鉴定题库-电力电缆工高级工30%试题
中英姓氏对应—不懂英文也能给自己取英文名字
A 艾——Ai安——Ann/An敖——Ao B 巴——Pa⽩——Pai包/鲍——Paul/Pao班——Pan贝——Pei毕——Pih卞——Bein⼘/薄——Po/Pu步——Poo百⾥——Pai-li C 蔡/柴——Tsia/Choi/Tsai曹/晁/巢——Chao/Chiao/Tsao岑——Cheng崔——Tsui查——Cha常——Chiong车——Che陈——Chen/Chan/Tan成/程——Cheng池——Chi褚/楚——Chu淳于——Chwen-yu D 戴/代——Day/Tai邓——Teng/Tang/Tung狄——Ti刁——Tiao丁——Ting/T董/东——Tung/Tong窦——Tou杜——To/Du/Too段——Tuan端⽊——Duan-mu东郭——Tung-kuo东⽅——Tung-fang E F 范/樊——Fan/Van房/⽅——Fang费——Fei冯/凤/封——Fung/Fong符/傅——Fu/Foo G 盖——Kai⽢——Kan⾼/郜——Gao/Kao葛——Keh耿——Keng⼸/宫/龚/恭——Kung勾——Kou古/⾕/顾——Ku/Koo桂——Kwei管/关——Kuan/Kwan郭/国——Kwok/Kuo公孙——Kung-sun公⽺——Kung-yang公冶——Kung-yeh⾕梁——Ku-liang H 海——Hay韩——Hon/Han杭——Hang郝——Hoa/Howe何/贺——Ho桓——Won侯——Hou洪——Hung胡/扈——Hu/Hoo花/华——Hua宦——Huan黄——Wong/Hwang霍——Huo皇甫——Hwang-fu呼延——Hu-yen I J 纪/翼/季/吉/嵇/汲/籍/姬——Chi居——Chu贾——Chia翦/简——Jen/Jane/Chieh蒋/姜/江/——Chiang/Kwong焦——Chiao ⾦/靳——Jin/King景/荆——King/Ching讦——Gan K 阚——Kan康——Kang柯——Kor/Ko孔——Kong/Kung寇——Ker蒯——Kuai匡——Kuang L 赖——Lai蓝——Lan郎——Long劳——Lao乐——Loh雷——Rae/Ray/Lei冷——Leng黎/郦/利/李——Lee/Li/Lai/Li连——Lien廖——Liu/Liao梁——Leung/Liang林/蔺——Lim/Lin凌——Lin柳/刘——Liu/Lau龙——Long楼/娄——Lou卢/路/陆鲁——Lu/Loo伦——Lun罗/骆——Loh/Lo/Law/Lam/Rowe吕——Lui/Lu令狐——Lin-hoo M 马/⿇——Ma麦——Mai/Mak满——Man/Mai⽑——Mao梅——Mei孟/蒙——Mong/Meng⽶/宓——Mi苗/缪——Miau/Miao 闵——Min穆/慕——Moo/Mo莫——Mok/Mo万俟——Moh-chi慕容——Mo-yung N 倪——Nee甯——Ning聂——Nieh⽜——New/Niu农——Long南宫——Nan-kung O 欧/区——Au/Ou欧阳——Ou-yang P 潘——Pang/Pan庞——Pang裴——Pei/Bae彭——Phang/Pong⽪——Pee平——Ping浦/蒲/⼘——Poo/Pu濮阳——Poo-yang Q 祁/戚/齐——Chi/Chyi/Chi/Chih钱——Chien乔——Chiao/Joe秦——Ching裘/仇/邱——Chiu屈/曲/瞿——Chiu/Chu R 冉——Yien饶——Yau任——Jen/Yum容/荣——Yung阮——Yuen芮——Nei S 司——Sze桑——Sang沙——Sa邵——Shao单/⼭——San尚/商——Sang/Shang沈/申——Shen盛——Shen史/施/师/⽯——Shih/Shi苏/宿/舒——Sue/Se/Soo/Hsu孙——Sun/Suen宋——Song/Soung司空——Sze-kung司马——Sze-ma司徒——Sze-to单于——San-yu上官——Sang-kuan申屠——Shen-tu T 谈——Tan汤/唐——Town/Towne/Tang邰——Tai谭——Tan/Tam陶——Tao藤——Teng⽥——Tien童——Tung屠——Tu澹台——Tan-tai拓拔——Toh-bah W 万——Wan王/汪——Wong魏/卫/韦——Wei温/⽂/闻——Wen/Chin/Vane/Man翁——Ong吴/伍/巫/武/邬/乌——Wu/NG/Woo X 奚/席——Hsi/Chi夏——Har/Hsia/（Summer） 肖/萧——Shaw/Siu/Hsiao项/向——Hsiang解/谢——Tse/Shieh⾟——Hsing刑——Hsing熊——Hsiung/Hsiun许/徐/荀——Shun/Hui/Hsu宣——Hsuan薛——Hsueh西门——See-men夏侯——Hsia-hou轩辕——Hsuan-yuen Y 燕/晏/阎/严/颜——Yim/Yen杨/⽺/养——Young/Yang姚——Yao/Yau叶——Yip/Yeh/Yih伊/易/羿——Yih/E殷/阴/尹——Yi/Yin/Ying应——Ying尤/游——Yu/You俞/庾/于/余/虞/郁/余/禹——Yue/Yu袁/元——Yuan/Yuen岳——Yue云——Wing尉迟——Yu-chi宇⽂——Yu-wen U V Z 藏——Chang曾/郑——Tsang/Cheng/Tseng訾——Zi宗——Chung左/卓——Cho/Tso翟——Chia詹——Chan甄——Chen湛——Tsan张/章——Cheung/Chang赵/肇/招——Chao/Chiu/Chiao/Chioa周/邹——Chau/Chou/Chow钟——Chung祖/竺/朱/诸/祝——Chu/Chuh庄——Chong钟离——Chung-li诸葛——Chu-keh。
德文输入对照表
一、德语输入法的安装方法1、windows XP上输入法是可以设置的。
方法2、每台电脑里都有德语输入法，在控制面板里，打开区域和语言选项，选择语言，详细信息里面就可以添加德语了二、Windows XP里德语键盘布局与标准键盘对照表第一列是标准键盘上的键名，第二和是德语键盘布局下（输入语言改成德语时）第三两列是按键和Shift+按键输入的文字。
` ^ °1 1 !2 2 "3 3 §4 4 $5 5 %6 6 &7 7 /8 8 (9 9 )0 0 =- ß ?= ′ `q q Qw w We e Er r Rt t Ty z Zu u Ui i Io o Op p P[ ü ü] + *\ # 'a a As s Sd d Df f Fg g Gh h Hj j Jk k Kl l L; ö Ö' ä Äz y Yx x Xc c Cv v Vb b Bn n Nm m M, , ;. . :/ - _1.比较常用的几个特殊字母输入。
键盘上-键代表ß，[键代表ü，'键代表ä, ;键代表ö；大写吗就是加上shift没有特别的。
2.字母y&z在德语输入法中是互换的，应为在德语中z的使用量要大于y;(个人认为这个理由没有什么说服力)3. é这个字母被我研究出来的时候还是很高兴了一番的，+&e4.无数错位的特殊符号了。
大家可以不停的在键盘上试验或者直接转化成英文属于法后使用。
举几个常用的吧。
一句话，乱得很！比如，&-是互换的（）不是shift 9.0 而是shift8.9´直接用+键/是shift7;是shift,:是shift.常用的差不多了。
一些广东字的输入方法
淠（ilgj、pei，11）= 量词；例如：一淠屎=一堆屎
渌（iviy、lu，11）= 用水煮东西；例如：渌生菜
靓（gemq、liang，12）= 漂亮或表示幼稚； 例如：靓仔=帅哥/小子
揾（rjlg、wen，12）= 找、挣；例如：揾食=谋生
嘢（kjfb、ge，14）= 东西；例如：有嘢睇=有东西看
嘥（kthh、sai，14）= 浪费、错过；例：咁好嘅机会嘥咗=这么好的机会错过了
嗻（kyao、zhe，14）= 语气助词；例如：我先至23嗻=我才23呀~
遮（yaop、zhe，14）= 雨伞
潎（iumt、pie，14）= 闪，走人
奀（gidu、en，7）= 瘦小
呃（kdbn、e，7）= 骗；例如：呃神骗鬼
鸠（vqyg、jiu，7）= 粗俗用语；例如：戆叱鸠=傻屄
抦（rgmw、bing，8）= 殴打；例如：我哋去抦果条友=我们去揍那个傢伙
拎（rwyc、lin，8）= 提、拿
唔（kgkg、en，10）= 不；例如：唔係=不是
哩（kjf、li，10）= 语气助词，或见呢
?（kkmw、无，10）= 语气助词
悭（njcf、qian，10）= 节省；例如：你悭啲啦=你省着点吧（你少来了）
俾rtf、bi，10） = 给，亦作畀；例如：你俾我啦~=你给我吧~
一些广东字的输入方法
乜（nnv、mie，2）= 什么； 例如：你有乜？=你有什么？
廿（aghg、nian，4）= 二十
卅（gkk、sa，4）= 三十
冇（dmb、mao，4）= 没有；例如：有冇钱？=有没有钱？
2014年12月日语N1真题.doc
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五笔二级简码
五笔二级简码五笔二笔简码理论上应该有25*25=625个，实际上有625-9=616个。
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【英语】山东省滕州市第五中学2014-2015学年高一上学期期末考试
山东省滕州市第五中学2014-2015学年高一上学期期末考试英语试题車毬分驚I世t透择鏈）和羯】［卷（韭透释通》两禰分全霑画分1海严彎削间1旳分钟・弟唱t豳®U1汾）務一祁分1听力理科（共两节・蓿停、阴］回缺斤力部分时.诱无将答富标直漬4「.诉力韶分紳甫轨你将肖两分斡的时间将悔的苔富转越到吝理芸迪卡上.弟一节禹，卜魁i*滞.常片人'分）听下裔三股対话.毋段科话管玮—个彳密从題申圧2的£ & C三个迭项中送出凝任時项.井标在试桂的相应位圻芫弩股衬话后、fWWt Wt的时间来回答有关个題和冋读T-小趣.闿段对话仅话一週.1. What does the woman want to do?A . To have an X-ray .B . To go to the hospital .C . To help the wounded man .2. Where and when will the meeting be held?A . Room 303, 3:00 p . m .B . Room 303, 2:00 p . m .C . Room 302, 2:00 p . m .3 . When would Thomas and Lily like to leave?A . Tomorrow .B . Next Monday or Tuesday .C . This Monday .4 . What is the man ' s choice?A . He prefers trains for trip .B . He doesn ' t like travelingC . Not mentioned .5 . According to the man, what should the woman do at first?A . She should ask about the flat on the pho ne .B ．She should read the advertisements for flats in the newspaper ．C ． She should phone and make an appointment ． 第二节（共 15小题；每题 1． 5分，满分 22．5分）听下面5段对话或独白。
机械设计课后答案
3-1已知材料的对称循环弯曲疲劳极限乙=200\吗循环基数叫=10了，加= 循环i 欠数N 分别为8000, 30000, 600000次时，材科的有限寿命弯曲疲劳极限.解・/VI ■当AJ8000时2・2已矢喋合金钢的机械性能为乙・7j0MP “ ＜7.,-460- 0.2 ＞试绘制此材 料的极卩融力国。
3-3已知一经过精车加工的圆轴轴肩尺寸三D = 70mm : d = 60nun : r = 3mm,材料为碳钢，其强度极限＜7严500兀必，试计算轴肩处的鸾曲应力综合影响系数K 。
・试计算=”0灯 107 600000= 2117材料的极跟应力閤如下;当 3^30000 时当 W600000 时①七=J解:t根据方程式C2-7),轴肩处的弯曲应力综合影响慕数Kj2. 查表定系换d SO d查附图LI®得弯曲时园角的肓趙应力集中^&<=1.77；查附图2七得零件的尺寸系魏耳=0-78,查附图2-8醤零件的加工表丽虎量系数0二0.9,3. 计算综合影响系数将竟得的各系数代入上式得穹圧应力综合影响系数；177礼= =2.520,72x0.$已知一经过粗车加工的IS轴轴肩尺寸；0=70^, <M5nm Hnrn如果使用题2-2中的村料，其强度极限込.二900A"e试绘带J该轴的极隈应力图.解:1 •根据已知条件确定脉动循环疲劳极限，由由平均应力折算为应力幅的系数,得 o •。
二竺-二竺竺二 767MPa ° 1+忆 1 + 0.2 2.确定弯曲应力综合彫响系数恳疔=1.27 ,陋 1.2< — <2 d查附图2・l （c ：得上，二2.06 查附閨2・5得耳=0.73查附图2・8得Q 二0.743.计算坐标值根据 心可以获得零件的对称循环极限应力2脉动循环极限应力幅分别为’和屈服极限应力点（6』）•绘制零件极限应力图:4. 绘制零件极P 艮应力图代入上式亀—M = 3.81=\2\MPa 2K 一 2xK 7672x3.81 = l)lMPa在乐-矶坐标中，利用零件的对称循环极限应力点（0, 心，脉幼循环极限应力点普2K 460 ?813-5如果题目3^4中危险戡ffi上的平均应力Q齐=口,应力幅兀= 36_W7b,试讒IB循环特性尸常数.求出该截面的计算安全系数亠沙解:1.根据危险截面的应力= 25 MP包和叭=充MPa,在零件擲艮应力图上画出工作应力点2.因尸=常甑M点/位于Q4D区域之内，M点捣极限应力鼠应搜磯劳强度计算安全系艱根据方程式S11八计算安全系数险九第三章3-1螺纹按牙形不同分有哪几种？各有何特点？各适用于何种场合?答：主要类型有：普通螺纹、管螺纹、矩形螺纹、梯形螺纹和锯齿形螺纹五种。
特点和差异——精选推荐
第二章 方 言第一节 特点和差异芜湖市辖区内各区乡的方言，在语音上的主要情况如下：一、共同特点1、来自中古音山摄合口一等桓韵字和三等仙韵知系声母字，芜湖话读成o ～韵母。
例如搬潘满端团暖乱钻酸官宽欢换碗，转篆砖穿船软。
2、来自中古音深、臻摄开口舒声韵的字，与曾、梗摄开口舒声韵的字，芜湖话读音混同，一律读成n 收尾的韵母。
例如：根＝庚，针＝真＝蒸＝征，彬＝冰＝兵，林＝鳞＝陵＝灵。
3、来自中古音见、晓组声母开口二等韵的字，芜湖话有文白两种读音。
白读为舌根音声母和零声母，韵母为洪音；文读为舌面前音声母，韵母为细音。
例如：家k a 31／t i a 31、敲k ‘31／t ‘i 31、鞋x ε35／i e 35、眼a ～313／i I ～313。
4、声调都是五个，阴平、阳平、上声、去声（包括古全浊上声和去声）、入声。
调值也基本相同，入声读高短调。
二、内部差异1、多数地方的话n 和l 声母同一音位。
来自中古音泥来母的字，大都分不清。
例如：南＝篮，年＝连，怒＝路，脑＝老。
只有杨王话和四山话能分辨n 和l 声母，将每组等号前面的字读n 声母，等号后面的字读l 声母。
2、多数地方的话将来自中古音知章组声母的字和庄组声母的大部分字，读成翘舌音t t ‘声母，与平舌音t s t s ‘s 声母不混。
例如：招≠糟，巢≠曹，烧≠臊，蒸≠增、争，虫≠从、崇，声≠僧、生。
可是，芜湖老城话和石头路话将这六组字的声母都读成平舌音tt s ‘s 声母。
3、“绕人日”等来自中古音日母开口洪音韵的字，多数地方话都读成翘舌音的声母但是十里牌、大桥、杨王、四山等地话却读成舌根浊擦音声母。
4、来自中古音假摄开口三等麻韵的一些字，多数地方话读成i 韵母，例如：爹＝低，姐＝几，谢＝细，夜＝异。
只有石头路和向阳路的话中这两类韵母的字不混。
“爹姐谢夜”读i e 韵母，“低几细异”读i 韵母。
5、多数地方话将宕、江摄的舒声字与咸山摄的舒声洪音韵字读音混同。
五笔代码
五笔代码1吖kuh 2阿bs 3啊kb 4锕qbs 5腌edj 6嗄kdht 7哎kaq 8哀yeu 9埃fct 10挨rct 11唉kct 12娭vct13欸ctdw 14锿qyey 15皑rmnn 16癌ukk 17毐fxdr 18嗳kep 19矮tdtv 20蔼ayj 21霭fyjn 22艾aqu 23砹daqy 24爱ep25硋dynw 26隘buw 27碍djg 28嗳kep 29嗌kuw 30嫒vepc 31瑷gepc 32叆fcec 33暧jep 34安pv 35咹kpv 36桉spv37氨rnp 38唵kdjn 39庵ydjn 40谙yuj 41媕vwga 42鹌djng 43鞍afp 44鞌pvaf 45盦wynl 46俺wdjn 47埯fdj 48咹kpv49铵qpv 50揞rujg 51犴qtfh 52岸mdfj 53按rpv 54胺epv 55案pvs 56晻jdjn 57暗ju 58黯lfoj 59肮eym 60昂jqb61枊sqb 62盎mdl 63凹mmgd 64熬gqto 65爊oyno 66敖gqty 67遨gqtp 68嗷kgqt 69廒ygq 70璈ggqt 71獒gqtd 72聱gqtb73螯gqtj 74翱rdfn 75鳌gqtg 76鏖ynjq 77拗rxl 78袄put 79媪vjl 80岙tdm 81坳fx 82傲wgqt 83奥tmo 84澳itm85懊ntm 86鏊gqtq 87八wty 88巴cnh 90叭kwy 91扒rwy 91芭ac 92吧kc 93岜mcb 94峇mwgk 95疤ucv 96捌rklj97笆tcb 98粑ocn 99茇adc 100拔rdc 101菝ard 102跋khdc 103魃rqcc 104把rcn 105钯qcn 106靶afc 107坝fmy 108爸wqc109耙dic 110罢lfc 111鲅qgdc 112霸faf 113灞ifa 114掰rwvr 115白rrr 116百dj 117伯wr 118佰wdj 119柏srg 120捭rrt121摆rlf 122呗kmy 123拜rdfh 124败mty 125稗trtf 126扳rrc 127班gyt 128般tem 129颁wvd 130斑gyg 131搬rte 132瘢utec133癍ugyg 134坂frc 135板src 136版thgc 137钣qrc 138舨terc 139蝂jthc 140办lw 141半uf 142扮rwv 143伴wuf 144坢fufh145拌rufh 146绊xuf 147靽afuf 148瓣urc 149邦dtb 150帮dt 151梆sdt 152浜irgw 153绑xdt 154榜sup 155膀eup 156蚌jdh157棒sdw 158棓sukg 159傍wup 160谤yup 161蒡aupy 162磅dup 163镑qup 164包qn 165苞aqn 166孢bqn 167枹sqn 168胞eqn169炮oqn 170剥vijh 171龅hwbn 172煲wkso 173雹fqn 175薄aig 175饱qnqn 176保wk 177鸨xfq 178葆awk 179堡wksf 180褓puws181报rb 182刨qnjh 183抱rqn 184趵khqy 185豹eeqy 186鲍qgq 187暴jaw 188虣gahm 189瀑ija 190曝jja 191爆oja 192陂bhc193杯sgi 194卑rtfj 195背uxe 196桮sgik 197椑srtf 198悲djdn 199碑drt 200鹎rtfg 201北ux 202贝mhny 203孛fpbf 204邶uxb205狈qtmy 206备tlf 207钡qmy 208倍wuk 209悖nfpb 210被puhc 211辈djdl 212惫tln 213焙ouk 214蓓awuk 215碚duk 216鞁afhc217骳meh 218褙puue 219糒oate 220鞴afae 221鐾nkuq 222臂nkue 223奔dfa 224贲fam 225栟suah 226锛qdf 227本sg 228苯asg229畚cdl 230夯dlb 231坋fwvn 232坌wvff 233笨tsg 234崩mee 235绷xee 236嘣kme 237甭gie 238泵diu 239迸uap 240蚌jdh241镚qmee 242蹦khme 243屄npwi 244逼gklp 245鎞qtlx 246鲾qgll 247荸afpb 248鼻thl 249匕xtn 250比xx 251吡kxx 252佊whc253沘ixxn 254妣vxx 255彼thc 256秕txx 257笔tt 258俾wrt 259舭tex 260鄙kfl 261币tmh 262必nt 263毕xxf 264闭uft265庇yxx 266邲ntbh 267诐yhc 268苾antr 269畀lgj 270閟unte 271泌int 272駜cntt 273珌gntt 274荜axxf 275柲sntt 276毖xxnt277哔kxxf 278毙xxgx 279铋qntt 280秘tn 281狴qtxf 282萆art 283梐sxxf 284庳yrt 285敝umi 286婢vrt 287赑mmmu 288筚txxf289湢igkl 290愊ngkl 291愎ntjt 292弼xdj 293蓖atl 294跸khxf 295腷egkl 296痹ulgj 297煏ogkl 298滗itt 299裨pur 300辟nku301碧grd 302蔽aum 303箅tlg 304弊umia 305獘umid 306薜ank 307觱dgke 308篦ttlx 309壁nkuf 310避nk 311髀merf 312濞ithj313奰llld 314璧nkuy 313边lp 315砭dtp 317笾tlp 318编xyna 319煸oyna 320蝙jyna 321箯twgq 322鞭afw 323贬mtp 324窆pwtp325扁ynma 326匾ayna 327碥dyna 328藊atya 329卞yhu 330弁caj 331苄ayh 332抃ryhy 333汴iyh 334忭nyhy 335变yo 336昪jcaj337便wgj 338遍ynm 339缏xwgq 340艑teya 341辨uyt 342辩uyu 343辫uxu 344杓sqyy 345标sfi 346飑mqqn 347驫cccu 348彪hame349骠cs 350膘esf 351熛osfi 352幖mhsi 353飙dddq 354镖qsf 355瘭usf 356藨ayno 357瀌iyno 358镳qyno 359穮tyno 360表ge361婊vgey 362裱puge 363鳔qgs 364瘪uthx 365憋umin 366鳖umig 367别klj 368蹩umih 369玢gwv 370宾pr 371彬sse 372傧wpr373斌yga 374滨ipr 375缤xpr 376槟spr 377镔qpr 378濒ihim 379豳eem 380摈rpr 381殡gqp 382膑epr 383髌mepw 384鬓depw385冰ui 386并ua 387兵rgw 388屏nua 389丙gmw 390邴gmwb 391秉tgv 392柄sgm 393昺jgmw 394饼qnu 395炳ogm 396禀ylk397病ugm 398摒rnua 399拨rnt 400波ihc 401玻ghc 402钵qsg 403饽qnfb 404菠aih 405播rtol 406嶓mtol 407驳cqq 408帛rmh409泊ir 410勃fpb 411钹qcdy 412铂qrg 413亳ypta 414浡ifpb 415舶ter 416脖efp 417博fge 418葧afpl 419鹁fpbg 420渤ifp421搏rgef 422馎qngf 423僰gmiw 424箔tir 425魄rrqc 426膊egef 427镈qgef 428薄aig 429馞tjfb 430髆megf 431欂saif 432襮puji433礴dai 434跛khhc 435簸tadc 436檗nkus 437擘nkur 438卜hhy 439啵kih 440逋gehp 441峬mgey 442庯ygey 443晡jget 444醭sgoy445卟khy 446补puh 447捕rge 448哺kge 449不i 450布dmh 451步hi 452吥kgiy 453怖ndm 454钚qgiy 455埔fgey 456部uk457埠fwn 458瓿ukg 459蔀aukb 460篰tukb 461簿tig 462拆rry 463擦rpwi 464嚓kpw 465礤daw 466偲wlny 467猜qtge 468才ft469材sft 470财mf 471裁fay 472采es 473彩ese 474睬hes 475踩khes 476菜ae 477蔡awf 478縩xwfi 479参cd 480骖ccd481餐hq 482残gqg 483蚕gdj 484惭nl 485惨ncd 486憯naqj 487灿om 488掺rcd 489孱nbb 490粲hqco 491璨ghq 492仓wbb493伧wwbn 494苍awb 495沧iwb 496鸧wbqg 497舱tew 498藏adnt 499操rkk 500糙otf 501曹gma 502嘈kgmj 503漕igmj 504槽sgmj505螬jmgj 506艚tegj 507草ajj 508懆nkks 509册mm 510厕dmjk 511侧wmj 512测imj 513恻nmj 514策tgm 515箣tgmj 516岑mwyn517涔imw 518噌kul 519层nfc 520曾ul 521蹭khuj 522叉cyi 523杈scyy 524臿tfvd 525差uda 526插rtf 527喳ksj 528馇qns529碴dsj 530锸qtfv 531艖teua 532茬adhf 533茶aws 534查sj 535搽raws 536嵖msjg 537猹qts 538楂ssj 539槎suda 540察pwfi541檫spwi 542衩puc 543镲qpwi 544汊icyy 545岔wvmj 546刹qsj 547诧ypta 548姹vpt 549钗qcy 550侪wyj 551柴hxs 552豺eef553虿dnju 554瘥uuda 555觇hkm 556梴sthp 557搀rqku 558幨mhqy 559襜puqy 560单ujfj 561鋋qthp 562谗yqk 563婵vuj 564馋qnqu565禅pyuf 566缠xyj 567蝉jujf 568僝wnbb 569廛yjf 570潺inbb 571澶iylg 572镡qsjh 573瀍iyjf 574蟾jqd 575巉mqky 576躔khyf577产u 578浐iutt 579谄yqvg 580啴kujf 581铲qut 582阐uuj 583蒇admt 584骣cnb 585冁ujfe 586忏ntfh 587颤ylkm 588羼nudd589伥wta 590昌jj 591菖ajjf 592猖qtjj 593阊ujjd 594娼vjj 595鲳qgjj 596长ta 597场fnrt 598苌ata 599肠enr 600尝ipf601倘wim 602常ipkh 603偿wi 604徜tim 605裳ipke 606嫦viph 607厂dgt 608昶ynij 609惝nim 610敞imkt 611氅imkn 612玚gnrt613怅nta 614畅jhnr 615倡wjjg 616鬯qob 617唱kjj 618抄rit 619吵kit 620怊nvk 621钞qit 622绰xhj 623超fhv 624焯ohj625剿vjsj 626晁jiqb 627巢vjs 628朝fje 629嘲kfj 630潮ifj 631炒oit 632耖diit 633车lg 634伡wlh 635砗dlh 636尺nyi637扯rhg 638彻tavn 639坼fry 640掣rmhr 641撤ryc 642澈iyct 643抻rjh 644郴ssb 645綝xssy 646琛gpw 647嗔kfhw 648瞋hfhw649臣ahn 650尘iff 651辰dfe 652沉ipm 653忱np 654陈ba 655宸pdfe 656梣smwn 657晨jd 658谌yadn 659趻khwn 660碜dcd661衬puf 662龀hwbx 663称tq 664趁fhwe 665榇sus 666蛏jcfg 667铛qiv 668偁wem 669赪foh 670撑rip 671瞠hip 672成dnnt673丞big 674呈kg 675枨sta 676郕dnnb 677诚ydn 678承bd 679城fd 680埕fkg 681晟jdn 682乘tux 683盛dnnl 684铖qdn685程tkgg 686惩tghn 687椉yqas 688裎puk 689塍eudf 690澄iwgu 691橙swgu 692逞kgp 693骋cmg 694秤tgu 695吃ktn 696郗qdmb697哧kfo 698蚩bhgj 699胵egcf 700鸱qayg 701絺xqdh 702眵hqq 703笞tck 704摛rybc 705嗤kbhj 706痴utdk 707媸vbh 708螭jybc709魑rqcc 710池ib 711弛xb 712驰cbn 713迟nyp 714坻fqa 715茌awff 716持rff 717匙jghx 718漦fiti 719墀fni 720篪trhm721齿hwb 722侈wqq 723哆kqq 724耻bh 725豉gkuc 726褫purm 727彳ttth 728叱kxn 729斥ryi 730赤fo 731饬qntl 732炽ok733翅fcn 734眙hck 735敕gkit 736啻upmk 737傺wwfi 738瘈udhd 739憏nwfi 740瘛udhn 741冲ukh 742充yc 743忡nkh 744茺ayc745涌ice 746舂dwv 747憃dwvn 748憧nujf 749艟teuf 750虫jhny 751种tkh 752重tgj 753崇mpf 754宠pdx 755铳qyc 756抽rm757瘳unwe 758仇wvn 759帱mhd 760惆nmf 761绸xmf 762畴ldt 763酬sgyh 764稠tmfk 765愁tonu 766筹tdtf 767裯pumk 768踌khdf769雠wyy 770丑nfd 771瞅hto 772臭thdu 773出bm 774初puv 775摴rffn 776刍qvf 777除bwt 778鉏qegg 779厨dgkf 780锄qegl781滁ibw 782蜍jwt 783雏qvw 784篨tbwt 785橱sdgf 786躇khaj 787处th 788杵stfh 789础dbm 790楮stfj 791储wyf 792楚ssn793褚pufj 794亍fhk 795怵nsy 796绌xbm 797柷skqn 798俶whic 799畜yxl 800諔yhic 801搐ryxl 802触qejy 803滀iyxl 804憷nss805歜lqjw 806黜lfom 807斶rlqj 808矗fhfh 809欻ooqw 810揣rmd 811搋rrhm 812啜kccc 813嘬kjb 814踹khmj 815膪eupk 816川kthh817氚rnkj 818穿pwat 819传wfny 820舡tea 821船temk 822遄mdm 823椽sxe 824篅tmdj 825舛qah 826荈aqah 827喘kmd 828僢weph829串kkh 830钏qkh 831创wbj 832疮uwb 833窗pwt 834床ysi 835噇kujf 836幢mhu 837闯ucd 838怆nwb 839吹kqw 840炊oqw841垂tga 842陲btgf 843捶rtgf 844棰stg 845椎swy 846圌lmdj 847槌swn 848锤qtgf 849旾gbnj 850春dw 851椿sdwj 852輴srfh853蝽jdwj 854纯xgb 855莼axb 856唇dfek 857淳iyb 858錞qybg 859鹑ybq 860漘idfe 861醇sgyb 862蠢dwjj 863踔khhj 864戳nwya865辵ehu 866娖vkhy 867惙nccc 868婼vadk 869婥vhj 870绰xhj 871辍lccc 872龊hwbh 873刺gmi 874呲khxn 875疵uhx 876粢uqwo877骴mehx 878词yngk 879茈ahx 880茨auqw 881兹uxx 882祠pynk 883瓷uqwn 884薋auqm 885辞tduh 886慈uxxn 887磁du 888雌hxw889鹚uxxg 890餈uqwe 891糍oux 892此hx 893泚ihxn 894次uqw 895伺wng 896佽wuqw 897赐mjq 898匆qry 899苁awwu 900囱tlqi901枞sww 902葱aqrn 903骢ctl 904璁gtl 905聪bukn 906熜otl 907从ww 908丛wwg 909淙ipfi 910悰npfi 911琮gpf 912賨pfim913藂abci 914凑udw 915辏ldw 916腠edw 917粗oe 918殂gqe 919卒ywwf 920促wkh 921猝qtyf 922酢sgtf 923蔟ayt 924醋sga925憱nyin 926簇tyt 927蹙dhih 928蹴khyn 929汆tyiu 930撺rpwh 931镩qpw 932蹿khph 933攒rtfm 934窜pwk 935篡thdc 936爨wfmo937衰ykge 938崔mwy 939催wmw 940缞xyke 941榱syk 942摧rmw 943漼imwy 944璀gmwy 945脆eqd 946萃ayw 947啐kyw 948淬iywf949悴nywf 950毳tfnn 951瘁uyw 952顇ywwm 953粹oyw 954翠nywf 955膵eayf 956村sf 957皴cwtc 958存dhb 959蹲khuf 960忖nfy961寸fghy 962搓rud 963磋dud 964撮rjb 965蹉khua 966嵯mud 967矬tdw 968痤uww 969鹾hlqa 970脞eww 971剒ajjh 972莝awwf973厝daj 974挫rww 975措raj 976锉qww 977错qaj 978咑krsh 979耷dbf 980哒kdp 981搭rawk 982嗒kawk 983答tw 984鎝qawk985褡pua 986打rs 987达dp 988沓ijf 989怛njg 990妲vjg 991炟ojgg 992笪tjgf 993阘ujnd 994靼afjg 995瘩uaw 996鞑afdp997大dd 998垯fdpy 999疸ujg 1000塔fawk1001跶khdp 1002呆ks 1003呔kdyy 1004待tffy 1005歹gqi 1006逮vip 1007傣wdw 1008代wa 1009甙aafd 1010岱wamj1011迨ckp 1012绐xck 1013骀cck 1014玳gwa 1015带gkp 1016殆gqc 1017贷wam 1018怠ckn 1019埭fvi 1020袋waye1021叇fcvp 1022戴falw 1023黛wal 1024黱eudo 1025丹myd 1026担rjg 1027眈hpq 1028耽bpq 1029郸ujfb 1030聃bmfg1031殚gqu 1032阐uujf 1033箪tujf 1034儋wqd 1035胆ej 1036疸ujg 1037掸rujf 1038赕moo 1039亶ylkg 1040石dgtg1041旦jgf 1042但wjg 1043担rjg 1044诞ythp 1045疍nhjg 1046萏aqvf 1047啖koo 1048淡io 1049惮nuj 1050弹xuj1051蛋nhj 1052氮rno 1053髧depq 1054嚪kuqv 1055澹iqdy 1056当iv 1057珰givg 1058铛qiv 1059裆puiv 1060筜tivf1061挡riv 1062党ipk 1063谠yip 1064凼ibk 1065砀dnr 1066宕pdf 1067垱fivg 1068荡ain 1069档si 1070菪apd1071刀vn 1072叨kvn 1073忉nvn 1074氘rnj 1075鱽qgvn 1076捯rgcj 1077导nf 1078岛qynm 1079捣rqym 1080倒wgc1081祷pyd 1082蹈khev 1083到gc 1084焘dtfo 1085盗uqwl 1086道uthp 1087稻tev 1088纛gxf 1089嘚ktjf 1090得tj1091锝qjgf 1092德tfl 1093地f 1094的r 1095底yqa 1096脦ean 1097扽rgbn 1098灯os 1099登wgku 1100噔kwgu1101镫qwgu 1102簦twgu 1103蹬khwu 1104等tffu 1105戥jtga 1106邓cb 1107凳wgkm 1108嶝mwgu 1109磴dwgu 1110瞪hwg1111氐qayn 1112低wqa 1113羝udq 1114堤fjgh 1115提rj 1116嘀kum 1117滴ium 1118樀sumd 1119镝qum 1120狄qtoy1121迪mp 1122籴tyo 1123荻aqto 1124敌tdt 1125涤its 1126頔mdmy 1127笛tmf 1128觌fnuq 1128嫡vum 1130翟nywf1131邸qayb 1132诋yqay 1133坻fqa 1134抵rqa 1135柢sqa 1136牴trqy 1137砥dqay 1138骶meqy 1139弟uxh 1140俤wuxt1141帝up 1142递uxhp 1143娣vux 1144菂arqy 1145第tx 1146谛yuph 1147蒂aup 1148棣svi 1149睇hux 1150缔xup1151禘pyuh 1152碲duph 1153墬bxef 1154嗲kwq 1155掂ryh 1156滇ifhw 1157颠fhwm 1158攧rfhm 1159巅mfh 1160癫ufhm1161典maw 1162点hko 1163碘dma 1164踮khyk 1165电jn 1166佃wl 1167甸ql 1168阽bhkg 1169坫fhkg 1170店yhk1171玷ghk 1172垫rvyf 1173钿qlg 1174淀ipgh 1175惦nyh 1176奠usgd 1177殿naw 1178靛gep 1179簟tsj 1180癜una1181刁ngd 1182叼kng 1183汈ingg 1184凋umf 1185貂eev 1186碉dmf 1187雕mfky 1188鲷qgm 1189鸟qyng 1190屌nkmh1191吊kmh 1192钓qqyy 1193窎pwqg 1194调ymf 1195掉rhj 1196铞qkmh 1197铫qiq 1198爹wqqq 1199跌khr 1200迭rwp1201垤fgc 1202绖xgcf 1203瓞rcyw 1204啑kgvh 1205谍yan 1206堞fan 1207耋ftxf 1208揲rans 1209喋kans 1210惵nans1211牒thgs 1212叠cccg 1213碟dan 1214蝶jan 1215艓teas 1216蹀khas 1217螲jpwf 1218鲽qga 1219丁sgh 1220仃wsh1221叮ksh 1222玎gsh 1223盯hs 1224钉qs 1225疔usk 1226耵bsh 1227酊sgs 1228靪afsh 1229顶sdm 1230鼎hnd1231订ys 1232饤qnsh 1233定pg 1234啶kpgh 1235铤ktfp 1236腚epg 1237碇dpgh 1238锭qp 1239丢tfc 1240铥qtfc1241东ai 1242冬tuu 1243咚ktuy 1244岽mai 1245氡rntu 1246鸫aiq 1247董atg 1248懂nat 1249动fcl 1250冻uai1251侗wmgk 1252垌fmg 1253栋sai 1254峒mmgk 1255胨eai 1256洞imgk 1257恫nmg 1258硐dmg 1259都ftjb 1260唗kfhy1261兜qrnq 1262蔸aqrq 1263斗ufk 1264抖rufh 1265陡bfh 1266蚪jufh 1267豆gku 1268逗gkup 1269读yfn 1270酘sgmc1271脰egku 1272痘ugku 1273窦pwfd 1274督hich 1275嘟kftb 1276毒gxgu 1277独qtj 1278顿gbnm 1279渎ifnd 1280椟sfn1281犊trfd 1282牍thgd 1283黩lfod 1284髑mel 1285肚efg 1286笃tcf 1287堵fft 1288赌mftj 1289睹hft 1290芏aff1291杜sfg 1292妒vynt 1293度ya 1294渡iya 1295镀qya 1296蠹gkhj 1297耑mdmj 1298端umd 1299短tdg 1300段wdm1301断on 1302塅fwdc 1303缎xwd 1304椴swd 1305煅owd 1306碫dwdc 1307锻qwd 1308簖tonr 1309堆fwy 1310队bw1311对cf 1312兑ukqb 1313怼cfn 1314敦ybt 1315碓dwyg 1316錞qybg 1317憝ybtn 1318镦qyb 1319吨kgb 1320惇nybg1321墩fyb 1322撴rybt 1323驐cybt 1324礅dyb 1325蹲khuf 1326盹hgb 1327趸dnk 1328囤lgb 1329沌igb 1330炖ogbn1331砘dgb 1332钝qgbn 1333盾rfh 1334遁rfhp 1335楯srfh 1336多qq 1337咄kbm 1338哆kqq 1339剟cccj 1340掇rcc1341裰pucc 1342夺df 1343铎qcf 1344踱khyc 1345朵ms 1346垛fms 1347哚kms 1348躲tmds 1349驮cdy 1350剁msj1351饳qnbm 1352垛fms 1353柮sbmh 1354柂stbn 1355舵tepx 1356堕bdef 1357惰bda 1358媠vdae 1359跺khm 1360妸vskg1361屙nbs 1362娿bskv 1363婀vbs 1364痾ubs 1365讹ywxn 1366吪kwxn 1367俄wtr 1368莪atr 1369哦ktr 1370峨mtr1371娥vtr 1372睋htrt 1373锇qtrt 1374鹅trng 1375蛾jtr 1376额ptkm 1377恶gogn 1378厄dbv 1379苊adb 1380扼rdb1381呃kdb 1382轭ldb 1383呝kynn 1384垩gogf 1385饿qnt 1386鄂kkfb 1387瘀uywu 1388谔ykkn 1389萼akkn 1390遏jqwp1391崿mkk 1392愕nkk 1393頞pvdm 1394搤ruwl 1395腭ekk 1396鹗kkfg 1397锷qkkn 1398颚kkfm 1399噩gkkk 1400鳄qgkn1401欸ctdw 1402奀gidu 1403恩ldn 1404蒽aldn 1405摁rld 1406儿qt 1407而dmj 1408洏idmj 1409栭sdmj 1410胹edmj1411鲕qgdj 1412尔qiu 1413耳bgh 1414迩qip 1415洱ibg 1416饵qnbg 1417珥gbg 1418铒qbg 1419二fg 1420佴wbg1421贰afm 1422樲safm 1423发v 1424乏tpi 1425伐wat 1426罚ly 1427垡waff 1428阀uwa 1429筏twa 1430法if1431砝dfcy 1432珐gfc 1433帆mhm 1434番tol 1435蕃ato 1436幡mhtl 1437繙xtol 1438藩aitl 1439翻toln 1440凡my1441矾dmy 1442钒qmyy 1443烦odm 1444墦ftol 1445樊sqqd 1446璠gtol 1447膰etol 1448燔oto 1449繁txgi 1450蹯khtl1451蘩atxi 1452反rc 1453返rcp 1454犯qtb 1455饭qnr 1456泛itp 1457范aib 1458贩mr 1459畈lrc 1460梵ssm1461嬔vqkg 1462方yy 1463邡ybh 1464坊fyn 1465芳ay 1466枋syn 1467钫qyn 1468防by 1469妨vy 1470肪eyn1471房yny 1472鲂qgyn 1473仿wyn 1474访yyn 1475彷tyn 1476纺xy 1477舫teyn 1478髣deyb 1479放yt 1480飞nui1481妃vnn 1482非djd 1483菲adj 1484啡kdj 1485騑cdjd 1486绯xdjd 1487扉yndd 1488蜚djdj 1489霏fdjd 1490鲱qgdd1491肥ec 1492淝iec 1493腓edjd 1494匪adjd 1495诽ydj 1496悱ndjd 1497棐djds 1498斐djdy 1499榧sadd 1500翡djdn1501芾agm 1502吠kdy 1503肺egm 1504狒qtx 1505废ynty 1506沸ixj 1507费xjm 1508痱udjd 1509镄qxj 1510分wv1511芬awv 1512吩kwv 1513纷xwv 1514氛rnw 1515棻awvs 1516酚sgw 1517坟fy 1518汾iwv 1519蚡jwvn 1520棼ssw1521焚sso 1522濆ifam 1523豮efam 1524鼢vnuv 1525粉ow 1526份wwv 1527奋dlf 1528忿wvnu 1529偾wfa 1530粪oawu1531愤nfa 1532鲼qgfm 1533瀵iol 1534丰dh 1535风mq 1536沣idh 1537沨imqy 1538枫smq 1539封fffy 1540砜dmqy1541疯umq 1542峰mtd 1543烽ot 1544葑afff 1545锋qtd 1546蜂jtd 1547酆dhdb 1548冯uc 1549逢tdh 1550缝xtdp1551讽ymq 1552覂stpu 1553唪kdw 1554凤mc 1555甮qrej 1556俸wdwh 1557赗mghg 1558覅svqr 1559佛wxj 1560缶rmk1561否gik 1562夫fw 1563呋kfw 1564肤efw 1565柎swfy 1566趺khf 1567稃tebg 1568痡ugey 1569鄜ynjb 1570孵qytb1571敷geht 1572弗xjk 1573伏wdy 1574凫qynm 1575芙afwu 1576芾agm 1577芣agiu 1578扶rfw 1579孚ebf 1580刜xjjh1581苻awfu 1582茀axjj 1583拂rxjh 1584彿txjh 1585服eb 1586怫nxj 1587绂xdc 1588绋xxj 1589茯awd 1590罘lgi1591氟rnx 1592俘web 1593郛ebb 1594洑iwdy 1595祓pydc 1596莩aebf 1597栿swdy 1598砩dxj 1599蚨jfw 1600浮ieb1601菔aebc 1602桴seb 1603符twf 1604匐qgk 1605涪iuk 1606袱puwd 1607艴xjq 1608幅mhg 1609罦lebf 1610辐lgk1611蜉jeb 1612福pyg 1613箙tebc 1614髴dexj 1615蝠jgkl 1616幞mho 1617黻oguc 1618襆puoy 1619父wqu 1620甫geh1621抚rfq 1622拊rwf 1623斧wqr 1624府ywf 1625俛wqkq 1626俯wyw 1627釜wqf 1628辅lgey 1629脯ege 1630滏iwq1631腐ywfw 1632鬴gkmy 1633簠tgel 1634黼oguy 1635讣yhy 1636付wfy 1637负qm 1638妇vv 1639附bwf 1640咐kwf1641阜wnnf 1642驸cwf 1643赴fhh 1644复tjt 1645副gkl 1646蝜jqmy 1647赋mga 1648傅wge 1649富pgk 1650腹etj1651鲋qgw 1652缚xge 1653赙mge 1654蝮jtjt 1655鳆qgtt 1656覆stt 1657馥tjtt 1658夹guw 1659旮vjf 1660伽wlk1661呷klh 1662咖klk 1663胳etk 1664嘎kdh 1665轧lnn 1666钆qnn 1667尜idi 1668噶kaj 1669尕eiu 1670尬dnw1671该yynw 1672垓fynw 1673荄aynw 1674赅myn 1675改nty 1676胲eynw 1677丐ghn 1678匄qynv 1679芥awj 1680钙qgh1681盖ugl 1682溉ivc 1683概svc 1684戤ecla 1685干fggh 1686甘afd 1687杆sfh 1688肝ef 1689坩fafg 1690苷aaf1691矸dfh 1692泔iaf 1693柑saf 1694竿tfj 1695酐sgfh 1696疳uaf 1697尴dnjl 1698秆tfh 1699赶fhfk 1700敢nb1701感dgkn 1702澉inb 1703橄snb 1704擀rfj 1705旰jfh 1706绀xaf 1707淦iqg 1708骭mefh 1709赣ujt 1710冈mqi1711扛rag 1712刚mqj 1713杠sag 1714 rmqy 1715岗mmq 1716肛ea 1717钢qmq 1718缸rma 1719罡lgh 1720堽flgh1721港iawn 1722筻tgjq 1723戆ujtn 1724皋rdfj 1725高ym 1726羔ugo 1727槔srd 1728睾tlff 1729膏ypk 1730篙tymk1731糕ougo 1732杲jsu 1733搞rym 1734缟xym 1735槁symk 1736暠jymk 1737镐qym 1738稿tym 1739藁ayms 1740告tfkf1741郜tfkb 1742诰ytfk 1743锆qtfk 1744戈agnt 1745仡wtn 1746圪ftn 1747屹mtnn 1748疙utn 1749咯ktk 1750格st1751哥sks 1752鸽wgkg 1753袼putk 1754搁rut 1755割pdhj 1756歌sksw 1757革af 1758茖atkf 1759閤uwgk 1760阁utk1761鬲gkmh 1762葛ajq 1763蛤jw 1764颌wgkm 1765隔bgk 1766塥fgk 1767嗝kgkh 1768搿rwgr 1769膈egk 1770骼met1771镉qgkh 1772轕lajn 1773个wh 1774合wgk 1775各tk 1776哿lksk 1777舸tes 1778虼jtn 1779硌dtk 1780铬qtk1781给xw 1782根sve 1783跟khv 1784哏kve 1785艮vei 1786亘gjg 1787茛ave 1788更gjq 1789庚yvw 1790耕dif1791浭igjq 1792赓yvwm 1793縆xngg 1794鹒yvwg 1795羹ugod 1796埂fgj 1797耿bo 1798哽kgj 1799绠xgj 1800梗sgjq1801颈cad 1802鲠qggq 1803暅jngg 1804工a 1805弓xng 1806公wc 1807功al 1808红xa 1809攻at 1810供waw1811肱edc 1812宫pk 1813恭awnu 1814蚣jwc 1815躬tmdx 1816龚dxa 1817塨fawn 1818觥qei 1819巩amy 1820汞aiu1821拱raw 1822珙gaw 1823蛬awju 1824贡am 1825唝kamy 1826勾qci 1827句qkd 1828佝wqk 1829沟iqc 1830枸sqk1831钩qqc 1832缑xwn 1833篝tfjf 1834鞲afff 1835苟aqkf 1836岣mqk 1837狗qtq 1838笱tqk 1839构sq 1840购mqc1841诟yrg 1842垢fr 1843姤vrgk 1844够qkqq 1845遘fjgp 1846彀fpgc 1847媾vfj 1848觏fjgq 1849估wd 1850苽arcy1851咕kdg 1852呱krc 1853沽idg 1854姑vd 1855孤br 1856轱ldg 1857骨me 1858鸪dqyg 1859菇avd 1860菰abr1861蛄jdg 1862蓇amef 1863辜duj 1864酤sgdg 1865觚qer 1866毂fpl 1867箍tra 1868古dgh 1869谷wwk 1870汩ijg1871诂ydg 1872股emc 1873牯trdg 1874贾smu 1875罟ldf 1876钴qdg 1877蛊jlf 1878鹄tfkg 1879馉qnme 1880鼓fkuc1881榾sme 1882榖fptc 1883嘏dnh 1884鹘meq 1885臌efkc 1886瞽fkuh 1887盬ahnl 1888濲ifpc 1889固ldd 1890故dty1891顾db 1892堌fldg 1893梏stfk 1894崮mld 1895牿trtk 1896雇ynwy 1897锢qldg 1898痼uld 1899鲴qgld 1900瓜rcy1901呱krc 1902刮tdjh 1903括rtd 1904胍erc 1905鸹tdq 1906剐kmwj 1907寡pde 1908卦ffhy 1909诖yffg 1910挂rffg1911絓xffg 1912褂pufh 1913乖tfu 1914掴rlgy 1915拐rkl 1916怪nc 1917关ud 1918观cm 1919纶xwx 1920官pn1921冠pfqf 1922矜cbtn 1923莞apfq 1924倌wpn 1925棺spn 1926鳏qgli 1927馆qnp 1928琯gpn 1929輨lpnn 1930筦tpfq1931痯upnn 1932管tp 1933鳤qgtn 1934贯xfm 1935掼rxf 1936涫ipn 1937惯nxf 1938祼pyjs 1939盥qgi 1940灌iak1941瓘gaky 1942鹳akkg 1943罐rmay 1944光iq 1945咣kiq 1946胱eiq 1947广yygt 1948犷qtyt 1949逛qtgp 1950归jv1951圭fff 1952龟qjn 1953妫vyl 1954规fwm 1955邽ffbh 1956皈rrcy 1957闺uffd 1958珪gffg 1959硅dff 1960傀wrq1961廆yrqc 1962瑰grq 1963鲑qgff 1964鬶fwmh 1965瓌gyle 1966宄pvb 1967轨lv 1968庋yfc 1969匦alv 1970佹wqdb1971诡yqd 1972垝fqdb 1973姽vqdb 1974癸wgd 1975晷jthk 1976簋tvel 1977柜san 1978炅jou 1979刽wfcj 1980炔onw1981贵khgm 1982桂sff 1983桧swf 1984硊dqdb 1985跪khqb 1986鳜qgdw 1987衮uceu 1988绲xjx 1989辊lj 1990滚iuc1991磙duc 1992鲧qgti 1993棍sjx 1994过fp 1995呙kmwu 1996埚fkm 1997郭ybb 1998涡ikm 1999崞myb 2000聒btd2001锅qkm 2002蝈jlg 2003国l 2004帼mhl 2005腘elg 2006虢efhm 2007馘uthg 2008果js 2009菓ajsu 2010馃qnjs2011椁syb 2012蜾jjs 2013裹yjse 2014哈kwg 2015铪qwgk 2016虾jghy 2017奤ddmd 2018咳kynw 2019嗨kitu 2020还gip2021孩bynw 2022骸mey 2023海itx 2024醢sgdl 2025亥yntw 2026骇cynw 2027氦rnyw 2028害pd 2029嗐kpdk 2030顸fdmy2031蚶jaf 2032憨nbtn 2033鼾thlf 2034邗fbh 2035汗ifh 2036含wynk 2037函bib 2038琀gwyk 2039晗jwyk 2040焓owy2041涵ibi 2042韩fjfh 2043寒pfj 2044罕pwf 2045喊kdgt 2046 kunt 2047汉ic 2048扞rfh 2049闬ufk 2050汗ifh2051旱jfj 2052垾fjfh 2053捍rjf 2054悍njf 2055菡abib 2056睅hjf 2057颔wynm 2058蔊aojf 2059撖rnbt 2060暵jakw2061熯oakw 2062翰fjw 2063撼rdgn 2064憾ndgn 2065瀚ifjn 2066亢ymb 2067行tf 2068吭kym 2069迒ymp 2070杭sym2071绗xtf 2072航tey 2073颃ymdm 2074沆iym 2075巷awn 2076蒿aym 2077薅avd 2078嚆kay 2079号kgn 2080蚝jtf2081毫ypt 2082嗥krd 2083貉eetk 2084豪ypeu 2085壕fyp 2086嚎kyp 2087濠iyp 2088好vb 2089郝fob 2090昊jgd2091耗ditn 2092浩itfk 2093淏ijgd 2094皓rtfk 2095鄗ymkb 2096滈iymk 2097暠jymk 2098皞rrdf 2099澔irtk 2100颢jyim2101灏ijym 2102诃ysk 2103呵ksk 2104喝kjq 2105嗬kawk 2106蠚adkj 2107禾ttt 2108纥xtnn 2109何wsk 2110訸yty2111和t 2112郃wgkb 2113劾yntl 2114河isk 2115曷jqwn 2116饸qnwk 2117阂uyn 2118盍fclf 2119荷awsk 2120核synw2121盉tlf 2122菏ais 2123龁hwbn 2124盒wgkl 2125颌wgkm 2126阖ufc 2127鹖jqwg2128翮gkmn 2129鞨afjn 2130吓kgh2131佫wtk 2132贺lkm 2133猲qtjn 2134愒njqn 2135赫fof 2136褐pujn 2137鹤pwy 2138翯nymk 2139壑hpg 2140黑lfo2141嘿klf 2142痕uve 2143很tve 2144狠qtv 2145恨nv 2146亨ybj 2147哼kyb 2148脝eybh 2149恒ngj 2150姮vgj2151珩gtf 2152桁stfh 2153鸻tfhg 2154横sam 2155衡tqdh 2156蘅atqh 2157啈kfuf 2158噷kujw 2159吽krhh 2160轰lcc2161哄kaw 2162訇qyd 2163烘oaw 2164薨alpx 2165弘xcy 2166吰kdcy 2167泓ixc 2168荭axa 2169虹ja 2170鈜qdcy2171竑udcy 2172洪iaw 2173翃dcn 2174鉷qawy 2175魟qgag 2176鸿iaqg 2177葓aiaw 2178蕻adaw 2179黉ipa 2180讧yag2181澒iadm 2182齁thlk 2183侯wnt 2184猴qtw 2185睺hwn 2186瘊uwn 2187骺mer 2188篌twn 2189糇own 2190喉kwn2191吼kbn 2192犼qtbn 2193后rg 2194郈rgkb 2195厚djb 2196垕rgkf 2197逅rgkp 2198候whn 2199堠fwnd 2200鲎iqpg2201鲘qgrk 2202乎tuh 2203戏ca 2204呼kt 2205忽qrn 2206轷ltuh 2207烀otu 2208唿kqrn 2209淴iqrn 2210惚nqr2211嘑khah 2212滹ihah 2213糊ode 2214囫lqr 2215狐qtr 2216弧xrc 2217胡de 2218壶fpo 2219斛qeu 2220葫adef2221搰rmeg 2222鹄tfkg 2223猢qtde 2224湖ide 2225瑚gde 2226煳ode 2227鹕deq 2228嘝kqe 2229蝴jde 2230衚tdeh2231縠fpgc 2232醐sgde 2233觳fpgc 2234虎ha 2235浒iytf 2236唬kham 2237琥gha 2238互gx 2239户yne 2240冱ugx2241护ryn 2242沪iyn 2243枑sgxg 2244岵mdg 2245怙ndg 2246戽ynu 2247祜pydg 2248笏tqr 2249瓠dfny 2250扈ynkc2251鄠ffnb 2252鹱qync 2253鳠qgac 2254化wx 2255花awx 2256砉dhdf 2257哗kwx 2258划aj 2259华wxf 2260骅cwx2261铧qwx 2262猾qtm 2263滑ime 2264搳rpdk 2265豁pdhk 2266画gl 2267话ytd 2268桦swx 2269婳vgl 2270怀ngi2271徊tlk 2272淮iwy 2273槐srq 2274踝khjs 2275耲diye 2276坏fgi 2277欢cqw 2278讙yaky 2279獾qtay 2280驩caky2281环ggi 2282郇qjb 2283荁agjg 2284洹igj 2285桓sgj 2286萑awy 2287貆eegg 2288锾qefc 2289圜llg 2290阛ulge2291澴ilge 2292寰plg 2293嬛vlg 2294缳xlge 2295轘llge 2296鹮lgkg 2297鬟del 2298缓xef 2299幻xnn 2300奂qmd2301宦pah 2302换rq 2303唤kqm 2304涣iqm 2305涣iqm 2306浣ipfq 2307患kkhn 2308焕oqm 2309逭pnhp 2310睆hpf2311痪uqm 2312豢ude 2313漶ikkn 2314鲩qgp 2315擐rlge 2316肓ynef 2317荒aynq 2318塃fayq 2319慌nay 2320皇rgf2321黄amw 2322凰mrg 2323隍brg 2324喤krgg 2325遑rgp 2326徨trg 2327湟irgg 2328惶nrgg 2329煌orgg 2330锽qrgg2331潢iam 2332璜gamw 2333蝗jr 2334篁trgf 2335艎terg 2336磺dam 2337鐄qamw 2338癀uam 2339蟥jam 2340簧tamw2341鳇qgr 2342怳nkq 2343恍niq 2344晃ji 2345谎yay 2346幌mhjq 2347滉iji 2348榥sjiq 2349皝rgiq 2350灰do2351诙ydo 2352虺gqji 2353挥rpl 2354咴kdo 2355恢ndo 2356袆pufh 2357珲gpl 2358豗gqei 2359晖jplh 2360辉ipql2361翚nplj 2362麾yssn 2363徽tmgt 2364隳bdan 2365回lkd 2366茴alkf 2367洄ilk 2368蛔jlk 2369鮰qglk 2370悔ntx2371毁va 2372卉faj 2373汇ian 2374会wf 2375讳yfnh 2376荟awfc 2377哕kmq 2378浍iwfc 2379诲ytx 2380绘xwf2381恚ffnu 2382桧swf 2383贿mde 2384烩owf 2385彗dhdv 2386晦jtx 2387秽tmq 2388惠gjh 2389喙kxe 2390阓ukhm2391缋xkh 2392翙mqn 2393殨gqkm 2394蔧adhv 2395嘒kdhv 2396僡wgjn 2397慧dhd 2398蕙agj 2399槥sdhv 2400潓igjn2401憓ngjn 2402靧dmjm 2403蟪jgjn 2404昏qajf 2405荦apr 2406阍uqa 2407惛nqa 2408婚vq 2409棔sqaj 2410浑ipl2411馄qnjx 2412混ijx 2413魂fcr 2414诨ypl 2415圂lei 2416溷iley 2417慁len 2418耠diw 2419騞cdhd 2420锪qqrn2421劐awyj 2422嚄kawc 2423攉rfwy 2424活itd 2425火ooo 2426伙wo 2427钬qoy 2428漷iybb 2429夥jsq 2430或ak2431货wxm 2432获aqt 2433祸pykw 2434惑akgn 2435霍fwyf 2436雘myac 2437镬qaw 2438藿afwy 2439嚯kfwy 2440蠖jawc2441臛efwy 2442几mt 2443讥ymn 2444击fmk 2445饥qnm 2446叽kmn 2447玑gmn 2448圾fe 2449芨aey 2450机sm2451乩hkn 2452肌em 2453矶dmn 2454鸡cqy 2455其adw 2456奇dskf 2457咭kfkg 2458剞dskj 2459唧kvcb 2460积tkw2461笄tgaj 2462屐ntfc 2463姬vah 2464基ad 2465期adwe 2466赍fww 2467犄trd 2468嵇tdnm 2469缉xkb 2470畸lds2471跻khyj 2472錤qadw 2473襀pugm 2474箕tad 2475稽tdnj 2476觭qedk 2477齑ydjj 2478畿xxa 2479墼gjff 2480激iry2481羁laf 2482及ey 2483伋weyy 2484吉fk 2485岌meyu 2486汲iey 2487忣neyy 2488级xe 2489极se 2490即vcb2491佶wfkg 2492诘yfk 2493亟bkc 2494笈teyu 2495急qvn 2496姞vfkg 2497疾utd 2498棘gmii 2499殛gqb 2500戢kbnt2501集wys 2502蒺aut 2503楫skb 2504辑lkb 2505嵴miw 2506嫉vut 2507蕺akbt 2508踖khaj 2509瘠uiw 2510鹡iweg2511藉adi 2512蹐khie 2513籍tdij 2514己nng 2515纪xn 2516虮jmn 2517挤ryj 2518济iyj 2519脊iwe 2520掎rds2521鱾qgnn 2522戟fja 2523麂ynjm 2524计yf 2525记yn 2526伎wfcy 2527齐yjj 2528芰afcu 2529技rfc 2530系txi2531忌nnu 2532际bf 2533妓vfc 2534季tb 2535剂yjjh 2536垍fthg 2537荠ayjj 2538哜kyj 2539迹yop 2540洎ithg2541既vca 2542勣gmlt 2543觊mnmq 2544继xo 2545偈wjq 2546徛tds 2547祭wfi 2548悸ntb 2549寄pds 2550寂ph2551绩xgm 2552惎adwn 2553塈vcaf 2554蓟aqgj 2555霁fyj 2556跽khnn 2557穊tvcq 2558鲚qgyj 2559漈iwfi 2560暨vcag2561稷tlw 2562鲫qgvb 2563髻defk 2564冀uxl 2565穄twfi 2566罽ldo 2567鱀vcag2568檵sxxn 2569鰶qgwi 2570骥cux2571加lk 2572夹guw 2573伽wlk 2574茄alkf 2575佳wffg 2576迦lkp 2577珈glk 2578枷slk 2579浃igu 2580痂ulkd2581家pe 2582笳tlkf 2583袈lky 2584葭anhc 2585跏khlk 2586傢wpey 2587嘉fkuk 2588镓qpe 2589郏guwb 2590荚aguw2591恝dhvn 2592戛dha 2593铗qguw 2594颊guwm 2595蛱jgu 2596跲khwk 2597甲lhnh 2598岬mlh 2599胛elh 2600钾qlh2601假wnh 2602斝kkp 2603嘏dnh 2604槚ssmy 2605榎sdht 2606瘕unh 2607价wwj 2608驾lkc 2609架lks 2610嫁vpe2611稼tpe 2612戋gggt 2613尖id 2614奸vfh 2615歼gqt 2616坚jcf 2617间uj 2618浅igt 2619肩yned 2620艰cv2621监jtyl 2622兼uvo 2623菅apnn 2624笺tgr 2625渐il 2626犍trv 2627湔iue 2628缄xdg 2629瑊gdgt 2630蒹auv2631椾suej 2632搛ruvo 2633煎uejo 2634缣xuv 2635鲣qgjf 2636鹣uvog 2637熸oaq 2638鞬afvp 2639鞯afa 2640鳒qguo2641櫼swwg 2642囝lb 2643絸xqmn 2644拣ranw 2645枧smqn 2646茧aju 2647柬gli 2648俭wwgi 2649捡rwgi 2650笕tmqb2651检sw 2652趼khga 2653减udg 2654剪uejv 2655硷dwgi 2656揃ruej 2657睑hwgi 2658锏qujg 2659裥puuj 2660暕jgli2661简tuj 2662谫yue 2663戬goga 2664碱ddg 2665翦uejn 2666蹇pfjh 2667謇pfjy 2668劗tfqj 2669鬋deuj 2670瀽ipfh2671见mqb 2672件wrh 2673饯qngt 2674建vfhp 2675荐adh 2676贱mgt 2677牮war 2678剑wgi 2679涧iujg 2680健wvf2681舰temq 2682谏ygl 2683楗svfp 2684瞷hujg 2685践khgt 2686锏qujg 2687毽tfnp 2688腱evfp 2689溅imgt 2690鉴jtyq2691键qvfp 2692槛sjt 2693僭waqj 2694踺khvp 2695箭tue 2696江ia 2697茳aia 2698将uqf 2699姜ugv 2700豇gkua2701浆uqi 2702僵wgl 2703螀uqfj 2704缰xgl 2705鳉qguf 2706礓dgl 2707疆xfg 2708讲yfj 2709奖uqd 2710桨uqs2711蒋auq 2712耩diff 2713膙exk 2714匠ar 2715降bt 2716洚ita 2717绛xtah 2718弶xyiy 2719强xk 2720酱uqsg2721犟xkjh 2722糨ox 2723艽avb 2724交uq 2725郊uqb 2726茭auqu 2727峧muqy 2728浇iat 2729娇vtdj 2730姣vuq2731骄ctdj 2732胶eu 2733教ftbt 2734鵁uqqg 2735椒shi 2736蛟juq 2737焦wyo 2738跤khuq 2739僬wwyo 2740鲛qguq2741蕉awy 2742疁lwne 2743嶕mwyo 2744礁dwy 2745鹪wyog 2746矫tdtj 2747嚼kel 2748角qe 2749侥watq 2750佼wuq2751挢rtdj 2752狡qtu 2753饺qunq 2754绞xuq 2755铰quq 2756皎ruq 2757脚efcb 2758搅ripq 2759筊tuqu 2760湫itoy2761敫ryty 2762剿vjsj 2763儌wryt 2764徼try 2765皦rrty 2766叫kn 2767峤mtdj 2768觉ipmq 2769珓guq 2770校suq2771窌pwqb 2772窖pwtk 2773滘ipwk 2774斠fjgf 2775酵sgtb 2776嘦kswv 2777漖iftt 2778噍kwyo 2779噭kry 2780藠arrr2781醮sgwo 2782节ab 2783阶bwj 2784疖ubk 2785皆xxr 2786结xf 2787接ruv 2788秸rfkg 2789痎uynw 2790揭rjq2791喈kxxr 2792嗟kuda 2793街tffh 2794湝ixxr 2795楷sx 2796孑bnhg 2797讦yfh 2798劫fcl 2799劼fkln 2800杰so2801疌gvhi 2802诘yfk 2803拮rfk 2804洁ifk 2805桔sfk 2806倢wgvh 2807桀qahs 2808捷rgv 2809袺pufk 2810婕vgv2811絜dhvi 2812颉fkd 2813楬sjqn 2814睫hgv 2815蜐jfcl 2816截faw 2817榤sqas 2818碣djq 2819鲒qgfk 2820竭ujqn2821羯udjn 2822姐veg 2823毑xgub 2824解qev 2825檞sqeh 2826介wj 2827戒aak 2828玠gwjh 2829届nm 2830界lwj2831疥uwj 2832诫yaah 2833蚧jwj 2834借waj 2835骱mew 2836褯puyh 2837巾mhk 2838斤rtt 2839今wynb 2840紟xwyn2841金qqqq 2842津ivfh 2843衿puwn 2844矜cbtn 2845筋telb 2846禁ssf 2847襟pus 2848仅wcy 2849尽nyu 2850卺bigb2851紧jc 2852堇akgf 2853锦qrm 2854谨yak 2855馑qnag 2856廑yakg 2857瑾gakg 2858槿sak 2859进fj 2860近rp2861妗vwy 2862劲cal 2863荩anyu 2864浕inyu 2865晋gogj 2866赆mny 2867烬ony 2868浸ivp 2869琎gfjp 2870唫kqg2871祲pyvc 2872靳afr 2873搢rgoj 2874溍igoj 2875缙xgo 2876瑨ggoj 2877墐fakg 2878觐akgq 2879殣gqag 2880噤kssi2881茎aca 2882京yiu 2883泾ica 2884经x 2885荆aga 2886菁agef 2887猄qtyi 2888旌ytgg 2889惊nyiy 2890晶jjj2891腈egeg 2892鶄geqg 2893睛hg 2894粳ogj 2895兢dqd 2896精oge 2897鲸qgy 2898麖ynji 2899鼱vnue 2900井fjk2901阱bfj 2902汫ifjh 2903刭cajh 2904肼efj 2905景jy 2906儆waq 2907憬njy 2908璟gjyi 2909警aqky 2910径tca2911净uqv 2912弪xcag 2913胫eca 2914倞wyiy 2915痉uca 2916竞ukqb 2917竟ujq 2918婧vge 2919靓gem 2920敬aqkt2921靖uge 2922静geq 2923境fuj 2924獍qtuq 2925镜quj 2926坰fmkg 2927駉cmk 2928扃ynmk 2929冏mwkd 2930迥mkp2931泂imkg 2932絅xmk 2933炯omk 2934煚jano 2935颎xodm 2936窘pwvk 2937勼qvv 2938纠xnh 2939鸠vqyg 2940究pwv2941赳fhnh 2942阄uqj 2943揪rto 2944啾kto 2945樛snwe 2946鬏deto 2947九vt 2948久qy 2949氿ivn 2950玖gqy2951灸qyo 2952韭djdg 2953酒isgg 2954旧hj 2955臼vth 2956咎thk 2957疚uqy 2958柩saqy 2959桕svg 2960厩dvc2961救fiyt 2962就yi 2963舅vl 2964僦wyi 2965鹫yidg 2966且eg 2967苴aeg 2968拘rqk 2969狙qteg 2970泃iqkg2971居nd 2972驹cqk 2973挶rnnk 2974俱whw 2975罝legf 2976疽ueg 2977梮snnk 2978掬rqo 2979据rnd 2980琚gnd2981趄fhe 2982椐snd 2983跔khqk 2984锔qnnk 2985腒endg 2986雎egw 2987鮈qgqk 2988裾pund 2989鞠afq 2990鞫afqy2991局nnk 2992侷wnnk 2993菊aqo 2994焗onn 2995湨ihdy 2996 hdqg 2997跼khnk 2998橘scbk 2999弆fcaj 3000咀keg3001沮ieg 3002莒akkf 3003矩tda 3004举iwf 3005鉏qegg 3006椇shw 3007筥tkkf3008蒟auqk 3009榉siw 3010龃hwbg3011踽khty 3012巨and 3013句qkd 3014讵yang 3015苣aan 3016拒ran 3017具hw 3018炬oan 3019钜qan 3020秬tang3021倨wnd 3022粔oang 3023剧ndj 3024距kha 3025惧nhw 3026犋trhw 3027飓mqh 3028虡haow 3029锯qnd 3030聚bct3031窭pwo 3032踞khnd 3033屦ntov 3034遽hae 3035澽ihae 3036瞿hhwy 3037鐻qhae 3038醵sghe 3039捐rke 3040涓ike3041娟vke 3042圈lud 3043朘ecw 3044鹃keq 3045镌qwye 3046蠲uwlj 3047卷udbb 3048帣udmh 3049锩qudb 3050隽wyeb3051倦wud 3052狷qtke 3053桊uds 3054绢xke 3055鄄sfb 3056眷udhf 3057睊hkeg 3058罥lkef 3059撅rduw 3060噘kdu3061孓byi 3062决un 3063诀ynwy 3064抉rnwy 3065駃cnwy 3066玦gnw 3067珏ggy 3068砄dnwy 3069鴃qynw 3070鴂nwq3071绝xqc 3072倔wnb 3073桷sqe 3074掘rnbm 3075崛mnbm 3076觖qen 3077厥dubw 3078傕wpwy 3079劂dub 3080谲ycbk3091蕨adu 3082獗qtdw 3083橛sdu 3084噱khae 3085鐍qcbk 3086爵elv 3087蹶khdw 3088矍hhw 3089爝oel 3090攫rhh3091鑺qhhy 3092军pl 3093均fqu 3094君vtk 3095钧qqug 3096莙avtk 3097菌alt 3098皲plh 3099筠tfqu 3100鲪qgvk3101麇ynjt 3102俊wcw 3103郡vtkb 3104捃rvt 3105峻mcw 3106馂qnct 3107浚icwt 3108骏ccw 3109珺gvtk 3110焌ocw3111畯lcwt 3112竣ucw 3113寯pwyn 3114咔khhy 3115喀kpt 3116擖rajn 3117卡hhu 3118佧whh 3119胩ehh 3120开ga3121揩rxxr 3122锎quga 3123剀mnj 3124凯mnm 3125垲fmn 3126恺nmn 3127铠qmn 3128蒈axxr 3129慨nvc 3130锴qxx3131忾nyn 3132欬yntw 3133刊fjh 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ON THE AXIOMATISATION OF BOOLEAN CATEGORIES WITH AND WITHOUT MEDIAL
Theory and Applications of Categories,Vol.18,No.18,2007,pp.536–601.ON THE AXIOMATISATION OF BOOLEAN CATEGORIESWITH AND WITHOUT MEDIALLUTZ STRASSBURGERAbstract.The term“Boolean category”should be used for describing an object thatis to categories what a Boolean algebra is to posets.More specifically,a Boolean categoryshould provide the abstract algebraic structure underlying the proofs in Boolean Logic,in the same sense as a Cartesian closed category captures the proofs in intuitionisticlogic and a∗-autonomous category captures the proofs in linear logic.However,recentwork has shown that there is no canonical axiomatisation of a Boolean category.In thiswork,we will see a series(with increasing strength)of possible such axiomatisations,allbased on the notion of∗-autonomous category.We will particularly focus on the medialmap,which has its origin in an inference rule in KS,a cut-free deductive system forBoolean logic in the calculus of structures.Finally,we will present a category of proofnets as a particularly well-behaved example of a Boolean category.1.IntroductionThe questions“What is a proof?”and“When are two proofs the same?”are fundamental for proof theory.But for the most prominent logic,Boolean(or classical)propositional logic,we still have no satisfactory answers.This is not only embarrassing for proof theory itself,but also for computer science, where Boolean propositional logic plays a major role in automated reasoning and logic programming.Also the design and verification of hardware is based on Boolean logic. Every area in which proof search is employed can benefit from a better understanding of the concept of proof in Boolean logic,and the famous NP-versus-coNP problem can be reduced to the question whether there is a short(i.e.,polynomial size)proof for every Boolean tautology[CR79].Usually proofs are studied as syntactic objects within some deductive system(e.g., tableaux,sequent calculus,resolution,...).This paper takes the point of view that these syntactic objects(also known as proof trees)should be considered as concrete represen-tations of certain abstract proof objects,and that such an abstract proof object can be represented by a resolution proof tree and a sequent calculus proof tree,or even by several different sequent calculus proof trees.From this point of view the motivation for this work is to provide an abstract algebraic Received by the editors2006-06-12and,in revised form,2007-10-05.Transmitted by R.Blute.Published on2007-10-18.Reference corrected2007-12-13.2000Mathematics Subject Classification:03B05,03G05,03F03,18D15,18D35.Key words and phrases:Boolean category,*-autonomous category,proof theory,classical logic,proof nets.c Lutz Straßburger,2007.Permission to copy for private use granted.536ON THE AXIOMATISATION OF BOOLEAN CATEGORIES537 theory of proofs.Already Lambek[Lam68,Lam69]observed that such an algebraic treat-ment can be provided by category theory.For this,it is necessary to accept the following postulates about proofs:•for every proof f of conclusion B from hypothesis A(denoted by f:A→B)and every proof g of conclusion C from hypothesis B(denoted by g:B→C)there is a uniquely defined composite proof g◦f of conclusion C from hypothesis A(denoted by g◦f:A→C),•this composition of proofs is associative,•for each formula A there is an identity proof1A:A→A such that for f:A→B we have f◦1A=f=1B◦f.Under these assumptions the proofs are the arrows in a category whose objects are the formulae of the logic.What remains is to provide the right axioms for the“category of proofs”.It seems thatfinding these axioms is particularly difficult for the case of Boolean logic.For intuitionistic logic,Prawitz[Pra71]proposed the notion of proof normalization for identifying proofs.It was soon discovered that this notion of identity coincides with the notion of identity that results from the axioms of a Cartesian closed category(see, e.g.,[LS86]).In fact,one can say that the proofs of intuitionistic logic are the arrows in the free(bi-)Cartesian closed category generated by the set of propositional variables.An alternative way of representing the arrows in that category is via terms in the simply-typed λ-calculus:arrow composition is normalization of terms.This observation is well-known as the Curry-Howard-correspondence[How80].In the case of linear logic,the relation to∗-autonomous categories[Bar79]was noticed immediately after its discovery[Laf88,See89].In the sequent calculus linear logic proofs are identified when they can be transformed into each other via“trivial”rule permutations [Laf95].For multiplicative linear logic this coincides with the proof identifications induced by the axioms of a∗-autonomous category[Blu93,SL04].Therefore,we can safely say that a proof in multiplicative linear logic is an arrow in the free∗-autonomous category generated by the propositional variables[BCST96,LS06,Hug05a].But for classical logic no such well-accepted category of proofs exists.We can dis-tinguish two main reasons.First,if we start from a Cartesian closed category and add an involutive negation1,we get the collapse into a Boolean algebra,i.e.,any two proofs f,g:A→B are identified.For every formula there would be at most one proof(see, e.g.,[LS86,p.67]or the appendix of[Gir91]for details).Alternatively,starting from a ∗-autonomous category and adding natural transformations A→A∧A and A→t,i.e., the proofs for weakening and contraction,yields the same collapse.21i.e.,a natural isomorphism between A and the double-negation of A(in this paper denoted by¯A) 2Since we are dealing with Boolean logic,we will use the symbols∧and t for the tensor operation (usually )and the unit(usually1or I)in a∗-autonomous category.538LUTZ STRASSBURGERThe second reason is that cut elimination in the sequent calculus for classical logic is not confluent.Since cut elimination is the usual way of composing proofs,this means that there is no canonical way of composing two proofs,let alone associativity of composition.Consequently,for avoiding these two problems,we have to accept that(i)Cartesian closed categories do not provide an abstract algebraic axiomatisation for proofs in classical logic,and that(ii)the sequent calculus is not the right framework for investigating the identity of proofs in classical logic.There have already been several accounts for a proof theory for classical logic based on the axioms of Cartesian closed categories.Thefirst were probably Parigot’sλµ-calculus[Par92]and Girard’s LC[Gir91].The work on polarized proof nets by Lau-rent[Lau99,Lau03]shows that there is in fact not much difference between the two. Later,the category-theoretic axiomatisations underlying this proof theory has been in-vestigated and the close relationship to continuations[Thi97,SR98]has been established, culminating in Selinger’s control categories[Sel01].However,by sticking to the axioms of Cartesian closed categories,one has to sacrifice the perfect symmetry of Boolean logic.In this paper,we will go the opposite way.In the attempt of going from a Boolean algebra to a Boolean category we insist on keeping the symmetry between∧and∨.By doing this we have to leave the realm of Cartesian closed categories.That this is very well possible has recently been shown by several authors[DP04,FP04c,LS05a].However,the fact that all three proposals considerably differ from each other suggests that there might be no canonical way of giving a categorical axiomatisation for proofs in classical logic.We will provide a series of possible such axiomatisations with increasing strength. They will all build on the structure of a∗-autonomous category in which every object has a monoid(and a comonoid)structure.In this respect it will closely follow the work of [FP04c]and[LS05a],but will differ from[DP04].The approach that we take here is mainly motivated by the investigation in the com-plexity of proofs.Eventually,a good theory of proof identification should never identify two proofs if one is exponentially bigger than the other.The main proof-theoretic inspiration for this work comes from the system SKS[BT01], which is a deductive system for Boolean logic within the formalism of the calculus of structures[Gug07,GS01,BT01].A remarkable feature of the cut-free version of SKS, which is called KS,is that it can(cut-free)polynomially simulate not only sequent calculus and tableaux systems but also resolution and Frege-Hilbert systems[Gug04a,BG07].This means that if a tautology has a polynomial size proof in any of these systems,then it hasa cut-free polynomial size proof in KS.This ability of KS is a consequence of two features:1.Deep inference:Instead of decomposing the formulae along their root connectivesinto subformulae during the construction of a proof,in KS inference rules are applied deep inside formulae in the same way as we know it from term rewriting.2.The two inference rules switch and medial,which look as follows:F{(A∨B)∧C} sF{A∨(B∧C)}andF{(A∧B)∨(C∧D)}mF{(A∨C)∧(B∨D)},(1)ON THE AXIOMATISATION OF BOOLEAN CATEGORIES 539where F {}stands for an arbitrary (positive)formula context and A ,B ,C ,and D are formula variables.From deep inference to algebra.Deep inference allows us to establish the rela-tionship between proof theory and algebra in a much cleaner way than this is possible with shallow inference formalisms like the sequent calculus.The reason is that from a derivation in a deep inference formalism one can directly “read offthe morphisms”.Take for example the following derivation in system KS :(A ∧B )∨(C ∧D )r(A ∧B )∨(C ∧D )m (A ∨C )∧(B ∨D )(2)where A ,A ,B ,C ,and D are arbitrary formulae,and r is any inference rule taking A toA .In category-theoretic language this would be written as a composition of maps:(A ∧B )∨(C ∧D )(r ∧B )∨(C ∧D )(A ∧B )∨(C ∧D )m A,B,C,D (A ∨C )∧(B ∨D )where m A,B,C,D :(A ∧B )∨(C ∧D )→(A ∨C )∧(B ∨D )is called the medial map ,and r :A →A is the map corresponding to the rule r .System KS also allows the derivation(A ∧B )∨(C ∧D )m(A ∨C )∧(B ∨D )r (A ∨C )∧(B ∨D )(3)From the proof-theoretic point of view it makes perfect sense to identify the two deriva-tions in (2)and (3)because they do “essentially”the same.This is what Guglielmi calls bureaucracy of type B [Gug04c].In the language of category theory,the identification of(2)and (3)is saying that the diagram(A ∧B )∨(C ∧D )m A ,B,C,D (A ∨C )∧(B ∨D )(r ∨C )∧(B ∨D )(A ∧B )∨(C ∧D )(r ∧B )∨(C ∧D )(A ∨C )∧(B ∨D )m A,B,C,D (4)has to commute,which exactly means that the medial map has to be natural.For deep inference,Guglielmi also introduces the notion of bureaucracy of type A[Gug04b],which is the formal distinction between the derivationsA ∧B r 2A ∧B r 1A ∧B and A ∧B r 1A ∧B r 2A ∧B (5)540LUTZ STRASSBURGERwhere rule r1takes A to A,and rule r2takes B to B.Proof-theoretically,the two deriva-tions in(5)are“essentially”the same,so it makes sense to identify them.Translating this into category theory means to say that the operation∧is a bifunctor.However,it is not always the case that the demands of algebra and proof theory coincide so nicely.Sometimes they contradict each other,which causes“creative tensions”[LS06].One example is the treatment of units.Proof-theoretically it might be desirable to distinguish between the following two proofs in the sequent calculus(here t stands for “truth”and f for“falsum”):axiom(true)t weakening andt,f axiom(identity)t,f(6)This distinction is made,for example,by the proof nets presented in[LS05b].From the algebraic point of view,this causes certain difficulties:In[LS05a]the concept of weak units has been introduced in order to give a clean algebraic treatment to the distinction in (6).However,in this paper we will depart from this and use proper units instead.This is from the algebraic point of view more reasonable and simplifies the theory considerably. But it forces the identification of the two proofs in(6).Some remarks about switch and medial.The inference rule switch in(1),or the switch map s A,B,C:(A∨B)∧C→A∨(B∧C)has already been well investigated from the viewpoint of proof theory[Gug07],as well as from the viewpoint of category theory,where it is also called weak distributivity[HdP93,CS97b],linear distributivity,or dissociativity [DP04].On the other hand,the medial rule or medial map m A,B,C,D:(A∧B)∨(C∧D)→(A∨C)∧(B∨D)has not yet been so thoroughly investigated.Only very recently Lamarche[Lam07]started to study the consequences of the presence of the medial map in a∗-autonomous category,and Doˇs en and Petri´c[DP07]investigate it under the name intermutation from the viewpoint of coherence(but without taking the switch map into account).Seen from the deductive point of view,the two rules switch and medial have certain similarities:•switch allows the reduction of the identity rule and the cut rule to atomic form,and medial allows the reduction of the contraction rule(and the cocontraction rule)to atomic form(see[BT01]for details),•switch and medial are both self-dual,and•they look similar,as can bee seen in(1).In fact,recent work shows that they can both be seen as instance of a single more general inference rule[Gug02,Gug05]. However,from the algebraic point of view,they are quite different:Switch is a consequence of more primitive properties,namely the associativity of∧and∨and the de MorganON THE AXIOMATISATION OF BOOLEAN CATEGORIES541 duality between the two operations3,whereas medial has to be put as additional primitive, if we want it in the category.4Outline of the paper.In this work we will present a series of axioms that seem reasonable(from the proof-theoretic as well as from the algebraic points of view)to have in a Boolean category.While introducing axioms,we will also show their consequences. Some of the axioms presented here coincide with axioms given in the accompanying paper [Lam07]which has been written at the same time as this paper and appears in the same issue of this journal.This overlap is certainly not surprising.However,there are two main differences between the two papers.First,while[Lam07]works in the minimal setting of a∗-autonomous category with medial(or with“linear logic plus medial”),we assume from the beginning full classical propositional logic,i.e.,the presence of weakening and contraction.Second(and more importantly)we are staying in the realm of syntax, whereas[Lam07]is primarily concerned with the construction of concrete models for classical proofs.It is in fact a problem of the subject in general that there are only very few concrete examples of(symmetric)models of classical proofs.One of them is the category Rel of sets and relations[Hyl04],but it has the common problem that it identifies disjunction and conjunction.From the proof-theoretic point of view this kind of degenerate model is not very interesting.In fact,the investigation of medial is pointless in this setting.5Here the work in[Lam07]provides some breakthroughs towards new kind of models in which disjunction and conjunction do not coincide.In the end of this paper,we will also give a concrete example of a Boolean cate-gory,namely a variation of the proof nets of[LS05b].Although this example might be considered“only syntactic”,it nonetheless shows that the axioms presented here do not lead to the collapse into a Boolean algebra.Furthermore,this last section can be read independently by the reader interested only in proof nets and not in category theory.This paper is another attempt to be accessible to both the category theorist and the proof theorist.Since it is mainly about algebra,we use here the language of category theory.Nonetheless,the seasoned proof theorist mightfind it easier to understand if he substitutes everywhere“object”by“formula”and“map”/“morphism”/“arrow”by “proof”.Every commuting diagram in the paper is nothing but an equation between proofs written in a deep inference formalism.In order to make the paper easier accessible to proof theorists,all statements are proved in more detail than the seasoned category theorist mightfind appropriate.3Nonetheless it has been investigated in[CS97b]from the category theoretic viewpoint under the assumption that negation(and therefore the de Morgan duality)is absent.4This fact raises an open problem:can wefind simple primitives from which medial arises naturally, in the same way as switch arises naturally from associativity and duality?5For this reason,we will leave it as an exercise to the reader to verify that Rel fulfills all the axioms presented in this paper.542LUTZ STRASSBURGER2.What is a Boolean Category?Recall the analogy mentioned in the abstract:A Boolean category should be for categories, what a Boolean algebra is for posets.This leads to the following definition:2.1.Definition.We say a category C is a B0-category if there is a Boolean algebra B and a mapping F:C→B from objects of C to elements of B,such that for all objects A and B in C,we have F(A)≤F(B)in B if and only if there is an arrow f:A→B in C.In other words,a B0-category is a category whose image under the forgetful functor from the category of categories to the category of posets is a Boolean algebra.From the proof-theoretic point of view one should have that there is a proof from A to B if and only if A⇒B is a valid implication.However,from the algebraic point of view there are many models,including the category Rel of sets and relations,as well as the models constructed in the in the accompanying paper[Lam07],which have a map between any two objects A and B.Note that these models are not ruled out by Definition2.1because there is the trivial one-element Boolean algebra.In any case,we can make the following (trivial)observation.2.2.Observation.In a B0-category,we can for any pair of objects A and B,provide objects A∧B and A∨B and¯A,and there are objects t and f,such that there are maps ˆαA,B,C:A∧(B∧C)→(A∧B)∧CˇαA,B,C:A∨(B∨C)→(A∨B)∨C ˆσA,B:A∧B→B∧AˇσA,B:A∨B→B∨Aˆ A:A∧t→Aˇ A:A∨f→Aˆλ:t∧A→AˇλA:f∨A→AAˆıA:A∧¯A→fˇıA:t→¯A∨A(7)s A,B,C:(A∨B)∧C→A∨(B∧C)m A,B,C,D:(A∧B)∨(C∧D)→(A∨C)∧(B∨D)∆A:A→A∧A∇A:A∨A→AΠA:A→t A:f→Afor all objects A,B,and C.This can easily be shown by verifying that all of them correspond to valid implications in Boolean logic.Conversely,a category in which every arrow can be given as a composite of the ones given above by using only the operations of∧,∨,and the usual arrow composition,is a B0-category.This is a consequence of the completeness of system SKS[BT01],which is a deep inference deductive system for Boolean logic incorporating the maps in(7)as inference rules.ON THE AXIOMATISATION OF BOOLEAN CATEGORIES543 Note that Definition2.1is neither enlightening nor useful.It is necessary to add some additional structure in order to obtain a“nicely behaved”theory of Boolean categories. However,as already mentioned in the introduction,the naive approach of adding struc-ture,namely adding the structure of a bi-Cartesian closed category(also called Heyting category)with an involutive negation leads to collapse:Every Boolean category in that strong sense is a Boolean algebra.The hom-sets are either singletons or empty.This ob-servation hasfirst been made by Andr´e Joyal,and the proof can be found,for example,in [LS86],page67.For the sake of completeness,we repeat the argument here:First,recall that in a Cartesian closed category,we have,among other properties,(i)binary products, that we(following the notation of this paper)denote by∧,(ii)a terminal object t with the property that t∧A∼=A for all objects A,and(iii)a natural bijection between the maps f:A∧B→C and f∗:A→B⇒C,where B⇒C denotes the exponential of B and C.Going from f to f∗is also known as currying.Adding an involutive negation means adding a contravariant endofunctor(−)such that there is a natural bijection be-tween maps f:A→B and¯f:¯B→¯A.It also means that there is an initial object f=¯t. Hence,we have in particular for all objects A and B,thatHom(A,B)∼=Hom(t∧A,B)∼=Hom(t,A⇒B)∼=Hom(A⇒B,f).(8) Now observe that whenever we have an object X such that the two projectionsπ1,π2:X∧X→Xare equal,then for all objects Y,any two maps f,g:Y→X are equal,becausef=π1◦ f,g =π2◦ f,g =g.(9) Now note that since f is initial,there is exactly one map f→f⇒f,hence,by uncurrying there is exactly one map f∧f→f.Therefore,for every Y,there is at most one map Y→f.By(8),for all A,B,we have Hom(A,B)is either singleton or empty.Recapitulating the situation,we have here two extremes of Boolean categories:no structure and too much structure.Neither of them is very interesting,neither for proof theory nor for category theory.But there is a whole universe between the two,which we will start to investigate now.On our path,we will stick to(8)and carefully avoid to have (9).This is what makes our approach different from control categories[Sel01],in which the equation f=π1◦ f,g holds,but the rightmost bijection in(8)is absent.3.∗-Autonomous categoriesLet us stress the fact that in a plain B0-category there is no relation between the maps listed in(7).In particular,there is no functoriality of∨and∧,no naturality ofˆα,ˆσ,..., and no de Morgan duality.Adding this structure means exactly adding the structure of a∗-autonomous category[Bar79].Since we are working in classical logic,we will here use the symbols∧,∨,t,f for the usual , ,1,⊥.544LUTZ STRASSBURGER3.1.Definition.A B0-category C is symmetric ∧-monoidal if the operation −∧−:C ×C →C is a bifunctor and the maps ˆαA,B,C ,ˆσA,B ,ˆ A ,ˆλA in (7)are natural isomorphisms that obey the following equations:A ∧(B ∧(C ∧D ))A ∧ˆαB,C,D A ∧((B ∧C )∧D )ˆαA,B ∧C,D(A ∧B )∧(C ∧D )ˆαA,B,C ∧D (A ∧(B ∧C ))∧D ˆαA,B,C ∧D n n n n n n n n n n n n n ((A ∧B )∧C )∧DˆαA ∧B,C,DA ∧(B ∧C )ˆαA,B,C A ∧(C ∧B )A ∧ˆσB,C (A ∧B )∧CˆσA ∧B,C (A ∧C )∧B ˆαA,C,BC ∧(A ∧B )ˆαC,A,B (C ∧A )∧B ˆσA,C ∧BA ∧(t ∧B )ˆαA,t ,B (A ∧t )∧B ˆ A ∧B q q q q q q q q q q q A ∧BA ∧ˆλB w w w w w w w w w w w A ∧B ˆσA,B B ∧A ˆσB,A tt t t t t t t t t A ∧B 1A ∧B t t t t t t t t t tThe notion of symmetric ∨-monoidal is defined in a similar way.An important property of symmetric monoidal categories is the coherence theorem[Mac63],which says that every diagram containing only natural isomorphisms built out of ˆα,ˆσ,ˆ ,ˆλ,and the identity 1via ∧and ◦must commute (for details,see [Mac71]and [Kel64]).6As a consequence of the coherence theorem,we can omit certain parentheses to ease the reading.For example,we will write A ∧B ∧C ∧D for (A ∧B )∧(C ∧D )as well as for A ∧((B ∧C )∧D ).This can be done because there is a uniquely defined “coherence isomorphism”between any two of these objects.6In [Kel64],Kelly provides some simplifications to MacLane’s conditions in [Mac63].For example,the equations ˆ t =ˆλt :t ∧t →t and ˆ A ◦ˆσt ,A =ˆλA :t ∧A →A follow from the ones in Definition 3.1.ON THE AXIOMATISATION OF BOOLEAN CATEGORIES545 Let us now turn our attention to a very important feature of Boolean logic:the duality between∧and∨.We can safely say that it is reasonable to ask for this duality also in a Boolean category.That means,we are asking for¯A∼=A and A∧B∼=¯A∨¯B.At the same time we ask for the possibility of transposition(or currying):The proofs of A∧B→C are in one-to-one correspondence with the proofs of A→¯B∨C.This is exactly what makes a monoidal category∗-autonomous.3.2.Definition.A B0-category C is∗-autonomous if it is symmetric∧-monoidal and is equipped with a contravariant functor(−):C→C,such that(−):C→C is a natural isomorphism and such that for any three objects A,B,C there is a natural bijectionHom C(A∧B,C)∼=Hom C(A,¯B∨C).( ) where the bifunctor−∨−is defined via A∨B=¯B∧¯A.7We also define f=¯t.Clearly,if a B0-category C is∗-autonomous,then it is also∨-monoidal withˇαA,B,C=ˆα¯C,¯B,¯A,ˇσA,B=ˆσ¯B,¯A,ˇ A=ˆλ¯A,ˇλA=ˆ ¯A.Note that our definition is not the original one,but it is not difficult to show the equivalence,and this was already done in[Bar79].For further information,see also [BW99,Bar91,Hug05a,LS06].Let us continue with stating some well-known facts about∗-autonomous categories(for proofs of these facts,see e.g.[LS06]).Via the bijection( )we can assign to every map f:A→B∨C a map g:A∧¯B→C,and vice versa.We say that f and g are transposes of each other if they determine each other via( ).We will use the term“transpose”in a very general sense:given objects A,B,C,D,E such that D∼=A∧B and E∼=¯B∨C, then any f:D→C uniquely determines a g:A→E,and vice versa.Also in that general case we will say that f and g are transposes of each other.For example,ˆλA:t∧A→A andˇ A:A→A∨f are transposes of each other,and another way of transposing them yields the mapsˇıA:t→¯A∨A andˆıA:A∧¯A→f.If we have f:A→B∨C and b:B →B,thenA∧B A∧b A∧B f C is transpose of A g¯B∨C¯b∨C B ∨C(10) where g is transpose of f.Let us now transpose the identity1B∨C:B∨C→B∨C.This yields the evaluation map eval:(B∨C)∧¯C→B.Taking the∧of this with1A:A→A and transposing back determines a map s A,B,C:A∧(B∨C)→(A∧B)∨C that is natural in all three arguments, and that we call the switch map[Gug07,BT01]8.In a similar fashion we obtain maps 7Although we live in the commutative world,we invert the order of the arguments when taking the negation.8To category theorists it is probably better known under the names weak distributivity[HdP93,CS97b] or linear distributivity.However,strictly speaking,it is not a form of distributivity.An alternative is the name dissociativity[DP04].(A ∨B )∧C →A ∨(B ∧C )and A ∧(B ∨C )→B ∨(A ∧C )and (A ∨B )∧C →(A ∧C )∨B .Alternatively these maps can be obtained from s by composing with ˆσand ˇσ.For this reason we will use the term “switch”for all of them,and denote them by s A,B,C if it is clear from context which one is meant,as for example in the two diagrams(A ∨B )∧(C ∨D )s A,B,C ∨DA ∨(B ∧(C ∨D ))A ∨s B,C,D((A ∨B )∧C )∨Ds A ∨B,C,DA ∨(B ∧C )∨Ds A,B,C ∨D (11)andA ∧(B ∨C )∧DA ∧s B,C,DA ∧(B ∨(C ∧D ))s A,B,C ∧D((A ∧B )∨C )∧Ds A,B,C ∧D(A ∧B )∨(C ∧D )s A ∧B,C,D(12)which commute in any ∗-autonomous category.Sometimes we will denote the map defined by (11)by ˆt A,B,C,D :(A ∨B )∧(C ∨D )→A ∨(B ∧C )∨D ,called the tensor map 9and the one of (12)by ˇt A,B,C,D :A ∧(B ∨C )∧D →(A ∧B )∨(C ∧D ),called the cotensor map .Note that the switch map is self-dual,while the two maps ˆt and ˇt are dual to each other,i.e.,(A ∧B )∨Cs A,B,CA ∧(B ∨C )∼=¯C∧(¯B ∨¯A )∼=(¯C ∧¯B )∨¯A s ¯C,¯B,¯A (13)and(A ∧B )∨(C ∧D )ˇt A,B,C,DA ∧(B ∨C )∧D∼=(¯D∨¯C )∧(¯B ∨¯A )∼=¯D ∨(¯C ∧¯B )∨¯A ˆt ¯D,¯C,¯B,¯A (14)where the vertical maps are the canonical isomorphisms determined by the ∗-autonomousstructure.Another property of switch that we will use later is the commutativity of the following diagrams:(A ∨B )∧ts A,B,tA ∨B ˆ A ∨B A ∨(B ∧t )A ∨ˆBk k k k k k k k (f ∨A )∧Bs f ,A,BA ∧B ˇλ−1A∧B k k kk kk k k f ∨(A ∧B )ˇλ−1A ∨B(15)9This map describes precisely the tensor rule in the sequent system for linear logic.4.Some remarks on mixIn this section we will recall what it means for a ∗-autonomous category to have mix.Although most of the material of this section can also be found in [CS97a],[FP04a],[DP04],and [Lam07],we give here a complete survey since the main result,Corollary 4.3,is rather crucial for the following sections.This corollary essentially says that the mix-rule in the sequent calculusΓ ∆mixΓ,∆is a consequence of the fact that false implies true.Although this is not a very deep result,it might be surprising for logicians that a property of sequents (if two sequents can be proved independently,then they can be proved together)which does not involve any units comes out of an algebraic property concerning only the units.4.1.Theorem.Let C be a ∗-autonomous category and e :f →t be a map in C .Thenf ∧fe ∧ft ∧f ˆλf f ∧tf ∧efˆ f(16)if and only iftˇλ−1tf ∨te ∨tt ∨fˇ −1tt ∨tt ∨e(17)if and only ifA ∧BA ∧ˇλ−1BA ∧(f ∨B )s A,f ,B(A ∧f )∨B(A ∧e )∨B(A ∨f )∧Bˇ −1A ∧B(A ∧t )∨Bˆ A ∨BA ∨(f ∧B )s A,f ,BA ∨(t ∧B )A ∨(e ∧B )A ∨BA ∨ˆλB (18)for all objects A and B .。
DIN5480花键孔及外花键标准
DIN5480花键孔及外花键标准分析与应用DIN5480花键标准是德国于1986年颁布实施的米制模数变位制花键标准；该标准于2005年及2006年做了修订，标准号为DIN5480-1及DIN5480-2，这两个新标准各包含不同的内容，共同构成新的标准。
与旧标准相比，新标准更为简明、实用（如取消了旧标准中内花键公法线及偏差的计算），新标准还取消了37.5o及45o压力角花键的内容，只保留了30o 压力角的花键规格。
白80年代以来，我国大规模引进了德国汽车及液压产品技术，DIN5480花键在我国已被广泛使用，除了采用定型刀具（主要是拉刀）大批量生产定型产品外，采用通用加工手段少量配制DIN花键的情况也日益增多。
与国标GB/T3478花键标准相比，DIN标准主要有三处差别：其一是模数系列较国标模数多了m0.6及m0.8两个规格；其二是精度级别，DIIN5480 规定了7、8、9、10、11 计5 个级别，新标准DIN5480-1 则规定了5、6、7、8、9、10、11、12计8个级别，其主要差别在于DIN 9 级精度相当于GB的5级精度，DIN 10级相当于GB 6级，其余类推（线切割的制齿精度一般为DIN 9级）；其三，DIN5480全是变位键，其外花键大多采用正变位，相配的内花键为负变位，少量外花键为负变位，与之相配的内花键则为正变位，同一规格花键的变位系数相同仅符号相反，即同一花键副的总变位系数为0,由此决定了一套内、外花键的分度圆在变位前、后均相同且重合；国标花键则全是非变位键。
依据齿轮（含花键）变位加工原理可知，采用标准模数的花键滚刀可直接滚切出DIN5480的变位外花键，而内花键在小批量加工时则只能采用插床（而不是插齿机）单刀插齿或数控线切割制齿；当采用单刀插制内花键时，也要先由线切割制出刀形模板，若直接以外花键做母板配磨刀具则齿形精度及侧隙配合精度均无法保证。
当采用线切割制齿或制刀形样板时，则首先要在计算机上绘制全齿花键图。
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一、判断题1.>当三相负载越接近对称时，中性线中的电流就越小。
( )答案：√2.>在纯电感单相交流电路中，电压超前电流90°相位角；在纯电容单相交流电路中，电压滞后电流90°相位角。
( )答案：√3.>空气断路器和交流接触器均能长时间过负荷运行。
( )答案：×4.>10kV系统一相接地后，非接地相的对地电压为线电压。
( )答案：√5.>电介质在电场作用下的物理现象主要有极化、电导、损耗和击穿。
( )答案：√6.>吸收比是判断电缆好坏的一个主要因素，吸收比越大电缆绝缘越好。
( )答案：√7.>按运行需要，测量敷设后的电缆的电气参数主要有：电容、交直流电阻及阻抗。
( )答案：√8.>绝缘材料的电阻随温度的升高而升高，金属导体的电阻随温度的升高而降低。
( )答案：×9.>单芯交流电缆的护层不可采用钢铠，应采用非磁性材料。
( )答案：√10.>110kV以上运行中的电缆其试验电压为5倍额定电压。
( )答案：×11.>我国目前生产的最高电压等级电缆为500kV。
( )答案：√12.>电缆在直流电压与交流电压作用下的绝缘相同。
( )答案：×13.>电缆的绝缘结构与电压等级有关，一般电压等级越高，绝缘越厚，但不成正比。
( )答案：√14.>电缆在恒定条件下，其输送容量一般是根据它的最高工作温度来确定的。
( )答案：√15.>由于铝的导电系数较铜为高，在同样的长度和电阻下，铝制导体的截面积约为铜的1.65倍。
( )答案：×16.>电缆二芯接地故障时，允许利用一芯的自身电容作声测试验。
( )答案：×17.>聚乙烯的绝缘性能不比聚氯乙烯好。
( )答案：×18.>电缆试验地点周围应设围栏，以防无关人员接近。
( )答案：√二、选择题1.>为防止电缆相互间的粘合及施工人员粘手，常在电缆皮上涂( )粉。
(A)石英；(B)白灰；(C)白垩；(D)石膏。
答案：C2.>在三相系统中，( )将三芯电缆中的一芯接地运行。
(A)不得；(B)可以；(C)应；(D)不应。
答案：A3.>对称的三相电源星形连接时，相电压是线电压的( )倍。
(A)1；(B)2；；答案：C4.>叠加原理、欧姆定律分别只适用于( )电路。
(A)线性、非线性；(B)线性线性；(C)非线性、线性；(D)线性、非线性。
答案：B5.>当电源频率增加后，图A-2中的( )亮度会增加。
图A-2(A)A灯；(B)B灯；(C)C灯；(D)D灯。
答案：A6.>三相对称负载的功率cosPϕ=，其中ϕ角是( )的相位角。
(A)线电压与线电流之间；(B)相电压与对应相电流之间；(C)线电压与相电流之间；(D)相电压与线电流之间。
答案：B7.>线圈中自感电动势的方向是( )。
(A)与原电流方向相反；(B)与原电流方向相同；(C)阻止原磁通的变化；(D)加强原磁通的变化。
答案：C8.>电缆线芯的功能主要是输送电流，线芯的损耗是由( )来决定。
(A)导体截面和电导系数；(B)电压高低；(C)电阻系数；(D)温度系数。
答案：A9.>当应力锥长度固定后，附加绝缘加大，会使轴向应力( )。
(A)减少；(B)增大；(C)略有减少；(D)略有增大。
答案：B10.>单芯电缆的铅皮只在一端接地时，在铅皮另一端上的正常感应电压一般不应超过( )V。
(A)50；(B)55；(C)60；(D)65。
答案：C11.>自容式充油电缆外护套和接头外护套的直流耐压值为()。
(A)6kV;(B)10kV;(C)25kV;(D)35kV。
答案：A12.>堤坝上的电缆敷设，其设置要求( )。
(A)与直埋电缆相同；(B)与沟内敷设相同；(C)与隧道敷设相同；(D)有特殊规定。
答案：A13.>半导体屏蔽纸除了起均匀电场作用外，也可以起( )作用。
(A)改善老化；(B)改善绝缘；(C)改善温度；(D)改善老化和绝缘性能。
答案：D14.>110kV电缆终端头出线应保持固定位置，其带电裸露部分至接地部分的距离应不少于( )。
(A)0.5～0.9m；(B)0.9～1.0m；(C)1.0～1.5m；(D)1.5～2.0m。
答案：B15.>电缆事故报告中，事故中止的时间指( )时间。
(A)汇报时间；(B)接头完毕；(C)试验完毕；(D)施工完毕。
答案：16.>低油压充油电缆的长期允许油压( )kg。
(A)0.5～3；(B)1～4；(C)2～5；(D)2.5～4。
答案：A17.>电缆线路的正常工作电压一般不应超过电缆额定电压的( )。
(A)5%；(B)10%；(C)15%；(D)20%。
答案：C18.>110kV的电缆进线段，要求在电缆与架空线的连接处装设( )。
(A)放电间隙；(B)管型避雷器；(C)阀型避雷器；(D)管型或阀型避雷器。
答案：C三、计算题1.>铜导线长度L＝100m，截面积S＝0.1mm2，温度T2＝50℃时求导线电阻R2?已知在T2＝50℃时铜导线电阻温度系数为α＝0.0041(1/℃)答案：解：先求常温下(T1＝20℃)的电阻R1R1＝×L/S＝0.0172×100/0.1＝17.2Ω(1分)R2＝R1［1＋α(T2－T1)］＝17.2×［1＋0.0041×(50－20)］＝17.2×1.1＝19.35Ω(3分)答：温度T2＝50℃时导线电阻为19.35Ω。
(1分)2.>如图D-3所示，C1＝0.2μF，C2＝0.3μF，C3＝0.8μF，C4＝0.2μF，求开关K断开与闭合时，AB两点间等效电容C AB。
图D-3答案：解：(1)K断开时，C1、C2串联，C3、C4串联，然后两者并联，则：C AB＝C1×C2/(C1＋C2)＋C3×C4/(C3＋C4)＝0.2×0.3/(0.2＋0.3)＋0.8×0.2/(0.8＋0.2)＝0.12＋0.16＝0.28μF (2分)(2)K闭合时，C1、C3并联，C2、C4并联然后两者串联，C AB＝(C1＋C3)(C2＋C4)/(C1＋C3＋C2＋C4)＝(0.2＋0.8)(0.3＋0.2)/(0.2＋0.8＋0.3＋0.2)＝0.33μF (2分)答：开关K断开与闭合时，AB两点间等效电容分别为0.28μF和0.33μF。
(1分) 3.>一条充油电缆线路两端落差为12m，问内部的电缆油产生油柱静压力为多少?答案：解：ρ＝0.1Hd(1分)低粘度矿物油或烷基苯合成油的密度为0.9g/cm3(1分)ρ＝0.1×1200×0.9×10－3＝0.108kg/cm2(2分)答：12m落差产生的电缆内静油压为0.108kg/cm2。
(1分)4.>进行一条电缆水下敷设，水深8m，在入电缆在水中的质量为150kg，入水角控制在45℃，求敷设张力多少?答案：解：()1cosT WDθ=-(2分)T＝150kg×8m×(1－cos45°)＝150×8(1－0.707)＝351.6kgm(2分)351.6×9.8＝3446(N)答：电缆敷设时的张力为3446N。
(1分)5.>设一条10kV纸绝缘电缆，在周围土壤温度15℃时，电流为325(A)，问当土壤温度30℃时，其载流量为多少?又后者为前者的几倍?答案：解：12II=(2分)20.82=266.5I I=⨯(A)(2分)答：该电缆30℃时载流量266.5A，后者为前者的82%。
(1分)6.>一条400mm2，35kVXLPE绝缘单芯电缆，其线芯屏蔽层外径为11.7mm，问其绝缘厚度为多少?绝缘层外半径为多少?答案：解：公式phm1231UK K KG∆=(1分)取phm 1.5U=⨯K1＝1.1 K2＝4 K3＝1.1根据公式：123213ILK K KG∆=(1分)G1取10kV/mm代入得51.14 1.1=10.23∆⨯⨯⨯(1分)R＝Y c＋Δ＝11.7＋10.23＝21.93mm (1分)答：绝缘厚度以工频电压下计Δ＝10.23mm，绝缘层外半径为21.93mm。
(1分)四、简答题1.>配电装置包括哪些设备?答案：用来接受和分配电能的电气设备称为配电装置。
(2分)包括控制电器(断路器、隔离开关、负荷开关)，保护电器(熔断器、继电器及避雷器等)，测量电器(电流互感器、电压互感器、电流表、电压表等)以及母线和载流导体。
(4分)2.>为什么互感器二次回路必须有可靠接地点?答案：因为互感器的一次绕组在高电压状态下运行，为了确保人身防护和电气设备安全；(2分)防止一、二次绕组间绝缘损坏进，一次侧电路中的高压加在测量仪表或继电器上，危及工作人员和设备的安全，所以必须有可靠接地点。
(4分)3.>振荡波试验方法有哪些优点?答案：(1)能有效地检测缺陷。
(1.5分)(2)与50Hz试验结果相一致。
(1.5分)(3)设备简单、便宜。
(1.5分)(4)没有电压限制。
(1.5分)4.>ZRA(C)-YJV-8/10kV 电缆表示它是什么结构?主要用于什么地方?答案：铜芯交联聚乙烯绝缘聚乙烯护套阻燃电力电缆。
(3分)用于核电站、地铁、通信站、高层建筑、石油冶金、发电厂、军事设施、隧道等地方。
(答出其中5项即可得3分)5.>假设一条大截面电缆与两根小截面电缆并联运行，同样能满足用户需要，忽略经济因素，您选用哪种方式?为什么?答案：选用单根大截面电缆。
(2分)由于电缆的导体电阻很小(约0.05Ω/km)，两根电缆并联运行时，如其中一个导体与母排发生松动，则接触电阻可能达到或超过导体电阻，那么流过与之并联的另一个导体的电流将成倍增加，从而导致热击穿。
(4分)6.>电缆的路径选择，应符合哪些规定?答案：(1)避免电缆遭受机械性外力、过热、腐蚀等危害。
(1.5分)(2)满足安全要求条件下使电缆较短。
(1分)(3)便于敷设、维护。
(1分)(4)避开将要挖掘施工的地方。
(1分)(5)充油电缆线路通过起伏地形时，使供油装置较合理配置。
(1.5分)五、绘图题1.>画出电动势e=40sin(ωt+60°)V的波形图。
答案：如图E-60。
图E-602.>如图E-61所示，分别画出(a)、(b)相应简化的等效电路图。
图E-61答案：如图E-62所示。
图E-623.>如图E-63，标出35kV交联聚乙烯中间接头结构的各部分名称及尺寸数据。