J........@.....S | ||
| 000011a0:· | 000011a0:·7461·636b·5f6f·7665·7266·6c6f·775a·0800··tack_overflowZ.. | ||
| 000011b0:·00 | 000011b0:·0020·0004·6140·9040·4041·0501·1ca0·b0a0··.·..a@.@@A...... | ||
| 000011c0:· | 000011c0:·0452·0501·1f04·b205·011f·4092·0407·a32d··.R........@....- | ||
| 000011d0:· | 000011d0:·4f75·745f·6f66·5f6d·656d·6f72·7953·0800··Out_of_memoryS.. | ||
| 000011e0:·00 | 000011e0:·0020·0004·6940·9040·4041·0501·24a0·b0a0··.·..i@.@@A..$... | ||
| 000011f0:· | 000011f0:·045a·0501·2704·ba05·0127·4092·0407·a32d··.Z..'....'@....- | ||
| 00001200:· | 00001200:·4d61·7463·685f·6661·696c·7572·6552·0800··Match_failureR.. | ||
| 00001210:·00 | 00001210:·0020·0004·7140·90a0·c092·a004·70a0·046c··.·..q@......p..l | ||
| 00001220:· | 00001220:·a004·6d40·0205·f5e1·0040·6340·4041·0501··..m@.....@c@@A.. | ||
| 00001230:· | 00001230:·32a0·b0a0·0468·0501·3504·c805·0135·4092··2....h..5....5@. | ||
| 00001240:· | 00001240:·040d·a325·6279·7465·7343·0800·0038·0040··...%bytesC...8.@ | ||
| 00001250:·4040·4 | 00001250:·4040·4140·4040·4040·0501·3940·4040·9204··@@A@@@@@..9@@@.. | ||
| 00001260:·0 | 00001260:·0390·9226·5374·646c·6962·9226·4f70·7469··...&Stdlib.&Opti | ||
| 00001270:· | 00001270:·6f6e·4041·0501·34a0·b09e·b005·0145·90a0··on@A..4......E.. | ||
| 00001280:· | 00001280:·a0a0·c091·b209·2620·7b36·2048·6173·6863··......&·{6·Hashc | ||
| 00001290:· | 00001290:·6f6e·7369·6e67·2066·756e·6374·6f72·6961··onsing·functoria | ||
| 000012a0:·6 | 000012a0:·6c20·696e·7465·7266·6163·657d·20b0·c005··l·interface}·... | ||
| 000012b0:· | 000012b0:·0140·4d01·02c2·0102·c2c0·0501·414d·0102··.@M.........AM.. | ||
| 000012c0:· | 000012c0:·c201·02ed·4040·0403·4040·4004·0340·0403··....@@..@@@..@.. | ||
| 000012d0:· | 000012d0:·0501·3f04·03a0·b098·d0b1·2e48·6173·6863··..?........Hashc | ||
| 000012e0:· | 000012e0:·6f6e·7365·6454·7970·6500·5643·a004·02b0··onsedType.VC.... | ||
| 000012f0:· | 000012f0:·c005·014a·4f01·02ef·0102·fbc0·0501·4b4f··...JO.........KO | ||
| 00001300:· | 00001300:·0102·ef01·0309·4090·d091·b0a0·b09e·b005··......@......... | ||
| 00001310:· | 00001310:·0162·90a0·a0a0·c091·b20a·0000·0245·207b··.b...........E·{ | ||
| 00001320:· | 00001320:·3620·4765·6e65·7269·6320·6861·7368·636f··6·Generic·hashco | ||
| 00001330:·6 | 00001330:·6e73·696e·6720·7369·676e·6174·7572·657d··nsing·signature} | ||
| 00001340:· | 00001340:·0a0a·2020·2020·2020·2020·4769·7665·6e20··..········Given· | ||
| 00001350:· | 00001350:·616e·2065·7175·6976·616c·656e·6365·2072··an·equivalence·r | ||
| 00001360:·656 | 00001360:·656c·6174·696f·6e20·5b65·715d·2c20·6120··elation·[eq],·a· | ||
| Max diff block lines reached; 29591/137326 bytes (21.55%) of diff not shown. | |||
| Offset 1, 1924 lines modified | Offset 1, 1924 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0000·781 | 00000010:·0000·7811·0000·1527·0000·5854·0000·56cc··..x....'..XT..V. | ||
| 00000020:·0800·0034·0024·4865·6170·91b0·a0b0·98d0··...4.$Heap...... | 00000020:·0800·0034·0024·4865·6170·91b0·a0b0·98d0··...4.$Heap...... | ||
| 00000030:·b127·4f72·6465·7265·6400·5342·a004·02b0··.'Ordered.SB.... | 00000030:·b127·4f72·6465·7265·6400·5342·a004·02b0··.'Ordered.SB.... | ||
| 00000040:·c02c·636c·6962·2f68·6561·702e·6d6c·4d01··.,clib/heap.mlM. | 00000040:·c02c·636c·6962·2f68·6561·702e·6d6c·4d01··.,clib/heap.mlM. | ||
| 00000050:·02b2·0102·bec0·0402·4d01·02b2·0102·c540··........M......@ | 00000050:·02b2·0102·bec0·0402·4d01·02b2·0102·c540··........M......@ | ||
| 00000060:·90d0·91b0·a0b0·a141·a008·0000·2800·b121··.......A....(..! | 00000060:·90d0·91b0·a0b0·a141·a008·0000·2800·b121··.......A....(..! | ||
| 00000070:·7400·5141·a004·02b0·c004·104e·0102·cc01··t.QA.......N.... | 00000070:·7400·5141·a004·02b0·c004·104e·0102·cc01··t.QA.......N.... | ||
| 00000080:·02d3·c004·114e·0102·cc01·02d4·4040·0800··.....N......@@.. | 00000080:·02d3·c004·114e·0102·cc01·02d4·4040·0800··.....N......@@.. | ||
| 00000090:·0038·0040·4040·4140·4040·4040·b0c0·0414··.8.@@@A@@@@@.... | 00000090:·0038·0040·4040·4140·4040·4040·b0c0·0414··.8.@@@A@@@@@.... | ||
| 000000a0:·4e01·02cc·0102·ce04·0440·4040·40a1·0421··N........@@@@..! | 000000a0:·4e01·02cc·0102·ce04·0440·4040·40a1·0421··N........@@@@..! | ||
| 000000b0:·4040·4041·4004·0340·4008·0000·3000·a040··@@@A@..@@...0..@ | 000000b0:·4040·4041·4004·0340·4008·0000·3000·a040··@@@A@..@@...0..@ | Diff chunk too large, falling back to line-by-line diff (1748 lines added, 1748 lines removed) | |
| 000000c0:·4004·0104·0104·0104·0104·0104·0104·0140··@..............@ | 000000c0:·4004·0104·0104·0104·0104·0104·0104·0140··@..............@ | ||
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| Offset 111, 588 lines modified | Offset 111, 587 lines modified | ||
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| 00000c70:· | 00000c70:·3805·0105·0498·0501·0540·9204·07a3·2746··8........@....'F | ||
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| 00000d10:·0 | 00000d10:·07a3·2d4f·7574·5f6f·665f·6d65·6d6f·7279··..-Out_of_memory | ||
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| 00000da0:·40 | 00000da0:·4092·0403·9092·2653·7464·6c69·6292·2349··@.....&Stdlib.#I | ||
| 00000db0:· | 00000db0:·6e74·4041·0501·34a0·b098·d0b1·274f·7264··nt@A..4.....'Ord | ||
| 00000dc0:· | 00000dc0:·6572·6564·0053·42a0·0402·b0c0·0501·3b4d··ered.SB.......;M | ||
| 00000dd0:· | 00000dd0:·0102·b201·02be·c005·013c·4d01·02b2·0102··.........<M..... | ||
| 00000de0:· | 00000de0:·c540·90d0·91b0·a0b0·a141·a008·0000·2800··.@.......A....(. | ||
| 00000df0:·b | 00000df0:·b121·7400·5141·a004·02b0·c005·014a·4e01··.!t.QA.......JN. | ||
| 00000e00:·0 | 00000e00:·02cc·0102·d3c0·0501·4b4e·0102·cc01·02d4··........KN...... | ||
| 00000e10:· | 00000e10:·4040·0800·0038·0040·4040·4140·4040·4040··@@...8.@@@A@@@@@ | ||
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| 00000e30:· | 00000e30:·4040·a105·0165·4040·4041·4004·0340·4008··@@...e@@@A@..@@. | ||
| 00000e40:· | 00000e40:·0000·3000·0501·4d05·014d·0501·4d05·014d··..0...M..M..M..M | ||
| 00000e50:·0501· | 00000e50:·0501·4d05·014d·0501·4d05·014d·4005·014c··..M..M..M..M@..L | ||
| 00000e60:· | 00000e60:·4041·b004·0404·0740·a0b0·90f0·a027·636f··@A.....@.....'co | ||
| 00000e70:· | 00000e70:·6d70·6172·6500·52a0·0402·b0c0·0501·5a4f··mpare.R.......ZO | ||
| 00000e80:· | 00000e80:·0102·d501·02db·c005·015b·4f01·02d5·0102··.........[O..... | ||
| 00000e90:· | 00000e90:·e240·d0b1·40d0·b390·041b·a090·2174·b0c0··.@..@.......!t.. | ||
| 00000ea0:· | 00000ea0:·0501·654f·0102·d501·02e5·c005·0166·4f01··..eO.........fO. | ||
| 00000eb0:·0 | 00000eb0:·02d5·0102·e640·40c0·b304·0940·9040·0205··.....@@....@.@.. | ||
| Max diff block lines reached; 16689/79572 bytes (20.97%) of diff not shown. | |||
| Offset 1, 1866 lines modified | Offset 1, 1866 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0000·747 | 00000010:·0000·7471·0000·14cc·0000·56aa·0000·557f··..tq......V...U. | ||
| 00000020:·0800·0034·0027·4953·7472·6561·6d91·b0a0··...4.'IStream... | 00000020:·0800·0034·0027·4953·7472·6561·6d91·b0a0··...4.'IStream... | ||
| 00000030:·b0a3·41a0·0800·0028·00b1·246e·6f64·6500··..A....(..$node. | 00000030:·b0a3·41a0·0800·0028·00b1·246e·6f64·6500··..A....(..$node. | ||
| 00000040:·5141·a004·02b0·c02f·636c·6962·2f69·5374··QA...../clib/iSt | 00000040:·5141·a004·02b0·c02f·636c·6962·2f69·5374··QA...../clib/iSt | ||
| 00000050:·7265·616d·2e6d·6c4b·0102·a401·02ac·c004··ream.mlK........ | 00000050:·7265·616d·2e6d·6c4b·0102·a401·02ac·c004··ream.mlK........ | ||
| 00000060:·024b·0102·a401·02b0·40a0·a0d0·9021·61c0··.K......@....!a. | 00000060:·024b·0102·a401·02b0·40a0·a0d0·9021·61c0··.K......@....!a. | ||
| 00000070:·9090·0404·0205·f5e1·0040·0103·1808·0000··.........@...... | 00000070:·9090·0404·0205·f5e1·0040·0103·1808·0000··.........@...... | Diff chunk too large, falling back to line-by-line diff (1767 lines added, 1767 lines removed) | |
| 00000080:·3000·a040·4004·0104·0104·0104·0104·0104··0..@@........... | 00000080:·3000·a040·4004·0104·0104·0104·0104·0104··0..@@........... | ||
| 00000090:·0104·0140·b1b1·aba | 00000090:·0104·0140·b1b1·aba7·b1b2·b2b2·b2b2·b2b2··...@............ | ||
| 000000a0:·b2b2·b2b2·b2b | 000000a0:·b2b2·b2b2·b2b1·b1b1·b1b1·b1b1·b1b1·b1b1··................ | ||
| 000000b0:·b1b1·b1b1·b1 | 000000b0:·b1b1·b1b1·b140·a32a·666c·6f61·7461·7272··.....@.*floatarr | ||
| 000000c0:· | 000000c0:·6179·5108·0000·3800·4040·4041·4040·4040··ayQ...8.@@@A@@@@ | ||
| 000000d0:·40 | 000000d0:·40b0·c026·5f6e·6f6e·655f·4040·00ff·0402··@..&_none_@@.... | ||
| 000000e0:· | 000000e0:·4140·4040·9204·06a3·3565·7874·656e·7369··A@@@....5extensi | ||
| 000000f0:·6 | 000000f0:·6f6e·5f63·6f6e·7374·7275·6374·6f72·5008··on_constructorP. | ||
| 00000100:·0 | 00000100:·0000·3800·4040·4041·4040·4040·4004·0740··..8.@@@A@@@@@..@ | ||
| 00000110:·4040· | 00000110:·4040·9204·03a3·2369·6e74·4108·0000·3800··@@....#intA...8. | ||
| 00000120:· | 00000120:·4040·4041·4040·4040·4004·0b40·4140·9204··@@@A@@@@@..@A@.. | ||
| 00000130:·0 | 00000130:·03a3·2463·6861·7242·0800·0038·0040·4040··..$charB...8.@@@ | ||
| 00000140:·4 | 00000140:·4140·4040·4040·040f·4041·4092·0403·a326··A@@@@@..@A@....& | ||
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| 00000200:· | 00000200:·6578·6e47·0800·0038·0040·4041·4140·4040··exnG...8.@@AA@@@ | ||
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| 00000220:· | 00000220:·7948·0800·0038·00a0·c090·4002·05f5·e100··yH...8....@..... | ||
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| 00000350:· | 00000350:·9204·03a3·2569·6e74·3332·4c08·0000·3800··....%int32L...8. | ||
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| 00000430:· | 00000430:·92a0·0417·a004·13a0·0414·4002·05f5·e100··..........@..... | ||
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| 00000550:· | 00000550:·b0a0·0441·04de·04a1·04de·4092·0408·a330··...A......@....0 | ||
| 00000560:· | 00000560:·496e·7661·6c69·645f·6172·6775·6d65·6e74··Invalid_argument | ||
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| 00000580:· | 00000580:·04e4·a0b0·a004·4a04·e704·aa04·e740·9204··......J......@.. | ||
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| 000005c0:· | 000005c0:·2d4f·7574·5f6f·665f·6d65·6d6f·7279·5308··-Out_of_memoryS. | ||
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| 00000670:· | 00000670:·0040·0103·1540·4140·4140·a047·40a0·4240··.@...@A@A@.G@.B@ | ||
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| 00000690:· | 00000690:·0148·4d01·02bb·0102·d040·4040·40a1·0501··.HM......@@@@... | ||
| 000006a0:· | 000006a0:·5640·b121·7400·5241·0800·0038·00a0·c004··V@.!t.RA...8.... | ||
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| 000006c0:· | 000006c0:·96c0·b390·040b·a0c0·9040·4240·0103·7840··.........@B@..x@ | ||
| 000006d0:·4240·0103·7 | 000006d0:·9040·4240·0103·7942·4001·0316·a047·40a0··.@B@..yB@....G@. | ||
| 000006e0:· | 000006e0:·4240·4040·b0c0·0501·5c4f·0102·d201·02d2··B@@@....\O...... | ||
| 000006f0:·01 | 000006f0:·c005·015d·4f01·02d2·0102·eb40·4040·40a1··...]O......@@@@. | ||
| 00000700:· | 00000700:·0501·6b41·4040·b0c0·0501·604b·0102·a401··..kA@@....`K.... | ||
| 00000710:· | 00000710:·02a9·c005·0161·4b01·02a4·0102·ab40·40a0··.....aK......@@. | ||
| 00000720:· | 00000720:·4241·4008·0000·3800·a005·015d·4041·a1a0··BA@...8....]@A.. | ||
| 00000730:· | 00000730:·e0a0·234e·696c·0053·9040·40b0·c005·016c··..#Nil.S.@@....l | ||
| 00000740:·4 | 00000740:·4c01·02b3·0102·b5c0·0501·6d4c·0102·b301··L.........mL.... | ||
| 00000750:· | 00000750:·02ba·4040·a105·017b·42a0·e0a0·2443·6f6e··..@@...{B...$Con | ||
| 00000760:· | 00000760:·7300·5490·a0c0·9605·016f·4340·0103·19a0··s.T......oC@.... | ||
| 00000770:· | 00000770:·c0b3·9004·31a0·c096·0501·7643·4001·031b··....1.....vC@... | ||
| 00000780:· | 00000780:·4090·d041·0406·0408·c096·c0b3·a1a1·044b··@..A...........K | ||
| 00000790:· | 00000790:·244c·617a·7921·74a0·c0b3·9005·0190·a0c0··$Lazy!t......... | ||
| 000007a0:·9 | 000007a0:·9605·0187·0205·f5e1·0040·0109·d840·9091··.........@...@.. | ||
| 000007b0:·0 | 000007b0:·0413·0205·f5e1·0040·0109·da40·90d0·4104··.......@...@..A. | ||
| 000007c0:· | 000007c0:·0f04·11c0·96c0·b3a1·9092·3043·616d·6c69··..........0Camli | ||
| 000007d0:· | 000007d0:·6e74·6572·6e61·6c4c·617a·7921·74a0·c096··nternalLazy!t... | ||
| Max diff block lines reached; 11775/256402 bytes (4.59%) of diff not shown. | |||
| Offset 140, 15 lines modified | Offset 140, 15 lines modified | ||
| 000008b0:·646c·6962·9030·2d08·2666·be7f·c2ba·916e··dlib.0-.&f.....n | 000008b0:·646c·6962·9030·2d08·2666·be7f·c2ba·916e··dlib.0-.&f.....n | ||
| 000008c0:·7233·3974·91df·a0a0·3843·616d·6c69·6e74··r39t....8Camlint | 000008c0:·7233·3974·91df·a0a0·3843·616d·6c69·6e74··r39t....8Camlint | ||
| 000008d0:·6572·6e61·6c46·6f72·6d61·7442·6173·6963··ernalFormatBasic | 000008d0:·6572·6e61·6c46·6f72·6d61·7442·6173·6963··ernalFormatBasic | ||
| 000008e0:·7390·30c4·b583·a727·ec28·f5bc·9ba3·6adc··s.0....'.(....j. | 000008e0:·7390·30c4·b583·a727·ec28·f5bc·9ba3·6adc··s.0....'.(....j. | ||
| 000008f0:·64cf·c740·8495·a6be·0000·0008·0000·0004··d..@............ | 000008f0:·64cf·c740·8495·a6be·0000·0008·0000·0004··d..@............ | ||
| 00000900:·0000·000b·0000·000b·a040·a041·a090·4040··.........@.A..@@ | 00000900:·0000·000b·0000·000b·a040·a041·a090·4040··.........@.A..@@ | ||
| 00000910:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000910:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000920:·0000·35 | 00000920:·0000·3577·0000·08af·0000·2485·0000·22ca··..5w......$...". | ||
| 00000930:·0800·0034·0027·4953·7472·6561·6d92·b0a0··...4.'IStream... | 00000930:·0800·0034·0027·4953·7472·6561·6d92·b0a0··...4.'IStream... | ||
| 00000940:·b09e·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | 00000940:·b09e·b0a0·2a6f·6361·6d6c·2e74·6578·74b0··....*ocaml.text. | ||
| 00000950:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | 00000950:·c026·5f6e·6f6e·655f·4040·00ff·0402·4190··.&_none_@@....A. | ||
| 00000960:·a0a0·a0c0·91b2·09c5·207b·3520·5075·7265··........·{5·Pure | 00000960:·a0a0·a0c0·91b2·09c5·207b·3520·5075·7265··........·{5·Pure | ||
| 00000970:·6c79·2066·756e·6374·696f·6e61·6c20·7374··ly·functional·st | 00000970:·6c79·2066·756e·6374·696f·6e61·6c20·7374··ly·functional·st | ||
| 00000980:·7265·616d·737d·0a0a·2020·436f·6e74·7261··reams}..··Contra | 00000980:·7265·616d·737d·0a0a·2020·436f·6e74·7261··reams}..··Contra | ||
| 00000990:·7269·6c79·2074·6f20·4f43·616d·6c20·6d6f··rily·to·OCaml·mo | 00000990:·7269·6c79·2074·6f20·4f43·616d·6c20·6d6f··rily·to·OCaml·mo |
| Offset 1, 3143 lines modified | Offset 1, 3142 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0000·c44 | 00000010:·0000·c440·0000·2143·0000·8aec·0000·8830··...@..!C.......0 | ||
| 00000020:·0800·0034·0025·4d6f·6e61·6491·b0a0·b09d··...4.%Monad..... | 00000020:·0800·0034·0025·4d6f·6e61·6491·b0a0·b09d··...4.%Monad..... | ||
| 00000030:·b0a0·2a6f·6361·6d6c·2e74·6578·74b0·c026··..*ocaml.text..& | 00000030:·b0a0·2a6f·6361·6d6c·2e74·6578·74b0·c026··..*ocaml.text..& | ||
| 00000040:·5f6e·6f6e·655f·4040·00ff·0402·4190·a0a0··_none_@@....A... | 00000040:·5f6e·6f6e·655f·4040·00ff·0402·4190·a0a0··_none_@@....A... | ||
| 00000050:·a0c0·91b2·0926·2043·6f6d·6269·6e61·746f··.....&·Combinato | 00000050:·a0c0·91b2·0926·2043·6f6d·6269·6e61·746f··.....&·Combinato | ||
| 00000060:·7273·206f·6e20·6d6f·6e61·6469·6320·636f··rs·on·monadic·co | 00000060:·7273·206f·6e20·6d6f·6e61·6469·6320·636f··rs·on·monadic·co | ||
| 00000070:·6d70·7574·6174·696f·6e73·2e20·b0c0·2d63··mputations.·..-c | 00000070:·6d70·7574·6174·696f·6e73·2e20·b0c0·2d63··mputations.·..-c | ||
| 00000080:·6c69·622f·6d6f·6e61·642e·6d6c·4c01·02a5··lib/monad.mlL... | 00000080:·6c69·622f·6d6f·6e61·642e·6d6c·4c01·02a5··lib/monad.mlL... | ||
| 00000090:·0102·a5c0·0402·4c01·02a5·0102·d040·4004··......L......@@. | 00000090:·0102·a5c0·0402·4c01·02a5·0102·d040·4004··......L......@@. | ||
| 000000a0:·0440·4040·0404·4004·0404·0408·0000·3000··.@@@..@.......0. | 000000a0:·0440·4040·0404·4004·0404·0408·0000·3000··.@@@..@.......0. | Diff chunk too large, falling back to line-by-line diff (2918 lines added, 2918 lines removed) | |
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| 00000110:· | 00000110:·5f63·6f6e·7374·7275·6374·6f72·5008·0000··_constructorP... | ||
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| 00000270:· | 00000270:·3800·a0c0·0409·0205·f5e1·0040·5040·41a1··8..........@P@A. | ||
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| 000002c0:·4040· | 000002c0:·4040·048c·4040·4092·041b·a326·6f70·7469··@@..@@@....&opti | ||
| 000002d0:·6 | 000002d0:·6f6e·4a08·0000·3800·a0c0·0425·0205·f5e1··onJ...8....%.... | ||
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| 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 | 00000a10:·3800·4040·4041·4040·4040·4005·0109·4040··8.@@@A@@@@@...@@ | ||
| 00000760:·9204·03a3·3a55·6e64·6566·696e·6564·5f72··....:Undefined_r | |||
| 00000770:·6563·7572·7369·7665·5f6d·6f64·756c·655d··ecursive_module] | |||
| Max diff block lines reached; 11100/46588 bytes (23.83%) of diff not shown. | |||
| Offset 1, 3029 lines modified | Offset 1, 3028 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0000·bd | 00000010:·0000·bd1f·0000·2196·0000·8a25·0000·8852··......!....%...R | ||
| 00000020:·0800·0034·0029·5072·6564·6963·6174·6591··...4.)Predicate. | 00000020:·0800·0034·0029·5072·6564·6963·6174·6591··...4.)Predicate. | ||
| 00000030:·b0a0·b098·d0b1·2b4f·7264·6572·6564·5479··......+OrderedTy | 00000030:·b0a0·b098·d0b1·2b4f·7264·6572·6564·5479··......+OrderedTy | ||
| 00000040:·7065·0053·42a0·0402·b0c0·3163·6c69·622f··pe.SB.....1clib/ | 00000040:·7065·0053·42a0·0402·b0c0·3163·6c69·622f··pe.SB.....1clib/ | ||
| 00000050:·7072·6564·6963·6174·652e·6d6c·5701·05d5··predicate.mlW... | 00000050:·7072·6564·6963·6174·652e·6d6c·5701·05d5··predicate.mlW... | ||
| 00000060:·0105·e1c0·0402·5701·05d5·0105·ec40·90d0··......W......@.. | 00000060:·0105·e1c0·0402·5701·05d5·0105·ec40·90d0··......W......@.. | ||
| 00000070:·91b0·a0b0·a141·a008·0000·2800·b121·7400··.....A....(..!t. | 00000070:·91b0·a0b0·a141·a008·0000·2800·b121·7400··.....A....(..!t. | ||
| 00000080:·5141·a004·02b0·c004·1059·0105·f501·05fe··QA.......Y...... | 00000080:·5141·a004·02b0·c004·1059·0105·f501·05fe··QA.......Y...... | ||
| 00000090:·c004·1159·0105·f501·05ff·4040·0800·0038··...Y......@@...8 | 00000090:·c004·1159·0105·f501·05ff·4040·0800·0038··...Y......@@...8 | ||
| 000000a0:·0040·4040·4140·4040·4040·b0c0·0414·5901··.@@@A@@@@@....Y. | 000000a0:·0040·4040·4140·4040·4040·b0c0·0414·5901··.@@@A@@@@@....Y. | ||
| 000000b0:·05f5·0105·f904·0440·4040·40a1·0421·4040··.......@@@@..!@@ | 000000b0:·05f5·0105·f904·0440·4040·40a1·0421·4040··.......@@@@..!@@ | Diff chunk too large, falling back to line-by-line diff (2839 lines added, 2839 lines removed) | |
| 000000c0:·4041·4004·0340·4008·0000·3000·a040·4004··@A@..@@...0..@@. | 000000c0:·4041·4004·0340·4008·0000·3000·a040·4004··@A@..@@...0..@@. | ||
| 000000d0:·0104·0104·0104·0104·0104·0104·0140·aba | 000000d0:·0104·0104·0104·0104·0104·0104·0140·aba7··.............@.. | ||
| 000000e0:· | 000000e0:·b1b2·b2b2·b2b2·b2b2·b2b2·b2b2·b2b1·b1b1··................ | ||
| 000000f0:·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b1 | 000000f0:·b1b1·b1b1·b1b1·b1b1·b1b1·b1b1·b140·a32a··.............@.* | ||
| 00000100:· | 00000100:·666c·6f61·7461·7272·6179·5108·0000·3800··floatarrayQ...8. | ||
| 00000110:· | 00000110:·4040·4041·4040·4040·40b0·c026·5f6e·6f6e··@@@A@@@@@..&_non | ||
| 00000120:·6 | 00000120:·655f·4040·00ff·0402·4140·4040·9204·06a3··e_@@....A@@@.... | ||
| 00000130:· | 00000130:·3565·7874·656e·7369·6f6e·5f63·6f6e·7374··5extension_const | ||
| 00000140:·7 | 00000140:·7275·6374·6f72·5008·0000·3800·4040·4041··ructorP...8.@@@A | ||
| 00000150:·4 | 00000150:·4040·4040·4004·0740·4040·9204·03a3·2369··@@@@@..@@@....#i | ||
| 00000160:·6 | 00000160:·6e74·4108·0000·3800·4040·4041·4040·4040··ntA...8.@@@A@@@@ | ||
| 00000170:·40 | 00000170:·4004·0b40·4140·9204·03a3·2463·6861·7242··@..@A@....$charB | ||
| 00000180:· | 00000180:·0800·0038·0040·4040·4140·4040·4040·040f··...8.@@@A@@@@@.. | ||
| 00000190:· | 00000190:·4041·4092·0403·a326·7374·7269·6e67·4f08··@A@....&stringO. | ||
| 000001a0:·0 | 000001a0:·0000·3800·4040·4041·4040·4040·4004·1340··..8.@@@A@@@@@..@ | ||
| 000001b0:·4040· | 000001b0:·4040·9204·03a3·2566·6c6f·6174·4408·0000··@@....%floatD... | ||
| 000001c0:· | 000001c0:·3800·4040·4041·4040·4040·4004·1740·4040··8.@@@A@@@@@..@@@ | ||
| 000001d0:· | 000001d0:·9204·03a3·2462·6f6f·6c45·0800·0038·0040··....$boolE...8.@ | ||
| 000001e0:·40 | 000001e0:·40a1·a0e0·a325·6661·6c73·655e·9040·4004··@....%false^.@@. | ||
| 000001f0:· | 000001f0:·2140·9204·03a0·e0a3·2474·7275·655f·9040··!@......$true_.@ | ||
| 00000200:·40 | 00000200:·4004·2740·9204·0340·4041·4040·4040·4004··@.'@...@@A@@@@@. | ||
| 00000210:· | 00000210:·2840·4140·9204·10a3·2475·6e69·7446·0800··(@A@....$unitF.. | ||
| 00000220:·00 | 00000220:·0038·0040·40a1·a0e0·a322·2829·6090·4040··.8.@@...."()`.@@ | ||
| 00000230:· | 00000230:·0432·4092·0403·4040·4140·4040·4040·0433··.2@...@@A@@@@@.3 | ||
| 00000240:· | 00000240:·4041·4092·040a·a323·6578·6e47·0800·0038··@A@....#exnG...8 | ||
| 00000250:· | 00000250:·0040·4041·4140·4040·4040·0437·4040·4092··.@@AA@@@@@.7@@@. | ||
| 00000260:· | 00000260:·0403·a325·6172·7261·7948·0800·0038·00a0··...%arrayH...8.. | ||
| 00000270:· | 00000270:·c090·4002·05f5·e100·404f·4041·4041·40a0··..@.....@O@A@A@. | ||
| 00000280:· | 00000280:·007f·40a0·4040·4040·0440·4040·4092·0408··..@.@@@@.@@@@... | ||
| 00000290:· | 00000290:·a324·6c69·7374·4908·0000·3800·a0c0·0409··.$listI...8..... | ||
| 000002a0:·0 | 000002a0:·0205·f5e1·0040·5040·41a1·a0e0·a322·5b5d··.....@P@A...."[] | ||
| 000002b0:· | 000002b0:·6190·4040·044c·4092·0403·a0e0·a322·3a3a··a.@@.L@......":: | ||
| 000002c0:· | 000002c0:·6290·a004·0ea0·c0b3·9004·16a0·0413·4090··b.............@. | ||
| 000002d0:· | 000002d0:·4002·05f5·e100·4051·4040·0459·4092·040a··@.....@Q@@.Y@... | ||
| 000002e0:· | 000002e0:·4040·4140·a059·40a0·4040·4040·045c·4040··@@A@.Y@.@@@@.\@@ | ||
| 000002f0:·40 | 000002f0:·4092·041b·a326·6f70·7469·6f6e·4a08·0000··@....&optionJ... | ||
| 00000300:· | 00000300:·3800·a0c0·0425·0205·f5e1·0040·5340·41a1··8....%.....@S@A. | ||
| 00000310:·a | 00000310:·a0e0·a324·4e6f·6e65·6390·4040·0468·4092··...$Nonec.@@.h@. | ||
| 00000320:· | 00000320:·0403·a0e0·a324·536f·6d65·6490·a004·0e40··.....$Somed....@ | ||
| 00000330:·40 | 00000330:·4004·6f40·9204·0440·4041·40a0·5940·a040··@.o@...@@A@.Y@.@ | ||
| 00000340:·4040·40 | 00000340:·4040·4004·7240·4040·9204·15a3·266c·617a··@@@.r@@@....&laz | ||
| 00000350:·7 | 00000350:·795f·744e·0800·0038·00a0·c004·3b02·05f5··y_tN...8....;... | ||
| 00000360:· | 00000360:·e100·4055·4041·4041·40a0·5940·a040·4040··..@U@A@A@.Y@.@@@ | ||
| 00000370:·40 | 00000370:·4004·7a40·4040·9204·07a3·296e·6174·6976··@.z@@@....)nativ | ||
| 00000380:· | 00000380:·6569·6e74·4b08·0000·3800·4040·4041·4040··eintK...8.@@@A@@ | ||
| 00000390:·4040·40 | 00000390:·4040·4004·7e40·4040·9204·03a3·2569·6e74··@@@.~@@@....%int | ||
| 000003a0:· | 000003a0:·3332·4c08·0000·3800·4040·4041·4040·4040··32L...8.@@@A@@@@ | ||
| 000003b0:·40 | 000003b0:·4004·8240·4040·9204·03a3·2569·6e74·3634··@..@@@....%int64 | ||
| 000003c0:· | 000003c0:·4d08·0000·3800·4040·4041·4040·4040·4004··M...8.@@@A@@@@@. | ||
| 000003d0:· | 000003d0:·8640·4040·9204·03a3·3a55·6e64·6566·696e··.@@@....:Undefin | ||
| 000003e0:·6 | 000003e0:·6564·5f72·6563·7572·7369·7665·5f6d·6f64··ed_recursive_mod | ||
| 000003f0:· | 000003f0:·756c·655d·0800·0020·0090·0457·4090·a0c0··ule]...·...W@... | ||
| 00000400:· | 00000400:·92a0·c0b3·9004·8340·9040·0205·f5e1·0040··.......@.@.....@ | ||
| 00000410:·4 | 00000410:·4aa0·c0b3·9004·9040·9040·0205·f5e1·0040··J......@.@.....@ | ||
| 00000420:·40 | 00000420:·40a0·0405·4002·05f5·e100·4056·4040·4104··@...@.....@V@@A. | ||
| 00000430:· | 00000430:·9aa0·b0a0·3d6f·6361·6d6c·2e77·6172·6e5f··....=ocaml.warn_ | ||
| 00000440:· | 00000440:·6f6e·5f6c·6974·6572·616c·5f70·6174·7465··on_literal_patte | ||
| 00000450:· | 00000450:·726e·049e·0461·049e·4092·0417·a32e·4173··rn...a..@.....As | ||
| 00000460:·73 | 00000460:·7365·7274·5f66·6169·6c75·7265·5c08·0000··sert_failure\... | ||
| 00000470:· | 00000470:·2000·0418·4090·a0c0·92a0·0417·a004·13a0···...@........... | ||
| 00000480:· | 00000480:·0414·4002·05f5·e100·4058·4040·4104·a9a0··..@.....@X@@A... | ||
| 00000490:· | 00000490:·b0a0·040f·04ac·046f·04ac·4092·040d·a330··.......o..@....0 | ||
| 000004a0:· | 000004a0:·4469·7669·7369·6f6e·5f62·795f·7a65·726f··Division_by_zero | ||
| 000004b0:· | 000004b0:·5908·0000·2000·0426·4090·4040·4104·b1a0··Y...·..&@.@@A... | ||
| 000004c0:· | 000004c0:·b0a0·0417·04b4·0477·04b4·4092·0407·a32b··.......w..@....+ | ||
| 000004d0:· | 000004d0:·456e·645f·6f66·5f66·696c·6558·0800·0020··End_of_fileX...· | ||
| 000004e0:· | 000004e0:·0004·2e40·9040·4041·04b9·a0b0·a004·1f04··...@.@@A........ | ||
| 000004f0:· | 000004f0:·bc04·7f04·bc40·9204·07a3·2953·7973·5f65··.....@....)Sys_e | ||
| 00000500:· | 00000500:·7272·6f72·5708·0000·2000·0436·4090·a004··rrorW...·..6@... | ||
| 00000510:· | 00000510:·3240·4041·04c2·a0b0·a004·2804·c504·8804··2@@A......(..... | ||
| 00000520:· | 00000520:·c540·9204·08a3·2e53·7973·5f62·6c6f·636b··.@.....Sys_block | ||
| 00000530:·6 | 00000530:·6564·5f69·6f5b·0800·0020·0004·3f40·9040··ed_io[...·..?@.@ | ||
| 00000540:·404 | 00000540:·4041·04ca·a0b0·a004·3004·cd04·9004·cd40··@A......0......@ | ||
| 00000550:· | 00000550:·9204·07a3·294e·6f74·5f66·6f75·6e64·5608··....)Not_foundV. | ||
| 00000560:·0 | 00000560:·0000·2000·0447·4090·4040·4104·d2a0·b0a0··..·..G@.@@A..... | ||
| 00000570:· | 00000570:·0438·04d5·0498·04d5·4092·0407·a327·4661··.8......@....'Fa | ||
| 00000580:·6 | 00000580:·696c·7572·6555·0800·0020·0004·4f40·90a0··ilureU...·..O@.. | ||
| 00000590:· | 00000590:·044b·4040·4104·dba0·b0a0·0441·04de·04a1··.K@@A......A.... | ||
| 000005a0:· | 000005a0:·04de·4092·0408·a330·496e·7661·6c69·645f··..@....0Invalid_ | ||
| 000005b0:· | 000005b0:·6172·6775·6d65·6e74·5408·0000·2000·0458··argumentT...·..X | ||
| 000005c0:· | 000005c0:·4090·a004·5440·4041·04e4·a0b0·a004·4a04··@...T@@A......J. | ||
| 000005d0:· | 000005d0:·e704·aa04·e740·9204·08a3·2e53·7461·636b··.....@.....Stack | ||
| 000005e0:· | 000005e0:·5f6f·7665·7266·6c6f·775a·0800·0020·0004··_overflowZ...·.. | ||
| 000005f0:· | 000005f0:·6140·9040·4041·04ec·a0b0·a004·5204·ef04··a@.@@A......R... | ||
| 00000600:· | 00000600:·b204·ef40·9204·07a3·2d4f·7574·5f6f·665f··...@....-Out_of_ | ||
| 00000610:· | 00000610:·6d65·6d6f·7279·5308·0000·2000·0469·4090··memoryS...·..i@. | ||
| 00000620:· | 00000620:·4040·4104·f4a0·b0a0·045a·04f7·04ba·04f7··@@A......Z...... | ||
| 00000630:· | 00000630:·4092·0407·a32d·4d61·7463·685f·6661·696c··@....-Match_fail | ||
| 00000640:· | 00000640:·7572·6552·0800·0020·0004·7140·90a0·c092··ureR...·..q@.... | ||
| 00000650:· | 00000650:·a004·70a0·046c·a004·6d40·0205·f5e1·0040··..p..l..m@.....@ | ||
| 00000660:· | 00000660:·6340·4041·0501·02a0·b0a0·0468·0501·0504··c@@A.......h.... | ||
| 00000670:· | 00000670:·c805·0105·4092·040d·a325·6279·7465·7343··....@....%bytesC | ||
| 00000680:· | 00000680:·0800·0038·0040·4040·4140·4040·4040·0501··...8.@@@A@@@@@.. | ||
| 00000690:·0 | 00000690:·0940·4040·9204·0390·9226·5374·646c·6962··.@@@.....&Stdlib | ||
| 000006a0:· | 000006a0:·9226·4f70·7469·6f6e·4041·b005·0136·0501··.&Option@A...6.. | ||
| 000006b0:· | 000006b0:·3940·a0b0·90f0·a027·636f·6d70·6172·6500··9@.....'compare. | ||
| 000006c0:· | 000006c0:·52a0·0402·b0c0·0501·525a·0106·0001·0608··R.......RZ...... | ||
| 000006d0:· | 000006d0:·c005·0153·5a01·0600·0106·0f40·d0b1·40d0··...SZ......@..@. | ||
| 000006e0:· | 000006e0:·b390·0501·4da0·9021·74b0·c005·015d·5a01··....M..!t....]Z. | ||
| 000006f0:· | 000006f0:·0600·0106·11c0·0501·5e5a·0106·0001·0612··........^Z...... | ||
| 00000700:·0 | 00000700:·4040·c0b3·0409·4090·4002·05f5·e100·4001··@@....@.@.....@. | ||
| 00000710:·0 | 00000710:·0315·0800·0030·0005·014c·0501·4c05·014c··.....0...L..L..L | ||
| 00000720:·0501·4 | 00000720:·0501·4c05·014c·0501·4c05·014c·0501·4c40··..L..L..L..L..L@ | ||
| 00000730:· | 00000730:·b105·014c·0501·5805·0152·4041·b004·0804··...L..X..R@A.... | ||
| 00000740:· | 00000740:·0740·40d0·b140·d0b3·9005·015e·a090·2174··.@@..@.....^..!t | ||
| 00000750:· | 00000750:·b0c0·0501·6e5a·0106·0001·0616·c005·016f··....nZ.........o | ||
| 00000760:· | 00000760:·5a01·0600·0106·1740·40c0·b304·0940·9040··Z......@@....@.@ | ||
| 00000770:· | 00000770:·0205·f5e1·0040·0103·1604·11b0·0406·0405··.....@.......... | ||
| 00000780:· | 00000780:·4040·d0b3·9005·0135·a090·2369·6e74·b0c0··@@.....5..#int.. | ||
| 00000790:· | 00000790:·0501·7b5a·0106·0001·061b·c005·017c·5a01··..{Z.........|Z. | ||
| 000007a0:· | 000007a0:·0600·0106·1e40·40c0·b304·0940·9040·0205··.....@@....@.@.. | ||
| 000007b0:· | 000007b0:·f5e1·0040·0103·1704·1eb0·0406·0405·4040··...@..........@@ | ||
| 000007c0:· | 000007c0:·c0c1·4004·1304·0640·0205·f5e1·0040·0103··..@....@.....@.. | ||
| 000007d0:· | 000007d0:·1804·21b0·0416·0408·4040·c0c1·4004·2704··..!.....@@..@.'. | ||
| Max diff block lines reached; 22413/415252 bytes (5.40%) of diff not shown. | |||
| Offset 1, 9 lines modified | Offset 1, 9 lines modified | ||
| 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | 00000000:·4361·6d6c·3139·3939·5430·3330·8495·a6be··Caml1999T030.... | ||
| 00000010:·0000·4d6 | 00000010:·0000·4d69·0000·0d77·0000·3871·0000·3730··..Mi...w..8q..70 | ||
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| 1287 | -rw-r--r--···0·root·········(0)·root·········(0)···12253 | 1287 | -rw-r--r--···0·root·········(0)·root·········(0)···122537·2022-11-29·08:28:08.000000·./usr/share/doc/libcoq-stdlib/html/Coq.Classes.RelationClasses.html |
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| 1290 | -rw-r--r--···0·root·········(0)·root·········(0)····3250 | 1290 | -rw-r--r--···0·root·········(0)·root·········(0)····32502·2022-11-29·08:28:08.000000·./usr/share/doc/libcoq-stdlib/html/Coq.Classes.SetoidDec.html |
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| 1294 | -rw-r--r--···0·root·········(0)·root·········(0)·····2933·2022-11-29·08:28:08.000000·./usr/share/doc/libcoq-stdlib/html/Coq.Compat.Coq815.html | 1294 | -rw-r--r--···0·root·········(0)·root·········(0)·····2933·2022-11-29·08:28:08.000000·./usr/share/doc/libcoq-stdlib/html/Coq.Compat.Coq815.html |
| 1295 | -rw-r--r--···0·root·········(0)·root·········(0)·····2701·2022-11-29·08:28:08.000000·./usr/share/doc/libcoq-stdlib/html/Coq.Compat.Coq816.html | 1295 | -rw-r--r--···0·root·········(0)·root·········(0)·····2701·2022-11-29·08:28:08.000000·./usr/share/doc/libcoq-stdlib/html/Coq.Compat.Coq816.html |
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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="lab1012"></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="lab1013"></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="lab1014"></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="lab | 57 | <a·id="lab1015"></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="lab954"></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="lab976"></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="lab977"></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/> |
| Offset 48, 27 lines modified | Offset 48, 27 lines modified | ||
| 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="lab948"></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="lab949"></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="lab966"></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="lab967"></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="lab968"></a><h1·class="section">Relationship·between·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1> |
| 71 | </div> | 71 | </div> |
| 72 | <div·class="code"> | 72 | <div·class="code"> |
| 73 | <br/> | 73 | <br/> |
| 74 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Lt·file·is·obsolete.·Use·the·bidirectional·version·Nat.le_succ_l·instead.")]<br/> | 74 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Lt·file·is·obsolete.·Use·the·bidirectional·version·Nat.le_succ_l·instead.")]<br/> |
| 75 | <span·class="id"·title="keyword">Notation</span>·<a·id="lt_le_S"·class="idref"·href="#lt_le_S"><span·class="id"·title="abbreviation">lt_le_S</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#lt_le_S_stt"><span·class="id"·title="definition">Arith_prebase.lt_le_S_stt</span></a>.<br/> | 75 | <span·class="id"·title="keyword">Notation</span>·<a·id="lt_le_S"·class="idref"·href="#lt_le_S"><span·class="id"·title="abbreviation">lt_le_S</span></a>·:=·<a·class="idref"·href="Coq.Arith.Arith_prebase.html#lt_le_S_stt"><span·class="id"·title="definition">Arith_prebase.lt_le_S_stt</span></a>.<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1"></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> | 49 | <a·id="lab1017"></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="lab1011"></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="lab1018"></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="lab1019"></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="lab1020"></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="lab998"></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="lab999"></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="lab1000"></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="lab1001"></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="lab1002"></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="lab1003"></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="lab1004"></a><h2·class="section">Distributivity</h2> |
| 93 | </div> | 93 | </div> |
| 94 | <div·class="code"> | 94 | <div·class="code"> |
| 95 | <br/> | 95 | <br/> |
| 96 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_add_distr_r·instead.")]<br/> | 96 | #[<span·class="id"·title="var">deprecated</span>(<span·class="id"·title="var">since</span>="8.16",<span·class="id"·title="var">note</span>="The·Arith.Mult·file·is·obsolete.·Use·Nat.mul_add_distr_r·instead.")]<br/> |
| 97 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_plus_distr_r"·class="idref"·href="#mult_plus_distr_r"><span·class="id"·title="abbreviation">mult_plus_distr_r</span></a>·:=<br/> | 97 | <span·class="id"·title="keyword">Notation</span>·<a·id="mult_plus_distr_r"·class="idref"·href="#mult_plus_distr_r"><span·class="id"·title="abbreviation">mult_plus_distr_r</span></a>·:=<br/> |
| Offset 144, 15 lines modified | Offset 144, 15 lines modified | ||
| 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="lab8"></a><h2·class="section">Remaining·constants·not·defined·in·Coq.Init.Nat</h2> | 149 | <a·id="lab984"></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="lab958"></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="lab959"></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="lab1080"></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="lab1081"></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/> |
| Offset 61, 15 lines modified | Offset 61, 15 lines modified | ||
| 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="lab1095"></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="lab1133"></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="lab1118"></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="lab1122"></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="lab1114"></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="lab1120"></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="lab1121"></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="lab1128"></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="lab1125"></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="lab1127"></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="lab1113"></a><h1·class="section">Morphism·instances·for·relations.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra | 00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra | ||
| 00000670:·7279·2043·6f71·2e43·6c61·7373·6573·2e52··ry·Coq.Classes.R | 00000670:·7279·2043·6f71·2e43·6c61·7373·6573·2e52··ry·Coq.Classes.R | ||
| 00000680:·656c·6174·696f·6e43·6c61·7373·6573·3c2f··elationClasses</ | 00000680:·656c·6174·696f·6e43·6c61·7373·6573·3c2f··elationClasses</ | ||
| 00000690:·6831·3e0a·0a3c·6469·7620·636c·6173·733d··h1>..<div·class= | 00000690:·6831·3e0a·0a3c·6469·7620·636c·6173·733d··h1>..<div·class= | ||
| 000006a0:·2263·6f64·6522·3e0a·0a3c·6272·2f3e·0a3c··"code">..<br/>.< | 000006a0:·2263·6f64·6522·3e0a·0a3c·6272·2f3e·0a3c··"code">..<br/>.< | ||
| 000006b0:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas | 000006b0:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas | Diff chunk too large, falling back to line-by-line diff (7550 lines added, 7550 lines removed) | |
| 000006c0:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id=" | 000006c0:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id=" | ||
| 000006d0:·6c61·623 | 000006d0:·6c61·6231·3131·3522·3e3c·2f61·3e3c·6831··lab1115"></a><h1 | ||
| 000006e0:·636c | 000006e0:·2063·6c61·7373·3d22·7365·6374·696f·6e22···class="section" | ||
| 000006f0:·5479 | 000006f0:·3e54·7970·6563·6c61·7373·2d62·6173·6564··>Typeclass-based | ||
| 00000700:·7265 | 00000700:·2072·656c·6174·696f·6e73·2c20·7461·6374···relations,·tact | ||
| 00000710:·6373 | 00000710:·6963·7320·616e·6420·7374·616e·6461·7264··ics·and·standard | ||
| 00000720:·696e | 00000720:·2069·6e73·7461·6e63·6573·3c2f·6831·3e0a···instances</h1>. | ||
| 00000730:·0a3c | 00000730:·0a0a·3c64·6976·2063·6c61·7373·3d22·7061··..<div·class="pa | ||
| 00000740:·6167 | 00000740:·7261·6772·6170·6822·3e20·3c2f·6469·763e··ragraph">·</div> | ||
| 00000750:·0a | 00000750:·0a0a·2020·2054·6869·7320·6973·2074·6865··..···This·is·the | ||
| 00000760:·6261 | 00000760:·2062·6173·6963·2074·6865·6f72·7920·6e65···basic·theory·ne | ||
| 00000770:·6465 | 00000770:·6564·6564·2074·6f20·666f·726d·616c·697a··eded·to·formaliz | ||
| 00000780:·206d | 00000780:·6520·6d6f·7270·6869·736d·7320·616e·6420··e·morphisms·and· | ||
| 00000790:·6574 | 00000790:·7365·746f·6964·732e·0a0a·3c64·6976·2063··setoids...<div·c | ||
| 000007a0:·6173 | 000007a0:·6c61·7373·3d22·7061·7261·6772·6170·6822··lass="paragraph" | ||
| 000007b0:·203c | 000007b0:·3e20·3c2f·6469·763e·0a0a·2020·2041·7574··>·</div>..···Aut | ||
| 000007c0:·6f72 | 000007c0:·686f·723a·204d·6174·7468·6965·7520·536f··hor:·Matthieu·So | ||
| 000007d0:·6561 | 000007d0:·7a65·6175·0a20·2020·496e·7374·6974·7574··zeau.···Institut | ||
| 000007e0:·6f6e | 000007e0:·696f·6e3a·204c·5249·2c20·434e·5253·2055··ion:·LRI,·CNRS·U | ||
| 000007f0:·5220 | 000007f0:·4d52·2038·3632·3320·2d20·556e·6976·6572··MR·8623·-·Univer | ||
| 00000800:·6974 | 00000800:·7369·7479·2050·6172·6973·2053·7564·0a0a··sity·Paris·Sud.. | ||
| 00000810:·2f64 | 00000810:·3c2f·6469·763e·0a3c·6469·7620·636c·6173··</div>.<div·clas | ||
| 00000820:·3d | 00000820:·733d·2263·6f64·6522·3e0a·0a3c·6272·2f3e··s="code">..<br/> | ||
| 00000830:·3c73 | 00000830:·0a3c·7370·616e·2063·6c61·7373·3d22·6964··.<span·class="id | ||
| 00000840:·2074 | 00000840:·2220·7469·746c·653d·226b·6579·776f·7264··"·title="keyword | ||
| 00000850:·3e52 | 00000850:·223e·5265·7175·6972·653c·2f73·7061·6e3e··">Require</span> | ||
| 00000860:·3c73 | 00000860:·203c·7370·616e·2063·6c61·7373·3d22·6964···<span·class="id | ||
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| 000008f0:·2f73 | 000008f0:·3c2f·7370·616e·3e3c·2f61·3e2e·3c62·722f··</span></a>.<br/ | ||
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| 00000a50:·5072 | 00000a50:·2e50·726f·6772·616d·2e54·6163·7469·6373··.Program.Tactics | ||
| 00000a60:·6874 | 00000a60:·2e68·746d·6c23·223e·3c73·7061·6e20·636c··.html#"><span·cl | ||
| 00000a70:·7373 | 00000a70:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | ||
| 00000a80:·6962 | 00000a80:·6c69·6272·6172·7922·3e43·6f71·2e50·726f··library">Coq.Pro | ||
| 00000a90:·7261 | 00000a90:·6772·616d·2e54·6163·7469·6373·3c2f·7370··gram.Tactics</sp | ||
| 00000aa0:·6e3e | 00000aa0:·616e·3e3c·2f61·3e2e·3c62·722f·3e0a·3c73··an></a>.<br/>.<s | ||
| 00000ab0:·616e | 00000ab0:·7061·6e20·636c·6173·733d·2269·6422·2074··pan·class="id"·t | ||
| 00000ac0:·746c | 00000ac0:·6974·6c65·3d22·6b65·7977·6f72·6422·3e52··itle="keyword">R | ||
| 00000ad0:·7175 | 00000ad0:·6571·7569·7265·3c2f·7370·616e·3e20·3c73··equire</span>·<s | ||
| 00000ae0:·616e | 00000ae0:·7061·6e20·636c·6173·733d·2269·6422·2074··pan·class="id"·t | ||
| 00000af0:·746c | 00000af0:·6974·6c65·3d22·6b65·7977·6f72·6422·3e49··itle="keyword">I | ||
| 00000b00:·706f | 00000b00:·6d70·6f72·743c·2f73·7061·6e3e·203c·6120··mport</span>·<a· | ||
| 00000b10:·6c61 | 00000b10:·636c·6173·733d·2269·6472·6566·2220·6872··class="idref"·hr | ||
| 00000b20:·663d | 00000b20:·6566·3d22·436f·712e·5265·6c61·7469·6f6e··ef="Coq.Relation | ||
| 00000b30:·2e52 | 00000b30:·732e·5265·6c61·7469·6f6e·5f44·6566·696e··s.Relation_Defin | ||
| 00000b40:·7469 | 00000b40:·6974·696f·6e73·2e68·746d·6c23·223e·3c73··itions.html#"><s | ||
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| 00000b60:·746c | 00000b60:·6974·6c65·3d22·6c69·6272·6172·7922·3e43··itle="library">C | ||
| 00000b70:·712e | 00000b70:·6f71·2e52·656c·6174·696f·6e73·2e52·656c··oq.Relations.Rel | ||
| 00000b80:·7469 | 00000b80:·6174·696f·6e5f·4465·6669·6e69·7469·6f6e··ation_Definition | ||
| 00000b90:·3c2f | 00000b90:·733c·2f73·7061·6e3e·3c2f·613e·2e3c·6272··s</span></a>.<br | ||
| 00000ba0:·3e0a | 00000ba0:·2f3e·0a0a·3c62·722f·3e0a·3c73·7061·6e20··/>..<br/>.<span· | ||
| 00000bb0:·6c61 | 00000bb0:·636c·6173·733d·2269·6422·2074·6974·6c65··class="id"·title | ||
| 00000bc0:·226b | 00000bc0:·3d22·6b65·7977·6f72·6422·3e47·656e·6572··="keyword">Gener | ||
| 00000bd0:·6c69 | 00000bd0:·616c·697a·6162·6c65·2056·6172·6961·626c··alizable·Variabl | ||
| 00000be0:·733c | 00000be0:·6573·3c2f·7370·616e·3e20·3c73·7061·6e20··es</span>·<span· | ||
| 00000bf0:·6c61 | 00000bf0:·636c·6173·733d·2269·6422·2074·6974·6c65··class="id"·title | ||
| 00000c00:·2276 | 00000c00:·3d22·7661·7222·3e41·3c2f·7370·616e·3e20··="var">A</span>· | ||
| 00000c10:·7370 | 00000c10:·3c73·7061·6e20·636c·6173·733d·2269·6422··<span·class="id" | ||
| 00000c20:·7469 | 00000c20:·2074·6974·6c65·3d22·7661·7222·3e42·3c2f···title="var">B</ | ||
| 00000c30:·7061 | 00000c30:·7370·616e·3e20·3c73·7061·6e20·636c·6173··span>·<span·clas | ||
| 00000c40:·3d | 00000c40:·733d·2269·6422·2074·6974·6c65·3d22·7661··s="id"·title="va | ||
| 00000c50:·223e | 00000c50:·7222·3e43·3c2f·7370·616e·3e20·3c73·7061··r">C</span>·<spa | ||
| 00000c60:·2063 | 00000c60:·6e20·636c·6173·733d·2269·6422·2074·6974··n·class="id"·tit | ||
| 00000c70:·653d | 00000c70:·6c65·3d22·7661·7222·3e44·3c2f·7370·616e··le="var">D</span | ||
| 00000c80:·203c | 00000c80:·3e20·3c73·7061·6e20·636c·6173·733d·2269··>·<span·class="i | ||
| 00000c90:·2220 | 00000c90:·6422·2074·6974·6c65·3d22·7661·7222·3e52··d"·title="var">R | ||
| 00000ca0:·2f73 | 00000ca0:·3c2f·7370·616e·3e20·3c73·7061·6e20·636c··</span>·<span·cl | ||
| 00000cb0:·7373 | 00000cb0:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | ||
| 00000cc0:·6172 | 00000cc0:·7661·7222·3e53·3c2f·7370·616e·3e20·3c73··var">S</span>·<s | ||
| 00000cd0:·616e | 00000cd0:·7061·6e20·636c·6173·733d·2269·6422·2074··pan·class="id"·t | ||
| 00000ce0:·746c | 00000ce0:·6974·6c65·3d22·7661·7222·3e54·3c2f·7370··itle="var">T</sp | ||
| 00000cf0:·6e3e | 00000cf0:·616e·3e20·3c73·7061·6e20·636c·6173·733d··an>·<span·class= | ||
| 00000d00:·6964 | 00000d00:·2269·6422·2074·6974·6c65·3d22·7661·7222··"id"·title="var" | ||
| 00000d10:·553c | 00000d10:·3e55·3c2f·7370·616e·3e20·3c73·7061·6e20··>U</span>·<span· | ||
| 00000d20:·6c61 | 00000d20:·636c·6173·733d·2269·6422·2074·6974·6c65··class="id"·title | ||
| 00000d30:·2276 | 00000d30:·3d22·7661·7222·3e6c·3c2f·7370·616e·3e20··="var">l</span>· | ||
| 00000d40:·7370 | 00000d40:·3c73·7061·6e20·636c·6173·733d·2269·6422··<span·class="id" | ||
| 00000d50:·7469 | 00000d50:·2074·6974·6c65·3d22·7661·7222·3e65·7141···title="var">eqA | ||
| 00000d60:·2f73 | 00000d60:·3c2f·7370·616e·3e20·3c73·7061·6e20·636c··</span>·<span·cl | ||
| 00000d70:·7373 | 00000d70:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | ||
| 00000d80:·6172 | 00000d80:·7661·7222·3e65·7142·3c2f·7370·616e·3e20··var">eqB</span>· | ||
| 00000d90:·7370 | 00000d90:·3c73·7061·6e20·636c·6173·733d·2269·6422··<span·class="id" | ||
| 00000da0:·7469 | 00000da0:·2074·6974·6c65·3d22·7661·7222·3e65·7143···title="var">eqC | ||
| 00000db0:·2f73 | 00000db0:·3c2f·7370·616e·3e20·3c73·7061·6e20·636c··</span>·<span·cl | ||
| 00000dc0:·7373 | 00000dc0:·6173·733d·2269·6422·2074·6974·6c65·3d22··ass="id"·title=" | ||
| 00000dd0:·6172 | 00000dd0:·7661·7222·3e65·7144·3c2f·7370·616e·3e2e··var">eqD</span>. | ||
| 00000de0:·6272 | 00000de0:·3c62·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64··<br/>..<br/>.</d | ||
| 00000df0:·763e | 00000df0:·6976·3e0a·0a3c·6469·7620·636c·6173·733d··iv>..<div·class= | ||
| 00000e00:·646f | 00000e00:·2264·6f63·223e·0a57·6520·616c·6c6f·7720··"doc">.We·allow· | ||
| 00000e10:·6f20 | 00000e10:·746f·2075·6e66·6f6c·6420·7468·6520·3c73··to·unfold·the·<s | ||
| 00000e20:·616e | 00000e20:·7061·6e20·636c·6173·733d·2269·6e6c·696e··pan·class="inlin | ||
| 00000e30:·636f | 00000e30:·6563·6f64·6522·3e3c·7370·616e·2063·6c61··ecode"><span·cla | ||
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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="lab1132"></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="lab1131"></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="lab1130"></a><h1·class="section">Decidable·setoid·equality·theory.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab1129"></a><h1·class="section">Tactics·for·typeclass-based·setoids.</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ···Author:·Matthieu·Sozeau | 51 | ···Author:·Matthieu·Sozeau |
| 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud | 52 | ···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud |
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| 47 | <br/> | 47 | <br/> |
| 48 | <br/> | 48 | <br/> |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="doc"> | 50 | <div·class="doc"> |
| 51 | <a·id="lab | 51 | <a·id="lab589"></a><h1·class="section">FMapAVL</h1> |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | ·This·module·implements·maps·using·AVL·trees. | 53 | ·This·module·implements·maps·using·AVL·trees. |
| 54 | ····It·follows·the·implementation·from·Ocaml's·standard·library. | 54 | ····It·follows·the·implementation·from·Ocaml's·standard·library. |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
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| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab560"></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="lab655"></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="lab684"></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="lab672"></a><h1·class="section">Finite·map·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface |
| 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>·using·lists·of·pairs·ordered·(increasing)·with·respect·to | 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.S</span></span>·using·lists·of·pairs·ordered·(increasing)·with·respect·to |
| 53 | ·left·projection.· | 53 | ·left·projection.· |
| 54 | </div> | 54 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab669"></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="lab578"></a><h1·class="section">Finite·map·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent·interface |
| 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.WS</span></span>·using·lists·of·pairs,·unordered·but·without·redundancy.· | 52 | ·<span·class="inlinecode"><span·class="id"·title="var">FMapInterface.WS</span></span>·using·lists·of·pairs,·unordered·but·without·redundancy.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab559"></a><h1·class="section">FSetAVL·:·Implementation·of·FSetInterface·via·AVL·trees</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·implements·finite·sets·using·AVL·trees. | 51 | ·This·module·implements·finite·sets·using·AVL·trees. |
| 52 | ····It·follows·the·implementation·from·Ocaml's·standard·library, | 52 | ····It·follows·the·implementation·from·Ocaml's·standard·library, |
| 53 | <div·class="paragraph">·</div> | 53 | <div·class="paragraph">·</div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab542"></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="lab | 49 | <a·id="lab634"></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="lab | 58 | <a·id="lab635"></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="lab639"></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="lab541"></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="lab536"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>.·These | 51 | ·This·functor·derives·additional·facts·from·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>.·These |
| 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>·written·using | 52 | ··facts·are·mainly·the·specifications·of·<span·class="inlinecode"><span·class="id"·title="var">FSetInterface.S</span></span>·written·using |
| 53 | ··different·styles:·equivalence·and·boolean·equalities. | 53 | ··different·styles:·equivalence·and·boolean·equalities. |
| 54 | ··Moreover,·we·prove·that·<span·class="inlinecode"><span·class="id"·title="var">E.Eq</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Equal</span></span>·are·setoid·equalities. | 54 | ··Moreover,·we·prove·that·<span·class="inlinecode"><span·class="id"·title="var">E.Eq</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Equal</span></span>·are·setoid·equalities. |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab690"></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="lab540"></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="lab545"></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="lab670"></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="lab671"></a><h1·class="section">Going·from·<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·with·usual·Leibniz·equality</h1> |
| 59 | ····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.· | 59 | ····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.· |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/> | 63 | <span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab668"></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="lab5 | 53 | <a·id="lab535"></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="lab5 | 54 | <a·id="lab518"></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="lab5 | 53 | <a·id="lab534"></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/> |
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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.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="lab5 | 53 | <a·id="lab519"></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="lab5 | 58 | <a·id="lab520"></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="lab5 | 53 | <a·id="lab530"></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="lab5 | 58 | <a·id="lab531"></a><h2·class="section">Inductive·specification·of·floating-point·numbers</h2> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | Similar·to·<span·class="inlinecode"><span·class="id"·title="var">Flocq.IEEE754.Binary.full_float</span></span>,·but·with·no·NaN·payload.· | 60 | Similar·to·<span·class="inlinecode"><span·class="id"·title="var">Flocq.IEEE754.Binary.full_float</span></span>,·but·with·no·NaN·payload.· |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab297"></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="lab304"></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="lab303"></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="lab328"></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·6120·6964·3d22·6c61·6232··oc">.<a·id="lab2 | ||
| 00000a50:·3 | 00000a50:·3939·223e·3c2f·613e·3c68·3120·636c·6173··99"></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="lab311"></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="lab327"></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="lab325"></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="lab298"></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="lab310"></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="lab48"></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="lab80"></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="lab79"></a><h1·class="section">Memoization</h1> |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
| 56 | ·Successive·outputs·of·a·given·function·<span·class="inlinecode"><span·class="id"·title="var">f</span></span>·are·stored·in | 56 | ·Successive·outputs·of·a·given·function·<span·class="inlinecode"><span·class="id"·title="var">f</span></span>·are·stored·in |
| 57 | ····a·stream·in·order·to·avoid·duplicated·computations.· | 57 | ····a·stream·in·order·to·avoid·duplicated·computations.· |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="code"> | 59 | <div·class="code"> |
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| 46 | <div·class="code"> | 46 | <div·class="code"> |
| 47 | <br/> | 47 | <br/> |
| 48 | </div> | 48 | </div> |
| 49 | <div·class="doc"> | 49 | <div·class="doc"> |
| 50 | Some·facts·and·definitions·concerning·choice·and·description·in | 50 | Some·facts·and·definitions·concerning·choice·and·description·in |
| 51 | ·······intuitionistic·logic.·<a·id="lab | 51 | ·······intuitionistic·logic.·<a·id="lab208"></a><h1·class="section">References:</h1> |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | <span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory, | 53 | <span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory, |
| 54 | unpublished. | 54 | unpublished. |
| 55 | <div·class="paragraph">·</div> | 55 | <div·class="paragraph">·</div> |
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| 129 | <br/> | 129 | <br/> |
| 130 | <span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/> | 130 | <span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/> |
| 131 | <br/> | 131 | <br/> |
| 132 | </div> | 132 | </div> |
| 133 | <div·class="doc"> | 133 | <div·class="doc"> |
| 134 | <a·id="lab | 134 | <a·id="lab286"></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="lab194"></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="lab292"></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="lab240"></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="lab237"></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="lab243"></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="lab285"></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="lab47 | 176 | <a·id="lab247"></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="lab287"></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="lab296"></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="lab295"></a><h1·class="section">The·Set·universe·seen·as·a·synonym·for·Type</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·After·loading·this·file,·Set·becomes·just·another·name·for·Type. | 51 | ·After·loading·this·file,·Set·becomes·just·another·name·for·Type. |
| 52 | ····This·allows·easily·performing·a·Set-to-Type·migration,·or·at·least | 52 | ····This·allows·easily·performing·a·Set-to-Type·migration,·or·at·least |
| 53 | ····test·whether·a·development·relies·or·not·on·specific·features·of | 53 | ····test·whether·a·development·relies·or·not·on·specific·features·of |
| 54 | ····Set:·simply·insert·some·Require·Export·of·this·file·at·starting | 54 | ····Set:·simply·insert·some·Require·Export·of·this·file·at·starting |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1220"></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="lab12 | 76 | <a·id="lab1254"></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="lab1 | 49 | <a·id="lab1291"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·module·proves·many·properties·of·finite·sets·that | 51 | ·This·module·proves·many·properties·of·finite·sets·that |
| 52 | ····are·consequences·of·the·axiomatization·in·<span·class="inlinecode"><span·class="id"·title="var">FsetInterface</span></span> | 52 | ····are·consequences·of·the·axiomatization·in·<span·class="inlinecode"><span·class="id"·title="var">FsetInterface</span></span> |
| 53 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FsetProperties</span></span>·it·uses | 53 | ····Contrary·to·the·functor·in·<span·class="inlinecode"><span·class="id"·title="var">FsetProperties</span></span>·it·uses |
| 54 | ····sets·operations·instead·of·predicates·over·sets,·i.e. | 54 | ····sets·operations·instead·of·predicates·over·sets,·i.e. |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1284"></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. |
| 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="lab11 | 49 | <a·id="lab1151"></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="lab11 | 49 | <a·id="lab1139"></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="lab11 | 49 | <a·id="lab1146"></a><h1·class="section">Finite·sets·library</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·This·file·proposes·an·implementation·of·the·non-dependent | 51 | ·This·file·proposes·an·implementation·of·the·non-dependent |
| 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>·using·strictly·ordered·list.· | 52 | ····interface·<span·class="inlinecode"><span·class="id"·title="var">MSetInterface.S</span></span>·using·strictly·ordered·list.· |
| 53 | </div> | 53 | </div> |
| 54 | <div·class="code"> | 54 | <div·class="code"> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab1 | 49 | <a·id="lab1270"></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>, |
| 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="lab1178"></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> |
| 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="lab1289"></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="lab1 | 58 | <a·id="lab1290"></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="lab1 | 49 | <a·id="lab1134"></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="lab21"></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> |
| Offset 54, 56 lines modified | Offset 54, 56 lines modified | ||
| 54 | <br/> | 54 | <br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab1 | 58 | <a·id="lab12"></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="lab1 | 68 | <a·id="lab13"></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="lab1 | 76 | <a·id="lab14"></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="lab1 | 86 | <a·id="lab15"></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="lab23"></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/> |
| 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="lab902"></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="lab9 | 49 | <a·id="lab936"></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="lab9 | 49 | <a·id="lab910"></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="lab9 | 58 | <a·id="lab909"></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="lab9 | 53 | <a·id="lab921"></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="lab91 | 65 | <a·id="lab912"></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="lab9 | 53 | <a·id="lab911"></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="lab9 | 49 | <a·id="lab935"></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="lab9 | 49 | <a·id="lab934"></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="lab90 | 49 | <a·id="lab908"></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="lab881"></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/> |
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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="lab905"></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="lab94 | 49 | <a·id="lab940"></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="lab872"></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="lab884"></a><h1·class="section">DecimalQ</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span> | 51 | ····Proofs·that·conversions·between·decimal·numbers·and·<span·class="inlinecode"><span·class="id"·title="var">Q</span></span> |
| 52 | ····are·bijections.· | 52 | ····are·bijections.· |
| 53 | </div> | 53 | </div> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab90 | 49 | <a·id="lab904"></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> |
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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.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="lab9 | 53 | <a·id="lab906"></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="lab883"></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="lab9 | 49 | <a·id="lab938"></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="lab939"></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="lab871"></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="lab882"></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="lab90 | 49 | <a·id="lab907"></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="lab9 | 49 | <a·id="lab901"></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="lab947"></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="lab9 | 49 | <a·id="lab903"></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="lab895"></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="lab889"></a><h1·class="section">Euclidean·Division·for·integers,·Euclid·convention</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ····We·use·here·the·"usual"·formulation·of·the·Euclid·Theorem | 55 | ····We·use·here·the·"usual"·formulation·of·the·Euclid·Theorem |
| 56 | ····<span·class="inlinecode"><span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span><>0</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="tactic">exists</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><span·class="id"·title="var">q</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode">=</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>*<span·class="id"·title="var">q</span>+<span·class="id"·title="var">r</span></span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">0</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><</span>·<span·class="inlinecode">|<span·class="id"·title="var">b</span>|</span>·<span·class="inlinecode"></span> | 56 | ····<span·class="inlinecode"><span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span><>0</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="tactic">exists</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><span·class="id"·title="var">q</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode">=</span>·<span·class="inlinecode"><span·class="id"·title="var">b</span>*<span·class="id"·title="var">q</span>+<span·class="id"·title="var">r</span></span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">0</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·<span·class="inlinecode"><</span>·<span·class="inlinecode">|<span·class="id"·title="var">b</span>|</span>·<span·class="inlinecode"></span> |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab896"></a><h1·class="section">Euclidean·Division·for·integers·(Floor·convention)</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ····We·use·here·the·convention·known·as·Floor,·or·Round-Toward-Bottom, | 55 | ····We·use·here·the·convention·known·as·Floor,·or·Round-Toward-Bottom, |
| 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·closest·integer·below·the·exact·fraction. | 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·closest·integer·below·the·exact·fraction. |
| 57 | ····It·can·be·summarized·by: | 57 | ····It·can·be·summarized·by: |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZDiv.html#"><span·class="id"·title="library">NZDiv</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab885"></a><h1·class="section">Euclidean·Division·for·integers·(Trunc·convention)</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ····We·use·here·the·convention·known·as·Trunc,·or·Round-Toward-Zero, | 55 | ····We·use·here·the·convention·known·as·Trunc,·or·Round-Toward-Zero, |
| 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·integer·with·the·largest·absolute·value·to | 56 | ····where·<span·class="inlinecode"><span·class="id"·title="var">a</span>/<span·class="id"·title="var">b</span></span>·is·the·integer·with·the·largest·absolute·value·to |
| 57 | ····be·between·zero·and·the·exact·fraction.·It·can·be·summarized·by: | 57 | ····be·between·zero·and·the·exact·fraction.·It·can·be·summarized·by: |
| Offset 48, 15 lines modified | Offset 48, 15 lines modified | ||
| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZGcd.html#"><span·class="id"·title="library">ZGcd</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivTrunc.html#"><span·class="id"·title="library">ZDivTrunc</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivFloor.html#"><span·class="id"·title="library">ZDivFloor</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZSgnAbs.html#"><span·class="id"·title="library">ZSgnAbs</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZGcd.html#"><span·class="id"·title="library">ZGcd</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivTrunc.html#"><span·class="id"·title="library">ZDivTrunc</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZDivFloor.html#"><span·class="id"·title="library">ZDivFloor</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab894"></a><h1·class="section">Least·Common·Multiple</h1> |
| 54 | <div·class="paragraph">·</div> | 54 | <div·class="paragraph">·</div> |
| 55 | ·Unlike·other·functions·around,·we·will·define·lcm·below·instead·of | 55 | ·Unlike·other·functions·around,·we·will·define·lcm·below·instead·of |
| 56 | ··axiomatizing·it.·Indeed,·there·is·no·"prior·art"·about·lcm·in·the | 56 | ··axiomatizing·it.·Indeed,·there·is·no·"prior·art"·about·lcm·in·the |
| 57 | ··standard·library·to·be·compliant·with,·and·the·generic·definition | 57 | ··standard·library·to·be·compliant·with,·and·the·generic·definition |
| 58 | ··of·lcm·via·gcd·is·quite·reasonable. | 58 | ··of·lcm·via·gcd·is·quite·reasonable. |
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| 48 | <br/> | 48 | <br/> |
| 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> | 49 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZAxioms.html#"><span·class="id"·title="library">ZAxioms</span></a>·<a·class="idref"·href="Coq.Numbers.Integer.Abstract.ZMulOrder.html#"><span·class="id"·title="library">ZMulOrder</span></a>·<a·class="idref"·href="Coq.Structures.GenericMinMax.html#"><span·class="id"·title="library">GenericMinMax</span></a>.<br/> |
| 50 | <br/> | 50 | <br/> |
| 51 | </div> | 51 | </div> |
| 52 | <div·class="doc"> | 52 | <div·class="doc"> |
| 53 | <a·id="lab | 53 | <a·id="lab893"></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="lab9 | 70 | <a·id="lab900"></a><h1·class="section">An·<span·class="inlinecode"><span·class="id"·title="var">order</span></span>·tactic·for·integers</h1> |
| 71 | </div> | 71 | </div> |
| 72 | <div·class="code"> | 72 | <div·class="code"> |
| 73 | <br/> | 73 | <br/> |
| 74 | <span·class="id"·title="keyword">Ltac</span>·<span·class="id"·title="var">z_order</span>·:=·<span·class="id"·title="var">Z.order</span>.<br/> | 74 | <span·class="id"·title="keyword">Ltac</span>·<span·class="id"·title="var">z_order</span>·:=·<span·class="id"·title="var">Z.order</span>.<br/> |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.List.html#"><span·class="id"·title="library">List</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Lists.List.html#"><span·class="id"·title="library">List</span></a>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab93 | 56 | <a·id="lab937"></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="lab877"></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/> |
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| 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="lab873"></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"> |
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| 414 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZPow.html#NZPowProp"><span·class="id"·title="module">NZPowProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>)<br/> | 414 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZPow.html#NZPowProp"><span·class="id"·title="module">NZPowProp</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>)<br/> |
| 415 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">F</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#NZLog2Prop"><span·class="id"·title="module">NZLog2Prop</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#C"><span·class="id"·title="module">C</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#E"><span·class="id"·title="module">E</span></a>).<br/> | 415 | (<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">F</span>·:·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#NZLog2Prop"><span·class="id"·title="module">NZLog2Prop</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#A"><span·class="id"·title="module">A</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#B"><span·class="id"·title="module">B</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#C"><span·class="id"·title="module">C</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#D"><span·class="id"·title="module">D</span></a>·<a·class="idref"·href="Coq.Numbers.NatInt.NZLog.html#E"><span·class="id"·title="module">E</span></a>).<br/> |
| 416 | <br/> | 416 | <br/> |
| 417 | </div> | 417 | </div> |
| 418 | <div·class="doc"> | 418 | <div·class="doc"> |
| 419 | <a·id="lab | 419 | <a·id="lab880"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">log2_up</span></span>·:·a·binary·logarithm·that·rounds·up·instead·of·down</h1> |
| 420 | <div·class="paragraph">·</div> | 420 | <div·class="paragraph">·</div> |
| 421 | ·For·once,·we·define·instead·of·axiomatizing,·thanks·to·log2· | 421 | ·For·once,·we·define·instead·of·axiomatizing,·thanks·to·log2· |
| 422 | </div> | 422 | </div> |
| 423 | <div·class="code"> | 423 | <div·class="code"> |
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| 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="lab876"></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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| 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="lab94 | 53 | <a·id="lab945"></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="lab94 | 53 | <a·id="lab944"></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="lab94 | 58 | <a·id="lab946"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">BinNat.N</span></span>·already·implements·<span·class="inlinecode"><span·class="id"·title="var">NAxiomSig</span></span></h1> |
| 59 | </div> | 59 | </div> |
| 60 | <div·class="code"> | 60 | <div·class="code"> |
| 61 | <br/> | 61 | <br/> |
| 62 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>·<:·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#NAxiomsSig"><span·class="id"·title="module">NAxiomsSig</span></a>·:=·<a·class="idref"·href="Coq.NArith.BinNat.html#N"><span·class="id"·title="module">N</span></a>.<br/> | 62 | <span·class="id"·title="keyword">Module</span>·<a·id="N"·class="idref"·href="#N"><span·class="id"·title="module">N</span></a>·<:·<a·class="idref"·href="Coq.Numbers.Natural.Abstract.NAxioms.html#NAxiomsSig"><span·class="id"·title="module">NAxiomsSig</span></a>·:=·<a·class="idref"·href="Coq.NArith.BinNat.html#N"><span·class="id"·title="module">N</span></a>.<br/> |
| Offset 53, 15 lines modified | Offset 53, 15 lines modified | ||
| 53 | <br/> | 53 | <br/> |
| 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.PArith.BinPosDef.html#"><span·class="id"·title="library">BinPosDef</span></a>.<br/> | 54 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.PArith.BinPosDef.html#"><span·class="id"·title="library">BinPosDef</span></a>.<br/> |
| 55 | <br/> | 55 | <br/> |
| 56 | </div> | 56 | </div> |
| 57 | <div·class="doc"> | 57 | <div·class="doc"> |
| 58 | <a·id="lab | 58 | <a·id="lab109"></a><h1·class="section">Binary·positive·numbers,·operations·and·properties</h1> |
| 59 | <div·class="paragraph">·</div> | 59 | <div·class="paragraph">·</div> |
| 60 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· | 60 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· |
| 61 | <div·class="paragraph">·</div> | 61 | <div·class="paragraph">·</div> |
| 62 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> | 62 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> |
| Offset 45, 15 lines modified | Offset 45, 15 lines modified | ||
| 45 | <div·class="code"> | 45 | <div·class="code"> |
| 46 | <br/> | 46 | <br/> |
| 47 | </div> | 47 | </div> |
| 48 | <div·class="doc"> | 48 | <div·class="doc"> |
| 49 | <a·id="lab | 49 | <a·id="lab82"></a><h1·class="section">Binary·positive·numbers,·operations</h1> |
| 50 | <div·class="paragraph">·</div> | 50 | <div·class="paragraph">·</div> |
| 51 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· | 51 | ·Initial·development·by·Pierre·Crégut,·CNET,·Lannion,·France· |
| 52 | <div·class="paragraph">·</div> | 52 | <div·class="paragraph">·</div> |
| 53 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> | 53 | ·The·type·<span·class="inlinecode"><span·class="id"·title="var">positive</span></span>·and·its·constructors·<span·class="inlinecode"><span·class="id"·title="var">xI</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xO</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">xH</span></span> |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">positive_scope</span>.<br/> | 52 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">positive_scope</span>.<br/> |
| 53 | <br/> | 53 | <br/> |
| 54 | </div> | 54 | </div> |
| 55 | <div·class="doc"> | 55 | <div·class="doc"> |
| 56 | <a·id="lab | 56 | <a·id="lab164"></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/> |
| Offset 43, 15 lines modified | Offset 43, 15 lines modified | ||
| 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="lab81"></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 |
| Offset 51, 15 lines modified | Offset 51, 15 lines modified | ||
| 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="lab355"></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="lab331"></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="lab329"></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="lab330"></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/> |
| 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.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="lab334"></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="lab336"></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="lab337"></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="lab338"></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="lab339"></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="lab340"></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="lab | 58 | <a·id="lab779"></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="lab807"></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:·6964·3d22·6c61·6238·3038·223e·3c2f·613e··id="lab808"></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="lab802"></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="lab855"></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:·6964·3d22·6c61·6238·3133·223e·3c2f·613e··id="lab813"></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="lab | 56 | <a·id="lab866"></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="lab785"></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="lab809"></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="lab713"></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="lab | 61 | <a·id="lab793"></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·6162·3 | 00001280:·3e0a·3c61·2069·643d·226c·6162·3734·3122··>.<a·id="lab741" | ||
| 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:·6964·3d22·6c61·623 | 000012e0:·6964·3d22·6c61·6237·3432·223e·3c2f·613e··id="lab742"></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="lab770"></a><h1·class="section">Field·operations</h1> |
| 71 | <div·class="paragraph">·</div> | 71 | <div·class="paragraph">·</div> |
| 72 | <a·id="lab | 72 | <a·id="lab771"></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="lab738"></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="lab869"></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="lab706"></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="lab766"></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="lab | 62 | <a·id="lab850"></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="lab733"></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="lab9 | 76 | <a·id="lab790"></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="lab811"></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="lab728"></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="lab704"></a><h1·class="section">Definition·of·sequence·and·properties</h1> |
| 60 | </div> | 60 | </div> |
| 61 | <div·class="code"> | 61 | <div·class="code"> |
| 62 | <br/> | 62 | <br/> |
| 63 | <span·class="id"·title="keyword">Section</span>·<a·id="sequence"·class="idref"·href="#sequence"><span·class="id"·title="section">sequence</span></a>.<br/> | 63 | <span·class="id"·title="keyword">Section</span>·<a·id="sequence"·class="idref"·href="#sequence"><span·class="id"·title="section">sequence</span></a>.<br/> |
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| 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Logic.Classical_Pred_Type.html#"><span·class="id"·title="library">Classical_Pred_Type</span></a>.<br/> | 55 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Logic.Classical_Pred_Type.html#"><span·class="id"·title="library">Classical_Pred_Type</span></a>.<br/> |
| 56 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> | 56 | <span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">R_scope</span>.<br/> |
| 57 | <br/> | 57 | <br/> |
| 58 | </div> | 58 | </div> |
| 59 | <div·class="doc"> | 59 | <div·class="doc"> |
| 60 | <a·id="lab | 60 | <a·id="lab787"></a><h1·class="section">General·definitions·and·propositions</h1> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="code"> | 62 | <div·class="code"> |
| 63 | <br/> | 63 | <br/> |
| 64 | <span·class="id"·title="keyword">Definition</span>·<a·id="included"·class="idref"·href="#included"><span·class="id"·title="definition">included</span></a>·(<a·id="D1:1"·class="idref"·href="#D1:1"><span·class="id"·title="binder">D1</span></a>·<a·id="D2:2"·class="idref"·href="#D2:2"><span·class="id"·title="binder">D2</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<span·class="id"·title="keyword">forall</span>·<a·id="x:3"·class="idref"·href="#x:3"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtopology.html#D1:1"><span·class="id"·title="variable">D1</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#D2:2"><span·class="id"·title="variable">D2</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>.<br/> | 64 | <span·class="id"·title="keyword">Definition</span>·<a·id="included"·class="idref"·href="#included"><span·class="id"·title="definition">included</span></a>·(<a·id="D1:1"·class="idref"·href="#D1:1"><span·class="id"·title="binder">D1</span></a>·<a·id="D2:2"·class="idref"·href="#D2:2"><span·class="id"·title="binder">D2</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<span·class="id"·title="keyword">Prop</span>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<span·class="id"·title="keyword">forall</span>·<a·id="x:3"·class="idref"·href="#x:3"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>,·<a·class="idref"·href="Coq.Reals.Rtopology.html#D1:1"><span·class="id"·title="variable">D1</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#D2:2"><span·class="id"·title="variable">D2</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:3"><span·class="id"·title="variable">x</span></a>.<br/> |
| 65 | <span·class="id"·title="keyword">Definition</span>·<a·id="disc"·class="idref"·href="#disc"><span·class="id"·title="definition">disc</span></a>·(<a·id="x:4"·class="idref"·href="#x:4"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·(<a·id="delta:5"·class="idref"·href="#delta:5"><span·class="id"·title="binder">delta</span></a>:<a·class="idref"·href="Coq.Reals.RIneq.html#posreal"><span·class="id"·title="record">posreal</span></a>)·(<a·id="y:6"·class="idref"·href="#y:6"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#Rabs"><span·class="id"·title="definition">Rabs</span></a>·(<a·class="idref"·href="Coq.Reals.Rtopology.html#y:6"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:4"><span·class="id"·title="variable">x</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#delta:5"><span·class="id"·title="variable">delta</span></a>.<br/> | 65 | <span·class="id"·title="keyword">Definition</span>·<a·id="disc"·class="idref"·href="#disc"><span·class="id"·title="definition">disc</span></a>·(<a·id="x:4"·class="idref"·href="#x:4"><span·class="id"·title="binder">x</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·(<a·id="delta:5"·class="idref"·href="#delta:5"><span·class="id"·title="binder">delta</span></a>:<a·class="idref"·href="Coq.Reals.RIneq.html#posreal"><span·class="id"·title="record">posreal</span></a>)·(<a·id="y:6"·class="idref"·href="#y:6"><span·class="id"·title="binder">y</span></a>:<a·class="idref"·href="Coq.Reals.Rdefinitions.html#RbaseSymbolsImpl.R"><span·class="id"·title="axiom">R</span></a>)·:·<span·class="id"·title="keyword">Prop</span>·:=·<a·class="idref"·href="Coq.Reals.Rbasic_fun.html#Rabs"><span·class="id"·title="definition">Rabs</span></a>·(<a·class="idref"·href="Coq.Reals.Rtopology.html#y:6"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#x:4"><span·class="id"·title="variable">x</span></a>)·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Reals.Rtopology.html#delta:5"><span·class="id"·title="variable">delta</span></a>.<br/> |
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| 160 | 1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> | 160 | 1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span·class="id"·title="notation"><></span></a>·0·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/> |
| 161 | <a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·(<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">/</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a>1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>.<br/> | 161 | <a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·(<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3e57074ab0a6a36b253e873a1bdac6ad"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">/</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">(</span></a>1·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#::R_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#x:52"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Reals.Rdefinitions.html#e1dae7ed53bffa765009199d058dd327"><span·class="id"·title="notation">*</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#tan"><span·class="id"·title="definition">tan</span></a>·<a·class="idref"·href="Coq.Reals.Rtrigo1.html#y:53"><span·class="id"·title="variable">y</span></a><a·class="idref"·href="Coq.Reals.Rdefinitions.html#3942bfc45fbeaeee49010b741755982b"><span·class="id"·title="notation">)</span></a>.<br/> |
| 162 | <br/> | 162 | <br/> |
| 163 | </div> | 163 | </div> |
| 164 | <div·class="doc"> | 164 | <div·class="doc"> |
| 165 | <a·id="lab | 165 | <a·id="lab799"></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="lab78"></a><h1·class="section">Definition·of·exponential</h1> | 54 | <a·id="lab782"></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="lab698"></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="lab44 | 49 | <a·id="lab44"></a><h1·class="section">Some·properties·of·the·operators·on·relations</h1> |
| 50 | <a·id="lab4 | 50 | <a·id="lab45"></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="lab25"></a><h1·class="section">Some·operators·on·relations</h1> |
| 50 | <a·id="lab | 50 | <a·id="lab26"></a><h1·class="section">Initial·authors:·Bruno·Barras,·Cristina·Cornes</h1> |
| 51 | <a·id="lab | 51 | <a·id="lab27"></a><h1·class="section">···································································</h1> |
| 52 | <a·id="lab | 52 | <a·id="lab28"></a><h1·class="section">Some·of·the·initial·definitions·were·taken·from·:</h1> |
| 53 | <a·id="lab | 53 | <a·id="lab29"></a><h1·class="section">Constructing·Recursion·Operators·in·Type·Theory</h1> |
| 54 | <a·id="lab | 54 | <a·id="lab30"></a><h1·class="section">L.·Paulson··JSC·(1986)·2,·325-355</h1> |
| 55 | <a·id="lab | 55 | <a·id="lab31"></a><h1·class="section">···································································</h1> |
| 56 | <a·id="lab | 56 | <a·id="lab32"></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="lab33"></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="lab1101"></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="lab1103"></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="lab1104"></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="lab1098"></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="lab1102"></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="lab1"></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="lab4 | 66 | <a·id="lab4"></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="lab1 | 61 | <a·id="lab1042"></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="lab1 | 69 | <a·id="lab1043"></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="lab1 | 77 | <a·id="lab1044"></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="lab1 | 58 | <a·id="lab1052"></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="lab1 | 70 | <a·id="lab1070"></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="lab1 | 80 | <a·id="lab1071"></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="lab1 | 56 | <a·id="lab1040"></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="lab1 | 53 | <a·id="lab1053"></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="lab1 | 55 | <a·id="lab1054"></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="lab1 | 58 | <a·id="lab1029"></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/> |
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| 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="lab1 | 51 | <a·id="lab1062"></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="lab1 | 57 | <a·id="lab1063"></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="lab1 | 54 | <a·id="lab1031"></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="lab1 | 56 | <a·id="lab1064"></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="lab1 | 59 | <a·id="lab1065"></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="lab1 | 67 | <a·id="lab1066"></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="lab1 | 57 | <a·id="lab1045"></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="lab1 | 57 | <a·id="lab1046"></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="lab1 | 56 | <a·id="lab1038"></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="lab1 | 54 | <a·id="lab1025"></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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| 00017040:· | 00017040:·6d6c·236e·3a31·3939·223e·3c73·7061·6e20··ml#n:199"><span· | ||
| 00017050:· | 00017050:·636c·6173·733d·2269·6422·2074·6974·6c65··class="id"·title | ||
| 00017060:· | 00017060:·3d22·7661·7269·6162·6c65·223e·6e3c·2f73··="variable">n</s | ||
| 00017070:·7 | 00017070:·7061·6e3e·3c2f·613e·203a·3d3c·6272·2f3e··pan></a>·:=<br/> | ||
| 00017080:· | 00017080:·0a26·6e62·7370·3b26·6e62·7370·3b3c·7370··. <sp | ||
| 00017090:· | 00017090:·616e·2063·6c61·7373·3d22·6964·2220·7469··an·class="id"·ti | ||
| 000170a0:· | 000170a0:·746c·653d·226b·6579·776f·7264·223e·6669··tle="keyword">fi | ||
| 000170b0:· | 000170b0:·783c·2f73·7061·6e3e·203c·7370·616e·2063··x</span>·<span·c | ||
| 000170c0:·6 | 000170c0:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title= | ||
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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="lab373"></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="lab388"></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="lab490"></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="lab486"></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="lab3 | 75 | <a·id="lab385"></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="lab468"></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="lab470"></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="lab471"></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="lab437"></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="lab481"></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="lab485"></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="lab447"></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="lab4 | 103 | <a·id="lab482"></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="lab4 | 113 | <a·id="lab483"></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="lab478"></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="lab461"></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="lab448"></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="lab449"></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="lab4 | 54 | <a·id="lab467"></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="lab488"></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="lab458"></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="lab428"></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="lab466"></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="lab167"></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="lab169"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab170"></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="lab171"></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="lab183"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab184"></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="lab185"></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="lab172"></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="lab173"></a><h1·class="section">into·equivalent·versions·in·Haskell's·Prelude·that·are·defined</h1> |
| 49 | <a·id="lab | 49 | <a·id="lab174"></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="lab175"></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="lab176"></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="lab177"></a><h1·class="section">Special·handling·of·ascii·and·strings·for·extraction·to·Haskell.</h1> |
| 48 | · | 48 | · |
| 49 | </div> | 49 | </div> |
| 50 | <div·class="code"> | 50 | <div·class="code"> |
| 51 | <br/> | 51 | <br/> |
| 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> | 52 | <span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.extraction.Extraction.html#"><span·class="id"·title="library">Coq.extraction.Extraction</span></a>.<br/> |
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| 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> | 58 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>.<br/> |
| 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellZNum.html#"><span·class="id"·title="library">ExtrHaskellZNum</span></a>.<br/> | 59 | <span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.extraction.ExtrHaskellZNum.html#"><span·class="id"·title="library">ExtrHaskellZNum</span></a>.<br/> |
| 60 | <br/> | 60 | <br/> |
| 61 | </div> | 61 | </div> |
| 62 | <div·class="doc"> | 62 | <div·class="doc"> |
| 63 | <a·id="lab | 63 | <a·id="lab186"></a><h1·class="section">Disclaimer:·trying·to·obtain·efficient·certified·programs</h1> |
| 64 | <a·id="lab | 64 | <a·id="lab187"></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="lab188"></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="lab189"></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="lab190"></a><h1·class="section">into·equivalent·versions·in·Haskell's·Prelude·that·are·defined</h1> |
| 49 | <a·id="lab | 49 | <a·id="lab191"></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="lab192"></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="lab193"></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="lab10 | 56 | <a·id="lab1079"></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 |
| Offset 558, 16 lines modified | Offset 558, 16 lines modified | ||
| 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.Program.Tactics.html">Tactics</a>·[library]<br/> | ||
| 566 | <a·href="Coq.Init.Tactics.html">Tactics</a>·[library]<br/> | 565 | <a·href="Coq.Init.Tactics.html">Tactics</a>·[library]<br/> |
| 566 | <a·href="Coq.Program.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/> |
| 571 | <a·href="Coq.ssr.ssrfun.html#Tagged">Tagged</a>·[definition,·in·<a·href="Coq.ssr.ssrfun.html">Coq.ssr.ssrfun</a>]<br/> | 571 | <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/> | 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/> |
| Offset 984, 16 lines modified | Offset 984, 16 lines modified | ||
| 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.Program.Wf.html">Wf</a>·[library]<br/> | ||
| 992 | <a·href="Coq.Init.Wf.html">Wf</a>·[library]<br/> | 991 | <a·href="Coq.Init.Wf.html">Wf</a>·[library]<br/> |
| 992 | <a·href="Coq.Program.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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| 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.Program.Tactics.html">Tactics</a>·<br/> | ||
| 560 | <a·href="Coq.Init.Tactics.html">Tactics</a>·<br/> | 559 | <a·href="Coq.Init.Tactics.html">Tactics</a>·<br/> |
| 560 | <a·href="Coq.Program.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"> |
| 566 | ········<div·class="block"> | 566 | ········<div·class="block"> |
| 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.Program.Wf.html">Wf</a>·<br/> | ||
| 563 | <a·href="Coq.Init.Wf.html">Wf</a>·<br/> | 562 | <a·href="Coq.Init.Wf.html">Wf</a>·<br/> |
| 563 | <a·href="Coq.Program.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> |