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/srv/reproducible-results/rbuild-debian/r-b-build.zYJO2SQH/b2/coq_8.20.1+dfsg-1_amd64.changes ¶

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coq_8.20.1+dfsg-1_amd64.deb ¶

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libcoq-core-ocaml-dev_8.20.1+dfsg-1_amd64.deb ¶

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/cMap.cmt ¶

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/cString.cmti ¶

Format-specific differences are supported for OCaml interface files but no file-specific differences were detected; falling back to a binary diff. file(1) reports: OCaml interface file (.cmi) (Version 035)

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/exninfo.cmti ¶

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libcoq-core-ocaml_8.20.1+dfsg-1_amd64.deb ¶

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/clib.cma ¶

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libcoq-stdlib_8.20.1+dfsg-1_amd64.deb ¶

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Offset 1, 9 lines modified		Offset 1, 9 lines modified

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022e32e0:·a32c·636f·6e74·696e·7561·7469·6f6e·4908··.,continuationI.		022e32e0:·2c63·6f6e·7469·6e75·6174·696f·6e49·0800··,continuationI..
022e32f0:·00~~00·~~3b00·a0c0·9040·0205·f5e1·0040·50a0··..;....@.....@P.		022e32f0:·003b·00a0·c090·4002·05f5·e100·4050·a0c0··.;....@.....@P..
022e3300:·c090·4002·05f5·e100·4051·4042·042f·4140··..@.....@Q@B./A@		022e3300:·9040·0205·f5e1·0040·5140·4204·2f41·40a0··.@.....@Q@B./A@.
022e3310:·a06e·a059·40a0·42a0·4240·4040·~~046d~~·4040··.n.Y@.B.B@@@.m@@		022e3310:·6ea0·5940·a042·a042·4040·4004·6c40·4040··n.Y@.B.B@@@.l@@@
022e3320:·4092·040d·a335·6578·7465·6e73·696f·6e5f··@....5extension_		022e3320:·9204·0da3·3565·7874·656e·7369·6f6e·5f63··....5extension_c
022e3330:·636f·6e73·7472·7563·746f·7252·0800·003b··constructorR...;		022e3330:·6f6e·7374·7275·6374·6f72·5208·0000·3b00··onstructorR...;.
022e3340:·0040·4004·3741·4040·4040·4004·7140·4040··.@@.7A@@@@@.q@@@		022e3340:·4040·0437·4140·4040·4040·0470·4040·4092··@@.7A@@@@@.p@@@.
022e3350:·9204·03a3·2566·6c6f·6174·4408·0000·3b00··....%floatD...;.		022e3350:·0403·a325·666c·6f61·7444·0800·003b·0040··...%floatD...;.@
022e3360:·4040·~~043b~~·4140·4040·4040·0475·4040·4092··@@.;A@@@@@.u@@@.		022e3360:·4004·3b41·4040·4040·4004·7440·4040·9204··@.;A@@@@@.t@@@..
022e3370:·0403·a32a·666c·6f61·7461·7272·6179·5308··...*floatarrayS.		022e3370:·03a3·2a66·6c6f·6174·6172·7261·7953·0800··..*floatarrayS..
022e3380:·00~~00·~~3b00·4040·043f·4140·4040·4040·0479··..;.@@.?A@@@@@.y		022e3380:·003b·0040·4004·3f41·4040·4040·4004·7840··.;.@@.?A@@@@@.x@
022e3390:·4040·4092·0403·a323·696e·7441·0800·003b··@@@....#intA...;		022e3390:·4040·9204·03a3·2369·6e74·4108·0000·3b00··@@....#intA...;.
022e33a0:·0040·4004·4341·4040·4040·4004~~·7d~~40·4140··.@@.CA@@@@@.}@A@		022e33a0:·4040·0443·4140·4040·4040·047c·4041·4092··@@.CA@@@@@.\|@A@.
022e33b0:·9204·03a3·2569·6e74·3332·4e08·0000·3b00··....%int32N...;.		022e33b0:·0403·a325·696e·7433·324e·0800·003b·0040··...%int32N...;.@
022e33c0:·4040·0447·4140·4040·4040·0481·4040·4092··@@.GA@@@@@..@@@.		022e33c0:·4004·4741·4040·4040·4004·8040·4040·9204··@.GA@@@@@..@@@..
022e33d0:·0403·a325·696e·7436·344f·0800·003b·0040··...%int64O...;.@		022e33d0:·03a3·2569·6e74·3634·4f08·0000·3b00·4040··..%int64O...;.@@
022e33e0:·4004·4b41·4040·4040·4004·8540·4040·9204··@.KA@@@@@..@@@..		022e33e0:·044b·4140·4040·4040·0484·4040·4092·0403··.KA@@@@@..@@@...
022e33f0:·03a3·266c·617a·795f·7450·0800·003b·00a0··..&lazy_tP...;..		022e33f0:·a326·6c61·7a79·5f74·5008·0000·3b00·a0c0··.&lazy_tP...;...
022e3400:·c090·4002·05f5·e100·4052·4041·0452·4140··..@.....@R@A.RA@		022e3400:·9040·0205·f5e1·0040·5240·4104·5241·40a0··.@.....@R@A.RA@.
022e3410:·~~a059~~·40a0·4040·4040·~~048e~~·4040·4092·0408··.Y@.@@@@..@@@...		022e3410:·5940·a040·4040·4004·8d40·4040·9204·08a3··Y@.@@@@..@@@....
022e3420:·a324·6c69·7374·4b08·0000·3b00·a0c0·9040··.$listK...;....@		022e3420:·246c·6973·744b·0800·003b·00a0·c090·4002··$listK...;....@.
022e3430:·0205·f5e1·0040·5340·41a2·a0e0·a322·5b5d··.....@S@A...."[]		022e3430:·05f5·e100·4053·4041·a2a0·e0a3·225b·5d64··....@S@A...."[]d
022e3440:·6490·4040·04~~9b·~~4092·0403·a0e0·a322·3a3a··d.@@..@......"::		022e3440:·9040·4004·9a40·9204·03a0·e0a3·223a·3a65··.@@..@......"::e
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022e3460:·4002·05f5·e100·4054·4040·04~~a8·~~4092·040a··@.....@T@@..@...		022e3460:·0205·f5e1·0040·5440·4004·a740·9204·0a40··.....@T@@..@...@
022e3470:·4040·4140·~~a059~~·40a0·4040·4040·~~04ab~~·4040··@@A@.Y@.@@@@..@@		022e3470:·4041·40a0·5940·a040·4040·4004·aa40·4040··@A@.Y@.@@@@..@@@
022e3480:·4092·041c·a329·6e61·7469·7665·696e·744d··@....)nativeintM		022e3480:·9204·1ca3·296e·6174·6976·6569·6e74·4d08··....)nativeintM.
022e3490:·0800·003b·0040·4004·7541·4040·4040·40~~04·~~·...;.@@.uA@@@@@.		022e3490:·0000·3b00·4040·0475·4140·4040·4040·04ae··..;.@@.uA@@@@@..
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022e34d0:·bc40·9204·03a0·e0a3·2453·6f6d·6567·90a0··.@......$Someg..		022e34d0:·4092·0403·a0e0·a324·536f·6d65·6790·a004··@......$Someg...
022e34e0:·040f·4040·04~~c3·~~4092·0404·4040·4140·a059··..@@..@...@@A@.Y		022e34e0:·0f40·4004·c240·9204·0440·4041·40a0·5940··.@@..@...@@A@.Y@
022e34f0:·40a0·4040·4040·~~04c6~~·4040·4092·0416·a326··@.@@@@..@@@....&		022e34f0:·a040·4040·4004·c540·4040·9204·16a3·2673··.@@@@..@@@....&s
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022e3540:·6e69·7446·0800·003b·0040·40a2·a0e0·a322··nitF...;.@@...."		022e3540:·6974·4608·0000·3b00·4040·a2a0·e0a3·2228··itF...;.@@...."(
022e3550:·2829·6390·4040·04~~d8·~~4092·0403·4040·4140··()c.@@..@...@@A@		022e3550:·2963·9040·4004·d740·9204·0340·4041·4040··)c.@@..@...@@A@@
022e3560:·4040·40~~40·~~04~~d9·~~4041·4092·040a·a32e·4173··@@@@..@A@.....As		022e3560:·4040·4004·d840·4140·9204·0aa3·2e41·7373··@@@..@A@.....Ass
022e3570:·7365·7274·5f66·6169·6c75·7265·5e08·0000··sert_failure^...		022e3570:·6572·745f·6661·696c·7572·655e·0800·0023··ert_failure^...#
022e3580:·2300·9004·8b40·90a0·c092·a0c0·b390·041f··#....@..........		022e3580:·0090·048b·4090·a0c0·92a0·c0b3·9004·1f40··....@..........@
022e3590:·4090·4002·05f5·e100·404a·a0c0·b390·0471··@.@.....@J.....q		022e3590:·9040·0205·f5e1·0040·4aa0·c0b3·9004·7140··.@.....@J.....q@
022e35a0:·4090·4002·05f5·e100·4040·a004·0540·0205··@.@.....@@...@..		022e35a0:·9040·0205·f5e1·0040·40a0·0405·4002·05f5··.@.....@@...@...
022e35b0:·f5e1·0040·5940·4041·04~~ed·~~a0b0·a03d·6f63··...@Y@@A.....=oc		022e35b0:·e100·4059·4040·4104·eca0·b0a0·3d6f·6361··..@Y@@A.....=oca
022e35c0:·616d·6c2e·7761·726e·5f6f·6e5f·6c69·7465··aml.warn_on_lite		022e35c0:·6d6c·2e77·6172·6e5f·6f6e·5f6c·6974·6572··ml.warn_on_liter
022e35d0:·7261·6c5f·7061·7474·6572·6e04~~·f1~~04·b704··ral_pattern.....		022e35d0:·616c·5f70·6174·7465·726e·04f0·04b7·04f0··al_pattern......
022e35e0:·f140·9204·17a3·3044·6976·6973·696f·6e5f··.@....0Division_		022e35e0:·4092·0417·a330·4469·7669·7369·6f6e·5f62··@....0Division_b
022e35f0:·6279·5f7a·6572·6f5b·0800·0023·0004·1840··by_zero[...#...@		022e35f0:·795f·7a65·726f·5b08·0000·2300·0418·4090··y_zero[...#...@.
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022e3610:·f940·9204·07a3·2b45·6e64·5f6f·665f·6669··.@....+End_of_fi		022e3610:·4092·0407·a32b·456e·645f·6f66·5f66·696c··@....+End_of_fil
022e3620:·6c65·5a08·0000·2300·0420·4090·4040·4104··leZ...#..·@.@@A.		022e3620:·655a·0800·0023·0004·2040·9040·4041·04fd··eZ...#..·@.@@A..
022e3630:·fea0·b0a0·0411·0501·0104·c705·01~~01·~~4092··..............@.		022e3630:·a0b0·a004·1105·0100·04c7·0501·0040·9204··.............@..
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022e3650:·0004·2840·90a0·0424·4040·4105·01~~07·~~a0b0··..(@...$@@A.....		022e3650:·0428·4090·a004·2440·4041·0501·06a0·b0a0··.(@...$@@A......
022e3660:·a004·1a05·01~~0a·~~04d0·0501~~·0a~~40·9204·08a3··...........@....		022e3660:·041a·0501·0904·d005·0109·4092·0408·a330··..........@....0
022e3670:·3049·6e76·616c·6964·5f61·7267·756d·656e··0Invalid_argumen		022e3670:·496e·7661·6c69·645f·6172·6775·6d65·6e74··Invalid_argument
022e3680:·7456·0800·0023·0004·3140·90a0·042d·4040··tV...#..1@...-@@		022e3680:·5608·0000·2300·0431·4090·a004·2d40·4041··V...#..1@...-@@A
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022e36a0:·1340·9204·08a3·2d4d·6174·6368·5f66·6169··.@....-Match_fai		022e36a0:·4092·0408·a32d·4d61·7463·685f·6661·696c··@....-Match_fail
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022e36c0:·92a0·0439·a004·35a0·0436·4002·05f5·e100··...9..5..6@.....		022e36c0:·a004·39a0·0435·a004·3640·0205·f5e1·0040··..9..5..6@.....@
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022e36e0:·04e7·0501~~·21~~40·9204·0da3·294e·6f74·5f66··....!@....)Not_f		022e36e0:·e705·0120·4092·040d·a329·4e6f·745f·666f··...·@....)Not_fo
022e36f0:·6f75·6e64·5808·0000·2300·0448·4090·4040··oundX...#..H@.@@		022e36f0:·756e·6458·0800·0023·0004·4840·9040·4041··undX...#..H@.@@A
022e3700:·4105·01~~26·~~a0b0·a004·3905·01~~29·~~04ef·0501··A..&....9..)....		022e3700:·0501·25a0·b0a0·0439·0501·2804·ef05·0128··..%....9..(....(
022e3710:·2940·9204·07a3·2d4f·7574·5f6f·665f·6d65··)@....-Out_of_me		022e3710:·4092·0407·a32d·4f75·745f·6f66·5f6d·656d··@....-Out_of_mem
022e3720:·6d6f·7279·5508·0000·2300·0450·4090·4040··moryU...#..P@.@@		022e3720:·6f72·7955·0800·0023·0004·5040·9040·4041··oryU...#..P@.@@A
022e3730:·4105·01~~2e·~~a0b0·a004·4105·01~~31·~~04f7·0501··A.......A..1....		022e3730:·0501·2da0·b0a0·0441·0501·3004·f705·0130··..-....A..0....0
022e3740:·3140·9204·07a3·2e53·7461·636b·5f6f·7665··1@.....Stack_ove		022e3740:·4092·0407·a32e·5374·6163·6b5f·6f76·6572··@.....Stack_over
022e3750:·7266·6c6f·775c·0800·0023·0004·5840·9040··rflow\...#..X@.@		022e3750:·666c·6f77·5c08·0000·2300·0458·4090·4040··flow\...#..X@.@@
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022e3770:·0139·4092·0407·a32e·5379·735f·626c·6f63··.9@.....Sys_bloc		022e3770:·3840·9204·07a3·2e53·7973·5f62·6c6f·636b··8@.....Sys_block
022e3780:·6b65·645f·696f·5d08·0000·2300·0460·4090··ked_io]...#..`@.		022e3780:·6564·5f69·6f5d·0800·0023·0004·6040·9040··ed_io]...#..`@.@
022e3790:·4040·4105·01~~3e·~~a0b0·a004·5105·01~~41·~~0501··@@A..>....Q..A..		022e3790:·4041·0501·3da0·b0a0·0451·0501·4005·0107··@A..=....Q..@...
022e37a0:·0705·0141·4092·0407·a329·5379·735f·6572··...A@....)Sys_er		022e37a0:·0501·4040·9204·07a3·2953·7973·5f65·7272··..@@....)Sys_err
022e37b0:·726f·7259·0800·0023·0004·6840·90a0·0464··rorY...#..h@...d		022e37b0:·6f72·5908·0000·2300·0468·4090·a004·6440··orY...#..h@...d@
022e37c0:·4040·4105·01~~47·~~a0b0·a004·5a05·01~~4a·~~0501··@@A..G....Z..J..		022e37c0:·4041·0501·46a0·b0a0·045a·0501·4905·0110··@A..F....Z..I...
022e37d0:·1005·01~~4a·~~4092·0408·a33a·556e·6465·6669··...J@....:Undefi		022e37d0:·0501·4940·9204·08a3·3a55·6e64·6566·696e··..I@....:Undefin
022e37e0:·6e65·645f·7265·6375·7273·6976·655f·6d6f··ned_recursive_mo		022e37e0:·6564·5f72·6563·7572·7369·7665·5f6d·6f64··ed_recursive_mod
022e37f0:·6475·6c65·5f08·0000·2300·0471·4090·a0c0··dule_...#..q@...		022e37f0:·756c·655f·0800·0023·0004·7140·90a0·c092··ule_...#..q@....
022e3800:·92a0·0470·a004·6ca0·046d·4002·05f5·e100··...p..l..m@.....		022e3800:·a004·70a0·046c·a004·6d40·0205·f5e1·0040··..p..l..m@.....@
022e3810:·4066·4040·4105·0155·a0b0·a004·6805·0158··@f@@A..U....h..X		022e3810:·6640·4041·0501·54a0·b0a0·0468·0501·5705··f@@A..T....h..W.
022e3820:·0501·1e05·01~~58·~~4092·040d·a33a·436f·6e74··.....X@....:Cont		022e3820:·011e·0501·5740·9204·0da3·3a43·6f6e·7469··....W@....:Conti
022e3830:·696e·7561·7469·6f6e·5f61·6c72·6561·6479··inuation_already		022e3830:·6e75·6174·696f·6e5f·616c·7265·6164·795f··nuation_already_
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022e3850:·4040·4105·01~~5d·~~a0b0·a004·7005·01~~60·~~0501··@@A..]....p..`..		022e3850:·4041·0501·5ca0·b0a0·0470·0501·5f05·0126··@A..\....p.._..&
022e3860:·2605·01~~60·~~4092·0407·9092·2653·7464·6c69··&..`@.....&Stdli		022e3860:·0501·5f40·9204·0790·9226·5374·646c·6962··.._@.....&Stdlib
022e3870:·~~6292·2349·6e74·~~9226·4f70·7469·6f6e·b12b··~~b.#Int~~.&Option.+		022e3870:·9226·4f70·7469·6f6e·b12b·4f72·6465·7265··.&Option.+Ordere
022e3880:·~~4f72·6465·7265·~~6454·7970·6501·0110·42c0··~~Ordere~~dType...B.		022e3880:·6454·7970·6501·0110·42c0·9091·a0c1·b121··dType...B......!
022e3890:·~~909~~1·~~a0c1~~·b121·7401·010e·4~~108~~·0000·~~3b0~~0··....~~.!t~~...A..~~.;.~~		022e3890:·7401·010e·4108·0000·3b00·4040·9040·4140··t...A...;.@@.@A@
022e38a0:·4040·9040~~·41~~40·4040·~~4040~~·~~b0c0~~·~~2c63~~·6c~~69·~~·~~@@.@A@~~@@@@..,cli		022e38a0:·4040·4040·b0c0·2c63·6c69·622f·634d·6170··@@@@..,clib/cMap
022e38b0:·~~622f~~·634d·6170·2~~e6d·6~~c4d·0102·c201·0~~2c4~~··~~b/c~~Map.~~mlM~~......		022e38b0:·2e6d·6c4d·0102·c201·02c4·c004·024d·0102··.mlM.........M..
022e38c0:·c004·024d·0102·~~c201~~·~~02ca~~·~~4040~~·4040·b105··...~~M....~~..@@@~~@..~~		022e38c0:·c201·02ca·4040·4040·b105·0176·4041·4140··....@@@@...v@AA@
022e38d0:·~~0179·4041·4140·~~a0b0·a027·636f·6d70·6172··~~.y@AA@~~...'compar		022e38d0:·a0b0·a027·636f·6d70·6172·6501·010f·d0c0··...'compare.....
Max diff block lines reached; 31937466/32684429 bytes (97.71%) of diff not shown.


Offset 2321727, 236890 lines modified		Offset 2321727, 236890 lines modified
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0236d3f0:·0000·004f·0000·0006·0000·0020·0000·0016··...O.......·....		0236d3f0:·0000·004f·0000·0006·0000·0020·0000·0016··...O.......·....
0236d400:·a02f·2f77·6f72·6b73·7061·6365·5f72·6f6f··.//workspace_roo		0236d400:·a02f·2f77·6f72·6b73·7061·6365·5f72·6f6f··.//workspace_roo
0236d410:·74a0·342f·776f·726b·7370·6163·655f·726f··t.4/workspace_ro		0236d410:·74a0·342f·776f·726b·7370·6163·655f·726f··t.4/workspace_ro
0236d420:·6f74·2f63·6c69·62a0·0924·2f77·6f72·6b73··ot/clib..$/works		0236d420:·6f74·2f63·6c69·62a0·0924·2f77·6f72·6b73··ot/clib..$/works
0236d430:·7061·6365·5f72·6f6f·742f·636c·6962·2f2e··pace_root/clib/.		0236d430:·7061·6365·5f72·6f6f·742f·636c·6962·2f2e··pace_root/clib/.
0236d440:·636c·6962·2e6f·626a·732f·6279·7465·4000··clib.objs/byte@.		0236d440:·636c·6962·2e6f·626a·732f·6279·7465·4000··clib.objs/byte@.
0236d450:·07e4·2884·95a6·be00·0157·a500·003e·4800··..(......W...>H.		0236d450:·07e4·2884·95a6·be00·0157·9f00·003e·4500··..(......W...>E.
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0236d470:·2443·4d61·70b0·c026·5f6e·6f6e·655f·4040··$CMap..&_none_@@		0236d470:·2443·4d61·70b0·c026·5f6e·6f6e·655f·4040··$CMap..&_none_@@
0236d480:·00ff·0402·4141·223f·3f41·40d0·4040·4040··....AA"??A@.@@@@		0236d480:·00ff·0402·4141·223f·3f41·40d0·4040·4040··....AA"??A@.@@@@
0236d490:·40a0·4040·4040·a008·0000·2f00·0121·9004··@.@@@@..../..!..		0236d490:·40a0·4040·4040·a008·0000·2f00·0121·9004··@.@@@@..../..!..
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Diff chunk too large, falling back to line-by-line diff (5404 lines added, 5404 lines removed)
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Offset 3, 25 lines modified		Offset 3, 25 lines modified
3	drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-15·14:19:14.000000·./usr/bin/	3	drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-15·14:19:14.000000·./usr/bin/
4	-rwxr-xr-x···0·root·········(0)·root·········(0)······892·2025-02-15·14:19:14.000000·./usr/bin/coq-makefile-make-both-single-timing-files	4	-rwxr-xr-x···0·root·········(0)·root·········(0)······892·2025-02-15·14:19:14.000000·./usr/bin/coq-makefile-make-both-single-timing-files
5	-rwxr-xr-x···0·root·········(0)·root·········(0)······932·2025-02-15·14:19:14.000000·./usr/bin/coq-makefile-make-both-time-files	5	-rwxr-xr-x···0·root·········(0)·root·········(0)······932·2025-02-15·14:19:14.000000·./usr/bin/coq-makefile-make-both-time-files
6	-rwxr-xr-x···0·root·········(0)·root·········(0)······698·2025-02-15·14:19:14.000000·./usr/bin/coq-makefile-make-one-time-file	6	-rwxr-xr-x···0·root·········(0)·root·········(0)······698·2025-02-15·14:19:14.000000·./usr/bin/coq-makefile-make-one-time-file
7	-rwxr-xr-x···0·root·········(0)·root·········(0)··1386744·2025-02-15·14:19:14.000000·./usr/bin/coq-tex	7	-rwxr-xr-x···0·root·········(0)·root·········(0)··1386744·2025-02-15·14:19:14.000000·./usr/bin/coq-tex
8	-rwxr-xr-x···0·root·········(0)·root·········(0)··2757688·2025-02-15·14:19:14.000000·./usr/bin/coq_makefile	8	-rwxr-xr-x···0·root·········(0)·root·········(0)··2757688·2025-02-15·14:19:14.000000·./usr/bin/coq_makefile
9	-rwxr-xr-x···0·root·········(0)·root·········(0)·20014136·2025-02-15·14:19:14.000000·./usr/bin/coqc	9	-rwxr-xr-x···0·root·········(0)·root·········(0)·20014136·2025-02-15·14:19:14.000000·./usr/bin/coqc
10	-rwxr-xr-x···0·root·········(0)·root·········(0)·40371869·2025-02-15·14:19:14.000000·./usr/bin/coqc.byte	10	-rwxr-xr-x···0·root·········(0)·root·········(0)·40371863·2025-02-15·14:19:14.000000·./usr/bin/coqc.byte
11	-rwxr-xr-x···0·root·········(0)·root·········(0)··6176536·2025-02-15·14:19:14.000000·./usr/bin/coqchk	11	-rwxr-xr-x···0·root·········(0)·root·········(0)··6176536·2025-02-15·14:19:14.000000·./usr/bin/coqchk
12	-rwxr-xr-x···0·root·········(0)·root·········(0)··3077752·2025-02-15·14:19:14.000000·./usr/bin/coqdep	12	-rwxr-xr-x···0·root·········(0)·root·········(0)··3077752·2025-02-15·14:19:14.000000·./usr/bin/coqdep
13	-rwxr-xr-x···0·root·········(0)·root·········(0)··2901816·2025-02-15·14:19:14.000000·./usr/bin/coqdoc	13	-rwxr-xr-x···0·root·········(0)·root·········(0)··2901816·2025-02-15·14:19:14.000000·./usr/bin/coqdoc
14	-rwxr-xr-x···0·root·········(0)·root·········(0)·40937854·2025-02-15·14:19:14.000000·./usr/bin/coqidetop.byte	14	-rwxr-xr-x···0·root·········(0)·root·········(0)·40937848·2025-02-15·14:19:14.000000·./usr/bin/coqidetop.byte
15	-rwxr-xr-x···0·root·········(0)·root·········(0)·20350808·2025-02-15·14:19:14.000000·./usr/bin/coqidetop.opt	15	-rwxr-xr-x···0·root·········(0)·root·········(0)·20350808·2025-02-15·14:19:14.000000·./usr/bin/coqidetop.opt
16	-rwxr-xr-x···0·root·········(0)·root·········(0)··6505784·2025-02-15·14:19:14.000000·./usr/bin/coqnative	16	-rwxr-xr-x···0·root·········(0)·root·········(0)··6505784·2025-02-15·14:19:14.000000·./usr/bin/coqnative
17	-rwxr-xr-x···0·root·········(0)·root·········(0)··1705752·2025-02-15·14:19:14.000000·./usr/bin/coqpp	17	-rwxr-xr-x···0·root·········(0)·root·········(0)··1705752·2025-02-15·14:19:14.000000·./usr/bin/coqpp
18	-rwxr-xr-x···0·root·········(0)·root·········(0)··2113784·2025-02-15·14:19:14.000000·./usr/bin/coqtimelog2html	18	-rwxr-xr-x···0·root·········(0)·root·········(0)··2113784·2025-02-15·14:19:14.000000·./usr/bin/coqtimelog2html
19	-rwxr-xr-x···0·root·········(0)·root·········(0)·20014136·2025-02-15·14:19:14.000000·./usr/bin/coqtop	19	-rwxr-xr-x···0·root·········(0)·root·········(0)·20014136·2025-02-15·14:19:14.000000·./usr/bin/coqtop
20	-rwxr-xr-x···0·root·········(0)·root·········(0)·64804192·2025-02-15·14:19:14.000000·./usr/bin/coqtop.byte	20	-rwxr-xr-x···0·root·········(0)·root·········(0)·64804186·2025-02-15·14:19:14.000000·./usr/bin/coqtop.byte
21	-rwxr-xr-x···0·root·········(0)·root·········(0)··1039784·2025-02-15·14:19:14.000000·./usr/bin/coqwc	21	-rwxr-xr-x···0·root·········(0)·root·········(0)··1039784·2025-02-15·14:19:14.000000·./usr/bin/coqwc
22	-rwxr-xr-x···0·root·········(0)·root·········(0)·20019160·2025-02-15·14:19:14.000000·./usr/bin/coqworker.opt	22	-rwxr-xr-x···0·root·········(0)·root·········(0)·20019160·2025-02-15·14:19:14.000000·./usr/bin/coqworker.opt
23	-rwxr-xr-x···0·root·········(0)·root·········(0)··1567640·2025-02-15·14:19:14.000000·./usr/bin/coqworkmgr	23	-rwxr-xr-x···0·root·········(0)·root·········(0)··1567640·2025-02-15·14:19:14.000000·./usr/bin/coqworkmgr
24	-rwxr-xr-x···0·root·········(0)·root·········(0)··2606648·2025-02-15·14:19:14.000000·./usr/bin/csdpcert	24	-rwxr-xr-x···0·root·········(0)·root·········(0)··2606648·2025-02-15·14:19:14.000000·./usr/bin/csdpcert
25	-rwxr-xr-x···0·root·········(0)·root·········(0)··1596280·2025-02-15·14:19:14.000000·./usr/bin/ocamllibdep	25	-rwxr-xr-x···0·root·········(0)·root·········(0)··1596280·2025-02-15·14:19:14.000000·./usr/bin/ocamllibdep
26	-rwxr-xr-x···0·root·········(0)·root·········(0)··1834136·2025-02-15·14:19:14.000000·./usr/bin/votour	26	-rwxr-xr-x···0·root·········(0)·root·········(0)··1834136·2025-02-15·14:19:14.000000·./usr/bin/votour
27	drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-15·14:19:14.000000·./usr/lib/	27	drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-15·14:19:14.000000·./usr/lib/


Offset 2711647, 330994 lines modified		Offset 2711647, 330994 lines modified
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029609b0:·6~~566~~·9040·4004·bc40·9204·03a0·e0a3·2453··ef.@@..@......$S		029609b0:·6690·4040·04bb·4092·0403·a0e0·a324·536f··f.@@..@......$So
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02960a40:·0403·4040·4140·4040·4040·04~~d9·~~4041·4092··..@@A@@@@@..@A@.		02960a40:·0340·4041·4040·4040·4004·d840·4140·9204··.@@A@@@@@..@A@..
02960a50:·040a·a32e·4173·7365·7274·5f66·6169·6c75··....Assert_failu		02960a50:·0aa3·2e41·7373·6572·745f·6661·696c·7572··...Assert_failur
02960a60:·7265·5e08·0000·2300·9004·8b40·90a0·c092··re^...#....@....		02960a60:·655e·0800·0023·0090·048b·4090·a0c0·92a0··e^...#....@.....
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02960ad0:·6973·696f·6e5f·6279·5f7a·6572·6f5b·0800··ision_by_zero[..		02960ad0:·7369·6f6e·5f62·795f·7a65·726f·5b08·0000··sion_by_zero[...
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02960b30:·6557·0800·0023·0004·2840·90a0·0424·4040··eW...#..(@...$@@		02960b30:·5708·0000·2300·0428·4090·a004·2440·4041··W...#..(@...$@@A
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02960b50:·0a40·9204·08a3·3049·6e76·616c·6964·5f61··.@....0Invalid_a		02960b50:·4092·0408·a330·496e·7661·6c69·645f·6172··@....0Invalid_ar
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02960c00:·5f6f·665f·6d65·6d6f·7279·5508·0000·2300··_of_memoryU...#.		02960c00:·6f66·5f6d·656d·6f72·7955·0800·0023·0004··of_memoryU...#..
02960c10:·0450·4090·4040·4105·01~~2e·~~a0b0·a004·4105··.P@.@@A.......A.		02960c10:·5040·9040·4041·0501·2da0·b0a0·0441·0501··P@.@@A..-....A..
02960c20:·0131·04f7·0501~~·31~~40·9204·07a3·2e53·7461··.1....1@.....Sta		02960c20:·3004·f705·0130·4092·0407·a32e·5374·6163··0....0@.....Stac
02960c30:·636b·5f6f·7665·7266·6c6f·775c·0800·0023··ck_overflow\...#		02960c30:·6b5f·6f76·6572·666c·6f77·5c08·0000·2300··k_overflow\...#.
02960c40:·0004·5840·9040·4041·0501~~·36~~a0·b0a0·0449··..X@.@@A..6....I		02960c40:·0458·4090·4040·4105·0135·a0b0·a004·4905··.X@.@@A..5....I.
02960c50:·0501~~·39~~04·ff05·01~~39·~~4092·0407·a32e·5379··..9....9@.....Sy		02960c50:·0138·04ff·0501·3840·9204·07a3·2e53·7973··.8....8@.....Sys
02960c60:·735f·626c·6f63·6b65·645f·696f·5d08·0000··s_blocked_io]...		02960c60:·5f62·6c6f·636b·6564·5f69·6f5d·0800·0023··_blocked_io]...#
02960c70:·2300·0460·4090·4040·4105·01~~3e·~~a0b0·a004··#..`@.@@A..>....		02960c70:·0004·6040·9040·4041·0501·3da0·b0a0·0451··..`@.@@A..=....Q
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Offset 72, 365 lines modified		Offset 72, 365 lines modified
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000005e0:·~~6cd2~~·~~44d8~~·0~~b09~~·ee84·~~190c~~·~~86a1~~·~~c6d3~~·~~b4c5~~··l.D.........~~...~~.		000005e0:·125b·161a·0c26·428d·e859·9389·276c·2696··.[...&B..Y..'l&.
000005f0:·~~b009~~·9~~361~~·729c·~~6119~~·6~~902~~·~~125c~~·1~~20a~~·~~6669~~··...ar.a.i..\..fi		000005f0:·234d·9c95·5890·d8a2·6c39·cdf3·147f·d896··#M..X...l9......
00000600:·9a38·~~086e~~·~~0996~~·7284·c970·~~9658~~·~~4963~~·829d·~~·.8~~.n..r..p.~~XIc~~..		00000600:·7431·47d8·4cf4·2233·c918·0c95·5a1a·b289··t1G.L."3....Z...
00000610:·514a·~~3fb8~~·025c·d152·~~d266~~·2686·~~2a61~~·~~5962~~··~~QJ?.~~.\.R.f&.*aYb		00000610:·5042·9449·d24e·11ca·8479·9185·e9d2·90cd··PB.I.N...y......
00000620:·~~22aa~~·~~f48~~3·~~1b3~~2·~~21ca~~·~~2c9f~~·~~f013~~·6~~c56·~~c250··"....2!.,...lV.P		00000620:·0816·a9d3·7412·9d60·3413·96d6·a6b8·6c2e··....t..`4.....l.
00000630:·~~1b94~~·~~16ee~~·~~a78a~~·~~8cb4~~·9533·~~185a~~·~~2605~~·~~c1dd~~··.........3.~~Z&.~~..		00000630:·2db3·b266·d260·a929·e587·ed03·7b58·9f9b··-..f.`.)....{X..


Offset 1, 13 lines modified		Offset 1, 13 lines modified
1	Package:·libcoq-stdlib	1	Package:·libcoq-stdlib
2	Source:·coq	2	Source:·coq
3	Version:·8.20.1+dfsg-1	3	Version:·8.20.1+dfsg-1
4	Architecture:·amd64	4	Architecture:·amd64
5	Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org>	5	Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org>
6	Installed-Size:·146164	6	Installed-Size:·146163
7	Recommends:·coq	7	Recommends:·coq
8	Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1),·coq-theories	8	Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1),·coq-theories
9	Replaces:·coq-libs·(<<·8.2.pl1),·coq-theories	9	Replaces:·coq-libs·(<<·8.2.pl1),·coq-theories
10	Provides:·coq-theories·(=·8.20.1+dfsg-1),·libcoq-stdlib-68yx1	10	Provides:·coq-theories·(=·8.20.1+dfsg-1),·libcoq-stdlib-68yx1
11	Section:·math	11	Section:·math
12	Priority:·optional	12	Priority:·optional
13	Homepage:·http://coq.inria.fr/	13	Homepage:·http://coq.inria.fr/


Offset 54, 15 lines modified		Offset 54, 15 lines modified
54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#"><span·class="id"·title="library">EqNat</span></a>.<br/>	54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#"><span·class="id"·title="library">EqNat</span></a>.<br/>
55	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Wf_nat.html#"><span·class="id"·title="library">Wf_nat</span></a>.<br/>	55	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Wf_nat.html#"><span·class="id"·title="library">Wf_nat</span></a>.<br/>

56	<br/>	56	<br/>
57	</div>	57	</div>

58	<div·class="doc">	58	<div·class="doc">
59	<a·id="lab1~~236~~"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">arith</span></span>·hint·database</h1>	59	<a·id="lab782"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">arith</span></span>·hint·database</h1>

60	</div>	60	</div>
61	<div·class="code">	61	<div·class="code">

62	<br/>	62	<br/>
63	#[<span·class="id"·title="var">global</span>]<br/>	63	#[<span·class="id"·title="var">global</span>]<br/>
64	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">eq_nat_refl</span>:·<span·class="id"·title="var">arith</span>.<br/>	64	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">eq_nat_refl</span>:·<span·class="id"·title="var">arith</span>.<br/>
Offset 76, 15 lines modified		Offset 76, 15 lines modified
76	#[<span·class="id"·title="var">global</span>]<br/>	76	#[<span·class="id"·title="var">global</span>]<br/>
77	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">in_int_Sp_q</span>·<span·class="id"·title="var">exists_le_S</span>·<span·class="id"·title="var">exists_S_le</span>:·<span·class="id"·title="var">arith</span>.<br/>	77	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">in_int_Sp_q</span>·<span·class="id"·title="var">exists_le_S</span>·<span·class="id"·title="var">exists_S_le</span>:·<span·class="id"·title="var">arith</span>.<br/>

78	<br/>	78	<br/>
79	</div>	79	</div>

80	<div·class="doc">	80	<div·class="doc">
81	<a·id="lab1237"></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>	81	<a·id="lab783"></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>

82	</div>	82	</div>
83	<div·class="code">	83	<div·class="code">
84	#[<span·class="id"·title="var">global</span>]<br/>	84	#[<span·class="id"·title="var">global</span>]<br/>
85	<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/>	85	<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/>
86	<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/>	86	<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/>
87	<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/>	87	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_0_l</span>·<span·class="id"·title="var">Nat.nle_succ_0</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>


Offset 52, 15 lines modified		Offset 52, 15 lines modified
52	<br/>	52	<br/>
53	</div>	53	</div>

54	<div·class="doc">	54	<div·class="doc">
55	Equality·on·natural·numbers·	55	Equality·on·natural·numbers·
56	<div·class="paragraph">·</div>	56	<div·class="paragraph">·</div>

57	<a·id="lab1~~253~~"></a><h1·class="section">Propositional·equality</h1>	57	<a·id="lab766"></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/>


Offset 62, 15 lines modified		Offset 62, 15 lines modified
62	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations"><span·class="id"·title="module">BoolNotations</span></a>.<br/>	62	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations"><span·class="id"·title="module">BoolNotations</span></a>.<br/>
63	<span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">typeclass_instances</span>·<span·class="id"·title="var">relations</span>.<br/>	63	<span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">typeclass_instances</span>·<span·class="id"·title="var">relations</span>.<br/>

64	<br/>	64	<br/>
65	</div>	65	</div>

66	<div·class="doc">	66	<div·class="doc">
67	<a·id="lab~~975~~"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1>	67	<a·id="lab436"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1>

68	</div>	68	</div>
69	<div·class="code">	69	<div·class="code">

70	<br/>	70	<br/>
71	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="lemma">le_refl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:1"·class="idref"·href="#b:1"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>.<br/>	71	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="lemma">le_refl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:1"·class="idref"·href="#b:1"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>.<br/>
72	·	72	·
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88	#[<span·class="id"·title="var">global</span>]<br/>	88	#[<span·class="id"·title="var">global</span>]<br/>
89	<span·class="id"·title="keyword">Instance</span>·<a·id="le_compat"·class="idref"·href="#le_compat"><span·class="id"·title="instance">le_compat</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.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.Init.Logic.html#iff"><span·class="id"·title="definition">iff</span></a>)·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>	89	<span·class="id"·title="keyword">Instance</span>·<a·id="le_compat"·class="idref"·href="#le_compat"><span·class="id"·title="instance">le_compat</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.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.Init.Logic.html#iff"><span·class="id"·title="definition">iff</span></a>)·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>
90	·	90	·
91	<br/>	91	<br/>
92	</div>	92	</div>

93	<div·class="doc">	93	<div·class="doc">
94	<a·id="lab~~976~~"></a><h1·class="section">Strict·order·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1>	94	<a·id="lab437"></a><h1·class="section">Strict·order·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1>

95	</div>	95	</div>
96	<div·class="code">	96	<div·class="code">

97	<br/>	97	<br/>
98	<span·class="id"·title="keyword">Lemma</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="lemma">lt_irrefl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:7"·class="idref"·href="#b:7"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>.<br/>	98	<span·class="id"·title="keyword">Lemma</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="lemma">lt_irrefl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:7"·class="idref"·href="#b:7"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>.<br/>
99	·	99	·
Offset 123, 15 lines modified		Offset 123, 15 lines modified
123	<br/>	123	<br/>
124	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_lteq"·class="idref"·href="#le_lteq"><span·class="id"·title="lemma">le_lteq</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:19"·class="idref"·href="#b1:19"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:20"·class="idref"·href="#b2:20"><span·class="id"·title="binder">b2</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</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#b2:20"><span·class="id"·title="variable">b2</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.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><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.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>.<br/>	124	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_lteq"·class="idref"·href="#le_lteq"><span·class="id"·title="lemma">le_lteq</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:19"·class="idref"·href="#b1:19"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:20"·class="idref"·href="#b2:20"><span·class="id"·title="binder">b2</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</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#b2:20"><span·class="id"·title="variable">b2</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.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><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.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>.<br/>
125	·	125	·
126	<br/>	126	<br/>
127	</div>	127	</div>

128	<div·class="doc">	128	<div·class="doc">
129	<a·id="lab~~977~~"></a><h1·class="section">Order·structures</h1>	129	<a·id="lab438"></a><h1·class="section">Order·structures</h1>

130	</div>	130	</div>
131	<div·class="code">	131	<div·class="code">

132	<br/>	132	<br/>
133	#[<span·class="id"·title="var">global</span>]<br/>	133	#[<span·class="id"·title="var">global</span>]<br/>
134	<span·class="id"·title="keyword">Instance</span>·<a·id="le_preorder"·class="idref"·href="#le_preorder"><span·class="id"·title="instance">le_preorder</span></a>·:·<a·class="idref"·href="Coq.Classes.RelationClasses.html#PreOrder"><span·class="id"·title="class">PreOrder</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>	134	<span·class="id"·title="keyword">Instance</span>·<a·id="le_preorder"·class="idref"·href="#le_preorder"><span·class="id"·title="instance">le_preorder</span></a>·:·<a·class="idref"·href="Coq.Classes.RelationClasses.html#PreOrder"><span·class="id"·title="class">PreOrder</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</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~~872~~"></a><h1·class="section">Typeclass-based·setoids.·Definitions·on·<span·class="inlinecode"><span·class="id"·title="var">Equivalence</span></span>.</h1>	49	<a·id="lab746"></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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00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra		00000660:·226c·6962·7469·746c·6522·3e4c·6962·7261··"libtitle">Libra
00000670:·7279·2043·6f71·2e43·6c61·7373·6573·2e52··ry·Coq.Classes.R		00000670:·7279·2043·6f71·2e43·6c61·7373·6573·2e52··ry·Coq.Classes.R
00000680:·656c·6174·696f·6e43·6c61·7373·6573·3c2f··elationClasses</		00000680:·656c·6174·696f·6e43·6c61·7373·6573·3c2f··elationClasses</
00000690:·6831·3e0a·0a3c·6469·7620·636c·6173·733d··h1>..<div·class=		00000690:·6831·3e0a·0a3c·6469·7620·636c·6173·733d··h1>..<div·class=
000006a0:·2263·6f64·6522·3e0a·0a3c·6272·2f3e·0a3c··"code">..<br/>.<		000006a0:·2263·6f64·6522·3e0a·0a3c·6272·2f3e·0a3c··"code">..<br/>.<
000006b0:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas		000006b0:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas
000006c0:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id="		000006c0:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id="
000006d0:·6c61·6238·3737·223e·3c2f·613e·3c68·3120··lab~~877~~"></a><h1·		000006d0:·6c61·6237·3336·223e·3c2f·613e·3c68·3120··lab736"></a><h1·
000006e0:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section">		000006e0:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section">
000006f0:·5479·7065·636c·6173·732d·6261·7365·6420··Typeclass-based·		000006f0:·5479·7065·636c·6173·732d·6261·7365·6420··Typeclass-based·
00000700:·7265·6c61·7469·6f6e·732c·2074·6163·7469··relations,·tacti		00000700:·7265·6c61·7469·6f6e·732c·2074·6163·7469··relations,·tacti
00000710:·6373·2061·6e64·2073·7461·6e64·6172·6420··cs·and·standard·		00000710:·6373·2061·6e64·2073·7461·6e64·6172·6420··cs·and·standard·
00000720:·696e·7374·616e·6365·733c·2f68·313e·0a0a··instances</h1>..		00000720:·696e·7374·616e·6365·733c·2f68·313e·0a0a··instances</h1>..
00000730:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par		00000730:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par
00000740:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>.		00000740:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>.
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00010640:·0a3c·2f64·6976·3e0a·0a3c·6469·7620·636c··.</div>..<div·cl		00010640:·0a3c·2f64·6976·3e0a·0a3c·6469·7620·636c··.</div>..<div·cl
00010650:·6173·733d·2264·6f63·223e·0a57·6520·6361··ass="doc">.We·ca		00010650:·6173·733d·2264·6f63·223e·0a57·6520·6361··ass="doc">.We·ca
00010660:·6e20·616c·7265·6164·7920·6475·616c·697a··n·already·dualiz		00010660:·6e20·616c·7265·6164·7920·6475·616c·697a··n·already·dualiz
00010670:·6520·616c·6c20·7468·6573·6520·7072·6f70··e·all·these·prop		00010670:·6520·616c·6c20·7468·6573·6520·7072·6f70··e·all·these·prop
00010680:·6572·7469·6573·2e20·0a3c·6469·7620·636c··erties.·.<div·cl		00010680:·6572·7469·6573·2e20·0a3c·6469·7620·636c··erties.·.<div·cl
00010690:·6173·733d·2270·6172·6167·7261·7068·223e··ass="paragraph">		00010690:·6173·733d·2270·6172·6167·7261·7068·223e··ass="paragraph">
000106a0:·203c·2f64·6976·3e0a·0a3c·6120·6964·3d22···</div>..<a·id="		000106a0:·203c·2f64·6976·3e0a·0a3c·6120·6964·3d22···</div>..<a·id="
000106b0:·6c61·6238·3738·223e·3c2f·613e·3c68·3120··lab~~878~~"></a><h1·		000106b0:·6c61·6237·3337·223e·3c2f·613e·3c68·3120··lab737"></a><h1·
000106c0:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section">		000106c0:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section">
000106d0:·5374·616e·6461·7264·2069·6e73·7461·6e63··Standard·instanc		000106d0:·5374·616e·6461·7264·2069·6e73·7461·6e63··Standard·instanc
000106e0:·6573·2e3c·2f68·313e·0a0a·3c2f·6469·763e··es.</h1>..</div>		000106e0:·6573·2e3c·2f68·313e·0a0a·3c2f·6469·763e··es.</h1>..</div>
000106f0:·0a3c·6469·7620·636c·6173·733d·2263·6f64··.<div·class="cod		000106f0:·0a3c·6469·7620·636c·6173·733d·2263·6f64··.<div·class="cod
00010700:·6522·3e0a·0a3c·6272·2f3e·0a3c·7370·616e··e">..<br/>.<span		00010700:·6522·3e0a·0a3c·6272·2f3e·0a3c·7370·616e··e">..<br/>.<span
00010710:·2063·6c61·7373·3d22·6964·2220·7469·746c···class="id"·titl		00010710:·2063·6c61·7373·3d22·6964·2220·7469·746c···class="id"·titl
00010720:·653d·226b·6579·776f·7264·223e·4c74·6163··e="keyword">Ltac		00010720:·653d·226b·6579·776f·7264·223e·4c74·6163··e="keyword">Ltac
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0001d3b0:·732e·6874·6d6c·2342·696e·6172·792e·4122··s.html#Binary.A"		0001d3b0:·732e·6874·6d6c·2342·696e·6172·792e·4122··s.html#Binary.A"
0001d3c0:·3e3c·7370·616e·2063·6c61·7373·3d22·6964··><span·class="id		0001d3c0:·3e3c·7370·616e·2063·6c61·7373·3d22·6964··><span·class="id
0001d3d0:·2220·7469·746c·653d·2276·6172·6961·626c··"·title="variabl		0001d3d0:·2220·7469·746c·653d·2276·6172·6961·626c··"·title="variabl
0001d3e0:·6522·3e41·3c2f·7370·616e·3e3c·2f61·3e29··e">A</span></a>)		0001d3e0:·6522·3e41·3c2f·7370·616e·3e3c·2f61·3e29··e">A</span></a>)
0001d3f0:·2e3c·6272·2f3e·0a20·0a3c·6272·2f3e·0a3c··.<br/>.·.<br/>.<		0001d3f0:·2e3c·6272·2f3e·0a20·0a3c·6272·2f3e·0a3c··.<br/>.·.<br/>.<
0001d400:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas		0001d400:·2f64·6976·3e0a·0a3c·6469·7620·636c·6173··/div>..<div·clas
0001d410:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id="		0001d410:·733d·2264·6f63·223e·0a3c·6120·6964·3d22··s="doc">.<a·id="
0001d420:·6c61·6238·3739·223e·3c2f·613e·3c68·3320··lab~~879~~"></a><h3·		0001d420:·6c61·6237·3338·223e·3c2f·613e·3c68·3320··lab738"></a><h3·
0001d430:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section">		0001d430:·636c·6173·733d·2273·6563·7469·6f6e·223e··class="section">
0001d440:·5061·7274·6961·6c20·4f72·6465·722e·3c2f··Partial·Order.</		0001d440:·5061·7274·6961·6c20·4f72·6465·722e·3c2f··Partial·Order.</
0001d450:·6833·3e0a·0a20·2020·4120·7061·7274·6961··h3>..···A·partia		0001d450:·6833·3e0a·0a20·2020·4120·7061·7274·6961··h3>..···A·partia
0001d460:·6c20·6f72·6465·7220·6973·2061·2070·7265··l·order·is·a·pre		0001d460:·6c20·6f72·6465·7220·6973·2061·2070·7265··l·order·is·a·pre
0001d470:·6f72·6465·7220·7768·6963·6820·6973·2061··order·which·is·a		0001d470:·6f72·6465·7220·7768·6963·6820·6973·2061··order·which·is·a
0001d480:·6464·6974·696f·6e61·6c6c·7920·616e·7469··dditionally·anti		0001d480:·6464·6974·696f·6e61·6c6c·7920·616e·7469··dditionally·anti
0001d490:·7379·6d6d·6574·7269·632e·0a20·2020·5765··symmetric..···We		0001d490:·7379·6d6d·6574·7269·632e·0a20·2020·5765··symmetric..···We


Offset 47, 15 lines modified		Offset 47, 15 lines modified

47	<br/>	47	<br/>

48	<br/>	48	<br/>
49	</div>	49	</div>

50	<div·class="doc">	50	<div·class="doc">
51	<a·id="lab~~776~~"></a><h1·class="section">FMapAVL</h1>	51	<a·id="lab849"></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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84	<span·class="id"·title="keyword">Notation</span>·<a·id="0ecba978737ceb119698c5cbbee795eb"·class="idref"·href="#0ecba978737ceb119698c5cbbee795eb"><span·class="id"·title="notation">"</span></a>s·#1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#fst"><span·class="id"·title="definition">fst</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#1'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>	84	<span·class="id"·title="keyword">Notation</span>·<a·id="0ecba978737ceb119698c5cbbee795eb"·class="idref"·href="#0ecba978737ceb119698c5cbbee795eb"><span·class="id"·title="notation">"</span></a>s·#1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#fst"><span·class="id"·title="definition">fst</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#1'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>
85	<span·class="id"·title="keyword">Notation</span>·<a·id="3434f18bfdd992fed799ae34b564c3f9"·class="idref"·href="#3434f18bfdd992fed799ae34b564c3f9"><span·class="id"·title="notation">"</span></a>s·#2"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#snd"><span·class="id"·title="definition">snd</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#2'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>	85	<span·class="id"·title="keyword">Notation</span>·<a·id="3434f18bfdd992fed799ae34b564c3f9"·class="idref"·href="#3434f18bfdd992fed799ae34b564c3f9"><span·class="id"·title="notation">"</span></a>s·#2"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#snd"><span·class="id"·title="definition">snd</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#2'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>

86	<br/>	86	<br/>
87	</div>	87	</div>

88	<div·class="doc">	88	<div·class="doc">
89	<a·id="lab~~777~~"></a><h1·class="section">The·Raw·functor</h1>	89	<a·id="lab850"></a><h1·class="section">The·Raw·functor</h1>


90	<div·class="paragraph">·</div>	90	<div·class="paragraph">·</div>

91	···Functor·of·pure·functions·+·separate·proofs·of·invariant	91	···Functor·of·pure·functions·+·separate·proofs·of·invariant
92	···preservation·	92	···preservation·
93	</div>	93	</div>
Offset 110, 19 lines modified		Offset 110, 19 lines modified
110	#[<span·class="id"·title="var">global</span>]<br/>	110	#[<span·class="id"·title="var">global</span>]<br/>
111	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Transparent</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.key"><span·class="id"·title="definition">key</span></a>·:·<span·class="id"·title="var">core</span>.<br/>	111	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Transparent</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.key"><span·class="id"·title="definition">key</span></a>·:·<span·class="id"·title="var">core</span>.<br/>

112	<br/>	112	<br/>
113	</div>	113	</div>

114	<div·class="doc">	114	<div·class="doc">
115	<a·id="lab~~778~~"></a><h1·class="section">Trees</h1>	115	<a·id="lab851"></a><h1·class="section">Trees</h1>

116	<div·class="paragraph">·</div>	116	<div·class="paragraph">·</div>

117	<a·id="lab~~779~~"></a><h1·class="section">Trees</h1>	117	<a·id="lab852"></a><h1·class="section">Trees</h1>


118	<div·class="paragraph">·</div>	118	<div·class="paragraph">·</div>

119	···The·fifth·field·of·<span·class="inlinecode"><span·class="id"·title="var">Node</span></span>·is·the·height·of·the·tree·	119	···The·fifth·field·of·<span·class="inlinecode"><span·class="id"·title="var">Node</span></span>·is·the·height·of·the·tree·
120	</div>	120	</div>
121	<div·class="code">	121	<div·class="code">
Offset 146, 15 lines modified		Offset 146, 15 lines modified
146	<br/>	146	<br/>
147	<span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>.<br/>	147	<span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>.<br/>

148	<br/>	148	<br/>
149	</div>	149	</div>

150	<div·class="doc">	150	<div·class="doc">
151	<a·id="lab~~780~~"></a><h1·class="section">Basic·functions·on·trees:·height·and·cardinal</h1>	151	<a·id="lab853"></a><h1·class="section">Basic·functions·on·trees:·height·and·cardinal</h1>

152	</div>	152	</div>
153	<div·class="code">	153	<div·class="code">

154	<br/>	154	<br/>
155	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.height"·class="idref"·href="#Raw.height"><span·class="id"·title="definition">height</span></a>·(<a·id="m:6"·class="idref"·href="#m:6"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>)·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.int"><span·class="id"·title="abbreviation">int</span></a>·:=<br/>	155	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.height"·class="idref"·href="#Raw.height"><span·class="id"·title="definition">height</span></a>·(<a·id="m:6"·class="idref"·href="#m:6"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>)·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.int"><span·class="id"·title="abbreviation">int</span></a>·:=<br/>
156	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:6"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	156	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:6"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
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169	\|·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Node"><span·class="id"·title="constructor">Node</span></a>·<span·class="id"·title="var">l</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">r</span>·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">l</span>·<a·class="idref"·href="Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">r</span>)<br/>	169	\|·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Node"><span·class="id"·title="constructor">Node</span></a>·<span·class="id"·title="var">l</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">r</span>·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">l</span>·<a·class="idref"·href="Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">r</span>)<br/>
170	<span·class="id"·title="keyword">end</span>.<br/>	170	<span·class="id"·title="keyword">end</span>.<br/>

171	<br/>	171	<br/>
172	</div>	172	</div>

173	<div·class="doc">	173	<div·class="doc">
174	<a·id="lab~~781~~"></a><h1·class="section">Empty·Map</h1>	174	<a·id="lab854"></a><h1·class="section">Empty·Map</h1>

175	</div>	175	</div>
176	<div·class="code">	176	<div·class="code">

177	<br/>	177	<br/>
178	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.empty"·class="idref"·href="#Raw.empty"><span·class="id"·title="definition">empty</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>·:=·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>.<br/>	178	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.empty"·class="idref"·href="#Raw.empty"><span·class="id"·title="definition">empty</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>·:=·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>.<br/>

179	<br/>	179	<br/>
180	</div>	180	</div>

181	<div·class="doc">	181	<div·class="doc">
182	<a·id="lab~~782~~"></a><h1·class="section">Emptyness·test</h1>	182	<a·id="lab855"></a><h1·class="section">Emptyness·test</h1>

183	</div>	183	</div>
184	<div·class="code">	184	<div·class="code">

185	<br/>	185	<br/>
186	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.is_empty"·class="idref"·href="#Raw.is_empty"><span·class="id"·title="definition">is_empty</span></a>·<a·id="m:11"·class="idref"·href="#m:11"><span·class="id"·title="binder">m</span></a>·:=·<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:11"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</span></a>·\|·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·<span·class="id"·title="keyword">end</span>.<br/>	186	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.is_empty"·class="idref"·href="#Raw.is_empty"><span·class="id"·title="definition">is_empty</span></a>·<a·id="m:11"·class="idref"·href="#m:11"><span·class="id"·title="binder">m</span></a>·:=·<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:11"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</span></a>·\|·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·<span·class="id"·title="keyword">end</span>.<br/>

187	<br/>	187	<br/>
188	</div>	188	</div>

189	<div·class="doc">	189	<div·class="doc">
190	<a·id="lab~~783~~"></a><h1·class="section">Membership</h1>	190	<a·id="lab856"></a><h1·class="section">Membership</h1>

191	<div·class="paragraph">·</div>	191	<div·class="paragraph">·</div>

192	·The·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·function·is·deciding·membership.·It·exploits·the·<span·class="inlinecode"><span·class="id"·title="var">bst</span></span>·property	192	·The·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·function·is·deciding·membership.·It·exploits·the·<span·class="inlinecode"><span·class="id"·title="var">bst</span></span>·property
193	····to·achieve·logarithmic·complexity.·	193	····to·achieve·logarithmic·complexity.·
194	</div>	194	</div>
195	<div·class="code">	195	<div·class="code">
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228	<span·class="id"·title="keyword">end</span><br/>	228	<span·class="id"·title="keyword">end</span><br/>
229	<span·class="id"·title="keyword">end</span>.<br/>	229	<span·class="id"·title="keyword">end</span>.<br/>

230	<br/>	230	<br/>
231	</div>	231	</div>

232	<div·class="doc">	232	<div·class="doc">
233	<a·id="lab~~784~~"></a><h1·class="section">Helper·functions</h1>	233	<a·id="lab857"></a><h1·class="section">Helper·functions</h1>

234	<div·class="paragraph">·</div>	234	<div·class="paragraph">·</div>

235	·<span·class="inlinecode"><span·class="id"·title="var">create</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·creates·a·node,·assuming·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>	235	·<span·class="inlinecode"><span·class="id"·title="var">create</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·creates·a·node,·assuming·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>
236	····to·be·balanced·and·<span·class="inlinecode">\|<span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode">-</span>·<span·class="inlinecode"><span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span>\|</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode">2</span>.·	236	····to·be·balanced·and·<span·class="inlinecode">\|<span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode">-</span>·<span·class="inlinecode"><span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span>\|</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode">2</span>.·
237	</div>	237	</div>
238	<div·class="code">	238	<div·class="code">
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291	<span·class="id"·title="keyword">else</span><br/>	291	<span·class="id"·title="keyword">else</span><br/>
292	<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.create"><span·class="id"·title="definition">create</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#l:25"><span·class="id"·title="variable">l</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#x:26"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#d:27"><span·class="id"·title="variable">d</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#r:28"><span·class="id"·title="variable">r</span></a>.<br/>	292	<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.create"><span·class="id"·title="definition">create</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#l:25"><span·class="id"·title="variable">l</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#x:26"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#d:27"><span·class="id"·title="variable">d</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#r:28"><span·class="id"·title="variable">r</span></a>.<br/>

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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="lab736"></a><h1·class="section">Overview</h1>	76	<a·id="lab936"></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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177	</pre>	177	</pre>

178	</li>	178	</li>
179	</ul>	179	</ul>

180	<div·class="paragraph">·</div>	180	<div·class="paragraph">·</div>

181	<a·id="lab737"></a><h1·class="section">Facts·and·Tactics·for·Propositional·Logic</h1>	181	<a·id="lab937"></a><h1·class="section">Facts·and·Tactics·for·Propositional·Logic</h1>

182	······These·lemmas·and·tactics·are·in·a·module·so·that·they·do	182	······These·lemmas·and·tactics·are·in·a·module·so·that·they·do
183	······not·affect·the·namespace·if·you·import·the·enclosing	183	······not·affect·the·namespace·if·you·import·the·enclosing
184	······module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.·	184	······module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.·
185	</div>	185	</div>
186	<div·class="code">	186	<div·class="code">
187	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetLogicalFacts"·class="idref"·href="#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>	187	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetLogicalFacts"·class="idref"·href="#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>
188	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Logic.Decidable.html#"><span·class="id"·title="module">Decidable</span></a>.<br/>	188	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Logic.Decidable.html#"><span·class="id"·title="module">Decidable</span></a>.<br/>
189	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="module">Setoid</span></a>.<br/>	189	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="module">Setoid</span></a>.<br/>

190	<br/>	190	<br/>
191	</div>	191	</div>

192	<div·class="doc">	192	<div·class="doc">
193	<a·id="lab738"></a><h2·class="section">Lemmas·and·Tactics·About·Decidable·Propositions</h2>	193	<a·id="lab938"></a><h2·class="section">Lemmas·and·Tactics·About·Decidable·Propositions</h2>

194	<div·class="paragraph">·</div>	194	<div·class="paragraph">·</div>

195	<a·id="lab739"></a><h2·class="section">Propositional·Equivalences·Involving·Negation</h2>	195	<a·id="lab939"></a><h2·class="section">Propositional·Equivalences·Involving·Negation</h2>

196	········These·are·all·written·with·the·unfolded·form·of	196	········These·are·all·written·with·the·unfolded·form·of
197	········negation,·since·I·am·not·sure·if·setoid·rewriting·will	197	········negation,·since·I·am·not·sure·if·setoid·rewriting·will
198	········always·perform·conversion.·	198	········always·perform·conversion.·
199	<div·class="paragraph">·</div>	199	<div·class="paragraph">·</div>

200	<a·id="lab740"></a><h2·class="section">Tactics·for·Negations</h2>	200	<a·id="lab940"></a><h2·class="section">Tactics·for·Negations</h2>

201	</div>	201	</div>
202	<div·class="code">	202	<div·class="code">

203	<br/>	203	<br/>
204	<span·class="id"·title="keyword">Tactic·Notation</span>·"fold"·"any"·"not"·:=<br/>	204	<span·class="id"·title="keyword">Tactic·Notation</span>·"fold"·"any"·"not"·:=<br/>
205	<span·class="id"·title="tactic">repeat</span>·(<br/>	205	<span·class="id"·title="tactic">repeat</span>·(<br/>
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454	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>	454	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>
455	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>	455	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>

456	<br/>	456	<br/>
457	</div>	457	</div>

458	<div·class="doc">	458	<div·class="doc">
459	<a·id="lab741"></a><h1·class="section">Auxiliary·Tactics</h1>	459	<a·id="lab941"></a><h1·class="section">Auxiliary·Tactics</h1>

460	······Again,·these·lemmas·and·tactics·are·in·a·module·so·that	460	······Again,·these·lemmas·and·tactics·are·in·a·module·so·that
461	······they·do·not·affect·the·namespace·if·you·import·the	461	······they·do·not·affect·the·namespace·if·you·import·the
462	······enclosing·module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.··	462	······enclosing·module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.··
463	</div>	463	</div>
464	<div·class="code">	464	<div·class="code">
465	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetDecideAuxiliary"·class="idref"·href="#WDecide_fun.FSetDecideAuxiliary"><span·class="id"·title="module">FSetDecideAuxiliary</span></a>.<br/>	465	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetDecideAuxiliary"·class="idref"·href="#WDecide_fun.FSetDecideAuxiliary"><span·class="id"·title="module">FSetDecideAuxiliary</span></a>.<br/>

466	<br/>	466	<br/>
467	</div>	467	</div>

468	<div·class="doc">	468	<div·class="doc">
469	<a·id="lab742"></a><h2·class="section">Generic·Tactics</h2>	469	<a·id="lab942"></a><h2·class="section">Generic·Tactics</h2>

470	········We·begin·by·defining·a·few·generic,·useful·tactics.·	470	········We·begin·by·defining·a·few·generic,·useful·tactics.·
471	<div·class="paragraph">·</div>	471	<div·class="paragraph">·</div>

472	·remove·logical·hypothesis·inter-dependencies·(fix·2136).·	472	·remove·logical·hypothesis·inter-dependencies·(fix·2136).·
473	</div>	473	</div>
474	<div·class="code">	474	<div·class="code">
Offset 584, 15 lines modified		Offset 584, 15 lines modified
584	\|·<span·class="id"·title="var">H</span>:·<span·class="id"·title="var">_</span>·\|-·<span·class="id"·title="var">_</span>·=>·<span·class="id"·title="tactic">progress</span>·(<span·class="id"·title="var">decompose</span>·<span·class="id"·title="var">record</span>·<span·class="id"·title="var">H</span>);·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>	584	\|·<span·class="id"·title="var">H</span>:·<span·class="id"·title="var">_</span>·\|-·<span·class="id"·title="var">_</span>·=>·<span·class="id"·title="tactic">progress</span>·(<span·class="id"·title="var">decompose</span>·<span·class="id"·title="var">record</span>·<span·class="id"·title="var">H</span>);·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>
585	<span·class="id"·title="keyword">end</span>).<br/>	585	<span·class="id"·title="keyword">end</span>).<br/>

586	<br/>	586	<br/>
587	</div>	587	</div>

588	<div·class="doc">	588	<div·class="doc">
589	<a·id="lab743"></a><h2·class="section">Discarding·Irrelevant·Hypotheses</h2>	589	<a·id="lab943"></a><h2·class="section">Discarding·Irrelevant·Hypotheses</h2>

590	········We·will·want·to·clear·the·context·of·any	590	········We·will·want·to·clear·the·context·of·any
591	········non-FSet-related·hypotheses·in·order·to·increase·the	591	········non-FSet-related·hypotheses·in·order·to·increase·the
592	········speed·of·the·tactic.··To·do·this,·we·will·need·to·be	592	········speed·of·the·tactic.··To·do·this,·we·will·need·to·be
593	········able·to·decide·which·are·relevant.··We·do·this·by·making	593	········able·to·decide·which·are·relevant.··We·do·this·by·making
594	········a·simple·inductive·definition·classifying·the	594	········a·simple·inductive·definition·classifying·the
595	········propositions·of·interest.·	595	········propositions·of·interest.·
Offset 664, 15 lines modified		Offset 664, 15 lines modified
664	<span·class="id"·title="keyword">else</span>·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>	664	<span·class="id"·title="keyword">else</span>·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>
665	<span·class="id"·title="keyword">end</span>).<br/>	665	<span·class="id"·title="keyword">end</span>).<br/>

666	<br/>	666	<br/>
667	</div>	667	</div>

668	<div·class="doc">	668	<div·class="doc">
669	<a·id="lab744"></a><h2·class="section">Turning·Set·Operators·into·Propositional·Connectives</h2>	669	<a·id="lab944"></a><h2·class="section">Turning·Set·Operators·into·Propositional·Connectives</h2>

670	········The·lemmas·from·<span·class="inlinecode"><span·class="id"·title="var">FSetFacts</span></span>·will·be·used·to·break·down	670	········The·lemmas·from·<span·class="inlinecode"><span·class="id"·title="var">FSetFacts</span></span>·will·be·used·to·break·down
671	········set·operations·into·propositional·formulas·built·over	671	········set·operations·into·propositional·formulas·built·over
672	········the·predicates·<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">E.eq</span></span>·applied·only·to	672	········the·predicates·<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">E.eq</span></span>·applied·only·to
673	········variables.··We·are·going·to·use·them·with·<span·class="inlinecode"><span·class="id"·title="tactic">autorewrite</span></span>.	673	········variables.··We·are·going·to·use·them·with·<span·class="inlinecode"><span·class="id"·title="tactic">autorewrite</span></span>.
674	········	674	········
675	</div>	675	</div>
Offset 690, 15 lines modified		Offset 690, 15 lines modified
690	<br/>	690	<br/>
691	<span·class="id"·title="keyword">Global·Hint·Rewrite</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetDecideAuxiliary.eq_refl_iff"><span·class="id"·title="lemma">eq_refl_iff</span></a>·:·<span·class="id"·title="var">set_eq_simpl</span>.<br/>	691	<span·class="id"·title="keyword">Global·Hint·Rewrite</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetDecideAuxiliary.eq_refl_iff"><span·class="id"·title="lemma">eq_refl_iff</span></a>·:·<span·class="id"·title="var">set_eq_simpl</span>.<br/>

692	<br/>	692	<br/>
693	</div>	693	</div>

694	<div·class="doc">	694	<div·class="doc">
695	<a·id="lab745"></a><h2·class="section">Decidability·of·FSet·Propositions</h2>	695	<a·id="lab945"></a><h2·class="section">Decidability·of·FSet·Propositions</h2>
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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~~712~~"></a><h1·class="section">Finite·sets·library·:·conversion·to·old·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span></h1>	49	<a·id="lab985"></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~~713~~"></a><h1·class="section">Going·from·<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·with·usual·Leibniz·equality</h1>	58	<a·id="lab986"></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 48, 15 lines modified		Offset 48, 15 lines modified
48	<br/>	48	<br/>
49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>.<br/>	49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>.<br/>

50	<br/>	50	<br/>
51	</div>	51	</div>

52	<div·class="doc">	52	<div·class="doc">
53	<a·id="lab~~363~~"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1>	53	<a·id="lab788"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1>

54	</div>	54	</div>
55	<div·class="code">	55	<div·class="code">

56	<br/>	56	<br/>
57	<span·class="id"·title="keyword">Notation</span>·<a·id="valid_binary"·class="idref"·href="#valid_binary"><span·class="id"·title="abbreviation">valid_binary</span></a>·:=·(<a·class="idref"·href="Coq.Floats.SpecFloat.html#valid_binary"><span·class="id"·title="definition">valid_binary</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>).<br/>	57	<span·class="id"·title="keyword">Notation</span>·<a·id="valid_binary"·class="idref"·href="#valid_binary"><span·class="id"·title="abbreviation">valid_binary</span></a>·:=·(<a·class="idref"·href="Coq.Floats.SpecFloat.html#valid_binary"><span·class="id"·title="definition">valid_binary</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>).<br/>


Offset 49, 15 lines modified		Offset 49, 15 lines modified
49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>·<a·class="idref"·href="Coq.Floats.FloatAxioms.html#"><span·class="id"·title="library">FloatAxioms</span></a>.<br/>	49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>·<a·class="idref"·href="Coq.Floats.FloatAxioms.html#"><span·class="id"·title="library">FloatAxioms</span></a>.<br/>
50	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Psatz.html#"><span·class="id"·title="library">Psatz</span></a>.<br/>	50	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Psatz.html#"><span·class="id"·title="library">Psatz</span></a>.<br/>

51	<br/>	51	<br/>
52	</div>	52	</div>

53	<div·class="doc">	53	<div·class="doc">
54	<a·id="lab~~362~~"></a><h1·class="section">Support·results·involving·frexp·and·ldexp</h1>	54	<a·id="lab789"></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.Numbers.Cyclic.Int63.PrimInt63.html#"><span·class="id"·title="library">PrimInt63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>	49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.PrimInt63.html#"><span·class="id"·title="library">PrimInt63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>

50	<br/>	50	<br/>
51	</div>	51	</div>

52	<div·class="doc">	52	<div·class="doc">
53	<a·id="lab~~369~~"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1>	53	<a·id="lab790"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1>


54	<div·class="paragraph">·</div>	54	<div·class="paragraph">·</div>

55	This·interface·provides·processor·operators·for·the·Binary64·format·of·the	55	This·interface·provides·processor·operators·for·the·Binary64·format·of·the
56	IEEE·754-2008·standard.·	56	IEEE·754-2008·standard.·
57	<div·class="paragraph">·</div>	57	<div·class="paragraph">·</div>

58	<a·id="lab~~370~~"></a><h2·class="section">Type·definition·for·the·co-domain·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h2>	58	<a·id="lab791"></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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73	<br/>	73	<br/>
74	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_class</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_class</span>.<br/>	74	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_class</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_class</span>.<br/>

75	<br/>	75	<br/>
76	</div>	76	</div>

77	<div·class="doc">	77	<div·class="doc">
78	<a·id="lab~~371~~"></a><h2·class="section">The·main·type</h2>	78	<a·id="lab792"></a><h2·class="section">The·main·type</h2>
79	·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>:·primitive·type·for·Binary64·floating-point·numbers.·	79	·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>:·primitive·type·for·Binary64·floating-point·numbers.·
80	</div>	80	</div>
81	<div·class="code">	81	<div·class="code">
82	<span·class="id"·title="keyword">Primitive</span>·<a·id="float"·class="idref"·href="#float"><span·class="id"·title="axiom">float</span></a>·:=·#<span·class="id"·title="var">float64_type</span>.<br/>	82	<span·class="id"·title="keyword">Primitive</span>·<a·id="float"·class="idref"·href="#float"><span·class="id"·title="axiom">float</span></a>·:=·#<span·class="id"·title="var">float64_type</span>.<br/>
83	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.float.type</span>.<br/>	83	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.float.type</span>.<br/>

84	<br/>	84	<br/>
Offset 91, 30 lines modified		Offset 91, 30 lines modified
91	<span·class="id"·title="keyword">Definition</span>·<a·id="printer"·class="idref"·href="#printer"><span·class="id"·title="definition">printer</span></a>·(<a·id="x:5"·class="idref"·href="#x:5"><span·class="id"·title="binder">x</span></a>·:·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrapper"><span·class="id"·title="record">float_wrapper</span></a>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrap"><span·class="id"·title="projection">float_wrap</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:5"><span·class="id"·title="variable">x</span></a>.<br/>	91	<span·class="id"·title="keyword">Definition</span>·<a·id="printer"·class="idref"·href="#printer"><span·class="id"·title="definition">printer</span></a>·(<a·id="x:5"·class="idref"·href="#x:5"><span·class="id"·title="binder">x</span></a>·:·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrapper"><span·class="id"·title="record">float_wrapper</span></a>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrap"><span·class="id"·title="projection">float_wrap</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:5"><span·class="id"·title="variable">x</span></a>.<br/>
92	<span·class="id"·title="keyword">Definition</span>·<a·id="parser"·class="idref"·href="#parser"><span·class="id"·title="definition">parser</span></a>·(<a·id="x:6"·class="idref"·href="#x:6"><span·class="id"·title="binder">x</span></a>·:·<span·class="id"·title="var">float</span>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:6"><span·class="id"·title="variable">x</span></a>.<br/>	92	<span·class="id"·title="keyword">Definition</span>·<a·id="parser"·class="idref"·href="#parser"><span·class="id"·title="definition">parser</span></a>·(<a·id="x:6"·class="idref"·href="#x:6"><span·class="id"·title="binder">x</span></a>·:·<span·class="id"·title="var">float</span>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:6"><span·class="id"·title="variable">x</span></a>.<br/>

93	<br/>	93	<br/>
94	</div>	94	</div>

95	<div·class="doc">	95	<div·class="doc">
96	<a·id="lab~~372~~"></a><h2·class="section">Syntax·support</h2>	96	<a·id="lab793"></a><h2·class="section">Syntax·support</h2>

97	</div>	97	</div>
98	<div·class="code">	98	<div·class="code">
99	<span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Import</span>·<a·id="PrimFloatNotationsInternalA"·class="idref"·href="#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>	99	<span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Import</span>·<a·id="PrimFloatNotationsInternalA"·class="idref"·href="#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>
100	<span·class="id"·title="keyword">Declare·Scope</span>·<span·class="id"·title="var">float_scope</span>.<br/>	100	<span·class="id"·title="keyword">Declare·Scope</span>·<span·class="id"·title="var">float_scope</span>.<br/>
101	<span·class="id"·title="keyword">Delimit</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>	101	<span·class="id"·title="keyword">Delimit</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>
102	<span·class="id"·title="keyword">Bind·Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>	102	<span·class="id"·title="keyword">Bind·Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>
103	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>	103	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>
104	<span·class="id"·title="keyword">Number·Notation</span>·<span·class="id"·title="var">float</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#parser"><span·class="id"·title="definition">parser</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#printer"><span·class="id"·title="definition">printer</span></a>·:·<span·class="id"·title="var">float_scope</span>.<br/>	104	<span·class="id"·title="keyword">Number·Notation</span>·<span·class="id"·title="var">float</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#parser"><span·class="id"·title="definition">parser</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#printer"><span·class="id"·title="definition">printer</span></a>·:·<span·class="id"·title="var">float_scope</span>.<br/>

105	<br/>	105	<br/>
106	</div>	106	</div>

107	<div·class="doc">	107	<div·class="doc">
108	<a·id="lab~~373~~"></a><h2·class="section">Floating-point·operators</h2>	108	<a·id="lab794"></a><h2·class="section">Floating-point·operators</h2>

109	</div>	109	</div>
110	<div·class="code">	110	<div·class="code">
111	<span·class="id"·title="keyword">Primitive</span>·<a·id="classify"·class="idref"·href="#classify"><span·class="id"·title="axiom">classify</span></a>·:=·#<span·class="id"·title="var">float64_classify</span>.<br/>	111	<span·class="id"·title="keyword">Primitive</span>·<a·id="classify"·class="idref"·href="#classify"><span·class="id"·title="axiom">classify</span></a>·:=·#<span·class="id"·title="var">float64_classify</span>.<br/>

112	<br/>	112	<br/>
113	<span·class="id"·title="keyword">Primitive</span>·<a·id="abs"·class="idref"·href="#abs"><span·class="id"·title="axiom">abs</span></a>·:=·#<span·class="id"·title="var">float64_abs</span>.<br/>	113	<span·class="id"·title="keyword">Primitive</span>·<a·id="abs"·class="idref"·href="#abs"><span·class="id"·title="axiom">abs</span></a>·:=·#<span·class="id"·title="var">float64_abs</span>.<br/>
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181	<span·class="id"·title="keyword">Notation</span>·<a·id="6e32edb097de448d43e1d8a509d61efe"·class="idref"·href="#6e32edb097de448d43e1d8a509d61efe"><span·class="id"·title="notation">"</span></a>x·/·y"·:=·(<span·class="id"·title="var">div</span>·<span·class="id"·title="var">x</span>·<span·class="id"·title="var">y</span>)·:·<span·class="id"·title="var">float_scope</span>.<br/>	181	<span·class="id"·title="keyword">Notation</span>·<a·id="6e32edb097de448d43e1d8a509d61efe"·class="idref"·href="#6e32edb097de448d43e1d8a509d61efe"><span·class="id"·title="notation">"</span></a>x·/·y"·:=·(<span·class="id"·title="var">div</span>·<span·class="id"·title="var">x</span>·<span·class="id"·title="var">y</span>)·:·<span·class="id"·title="var">float_scope</span>.<br/>
182	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalB"><span·class="id"·title="module">PrimFloatNotationsInternalB</span></a>.<br/>	182	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalB"><span·class="id"·title="module">PrimFloatNotationsInternalB</span></a>.<br/>

183	<br/>	183	<br/>
184	</div>	184	</div>

185	<div·class="doc">	185	<div·class="doc">
186	<a·id="lab~~374~~"></a><h2·class="section">Conversions</h2>	186	<a·id="lab795"></a><h2·class="section">Conversions</h2>

187	<div·class="paragraph">·</div>	187	<div·class="paragraph">·</div>

188	·<span·class="inlinecode"><span·class="id"·title="var">of_uint63</span></span>:·convert·a·primitive·unsigned·integer·into·a·float·value.	188	·<span·class="inlinecode"><span·class="id"·title="var">of_uint63</span></span>:·convert·a·primitive·unsigned·integer·into·a·float·value.
189	····The·value·is·rounded·if·need·be.·	189	····The·value·is·rounded·if·need·be.·
190	</div>	190	</div>
191	<div·class="code">	191	<div·class="code">
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220	<div·class="code">	220	<div·class="code">
221	<span·class="id"·title="keyword">Primitive</span>·<a·id="normfr_mantissa"·class="idref"·href="#normfr_mantissa"><span·class="id"·title="axiom">normfr_mantissa</span></a>·:=·#<span·class="id"·title="var">float64_normfr_mantissa</span>.<br/>	221	<span·class="id"·title="keyword">Primitive</span>·<a·id="normfr_mantissa"·class="idref"·href="#normfr_mantissa"><span·class="id"·title="axiom">normfr_mantissa</span></a>·:=·#<span·class="id"·title="var">float64_normfr_mantissa</span>.<br/>

222	<br/>	222	<br/>
223	</div>	223	</div>

224	<div·class="doc">	224	<div·class="doc">
225	<a·id="lab~~375~~"></a><h2·class="section">Exponent·manipulation·functions</h2>	225	<a·id="lab796"></a><h2·class="section">Exponent·manipulation·functions</h2>
226	·<span·class="inlinecode"><span·class="id"·title="var">frshiftexp</span></span>:·convert·a·float·to·fractional·part·in·[0.5,·1.)	226	·<span·class="inlinecode"><span·class="id"·title="var">frshiftexp</span></span>:·convert·a·float·to·fractional·part·in·[0.5,·1.)
227	and·integer·part.·	227	and·integer·part.·
228	</div>	228	</div>
229	<div·class="code">	229	<div·class="code">
230	<span·class="id"·title="keyword">Primitive</span>·<a·id="frshiftexp"·class="idref"·href="#frshiftexp"><span·class="id"·title="axiom">frshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_frshiftexp</span>.<br/>	230	<span·class="id"·title="keyword">Primitive</span>·<a·id="frshiftexp"·class="idref"·href="#frshiftexp"><span·class="id"·title="axiom">frshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_frshiftexp</span>.<br/>

231	<br/>	231	<br/>
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240	<div·class="code">	240	<div·class="code">
241	<span·class="id"·title="keyword">Primitive</span>·<a·id="ldshiftexp"·class="idref"·href="#ldshiftexp"><span·class="id"·title="axiom">ldshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_ldshiftexp</span>.<br/>	241	<span·class="id"·title="keyword">Primitive</span>·<a·id="ldshiftexp"·class="idref"·href="#ldshiftexp"><span·class="id"·title="axiom">ldshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_ldshiftexp</span>.<br/>

242	<br/>	242	<br/>
243	</div>	243	</div>

244	<div·class="doc">	244	<div·class="doc">
245	<a·id="lab~~376~~"></a><h2·class="section">Predecesor/Successor·functions</h2>	245	<a·id="lab797"></a><h2·class="section">Predecesor/Successor·functions</h2>

246	<div·class="paragraph">·</div>	246	<div·class="paragraph">·</div>

247	·<span·class="inlinecode"><span·class="id"·title="var">next_up</span></span>:·return·the·next·float·towards·positive·infinity.·	247	·<span·class="inlinecode"><span·class="id"·title="var">next_up</span></span>:·return·the·next·float·towards·positive·infinity.·
248	</div>	248	</div>
249	<div·class="code">	249	<div·class="code">
250	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_up"·class="idref"·href="#next_up"><span·class="id"·title="axiom">next_up</span></a>·:=·#<span·class="id"·title="var">float64_next_up</span>.<br/>	250	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_up"·class="idref"·href="#next_up"><span·class="id"·title="axiom">next_up</span></a>·:=·#<span·class="id"·title="var">float64_next_up</span>.<br/>
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262	<div·class="code">	262	<div·class="code">
263	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_down"·class="idref"·href="#next_down"><span·class="id"·title="axiom">next_down</span></a>·:=·#<span·class="id"·title="var">float64_next_down</span>.<br/>	263	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_down"·class="idref"·href="#next_down"><span·class="id"·title="axiom">next_down</span></a>·:=·#<span·class="id"·title="var">float64_next_down</span>.<br/>

264	<br/>	264	<br/>
265	</div>	265	</div>

266	<div·class="doc">	266	<div·class="doc">
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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~~388~~"></a><h1·class="section">Datatypes·with·zero·and·one·element</h1>	58	<a·id="lab499"></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 87, 15 lines modified		Offset 87, 15 lines modified
87	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">unit</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.type</span>.<br/>	87	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">unit</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.type</span>.<br/>
88	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">tt</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.tt</span>.<br/>	88	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">tt</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.tt</span>.<br/>

89	<br/>	89	<br/>
90	</div>	90	</div>

91	<div·class="doc">	91	<div·class="doc">
92	<a·id="lab~~389~~"></a><h1·class="section">The·boolean·datatype</h1>	92	<a·id="lab500"></a><h1·class="section">The·boolean·datatype</h1>

93	<div·class="paragraph">·</div>	93	<div·class="paragraph">·</div>

94	·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·is·the·datatype·of·the·boolean·values·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>·	94	·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·is·the·datatype·of·the·boolean·values·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>·
95	</div>	95	</div>
96	<div·class="code">	96	<div·class="code">

Offset 258, 15 lines modified		Offset 258, 15 lines modified
258	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecT</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecT</span>.<br/>	258	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecT</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecT</span>.<br/>
259	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecF</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecF</span>.<br/>	259	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecF</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecF</span>.<br/>

260	<br/>	260	<br/>
261	</div>	261	</div>

262	<div·class="doc">	262	<div·class="doc">
263	<a·id="lab~~390~~"></a><h1·class="section">Peano·natural·numbers</h1>	263	<a·id="lab501"></a><h1·class="section">Peano·natural·numbers</h1>

264	<div·class="paragraph">·</div>	264	<div·class="paragraph">·</div>

265	·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·is·the·datatype·of·natural·numbers·built·from·<span·class="inlinecode"><span·class="id"·title="var">O</span></span>·and·successor·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>;	265	·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·is·the·datatype·of·natural·numbers·built·from·<span·class="inlinecode"><span·class="id"·title="var">O</span></span>·and·successor·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>;
266	····note·that·the·constructor·name·is·the·letter·O.	266	····note·that·the·constructor·name·is·the·letter·O.
267	····Numbers·in·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·can·be·denoted·using·a·decimal·notation;	267	····Numbers·in·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·can·be·denoted·using·a·decimal·notation;
268	····e.g.·<span·class="inlinecode">3%<span·class="id"·title="var">nat</span></span>·abbreviates·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode"><span·class="id"·title="var">O</span>))</span>·	268	····e.g.·<span·class="inlinecode">3%<span·class="id"·title="var">nat</span></span>·abbreviates·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode"><span·class="id"·title="var">O</span>))</span>·
Offset 293, 15 lines modified		Offset 293, 15 lines modified
293	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">O</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.O</span>.<br/>	293	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">O</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.O</span>.<br/>
294	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">S</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.S</span>.<br/>	294	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">S</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.S</span>.<br/>

295	<br/>	295	<br/>
296	</div>	296	</div>

297	<div·class="doc">	297	<div·class="doc">
298	<a·id="lab~~391~~"></a><h1·class="section">Container·datatypes</h1>	298	<a·id="lab502"></a><h1·class="section">Container·datatypes</h1>

299	</div>	299	</div>
300	<div·class="code">	300	<div·class="code">

301	<br/>	301	<br/>

302	<br/>	302	<br/>
Offset 498, 15 lines modified		Offset 498, 15 lines modified

498	<br/>	498	<br/>

499	<br/>	499	<br/>
500	</div>	500	</div>

501	<div·class="doc">	501	<div·class="doc">
502	<a·id="lab~~392~~"></a><h1·class="section">The·comparison·datatype</h1>	502	<a·id="lab503"></a><h1·class="section">The·comparison·datatype</h1>

503	</div>	503	</div>
504	<div·class="code">	504	<div·class="code">

505	<br/>	505	<br/>
506	<span·class="id"·title="keyword">Inductive</span>·<a·id="comparison"·class="idref"·href="#comparison"><span·class="id"·title="definition,·inductive"><span·id="comparison_rect"·class="id"><span·id="comparison_ind"·class="id"><span·id="comparison_rec"·class="id"><span·id="comparison_sind"·class="id">comparison</span></span></span></span></span></a>·:·<span·class="id"·title="keyword">Set</span>·:=<br/>	506	<span·class="id"·title="keyword">Inductive</span>·<a·id="comparison"·class="idref"·href="#comparison"><span·class="id"·title="definition,·inductive"><span·id="comparison_rect"·class="id"><span·id="comparison_ind"·class="id"><span·id="comparison_rec"·class="id"><span·id="comparison_sind"·class="id">comparison</span></span></span></span></span></a>·:·<span·class="id"·title="keyword">Set</span>·:=<br/>
507	\|·<a·id="Eq"·class="idref"·href="#Eq"><span·class="id"·title="constructor">Eq</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#comparison:103"><span·class="id"·title="inductive">comparison</span></a><br/>	507	\|·<a·id="Eq"·class="idref"·href="#Eq"><span·class="id"·title="constructor">Eq</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#comparison:103"><span·class="id"·title="inductive">comparison</span></a><br/>
Offset 616, 15 lines modified		Offset 616, 15 lines modified
616	<span·class="id"·title="keyword">Lemma</span>·<a·id="CompSpec2Type"·class="idref"·href="#CompSpec2Type"><span·class="id"·title="lemma">CompSpec2Type</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="A:138"·class="idref"·href="#A:138"><span·class="id"·title="binder">A</span></a>·(<a·id="eq:139"·class="idref"·href="#eq:139"><span·class="id"·title="binder">eq</span></a>·<a·id="lt:140"·class="idref"·href="#lt:140"><span·class="id"·title="binder">lt</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#A:138"><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.Init.Datatypes.html#A:138"><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><span·class="id"·title="keyword">Prop</span>)·<a·id="x:141"·class="idref"·href="#x:141"><span·class="id"·title="binder">x</span></a>·<a·id="y:142"·class="idref"·href="#y:142"><span·class="id"·title="binder">y</span></a>·<a·id="c:143"·class="idref"·href="#c:143"><span·class="id"·title="binder">c</span></a>,<br/>	616	<span·class="id"·title="keyword">Lemma</span>·<a·id="CompSpec2Type"·class="idref"·href="#CompSpec2Type"><span·class="id"·title="lemma">CompSpec2Type</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="A:138"·class="idref"·href="#A:138"><span·class="id"·title="binder">A</span></a>·(<a·id="eq:139"·class="idref"·href="#eq:139"><span·class="id"·title="binder">eq</span></a>·<a·id="lt:140"·class="idref"·href="#lt:140"><span·class="id"·title="binder">lt</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#A:138"><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.Init.Datatypes.html#A:138"><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><span·class="id"·title="keyword">Prop</span>)·<a·id="x:141"·class="idref"·href="#x:141"><span·class="id"·title="binder">x</span></a>·<a·id="y:142"·class="idref"·href="#y:142"><span·class="id"·title="binder">y</span></a>·<a·id="c:143"·class="idref"·href="#c:143"><span·class="id"·title="binder">c</span></a>,<br/>
617	<a·class="idref"·href="Coq.Init.Datatypes.html#CompSpec"><span·class="id"·title="definition">CompSpec</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</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#CompSpecT"><span·class="id"·title="definition">CompSpecT</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</span></a>.<br/>	617	<a·class="idref"·href="Coq.Init.Datatypes.html#CompSpec"><span·class="id"·title="definition">CompSpec</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</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#CompSpecT"><span·class="id"·title="definition">CompSpecT</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</span></a>.<br/>
618	·	618	·
619	<br/>	619	<br/>
620	</div>	620	</div>

621	<div·class="doc">	621	<div·class="doc">
622	<a·id="lab~~393~~"></a><h1·class="section">Misc·Other·Datatypes</h1>	622	<a·id="lab504"></a><h1·class="section">Misc·Other·Datatypes</h1>

623	<div·class="paragraph">·</div>	623	<div·class="paragraph">·</div>

624	·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·is·the·family·of·datatypes·on·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·whose·sole·non-empty	624	·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·is·the·family·of·datatypes·on·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·whose·sole·non-empty
625	····member·is·the·singleton·datatype·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·whose	625	····member·is·the·singleton·datatype·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·whose
626	····sole·inhabitant·is·denoted·<span·class="inlinecode"><span·class="id"·title="var">identity_refl</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·	626	····sole·inhabitant·is·denoted·<span·class="inlinecode"><span·class="id"·title="var">identity_refl</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·
627	</div>	627	</div>


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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·6120·6964·3d22·6c61·6233··oc">.<a·id="lab3		00000a40:·6f63·223e·0a3c·6120·6964·3d22·6c61·6235··oc">.<a·id="lab5
00000a50:·3832·223e·3c2f·613e·3c68·3120·636c·6173··82"></a><h1·clas		00000a50:·3233·223e·3c2f·613e·3c68·3120·636c·6173··23"></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 4442, 15 lines modified		Offset 4442, 15 lines modified
00011590:·3436·3735·6130·6163·6633·6561·3932·6161··4675a0acf3ea92aa		00011590:·3436·3735·6130·6163·6633·6561·3932·6161··4675a0acf3ea92aa
000115a0:·6261·6464·223e·3c73·7061·6e20·636c·6173··badd"><span·clas		000115a0:·6261·6464·223e·3c73·7061·6e20·636c·6173··badd"><span·clas
000115b0:·733d·2269·6422·2074·6974·6c65·3d22·6e6f··s="id"·title="no		000115b0:·733d·2269·6422·2074·6974·6c65·3d22·6e6f··s="id"·title="no
000115c0:·7461·7469·6f6e·223e·293c·2f73·7061·6e3e··tation">)</span>		000115c0:·7461·7469·6f6e·223e·293c·2f73·7061·6e3e··tation">)</span>
000115d0:·3c2f·613e·2e3c·6272·2f3e·0a0a·3c62·722f··</a>.<br/>..<br/		000115d0:·3c2f·613e·2e3c·6272·2f3e·0a0a·3c62·722f··</a>.<br/>..<br/
000115e0:·3e0a·3c2f·6469·763e·0a0a·3c64·6976·2063··>.</div>..<div·c		000115e0:·3e0a·3c2f·6469·763e·0a0a·3c64·6976·2063··>.</div>..<div·c
000115f0:·6c61·7373·3d22·646f·6322·3e0a·3c61·2069··lass="doc">.<a·i		000115f0:·6c61·7373·3d22·646f·6322·3e0a·3c61·2069··lass="doc">.<a·i
00011600:·643d·226c·6162·3338·3322·3e3c·2f61·3e3c··d="lab~~383~~"></a><		00011600:·643d·226c·6162·3532·3422·3e3c·2f61·3e3c··d="lab524"></a><
00011610:·6831·2063·6c61·7373·3d22·7365·6374·696f··h1·class="sectio		00011610:·6831·2063·6c61·7373·3d22·7365·6374·696f··h1·class="sectio
00011620:·6e22·3e46·6972·7374·2d6f·7264·6572·2071··n">First-order·q		00011620:·6e22·3e46·6972·7374·2d6f·7264·6572·2071··n">First-order·q
00011630:·7561·6e74·6966·6965·7273·3c2f·6831·3e0a··uantifiers</h1>.		00011630:·7561·6e74·6966·6965·7273·3c2f·6831·3e0a··uantifiers</h1>.
00011640:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par		00011640:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par
00011650:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>.		00011650:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>.
00011660:·0a20·3c73·7061·6e20·636c·6173·733d·2269··.·<span·class="i		00011660:·0a20·3c73·7061·6e20·636c·6173·733d·2269··.·<span·class="i
00011670:·6e6c·696e·6563·6f64·6522·3e3c·7370·616e··nlinecode"><span		00011670:·6e6c·696e·6563·6f64·6522·3e3c·7370·616e··nlinecode"><span
Offset 6440, 15 lines modified		Offset 6440, 15 lines modified
00019270:·2063·6c61·7373·3d22·6964·2220·7469·746c···class="id"·titl		00019270:·2063·6c61·7373·3d22·6964·2220·7469·746c···class="id"·titl
00019280:·653d·2273·6563·7469·6f6e·223e·756e·6976··e="section">univ		00019280:·653d·2273·6563·7469·6f6e·223e·756e·6976··e="section">univ
00019290:·6572·7361·6c5f·7175·616e·7469·6669·6361··ersal_quantifica		00019290:·6572·7361·6c5f·7175·616e·7469·6669·6361··ersal_quantifica
000192a0:·7469·6f6e·3c2f·7370·616e·3e3c·2f61·3e2e··tion</span></a>.		000192a0:·7469·6f6e·3c2f·7370·616e·3e3c·2f61·3e2e··tion</span></a>.
000192b0:·3c62·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64··<br/>..<br/>.</d		000192b0:·3c62·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64··<br/>..<br/>.</d
000192c0:·6976·3e0a·0a3c·6469·7620·636c·6173·733d··iv>..<div·class=		000192c0:·6976·3e0a·0a3c·6469·7620·636c·6173·733d··iv>..<div·class=
000192d0:·2264·6f63·223e·0a3c·6120·6964·3d22·6c61··"doc">.<a·id="la		000192d0:·2264·6f63·223e·0a3c·6120·6964·3d22·6c61··"doc">.<a·id="la
000192e0:·6233·3834·223e·3c2f·613e·3c68·3120·636c··b~~384~~"></a><h1·cl		000192e0:·6235·3235·223e·3c2f·613e·3c68·3120·636c··b525"></a><h1·cl
000192f0:·6173·733d·2273·6563·7469·6f6e·223e·4571··ass="section">Eq		000192f0:·6173·733d·2273·6563·7469·6f6e·223e·4571··ass="section">Eq
00019300:·7561·6c69·7479·3c2f·6831·3e0a·0a3c·6469··uality</h1>..<di		00019300:·7561·6c69·7479·3c2f·6831·3e0a·0a3c·6469··uality</h1>..<di
00019310:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra		00019310:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra
00019320:·7068·223e·203c·2f64·6976·3e0a·0a20·3c73··ph">·</div>..·<s		00019320:·7068·223e·203c·2f64·6976·3e0a·0a20·3c73··ph">·</div>..·<s
00019330:·7061·6e20·636c·6173·733d·2269·6e6c·696e··pan·class="inlin		00019330:·7061·6e20·636c·6173·733d·2269·6e6c·696e··pan·class="inlin
00019340:·6563·6f64·6522·3e3c·7370·616e·2063·6c61··ecode"><span·cla		00019340:·6563·6f64·6522·3e3c·7370·616e·2063·6c61··ecode"><span·cla
00019350:·7373·3d22·6964·2220·7469·746c·653d·2276··ss="id"·title="v		00019350:·7373·3d22·6964·2220·7469·746c·653d·2276··ss="id"·title="v
Offset 16761, 15 lines modified		Offset 16761, 15 lines modified
00041780:·3a74·7970·655f·7363·6f70·653a·785f·272d··:type_scope:x_'-		00041780:·3a74·7970·655f·7363·6f70·653a·785f·272d··:type_scope:x_'-
00041790:·2667·743b·275f·7822·3e3c·7370·616e·2063··>'_x"><span·c		00041790:·2667·743b·275f·7822·3e3c·7370·616e·2063··>'_x"><span·c
000417a0:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title=		000417a0:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title=
000417b0:·226e·6f74·6174·696f·6e22·3e29·3c2f·7370··"notation">)</sp		000417b0:·226e·6f74·6174·696f·6e22·3e29·3c2f·7370··"notation">)</sp
000417c0:·616e·3e3c·2f61·3e2e·3c62·722f·3e0a·0a3c··an></a>.<br/>..<		000417c0:·616e·3e3c·2f61·3e2e·3c62·722f·3e0a·0a3c··an></a>.<br/>..<
000417d0:·6272·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469··br/>.</div>..<di		000417d0:·6272·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469··br/>.</div>..<di
000417e0:·7620·636c·6173·733d·2264·6f63·223e·0a3c··v·class="doc">.<		000417e0:·7620·636c·6173·733d·2264·6f63·223e·0a3c··v·class="doc">.<
000417f0:·6120·6964·3d22·6c61·6233·3835·223e·3c2f··a·id="lab~~385~~"></		000417f0:·6120·6964·3d22·6c61·6235·3236·223e·3c2f··a·id="lab526"></
00041800:·613e·3c68·3120·636c·6173·733d·2273·6563··a><h1·class="sec		00041800:·613e·3c68·3120·636c·6173·733d·2273·6563··a><h1·class="sec
00041810:·7469·6f6e·223e·4265·696e·6720·696e·6861··tion">Being·inha		00041810:·7469·6f6e·223e·4265·696e·6720·696e·6861··tion">Being·inha
00041820:·6269·7465·643c·2f68·313e·0a0a·3c64·6976··bited</h1>..<div		00041820:·6269·7465·643c·2f68·313e·0a0a·3c64·6976··bited</h1>..<div
00041830:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap		00041830:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap
00041840:·6822·3e20·3c2f·6469·763e·0a0a·2054·6865··h">·</div>..·The		00041840:·6822·3e20·3c2f·6469·763e·0a0a·2054·6865··h">·</div>..·The
00041850:·2070·7265·6469·6361·7465·203c·7370·616e···predicate·<span		00041850:·2070·7265·6469·6361·7465·203c·7370·616e···predicate·<span
00041860:·2063·6c61·7373·3d22·696e·6c69·6e65·636f···class="inlineco		00041860:·2063·6c61·7373·3d22·696e·6c69·6e65·636f···class="inlineco


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~~395~~"></a><h1·class="section">Peano·natural·numbers,·definitions·of·operations</h1>	55	<a·id="lab509"></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>
Offset 67, 15 lines modified		Offset 67, 15 lines modified
67	<br/>	67	<br/>
68	<span·class="id"·title="keyword">Definition</span>·<a·id="t"·class="idref"·href="#t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/>	68	<span·class="id"·title="keyword">Definition</span>·<a·id="t"·class="idref"·href="#t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/>

69	<br/>	69	<br/>
70	</div>	70	</div>

71	<div·class="doc">	71	<div·class="doc">
72	<a·id="lab~~396~~"></a><h2·class="section">Constants</h2>	72	<a·id="lab510"></a><h2·class="section">Constants</h2>

73	</div>	73	</div>
74	<div·class="code">	74	<div·class="code">

75	<br/>	75	<br/>
76	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'0'"·class="idref"·href="#:::'0'"><span·class="id"·title="notation">"</span></a>0"·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>.<br/>	76	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'0'"·class="idref"·href="#:::'0'"><span·class="id"·title="notation">"</span></a>0"·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>.<br/>
77	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>).<br/>	77	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>).<br/>
Offset 86, 15 lines modified		Offset 86, 15 lines modified
86	<span·class="id"·title="keyword">Definition</span>·<a·id="one"·class="idref"·href="#one"><span·class="id"·title="definition">one</span></a>·:=·1.<br/>	86	<span·class="id"·title="keyword">Definition</span>·<a·id="one"·class="idref"·href="#one"><span·class="id"·title="definition">one</span></a>·:=·1.<br/>
87	<span·class="id"·title="keyword">Definition</span>·<a·id="two"·class="idref"·href="#two"><span·class="id"·title="definition">two</span></a>·:=·2.<br/>	87	<span·class="id"·title="keyword">Definition</span>·<a·id="two"·class="idref"·href="#two"><span·class="id"·title="definition">two</span></a>·:=·2.<br/>

88	<br/>	88	<br/>
89	</div>	89	</div>

90	<div·class="doc">	90	<div·class="doc">
91	<a·id="lab~~397~~"></a><h2·class="section">Basic·operations</h2>	91	<a·id="lab511"></a><h2·class="section">Basic·operations</h2>

92	</div>	92	</div>
93	<div·class="code">	93	<div·class="code">

94	<br/>	94	<br/>
95	<span·class="id"·title="keyword">Definition</span>·<a·id="succ"·class="idref"·href="#succ"><span·class="id"·title="definition">succ</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>.<br/>	95	<span·class="id"·title="keyword">Definition</span>·<a·id="succ"·class="idref"·href="#succ"><span·class="id"·title="definition">succ</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>.<br/>

Offset 155, 15 lines modified		Offset 155, 15 lines modified
155	<br/>	155	<br/>
156	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">sub</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.sub</span>.<br/>	156	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">sub</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.sub</span>.<br/>

157	<br/>	157	<br/>
158	</div>	158	</div>

159	<div·class="doc">	159	<div·class="doc">
160	<a·id="lab~~398~~"></a><h2·class="section">Comparisons</h2>	160	<a·id="lab512"></a><h2·class="section">Comparisons</h2>

161	</div>	161	</div>
162	<div·class="code">	162	<div·class="code">

163	<br/>	163	<br/>
164	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="eqb"·class="idref"·href="#eqb"><span·class="id"·title="definition">eqb</span></a>·<a·id="n:17"·class="idref"·href="#n:17"><span·class="id"·title="binder">n</span></a>·<a·id="m:18"·class="idref"·href="#m:18"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>	164	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="eqb"·class="idref"·href="#eqb"><span·class="id"·title="definition">eqb</span></a>·<a·id="n:17"·class="idref"·href="#n:17"><span·class="id"·title="binder">n</span></a>·<a·id="m:18"·class="idref"·href="#m:18"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>
165	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:17"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:18"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	165	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:17"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:18"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
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201	<br/>	201	<br/>
202	<span·class="id"·title="keyword">Infix</span>·<a·id="4a243c6f4b9487508e7110341eeedb0b"·class="idref"·href="#4a243c6f4b9487508e7110341eeedb0b"><span·class="id"·title="notation">"</span></a>?="·:=·<a·class="idref"·href="Coq.Init.Nat.html#compare"><span·class="id"·title="definition">compare</span></a>·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·70)·:·<span·class="id"·title="var">nat_scope</span>.<br/>	202	<span·class="id"·title="keyword">Infix</span>·<a·id="4a243c6f4b9487508e7110341eeedb0b"·class="idref"·href="#4a243c6f4b9487508e7110341eeedb0b"><span·class="id"·title="notation">"</span></a>?="·:=·<a·class="idref"·href="Coq.Init.Nat.html#compare"><span·class="id"·title="definition">compare</span></a>·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·70)·:·<span·class="id"·title="var">nat_scope</span>.<br/>

203	<br/>	203	<br/>
204	</div>	204	</div>

205	<div·class="doc">	205	<div·class="doc">
206	<a·id="lab~~399~~"></a><h2·class="section">Minimum,·maximum</h2>	206	<a·id="lab513"></a><h2·class="section">Minimum,·maximum</h2>

207	</div>	207	</div>
208	<div·class="code">	208	<div·class="code">

209	<br/>	209	<br/>
210	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="max"·class="idref"·href="#max"><span·class="id"·title="definition">max</span></a>·<a·id="n:34"·class="idref"·href="#n:34"><span·class="id"·title="binder">n</span></a>·<a·id="m:35"·class="idref"·href="#m:35"><span·class="id"·title="binder">m</span></a>·:=<br/>	210	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="max"·class="idref"·href="#max"><span·class="id"·title="definition">max</span></a>·<a·id="n:34"·class="idref"·href="#n:34"><span·class="id"·title="binder">n</span></a>·<a·id="m:35"·class="idref"·href="#m:35"><span·class="id"·title="binder">m</span></a>·:=<br/>
211	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:34"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:35"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	211	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:34"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:35"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 226, 15 lines modified		Offset 226, 15 lines modified
226	\|·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">n'</span>,·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">m'</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#min:41"><span·class="id"·title="definition">min</span></a>·<span·class="id"·title="var">n'</span>·<span·class="id"·title="var">m'</span>)<br/>	226	\|·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">n'</span>,·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">m'</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#min:41"><span·class="id"·title="definition">min</span></a>·<span·class="id"·title="var">n'</span>·<span·class="id"·title="var">m'</span>)<br/>
227	<span·class="id"·title="keyword">end</span>.<br/>	227	<span·class="id"·title="keyword">end</span>.<br/>

228	<br/>	228	<br/>
229	</div>	229	</div>

230	<div·class="doc">	230	<div·class="doc">
231	<a·id="lab~~400~~"></a><h2·class="section">Parity·tests</h2>	231	<a·id="lab514"></a><h2·class="section">Parity·tests</h2>

232	</div>	232	</div>
233	<div·class="code">	233	<div·class="code">

234	<br/>	234	<br/>
235	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="definition">even</span></a>·<a·id="n:44"·class="idref"·href="#n:44"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>	235	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="definition">even</span></a>·<a·id="n:44"·class="idref"·href="#n:44"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>
236	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:44"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>	236	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:44"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 246, 15 lines modified		Offset 246, 15 lines modified
246	<br/>	246	<br/>
247	<span·class="id"·title="keyword">Definition</span>·<a·id="odd"·class="idref"·href="#odd"><span·class="id"·title="definition">odd</span></a>·<a·id="n:47"·class="idref"·href="#n:47"><span·class="id"·title="binder">n</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#negb"><span·class="id"·title="definition">negb</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#even"><span·class="id"·title="definition">even</span></a>·<a·class="idref"·href="Coq.Init.Nat.html#n:47"><span·class="id"·title="variable">n</span></a>).<br/>	247	<span·class="id"·title="keyword">Definition</span>·<a·id="odd"·class="idref"·href="#odd"><span·class="id"·title="definition">odd</span></a>·<a·id="n:47"·class="idref"·href="#n:47"><span·class="id"·title="binder">n</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#negb"><span·class="id"·title="definition">negb</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#even"><span·class="id"·title="definition">even</span></a>·<a·class="idref"·href="Coq.Init.Nat.html#n:47"><span·class="id"·title="variable">n</span></a>).<br/>

248	<br/>	248	<br/>
249	</div>	249	</div>

250	<div·class="doc">	250	<div·class="doc">
251	<a·id="lab~~401~~"></a><h2·class="section">Power</h2>	251	<a·id="lab515"></a><h2·class="section">Power</h2>

252	</div>	252	</div>
253	<div·class="code">	253	<div·class="code">

254	<br/>	254	<br/>
255	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="pow"·class="idref"·href="#pow"><span·class="id"·title="definition">pow</span></a>·<a·id="n:48"·class="idref"·href="#n:48"><span·class="id"·title="binder">n</span></a>·<a·id="m:49"·class="idref"·href="#m:49"><span·class="id"·title="binder">m</span></a>·:=<br/>	255	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="pow"·class="idref"·href="#pow"><span·class="id"·title="definition">pow</span></a>·<a·id="n:48"·class="idref"·href="#n:48"><span·class="id"·title="binder">n</span></a>·<a·id="m:49"·class="idref"·href="#m:49"><span·class="id"·title="binder">m</span></a>·:=<br/>
256	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#m:49"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	256	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#m:49"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
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264	<br/>	264	<br/>
265	<span·class="id"·title="keyword">where</span>·<a·id="81fd94e251a61ee523cdd7855774ae7c"·class="idref"·href="#81fd94e251a61ee523cdd7855774ae7c"><span·class="id"·title="notation">"</span></a>n·^·m"·:=·(<a·class="idref"·href="Coq.Init.Nat.html#pow:50"><span·class="id"·title="definition">pow</span></a>·<span·class="id"·title="var">n</span>·<span·class="id"·title="var">m</span>)·:·<span·class="id"·title="var">nat_scope</span>.<br/>	265	<span·class="id"·title="keyword">where</span>·<a·id="81fd94e251a61ee523cdd7855774ae7c"·class="idref"·href="#81fd94e251a61ee523cdd7855774ae7c"><span·class="id"·title="notation">"</span></a>n·^·m"·:=·(<a·class="idref"·href="Coq.Init.Nat.html#pow:50"><span·class="id"·title="definition">pow</span></a>·<span·class="id"·title="var">n</span>·<span·class="id"·title="var">m</span>)·:·<span·class="id"·title="var">nat_scope</span>.<br/>

266	<br/>	266	<br/>
267	</div>	267	</div>

268	<div·class="doc">	268	<div·class="doc">
269	<a·id="lab~~402~~"></a><h2·class="section">Tail-recursive·versions·of·<span·class="inlinecode"><span·class="id"·title="var">add</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">mul</span></span></h2>	269	<a·id="lab516"></a><h2·class="section">Tail-recursive·versions·of·<span·class="inlinecode"><span·class="id"·title="var">add</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">mul</span></span></h2>

270	</div>	270	</div>
271	<div·class="code">	271	<div·class="code">

272	<br/>	272	<br/>
273	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="tail_add"·class="idref"·href="#tail_add"><span·class="id"·title="definition">tail_add</span></a>·<a·id="n:52"·class="idref"·href="#n:52"><span·class="id"·title="binder">n</span></a>·<a·id="m:53"·class="idref"·href="#m:53"><span·class="id"·title="binder">m</span></a>·:=<br/>	273	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="tail_add"·class="idref"·href="#tail_add"><span·class="id"·title="definition">tail_add</span></a>·<a·id="n:52"·class="idref"·href="#n:52"><span·class="id"·title="binder">n</span></a>·<a·id="m:53"·class="idref"·href="#m:53"><span·class="id"·title="binder">m</span></a>·:=<br/>
274	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:52"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>	274	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:52"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>
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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~~380~~"></a><h1·class="section">Useful·tactics</h1>	56	<a·id="lab506"></a><h1·class="section">Useful·tactics</h1>

57	<div·class="paragraph">·</div>	57	<div·class="paragraph">·</div>

58	·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal.	58	·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal.
59	····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.·	59	····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.·
60	</div>	60	</div>
61	<div·class="code">	61	<div·class="code">
Offset 372, 15 lines modified		Offset 372, 15 lines modified
372	<span·class="id"·title="keyword">Tactic·Notation</span>·"dependent"·"destruction"·<span·class="id"·title="var">ident</span>(<span·class="id"·title="var">H</span>)·:=<br/>	372	<span·class="id"·title="keyword">Tactic·Notation</span>·"dependent"·"destruction"·<span·class="id"·title="var">ident</span>(<span·class="id"·title="var">H</span>)·:=<br/>
373	<span·class="id"·title="tactic">fail</span>·"To·use·dependent·destruction,·first·[Require·Import·Coq.Program.Equality.]".<br/>	373	<span·class="id"·title="tactic">fail</span>·"To·use·dependent·destruction,·first·[Require·Import·Coq.Program.Equality.]".<br/>

374	<br/>	374	<br/>
375	</div>	375	</div>

376	<div·class="doc">	376	<div·class="doc">
377	<a·id="lab~~381~~"></a><h3·class="section"><span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span></h3>	377	<a·id="lab507"></a><h3·class="section"><span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span></h3>
378	·The·built-in·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·will·frequently·leave·equalities·of	378	·The·built-in·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·will·frequently·leave·equalities·of
379	····dependent·pairs.··When·the·first·type·in·the·pair·is·an·hProp·or	379	····dependent·pairs.··When·the·first·type·in·the·pair·is·an·hProp·or
380	····otherwise·simplifies,·<span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span>·is·useful;·it·will·replace	380	····otherwise·simplifies,·<span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span>·is·useful;·it·will·replace
381	····the·equality·of·pairs·with·a·pair·of·equalities,·one·involving·a	381	····the·equality·of·pairs·with·a·pair·of·equalities,·one·involving·a
382	····term·casted·along·the·other.··This·might·also·prove·useful·for	382	····term·casted·along·the·other.··This·might·also·prove·useful·for
383	····writing·a·version·of·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·/·<span·class="inlinecode"><span·class="id"·title="tactic">dependent</span></span>·<span·class="inlinecode"><span·class="id"·title="tactic">destruction</span></span>·which	383	····writing·a·version·of·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·/·<span·class="inlinecode"><span·class="id"·title="tactic">dependent</span></span>·<span·class="inlinecode"><span·class="id"·title="tactic">destruction</span></span>·which
384	····does·not·lose·information,·i.e.,·does·not·turn·a·goal·which·is	384	····does·not·lose·information,·i.e.,·does·not·turn·a·goal·which·is


Offset 46, 15 lines modified		Offset 46, 15 lines modified
46	<div·class="code">	46	<div·class="code">

47	<br/>	47	<br/>
48	</div>	48	</div>

49	<div·class="doc">	49	<div·class="doc">
50	Some·facts·and·definitions·concerning·choice·and·description·in	50	Some·facts·and·definitions·concerning·choice·and·description·in
51	·······intuitionistic·logic.·<a·id="lab264"></a><h1·class="section">References:</h1>	51	·······intuitionistic·logic.·<a·id="lab1162"></a><h1·class="section">References:</h1>

52	<div·class="paragraph">·</div>	52	<div·class="paragraph">·</div>

53	<span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory,	53	<span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory,
54	unpublished.	54	unpublished.

55	<div·class="paragraph">·</div>	55	<div·class="paragraph">·</div>
Offset 85, 15 lines modified		Offset 85, 15 lines modified
85	<br/>	85	<br/>
86	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>	86	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>

87	<br/>	87	<br/>
88	</div>	88	</div>

89	<div·class="doc">	89	<div·class="doc">
90	<a·id="lab265"></a><h1·class="section">Definitions</h1>	90	<a·id="lab1163"></a><h1·class="section">Definitions</h1>

91	<div·class="paragraph">·</div>	91	<div·class="paragraph">·</div>

92	·Choice,·reification·and·description·schemes·	92	·Choice,·reification·and·description·schemes·
93	<div·class="paragraph">·</div>	93	<div·class="paragraph">·</div>

94	·We·make·them·all·polymorphic.·Most·of·them·have·existentials·as·conclusion	94	·We·make·them·all·polymorphic.·Most·of·them·have·existentials·as·conclusion
Offset 112, 15 lines modified		Offset 112, 15 lines modified
112	<br/>	112	<br/>
113	<span·class="id"·title="keyword">Variable</span>·<a·id="ChoiceSchemes.P"·class="idref"·href="#ChoiceSchemes.P"><span·class="id"·title="variable">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><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><span·class="id"·title="keyword">Prop</span>.<br/>	113	<span·class="id"·title="keyword">Variable</span>·<a·id="ChoiceSchemes.P"·class="idref"·href="#ChoiceSchemes.P"><span·class="id"·title="variable">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><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><span·class="id"·title="keyword">Prop</span>.<br/>

114	<br/>	114	<br/>
115	</div>	115	</div>

116	<div·class="doc">	116	<div·class="doc">
117	<a·id="lab266"></a><h2·class="section">Constructive·choice·and·description</h2>	117	<a·id="lab1164"></a><h2·class="section">Constructive·choice·and·description</h2>

118	<div·class="paragraph">·</div>	118	<div·class="paragraph">·</div>

119	·AC_rel·=·relational·form·of·the·(non·extensional)·axiom·of·choice	119	·AC_rel·=·relational·form·of·the·(non·extensional)·axiom·of·choice
120	·············(a·"set-theoretic"·axiom·of·choice)·	120	·············(a·"set-theoretic"·axiom·of·choice)·
121	</div>	121	</div>
122	<div·class="code">	122	<div·class="code">
Offset 342, 15 lines modified		Offset 342, 15 lines modified
342	<span·class="id"·title="keyword">forall</span>·<a·id="P:100"·class="idref"·href="#P:100"><span·class="id"·title="binder">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><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><span·class="id"·title="keyword">Prop</span>,<br/>	342	<span·class="id"·title="keyword">forall</span>·<a·id="P:100"·class="idref"·href="#P:100"><span·class="id"·title="binder">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><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><span·class="id"·title="keyword">Prop</span>,<br/>
343	<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#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">exists</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">!</span></a>·<a·id="x:101"·class="idref"·href="#x:101"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">,</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:101"><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.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">{</span></a>·<a·id="x:102"·class="idref"·href="#x:102"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">:</span></a><a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">\|</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:102"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">}</span></a>.<br/>	343	<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#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">exists</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">!</span></a>·<a·id="x:101"·class="idref"·href="#x:101"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">,</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:101"><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.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">{</span></a>·<a·id="x:102"·class="idref"·href="#x:102"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">:</span></a><a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">\|</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:102"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">}</span></a>.<br/>

344	<br/>	344	<br/>
345	</div>	345	</div>

346	<div·class="doc">	346	<div·class="doc">
347	<a·id="lab267"></a><h2·class="section">Weakly·classical·choice·and·description</h2>	347	<a·id="lab1165"></a><h2·class="section">Weakly·classical·choice·and·description</h2>

348	<div·class="paragraph">·</div>	348	<div·class="paragraph">·</div>

349	·GAC_rel·=·guarded·relational·form·of·the·(non·extensional)·axiom·of·choice·	349	·GAC_rel·=·guarded·relational·form·of·the·(non·extensional)·axiom·of·choice·
350	</div>	350	</div>
351	<div·class="code">	351	<div·class="code">

Offset 621, 15 lines modified		Offset 621, 15 lines modified

621	<div·class="paragraph">·</div>	621	<div·class="paragraph">·</div>

622	with·no·prerequisite·on·the·non-emptiness·of·domains	622	with·no·prerequisite·on·the·non-emptiness·of·domains

623	<div·class="paragraph">·</div>	623	<div·class="paragraph">·</div>

624	<a·id="lab268"></a><h1·class="section">Table·of·contents</h1>	624	<a·id="lab1166"></a><h1·class="section">Table·of·contents</h1>

625	</div>	625	</div>
626	<div·class="code">	626	<div·class="code">

627	<br/>	627	<br/>
628	</div>	628	</div>

Offset 678, 15 lines modified		Offset 678, 15 lines modified

678	<div·class="paragraph">·</div>	678	<div·class="paragraph">·</div>

679	9.2.·AC_fun_setoid·=·AC_fun·+·Ext_pred_repr·+·PI	679	9.2.·AC_fun_setoid·=·AC_fun·+·Ext_pred_repr·+·PI
680	·	680	·
681	<div·class="paragraph">·</div>	681	<div·class="paragraph">·</div>

682	<a·id="lab269"></a><h1·class="section">AC_rel·+·AC!·=·AC_fun</h1>	682	<a·id="lab1167"></a><h1·class="section">AC_rel·+·AC!·=·AC_fun</h1>


683	<div·class="paragraph">·</div>	683	<div·class="paragraph">·</div>

684	···We·show·that·the·functional·formulation·of·the·axiom·of·Choice	684	···We·show·that·the·functional·formulation·of·the·axiom·of·Choice
685	···(usual·formulation·in·type·theory)·is·equivalent·to·its·relational	685	···(usual·formulation·in·type·theory)·is·equivalent·to·its·relational
686	···formulation·(only·formulation·of·set·theory)·+·functional·relation	686	···formulation·(only·formulation·of·set·theory)·+·functional·relation
Offset 718, 26 lines modified		Offset 718, 26 lines modified
718	<span·class="id"·title="keyword">forall</span>·<a·id="A:214"·class="idref"·href="#A:214"><span·class="id"·title="binder">A</span></a>·<a·id="B:215"·class="idref"·href="#B:215"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>,·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalChoice_on"><span·class="id"·title="definition">FunctionalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><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><br/>	718	<span·class="id"·title="keyword">forall</span>·<a·id="A:214"·class="idref"·href="#A:214"><span·class="id"·title="binder">A</span></a>·<a·id="B:215"·class="idref"·href="#B:215"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>,·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalChoice_on"><span·class="id"·title="definition">FunctionalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><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><br/>
719	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice_on"><span·class="id"·title="definition">RelationalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalRelReification_on"><span·class="id"·title="definition">FunctionalRelReification_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>.<br/>	719	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice_on"><span·class="id"·title="definition">RelationalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalRelReification_on"><span·class="id"·title="definition">FunctionalRelReification_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>.<br/>

720	<br/>	720	<br/>
721	</div>	721	</div>

722	<div·class="doc">	722	<div·class="doc">
723	<a·id="lab~~270~~"></a><h1·class="section">Connection·between·the·guarded,·non·guarded·and·omniscient·choices</h1>	723	<a·id="lab1168"></a><h1·class="section">Connection·between·the·guarded,·non·guarded·and·omniscient·choices</h1>

724	<div·class="paragraph">·</div>	724	<div·class="paragraph">·</div>

725	·We·show·that·the·guarded·formulations·of·the·axiom·of·choice	725	·We·show·that·the·guarded·formulations·of·the·axiom·of·choice
726	···are·equivalent·to·their·"omniscient"·variant·and·comes·from·the·non·guarded	726	···are·equivalent·to·their·"omniscient"·variant·and·comes·from·the·non·guarded
727	···formulation·in·presence·either·of·the·independence·of·general·premises	727	···formulation·in·presence·either·of·the·independence·of·general·premises
728	···or·subset·types·(themselves·derivable·from·subtypes·thanks·to·proof-	728	···or·subset·types·(themselves·derivable·from·subtypes·thanks·to·proof-
729	···irrelevance)·	729	···irrelevance)·
730	<div·class="paragraph">·</div>	730	<div·class="paragraph">·</div>

731	<a·id="lab~~271~~"></a><h2·class="section">AC_rel·+·PI·->·GAC_rel·and·AC_rel·+·IGP·->·GAC_rel·and·GAC_rel·=·OAC_rel</h2>	731	<a·id="lab1169"></a><h2·class="section">AC_rel·+·PI·->·GAC_rel·and·AC_rel·+·IGP·->·GAC_rel·and·GAC_rel·=·OAC_rel</h2>

732	</div>	732	</div>
733	<div·class="code">	733	<div·class="code">

734	<br/>	734	<br/>
735	<span·class="id"·title="keyword">Lemma</span>·<a·id="rel_choice_and_proof_irrel_imp_guarded_rel_choice"·class="idref"·href="#rel_choice_and_proof_irrel_imp_guarded_rel_choice"><span·class="id"·title="lemma">rel_choice_and_proof_irrel_imp_guarded_rel_choice</span></a>·:<br/>	735	<span·class="id"·title="keyword">Lemma</span>·<a·id="rel_choice_and_proof_irrel_imp_guarded_rel_choice"·class="idref"·href="#rel_choice_and_proof_irrel_imp_guarded_rel_choice"><span·class="id"·title="lemma">rel_choice_and_proof_irrel_imp_guarded_rel_choice</span></a>·:<br/>
736	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice"><span·class="id"·title="abbreviation">RelationalChoice</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.ChoiceFacts.html#ProofIrrelevance"><span·class="id"·title="definition">ProofIrrelevance</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.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</span></a>.<br/>	736	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice"><span·class="id"·title="abbreviation">RelationalChoice</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.ChoiceFacts.html#ProofIrrelevance"><span·class="id"·title="definition">ProofIrrelevance</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.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</span></a>.<br/>
Offset 767, 15 lines modified		Offset 767, 15 lines modified
767	<span·class="id"·title="keyword">Corollary</span>·<a·id="guarded_iff_omniscient_rel_choice"·class="idref"·href="#guarded_iff_omniscient_rel_choice"><span·class="id"·title="lemma">guarded_iff_omniscient_rel_choice</span></a>·:<br/>	767	<span·class="id"·title="keyword">Corollary</span>·<a·id="guarded_iff_omniscient_rel_choice"·class="idref"·href="#guarded_iff_omniscient_rel_choice"><span·class="id"·title="lemma">guarded_iff_omniscient_rel_choice</span></a>·:<br/>
768	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</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.ChoiceFacts.html#OmniscientRelationalChoice"><span·class="id"·title="abbreviation">OmniscientRelationalChoice</span></a>.<br/>	768	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</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.ChoiceFacts.html#OmniscientRelationalChoice"><span·class="id"·title="abbreviation">OmniscientRelationalChoice</span></a>.<br/>

769	<br/>	769	<br/>
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Offset 129, 15 lines modified		Offset 129, 15 lines modified
129	<br/>	129	<br/>
130	<span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/>	130	<span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/>

131	<br/>	131	<br/>
132	</div>	132	</div>

133	<div·class="doc">	133	<div·class="doc">
134	<a·id="lab260"></a><h3·class="section">Weaker·lemmas·(compatibility·lemmas)</h3>	134	<a·id="lab1104"></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:38"·class="idref"·href="#A:38"><span·class="id"·title="binder">A</span></a>·<a·id="B:39"·class="idref"·href="#B:39"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:40"·class="idref"·href="#R:40"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:38"><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:39"><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:38"·class="idref"·href="#A:38"><span·class="id"·title="binder">A</span></a>·<a·id="B:39"·class="idref"·href="#B:39"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:40"·class="idref"·href="#R:40"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:38"><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:39"><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/>


Offset 110, 15 lines modified		Offset 110, 15 lines modified

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~~342~~"></a><h1·class="section">Prop·degeneracy·=·excluded-middle·+·prop·extensionality</h1>	113	<a·id="lab1110"></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>

Offset 186, 19 lines modified		Offset 186, 19 lines modified
186	<span·class="id"·title="keyword">Lemma</span>·<a·id="provable_prop_ext"·class="idref"·href="#provable_prop_ext"><span·class="id"·title="lemma">provable_prop_ext</span></a>·:<br/>	186	<span·class="id"·title="keyword">Lemma</span>·<a·id="provable_prop_ext"·class="idref"·href="#provable_prop_ext"><span·class="id"·title="lemma">provable_prop_ext</span></a>·:<br/>
187	<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#prop_extensionality"><span·class="id"·title="definition">prop_extensionality</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.ClassicalFacts.html#provable_prop_extensionality"><span·class="id"·title="definition">provable_prop_extensionality</span></a>.<br/>	187	<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#prop_extensionality"><span·class="id"·title="definition">prop_extensionality</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.ClassicalFacts.html#provable_prop_extensionality"><span·class="id"·title="definition">provable_prop_extensionality</span></a>.<br/>

188	<br/>	188	<br/>
189	</div>	189	</div>

190	<div·class="doc">	190	<div·class="doc">
191	<a·id="lab~~343~~"></a><h1·class="section">Classical·logic·and·proof-irrelevance</h1>	191	<a·id="lab1111"></a><h1·class="section">Classical·logic·and·proof-irrelevance</h1>

192	<div·class="paragraph">·</div>	192	<div·class="paragraph">·</div>

193	<a·id="lab~~344~~"></a><h2·class="section">CC·\|-·prop·ext·+·A·inhabited·->·(A·=·A->A)·->·A·has·fixpoint</h2>	193	<a·id="lab1112"></a><h2·class="section">CC·\|-·prop·ext·+·A·inhabited·->·(A·=·A->A)·->·A·has·fixpoint</h2>

194	<div·class="paragraph">·</div>	194	<div·class="paragraph">·</div>

195	·We·successively·show·that:	195	·We·successively·show·that:

196	<div·class="paragraph">·</div>	196	<div·class="paragraph">·</div>

Offset 243, 15 lines modified		Offset 243, 15 lines modified

243	<div·class="doc">	243	<div·class="doc">
244	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma·<span·class="inlinecode"><span·class="id"·title="var">ext_prop_fixpoint</span></span>	244	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma·<span·class="inlinecode"><span·class="id"·title="var">ext_prop_fixpoint</span></span>
245	····by·the·weakest·property·<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.	245	····by·the·weakest·property·<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.

246	<div·class="paragraph">·</div>	246	<div·class="paragraph">·</div>

247	<a·id="lab~~345~~"></a><h2·class="section">CC·\|-·prop_ext·/\·dep·elim·on·bool·->·proof-irrelevance</h2>	247	<a·id="lab1113"></a><h2·class="section">CC·\|-·prop_ext·/\·dep·elim·on·bool·->·proof-irrelevance</h2>

248	<div·class="paragraph">·</div>	248	<div·class="paragraph">·</div>

249	·<span·class="inlinecode"><span·class="id"·title="var">proof_irrelevance</span></span>·asserts·equality·of·all·proofs·of·a·given·formula·	249	·<span·class="inlinecode"><span·class="id"·title="var">proof_irrelevance</span></span>·asserts·equality·of·all·proofs·of·a·given·formula·
250	</div>	250	</div>
251	<div·class="code">	251	<div·class="code">
252	<span·class="id"·title="keyword">Definition</span>·<a·id="proof_irrelevance"·class="idref"·href="#proof_irrelevance"><span·class="id"·title="definition">proof_irrelevance</span></a>·:=·<span·class="id"·title="keyword">forall</span>·(<a·id="A:42"·class="idref"·href="#A:42"><span·class="id"·title="binder">A</span></a>:<span·class="id"·title="keyword">Prop</span>)·(<a·id="a1:43"·class="idref"·href="#a1:43"><span·class="id"·title="binder">a1</span></a>·<a·id="a2:44"·class="idref"·href="#a2:44"><span·class="id"·title="binder">a2</span></a>:<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#A:42"><span·class="id"·title="variable">A</span></a>),·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a1:43"><span·class="id"·title="variable">a1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a2:44"><span·class="id"·title="variable">a2</span></a>.<br/>	252	<span·class="id"·title="keyword">Definition</span>·<a·id="proof_irrelevance"·class="idref"·href="#proof_irrelevance"><span·class="id"·title="definition">proof_irrelevance</span></a>·:=·<span·class="id"·title="keyword">forall</span>·(<a·id="A:42"·class="idref"·href="#A:42"><span·class="id"·title="binder">A</span></a>:<span·class="id"·title="keyword">Prop</span>)·(<a·id="a1:43"·class="idref"·href="#a1:43"><span·class="id"·title="binder">a1</span></a>·<a·id="a2:44"·class="idref"·href="#a2:44"><span·class="id"·title="binder">a2</span></a>:<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#A:42"><span·class="id"·title="variable">A</span></a>),·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a1:43"><span·class="id"·title="variable">a1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a2:44"><span·class="id"·title="variable">a2</span></a>.<br/>
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340	<div·class="doc">	340	<div·class="doc">
341	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma	341	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma
342	····<span·class="inlinecode"><span·class="id"·title="var">ext_prop_dep_proof_irrel_gen</span></span>·by·the·weakest·property	342	····<span·class="inlinecode"><span·class="id"·title="var">ext_prop_dep_proof_irrel_gen</span></span>·by·the·weakest·property
343	····<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.	343	····<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.

344	<div·class="paragraph">·</div>	344	<div·class="paragraph">·</div>

345	<a·id="lab~~346~~"></a><h2·class="section">CIC·\|-·prop.·ext.·->·proof-irrelevance</h2>	345	<a·id="lab1114"></a><h2·class="section">CIC·\|-·prop.·ext.·->·proof-irrelevance</h2>

346	<div·class="paragraph">·</div>	346	<div·class="paragraph">·</div>

347	·In·the·Calculus·of·Inductive·Constructions,·inductively·defined·booleans	347	·In·the·Calculus·of·Inductive·Constructions,·inductively·defined·booleans
348	····enjoy·dependent·case·analysis,·hence·directly·proof-irrelevance·from	348	····enjoy·dependent·case·analysis,·hence·directly·proof-irrelevance·from
349	····propositional·extensionality.	349	····propositional·extensionality.

Offset 394, 15 lines modified		Offset 394, 15 lines modified

394	··<span·class="inlinecode">[<span·class="id"·title="var">Berardi90</span>]</span>·Stefano·Berardi,·"Type·dependence·and·constructive	394	··<span·class="inlinecode">[<span·class="id"·title="var">Berardi90</span>]</span>·Stefano·Berardi,·"Type·dependence·and·constructive
395	··mathematics",·Ph.·D.·thesis,·Dipartimento·Matematica,·Università·di	395	··mathematics",·Ph.·D.·thesis,·Dipartimento·Matematica,·Università·di
396	··Torino,·1990.	396	··Torino,·1990.

397	<div·class="paragraph">·</div>	397	<div·class="paragraph">·</div>

398	<a·id="lab~~347~~"></a><h2·class="section">CC·\|-·excluded-middle·+·dep·elim·on·bool·->·proof-irrelevance</h2>	398	<a·id="lab1115"></a><h2·class="section">CC·\|-·excluded-middle·+·dep·elim·on·bool·->·proof-irrelevance</h2>

399	<div·class="paragraph">·</div>	399	<div·class="paragraph">·</div>

400	·This·is·a·proof·in·the·pure·Calculus·of·Construction·that	400	·This·is·a·proof·in·the·pure·Calculus·of·Construction·that
401	····classical·logic·in·<span·class="inlinecode"><span·class="id"·title="keyword">Prop</span></span>·+·dependent·elimination·of·disjunction·entails	401	····classical·logic·in·<span·class="inlinecode"><span·class="id"·title="keyword">Prop</span></span>·+·dependent·elimination·of·disjunction·entails
402	····proof-irrelevance.	402	····proof-irrelevance.

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566	<br/>	566	<br/>
567	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#Proof_irrelevance_WEM_CC"><span·class="id"·title="section">Proof_irrelevance_WEM_CC</span></a>.<br/>	567	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#Proof_irrelevance_WEM_CC"><span·class="id"·title="section">Proof_irrelevance_WEM_CC</span></a>.<br/>

568	<br/>	568	<br/>
569	</div>	569	</div>

570	<div·class="doc">	570	<div·class="doc">
571	<a·id="lab~~348~~"></a><h2·class="section">CIC·\|-·excluded-middle·->·proof-irrelevance</h2>	571	<a·id="lab1116"></a><h2·class="section">CIC·\|-·excluded-middle·->·proof-irrelevance</h2>

572	<div·class="paragraph">·</div>	572	<div·class="paragraph">·</div>


573	<div·class="paragraph">·</div>	573	<div·class="paragraph">·</div>

574	····Since,·dependent·elimination·is·derivable·in·the·Calculus·of	574	····Since,·dependent·elimination·is·derivable·in·the·Calculus·of
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631	Remark:·in·the·Set-impredicative·CCI,·Hurkens'·paradox·still·holds·with	631	Remark:·in·the·Set-impredicative·CCI,·Hurkens'·paradox·still·holds·with
632	····<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·in·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>·and·since·<span·class="inlinecode">~<span·class="id"·title="var">true</span>=<span·class="id"·title="var">false</span></span>·for·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>	632	····<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·in·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>·and·since·<span·class="inlinecode">~<span·class="id"·title="var">true</span>=<span·class="id"·title="var">false</span></span>·for·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>
633	····in·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·from·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>,·we·get·the·inconsistency·of	633	····in·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·from·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>,·we·get·the·inconsistency·of
634	····<span·class="inlinecode"><span·class="id"·title="var">em</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·class="id"·title="keyword">Prop</span>,</span>·<span·class="inlinecode">{<span·class="id"·title="var">A</span>}+{~<span·class="id"·title="var">A</span>}</span>·in·the·Set-impredicative·CCI.	634	····<span·class="inlinecode"><span·class="id"·title="var">em</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·class="id"·title="keyword">Prop</span>,</span>·<span·class="inlinecode">{<span·class="id"·title="var">A</span>}+{~<span·class="id"·title="var">A</span>}</span>·in·the·Set-impredicative·CCI.

635	<div·class="paragraph">·</div>	635	<div·class="paragraph">·</div>

636	<a·id="lab~~349~~"></a><h1·class="section">Weak·classical·axioms</h1>	636	<a·id="lab1117"></a><h1·class="section">Weak·classical·axioms</h1>

637	<div·class="paragraph">·</div>	637	<div·class="paragraph">·</div>

638	·We·show·the·following·increasing·in·the·strength·of·axioms:	638	·We·show·the·following·increasing·in·the·strength·of·axioms:
639	<ul·class="doclist">	639	<ul·class="doclist">
640	<li>·weak·excluded-middle·and·classical·De·Morgan's·law	640	<li>·weak·excluded-middle·and·classical·De·Morgan's·law

Offset 653, 15 lines modified		Offset 653, 15 lines modified
653	<li>·excluded-middle	653	<li>·excluded-middle

654	</li>	654	</li>
655	</ul>	655	</ul>

656	<div·class="paragraph">·</div>	656	<div·class="paragraph">·</div>

657	<a·id="lab~~350~~"></a><h2·class="section">Weak·excluded-middle</h2>	657	<a·id="lab1118"></a><h2·class="section">Weak·excluded-middle</h2>

658	<div·class="paragraph">·</div>	658	<div·class="paragraph">·</div>

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Offset 104, 15 lines modified		Offset 104, 15 lines modified
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="lab293"></a><h1·class="section">Pred.·Ext.·+·Rel.·Axiom·of·Choice·->·Excluded-Middle</h1>	109	<a·id="lab1196"></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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200	<br/>	200	<br/>
201	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/>	201	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/>

202	<br/>	202	<br/>
203	</div>	203	</div>

204	<div·class="doc">	204	<div·class="doc">
205	<a·id="lab294"></a><h1·class="section">Proof-Irrel.·+·Rel.·Axiom·of·Choice·->·Excl.-Middle·for·Equality</h1>	205	<a·id="lab1197"></a><h1·class="section">Proof-Irrel.·+·Rel.·Axiom·of·Choice·->·Excl.-Middle·for·Equality</h1>

206	<div·class="paragraph">·</div>	206	<div·class="paragraph">·</div>

207	·This·is·an·adaptation·of·Diaconescu's·theorem,·exploiting·the	207	·This·is·an·adaptation·of·Diaconescu's·theorem,·exploiting·the
208	····form·of·extensionality·provided·by·proof-irrelevance·	208	····form·of·extensionality·provided·by·proof-irrelevance·
209	</div>	209	</div>
210	<div·class="code">	210	<div·class="code">
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307	<br/>	307	<br/>
308	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#ProofIrrel_RelChoice_imp_EqEM"><span·class="id"·title="section">ProofIrrel_RelChoice_imp_EqEM</span></a>.<br/>	308	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#ProofIrrel_RelChoice_imp_EqEM"><span·class="id"·title="section">ProofIrrel_RelChoice_imp_EqEM</span></a>.<br/>

309	<br/>	309	<br/>
310	</div>	310	</div>

311	<div·class="doc">	311	<div·class="doc">
312	<a·id="lab295"></a><h1·class="section">Extensional·Hilbert's·epsilon·description·operator·->·Excluded-Middle</h1>	312	<a·id="lab1198"></a><h1·class="section">Extensional·Hilbert's·epsilon·description·operator·->·Excluded-Middle</h1>

313	<div·class="paragraph">·</div>	313	<div·class="paragraph">·</div>

314	·Proof·sketch·from·Bell·<span·class="inlinecode">[<span·class="id"·title="var">Bell93</span>]</span>·(with·thanks·to·P.·Castéran)·	314	·Proof·sketch·from·Bell·<span·class="inlinecode">[<span·class="id"·title="var">Bell93</span>]</span>·(with·thanks·to·P.·Castéran)·
315	</div>	315	</div>
316	<div·class="code">	316	<div·class="code">