"` ؄  L)ZAddOrder(Abstract'Integer'Numbers#Coq@|)Decidable%Logic#Coq@2Relation_Operators)Relations#Coq@4Operators_Properties)Relations#Coq@)Relations#Coq@*Equalities*Structures#Coq@&Orders*Structures#Coq@4Relation_Definitions)Relations#Coq@-SetoidTactics'Classes#Coq@&Setoid'Setoids#Coq@$Init'Classes#Coq@/RelationClasses'Classes#Coq@)Morphisms'Classes#Coq@.Morphisms_Prop'Classes#Coq@*NumPrelude'Numbers#Coq@)Notations$Init#Coq@%Logic$Init#Coq@*Logic_Type$Init#Coq@)Datatypes$Init#Coq@&Specif$Init#Coq@%Peano$Init#Coq@"Wf$Init#Coq@'Tactics$Init#Coq@%Tauto$Init#Coq@'Prelude$Init#Coq@-GenericMinMax*Structures#Coq@(NZAxioms&NatInt'Numbers#Coq@'ZAxioms(Abstract'Integer'Numbers#Coq@%ZBase(Abstract'Integer'Numbers#Coq@$ZAdd(Abstract'Integer'Numbers#Coq@$ZMul(Abstract'Integer'Numbers#Coq@#ZLt(Abstract'Integer'Numbers#Coq@)Notations$Init#Coq@0&v!D]hwnv %Logic$Init#Coq@0\͉!Ig)Datatypes$Init#Coq@0.i bYN Z*Logic_Type$Init#Coq@0 1jc6&Specif$Init#Coq@0;RWMi\N'Decimal$Init#Coq@0C涳N*ua#Nat$Init#Coq@0eʤģPSR蠠%Peano$Init#Coq@0 jha|ؠ"Wf$Init#Coq@0q+W,J+'Tactics$Init#Coq@0/9m+ a%Tauto$Init#Coq@0̂"&/r'Prelude$Init#Coq@0JqTttֱ)Decidable%Logic#Coq@0ND걸풬/Oߠ&Basics'Program#Coq@0!bs߯? :VU$Init'Classes#Coq@0](p{yOh.'Tactics'Program#Coq@03u%+Equivalence'Classes#Coq@07;ꮹ-SetoidTactics'Classes#Coq@0S_`nOU$&Setoid'Setoids#Coq@0D9AsWE!>*Equalities*Structures#Coq@0όe얟)H.Ƞ2Relation_Operators)Relations#Coq@0%s鯰s4Operators_Properties)Relations#Coq@0U3y#h&)Relations#Coq@0r砠*NZMulOrder&NatInt'Numbers#Coq@0}\^ !"k}@R(NZParity&NatInt'Numbers#Coq@0H>ca'^^%NZPow&NatInt'Numbers#Coq@0)6*9 B:vȻ&NZSqrt&NatInt'Numbers#Coq@0` .%m%NZLog&NatInt'Numbers#Coq@0ꔉ .uV%NZGcd&NatInt'Numbers#Coq@0KgT7|%NZDiv&NatInt'Numbers#Coq@0$ |J?d (w&NZBits&NatInt'Numbers#Coq@0MlIpKt'ZAxioms(Abstract'Integer'Numbers#Coq@0\}zK=PK,NZProperties&NatInt'Numbers#Coq@01D%E`|3x%ZBase(Abstract'Integer'Numbers#Coq@0 TL}2_} ؠ$ZAdd(Abstract'Integer'Numbers#Coq@0Y9Vgⲫ򐠠$ZMul(Abstract'Integer'Numbers#Coq@0Mtr\b"H0xK|j2[&=}I =}H;jjnfYaР)ZAddOrder(Abstract'Integer'Numbers#Coq@ A-ZAddOrderPropA@!Z@!t @J %'A*Equalities*Structures#Coq@@@@@ %'@ @A@A@@@AA@@A@"eq @^@)@%*@.@**A@@@@@(eq_equiv @@/RelationClasses'Classes1@+Equivalence~@J@F*N@/ަ@@@FA݋X_A/RelationClasses'Classes#Coq@@A@@A$zero @Tf@!t*@@@@@AA@@A@$succ @r@y@*}@*@@@@@$pred @d@@&*@**@@@@@)'succ_wd @@)Morphisms'Classes#Coq@@&ProperL@@F*@J*@*respectful%WO?@U*@Y*@"eqަ@ަ@]ь@@@AviA4'Classes#Coq@@A@@n'pred_wd @@E@@*@*9@*@*@4ަ@8ަ@}8+@@@A3@A@@)pred_succ @@!n@*@Sަ@8+"@ьA@@@@@,bi_induction @@!A@5@*A@@?@* E@*I@ަ%Logic$Init@#iffС)0@B[@2ƀ@Ra@*DCl@ьJbq@ *EQ@@@kA@A@@#one @T@*@@@@@#two @T@+*@@@@@*(one_succ @@@ަ@*{̀@:ь@Q2ƀ@@@@@G(two_succ @@@ަ@9Ń@Wь@Q{̀@@@@@d"lt @h@@!t*@@*A@@@@@AA@@A@"le @h@@*@@*@@@@@%lt_wd @@i@ @*@@*٩^@*@@*䚠@Zަm#@*'@dަޚ+@"lt.@@@$A`@A@@Ǡ+lt_eq_cases @@-<@*!mC@*J@`(@"or @X@-.]@ަ@@@@@)lt_irrefl @@\k@*#@#notШw@L.UU@@@@@)lt_succ_r @@w@ *J@%*C@g.>@)ьv@S@@@@@6#add @d@@F*@@K*@O*@@@@@N#sub @d@@^*@@c*@g*@<@@e#mul @d@@u*@@z*@~*@S@@|&add_wd @@S@@*@@*@*L@*@@* @*@Lަ_@*@*@Zަ!@^ަ%@gO@@@AY@A@@&sub_wd @@@7@*@<@*@@*F@*@K@*O@*S@ަY@*]@*a@ަe@ަi@Ł@@@aA@A@@&mul_wd @@۰@{@*@@*@*@$*@@)*@-*@ަ@7*@;*@ަ@ަ@y@@@A@A@@H'add_0_l @@@W*@ަ@&gO@p2ƀ@@@@@f*add_succ_l @@̚@u*@z*@"ަ@IgO@ьϩ@ь@[gO@@@@@'sub_0_r @@ @*@Mަ@\Ł@2ƀ@@@@@*sub_succ_r @@)@*횠.@*3@pަ9@Ł婚?@ьD@8+J@Ł@@@@@㠠'mul_0_l @@IX@*]@ަc@yh@ 2ƀEl@2ƀ@@@@@*mul_succ_l @@kz@*>@*@ަ@y@!ьm@gO@yQx@@@@@4#max @d@@!t*@@*@ *@@@@@AA@@A@%max_l @@!x@*!y@$*@@"leAB@"eqަ@DxC@@@@@2%max_r @@/@J*-@O*@@+)*@(ަ @kx'4@@@@@W#min @d@@p*@@u*#@y*@@@@@o%min_l @@l1@*j6@*@<@h=A@dަG@7xcb@@@@@%min_r @@U@*Z@*@`@e@ަk@[x@@@@@)succ_pred @@!n{@*@"eqަ@ь@ 8+A@@@@@AA@@A@#opp @d@@;*@?*@@@@@&opp_wd @@)Morphisms'Classes#Coq@@&ProperL@@[*@_*@*respectful%WO?@j*@n*@Xަ@\ަ@I{@@@ %'A*Equalities*StructuresHAviA9'Classes#Coq@@A@@`%opp_0 @@@}ަ@@@s{ @$zero2ƀ@@@@@(opp_succ @@@*#@ަ)@{/@ь4@8+:@{@@@@@@'ZAxioms(Abstract'Integer'Numbers#Coq@/ZAxiomsMiniSig'@@@@*@U@Q*J@@A@Aަ@X@9ަ^@@A@@\@rh@@A@AB.@_@.h@@A@gO@d@gOd@@A@Ax@g@xd@@A@x@k@[xd@@A@ABCy@n@yd@@A@{@t@{̠T@@A@A{@w@{d@@A@Ł@{@Łd@@A@ABŃ@~@ŃT@@A@8+@@8+d@@A@ь@@ьr@@A@2@@-2ƠT@@A@ABCDE@'eq_refl @/RelationClasses'Classes#Coq@@5Equivalence_Reflexive@@#max<>d@@A@<@@#min<̠d@@A@ABC<@!@#mul<d@@A@<@(@#one@$zero ɠT@@A@ABCDE@A@&NZBase&NatInt'Numbers#Coq@*NZBasePropbl@@@A@@@@"NZ&NZBase&NatInt'Numbers#Coq@t@@’H@|@!t’HJ@@A@A™@@"eq™Ġ^@@A@š=@ @"leš=h@@A@ABšL@@"ltšLh@@A@"m@@#add"md@@A@A34@@#max34d@@A@3@@#min3 d@@A@ABC4@!@#mul4d@@A@6@'@#one6T@@A@A=@+@#sub=d@@A@>@1@#two>T@@A@AI@5@$predId@@A@G@:@$succGr@@A@@?@$zeroT@@A@ABCDE@A@@M  @]@@@d@!tРJ@@A@A L@@"eq L^@@A@r@ @#onerT@@A@AB4@@#two4T@@A@b@@$predbѠd@@A@F@@$succFr@@A@]l@@$zero]lT@@A@ABCD@A@@L:.+*@A@4*BackportEq@@@A@@@@ %'A*Equalities*StructuresIA݋X_AO@A@@AA@@A@&eq_sym @_@5Equivalence_Symmetric. UX?/(ˠĠg`@o@)SymmetricFgqD@_*H@^ަT|L@*. U\~Dl\U0)xbQhd@@[@@Q(eq_trans @@6Equivalence_TransitiveW}vQJ;@@*Transitive++-@*@ަԐTʐL@*W\̐DF?۠~whƠJ9@@@@*eq_sym_iff @ՠŠaZˠeT@P@!x@*!y@*%Logic$Init@#iffС)@ ަBA@ަ@@@@AA@@A@'neq_sym @ A:ݠ֠Ǡ%@A!n@:*!m&@A*@E@#notШ3@Iަ@ :@PަGC@@*@@@(eq_stepl @VF?۠~wbL;@BZ@u*_@z*!zf@*@l@ަ2y@s@ަDz@ަ@@@@iAۓό^)Morphisms@o@AB@@(succ_inj @(!Ġ`YJ,@C"n1@*"n2@*@@ަ@Oь©@Uьǩ@ަU@@@AviAG@AAۀߠKJ@N@@ABC@@Ϡ+succ_inj_wd @ՠΠqj ۠ʠud@`DI@*G@ * @ ަ@ьF@ьE@ަ@@@C@A8ӠA4Relation_Definitions)Relations=@AB@@/succ_inj_wd_neg @(PI֠4 @E,@G* 1@L*O :@Pަ@@ьE@ьL@bަY@@@@@P1central_induction @fVO*#r\K@F!A@m@*A$A_wd@&ProperL@@* @*respectful%WO?@*@ަ/@*@Zڶ@~@*E@@ь@*F@@@@@'add_0_r @ǠSLuӠ@@A&NZBase%NZMul&NatInt'Numbers#Coq@VT@A@%NZMul&NatInt'Numbers#Coq@)NZMulPropec@A@@@@"NZZY@A@%NZAdd&NatInt'Numbers#Coq@)NZAddProp/@@@A@@@A&NZBase%NZAdd&NatInt'Numbers#Coq@A@A@@@"NZ+@@,R@2@!t,RJ@@A@A,Y@@"eq,Y^@@A@,@@ @#add,@d@@A@AB,w@@#mul,wd@@A@,@@#one,T@@A@A,@@#sub,d@@A@,@@#two,T@@A@A,@"@$pred,d@@A@,@'@$succ,r@@A@,筷@,@$zero,筷T@@A@ABCDE@A@E@+H@c*M@cަS@gO_X@2ƀc@@@@@H@ABC@@AA@@A@*add_succ_r @xha<5&n]r@}Ac@*a@*@ަ@PgO@0ь۩@5ь@agO@@I@@F-add_succ_comm @G@ܠxiǠ٠Ԡà@B@*@*@ަ@gO@sь$䩚@gO멚@~ь)@@@@(add_comm @&  7*@C隠@!*皠 @&*@&ަ@gO#@gO @@@@'add_1_l @5%]VA+naSN=/@:D =@X*B@XަH@gOM@{̀XR@ь@@@@'add_1_r @l\U0)xbQtf@qEWt@*y@ަ@?gO@1{̀@#ьΐ@@1@@.)add_assoc @/(ˠĠg`QܠϠ@F@*@*!p@*@ަ@gO@gOթ@gO@gOgސ@@{@@x,add_cancel_l @ݠ֠yrҠ& @Gؚ@*֚@*J@*@ަ @gOҩ@gO@-ަ@@@@,add_cancel_r @1!YRߠ='j]OJ9+@6H9@T*>@Y*C@^*aJ@`ަP@gO\V@gOc[@qަ@@@@,add_shuffle0 @ue^92#kZ}o@zI`}@*^@*Қ@*@ަ@RgO@XgO,@]gO@cgOS@@K@@H,add_shuffle1 @IBޠzkɠ۠֠Š@J@*@*@*!q@*@ަ@gO@gOt@gO@gO@gO@gO@@@@,add_shuffle2 @   ?8۠ԠŠ# PC50@K @ :* $@ ?*t )@ D*Z .@ I* 3@ Iަ 9@gO ?@gOX D@gOQ I@ gO O@gO[ U@gO@@@@,add_shuffle3 @ o _ X3,{eTwi@tLZ w@ *X |@ *̚ @ * @ ަ @LgOR @RgO @WgO @]gOM@@E@@B'sub_1_r @    C <ߠؠ{teàՠР@M @ * @ ަ @sŁ֚ @|{̀ @s8+@@|@@y'mul_0_r @  ޠ נ z s  Ӡ' @%@֚ @ * @ ަ @y  @2ƀ @2ƀ@@@@AA@@A@*mul_succ_r @ $     L EҠ0 ]PB@[A  )@ D*  .@ I* 3@ Iަ 9@yF ?@ь D@gO J@yWV@@@@C(mul_comm @ d T M     ( ! pZI@BL i@ *J n@ * s@ ަ y@,y ~@1y@@&@@w/mul_add_distr_r @    $    \ U F}ѠĠ@C @ *~ @ *򚠐 @ * @ ަ @ey @xgOLé @}gO @vys @{yՐ@@p@@/mul_add_distr_l @  Ҡ ˠ n g     ؠǠ@Dʚ @ *Ț @ *< @ * @ ަ @y© @gO @gO @y @y@@@@ )mul_assoc @ ,     T M   ڠ 8 " eXJ@cE 1@ L* 6@ Q* ;@ V* @@ Vަ F@y  L@yY Q@ y W@ yb@@@@P'mul_1_l @ q a Z     5 .  } g V@FY v@ * {@ ަ @ 4y @ 3{̀@@.@@'mul_1_r @    , % Ƞ  d ] N   ٠̠@G @ * @ ަ @ cy @ b{̀@@]@@,mul_shuffle0 @ Ϡ   [ T     } ۠ Š @H @ * @ *) @ * @ ަ @ y @ y @ y @ y@@@@,mul_shuffle1 @      < 5 ؠ Ѡ     M@2@KI @ 4* @ 9*n #@ >*T (@ C* -@ Cަ 3@ y 9@ yR >@ yK C@ y I@ y֩ N@ y@@@@G,mul_shuffle2 @ h X Q     , %  t ^ M@JP m@ *N r@ *š w@ * |@ * @ ަ @ :y @ @y @ Ey @ Jy @ PyN @ Uy6@@J@@,mul_shuffle3 @    H A  ݠ  y j Ƞ  @K @ * @ * @ * @ ަ @ y @ y驚 @ y @ y@@@@ࠠ%le_wd @      ) " Š     栠@@A&NZBase'NZOrder&NatInt'Numbers#Coq@  @A@'NZOrder&NatInt'Numbers#Coq@+NZOrderProp  @A@@@@"NZ  @A@@@ (@ C*@ -@ H* 3@ N*@ 8@ S*˚ <@ Rަ B@ ]*՚ F@ \ަ c J@ @@@@AA@@A@*lt_le_incl @ g W P     + $  s ] LfYK@dA O l@ * M q@ *@ w@"lt.  }@ G @@@@@3'le_refl @    #    [ T E   |{@B  @ * @ k  @@ @@X.lt_succ_diag_r @    H A  ݠ  y j Ƞ  @C  @ * @O. ҩ @ fь @@@@@.le_succ_diag_r @  ֠ Ϡ r k       ܠ ˠؠ@D Κ @* @   @ ь ;@@@@@/neq_succ_diag_l @    8 1 Ԡ ͠    @ E @0* @2ަ"@ ь g -@@ a@@ؠ/neq_succ_diag_r @<,% Ƞ  d ]    H 2 !;. @9F $A@\* H@^ަ TN@ ь @@@@@/nlt_succ_diag_l @hXQ     , %  t ^ MgZL@eG Pm@* Ft@.z@ ь  @@@@@0/nle_succ_diag_l @}    X Q B   yx@H |@* r@ j@ @ь  @@@@@\)le_succ_l @LE    } n ̠  @I @* @* 蠩@ @ qь " ⩚@e. @@ @@+le_gt_cases @ߠ{      ۠@J ޚ@* ܚ@* @"or @ @  @. @@@@Ƞ-lt_trichotomy @,TM   ڠ 8 " +@)K 1@L* 6@Q*6=@. J=D@Zަ QI@. N@@@@@(lt_asymm @cSL'  o Y HbUG@`L Kh@* Im@*@s@.  Lz@. @@@ @@0(lt_trans @} XQB   yx@M |@* z@*@*@@2. =@@8. C@=.  z@@ @@k(le_trans @Ϡ[T} ۠ Š Π@N @* @*)@*@@  x@@  ~@ ;@@ @@+lt_strorder @ 1*͠Ơ  @O@@+StrictOrder.[@@0*@.@@ @@Ϡ+le_preorder @3#[T?)2%@0Pj@(PreOrder@?@Z*C@ @@ 1@@/le_partialorder @]MF~! iSB\OA@ZQ@,PartialOrder5͛Кh@*l@ަp@!xt@>F@Z; _@@@ @A @ h@ iAhC@ABC@@5)lt_compat @@%lt_wd*#Ơb[L@ &@@*@@* 7 %@*@@* B@ަ 4@* L@ަ ښ@J.@ŠĠà@@%lt_wd']ɠݠ͠ƠibӠ ܠϠ@@@@@(lt_total @ @3v䀠%@ ߚ@* ݚ@*@. @%ަ @. S@aǠ+SL٠7!*@@@@@ࠠ'le_lteq @\@+lt_eq_cases*ȀI92ՠΠqj @ +H@c* )M@h* kT@ aT[@. h`@vަ m@+lt_eq_cases&yvf_:3$l[uhZ@@@@@+0Private_OrderTacLA'IsTotalT A @)"ŠaZK@@+Equivalence~@@*@ަ@(eq_equiv7LIBޠzkɠ@@@@@r͐ @@;{٠ɠ e^@ʠ@*@f.@,Q^۠~wנ@@@@@r @@ze 6/Ҡˠ@ @@**@@/*  @5*@@:* #@9ަ )@D* -@CަJ1@.pD4-РɠleP:)C6(@@@@@g @f@o/C0`PI$@B_@z*@d@*dk@.xkr@ަw@.|czsNG8o|n@@@@@?_ @@gY2+ΠǠjcT@@*@*Ƞ@{@A.ũ@ަʐ]Р\U~ܠƠϠ @@@@@@@@@ː@#TacaA @۠~wנ֠@@D!P)OrdersTac*Structures#Coq@@@@A@@C!O@@&@@!t&J@@A@A&a@@"eq&a^@@A@&@ @"le&ڠh@@A@AB&@@"lt&h@@A@&̌@@#one&̌T@@A@A&̔I@@#two&̔IT@@A@&H@@$pred&Hd@@A@&؝[@"@$succ&؝[r@@A@&C@'@$zero&CT@@A@ABCDE@A@G,MakeOrderTacBk?@A@@@BONR@@@A@@AEUH@@@N@!tJ@@A@A$:@@"eq$:^@@A@$@ @"le$h@@A@$@@"lt$ h@@A@ABC@A@+*OrderFacts.#@A@@s@o@*t@ަӐ@@b@@AA@@A@ @yTM>utYRL1@Aš@*@j@@@@,)lt_irrefl @F?۠~whƠȠ|v[@B욠@*@U.+@@@@X @ՠΠqj ۠ʠנɠ@C@ *@*@@ަ@ަƐ@@@@*le_antisym @ B;ޠנȠ& (ܠ֠@)DL%@@*J*@E*@0@=@6@;@Qަ T@@)@@Ǡ< @TD=٠|u`J9SF8bR@cE_@z*d@*@>l@ަyEs@ަ9@@@@@*interp_ord @!o)OrdersTacm@#ordȠ@ @@@@@@@@@@@D@@%@ t1@et"70Ӡ̠ohY@(&@@*@@*C +k= R'6'6'6''+k+7T'+7T'6'ࠒAƠŠ@@"lt\ @r@"eq8L@tlР@RȠ@@ONM@@A@B@A@xA܀ݠy٠ؠ@@@@@%trans @;4נР !ܠՠϠ@"F"o'I"@=*G'@B*Ț,@G*@^@6 ƀA@ g@6 ƀ n@6 ƀ@)trans_ord-0uG@@@@٠ @|v@d МA+rqvf_:3$@u@*z@*@*@@ަ@@ަ@ަܐ L+k() LL7#'5@$t:3֠Ϡrk\wpjO@@@@@? @_fМBCϠԠĠ`Y@@*@*y@*@@w@@}@: L+k() NN7#'^t0)̠Š ѠʠĠ@@@@@ @B?<@AМB*).VO@T-@H*R2@M*Ӛ7@R*@=@.Ѷ@C@.שH@. L+k() MM7#'tbRK&nXG a T Fp`+$@@@@@+le_lt_trans @[;vo JC4@@*@*+@*@@_)@@ $./@ ).쐐 L+k() MN7#'tF?۠~whƠ   Ƞ|v[@@@@@K+lt_le_trans @ls ٠ޠΠǠjc@@*@*@*@@ v.@@@ .D L+k() NM7#'ht:3֠Ϡ    ۠ԠΠ@@@@@%eq_lt @MJGKL 216& ^W@\5@P*Z:@U*ۚ?@Z*@E@[ަٶ@K@ .ߩP@ . L+k() ML7#'tjZS.'v`O i \ Nxh3,& @@@@@%lt_eq @#cC~wRK<@@*@*3@*@@ &.1@@ަ7@ 1. L+k() LM7#'t NGpΠ   Р~c@@@@@S%eq_le @t{s֠Ϡrk@ @* @*@ *@@ ަ@@@L L+k() NL7#'pt B;ޠנȠ&   (ܠ֠@@@@@%le_eq @UROSTm:9>.'ʠàf_@d=@X*bB@]*㚠G@b*@M@@S@iަ穚X@" L+k() LN7#'Ȑtrb[6/ ~hW q d Vp;4.@@@@@&eq_neq @zUN?v   uZSM2@GÚ@*@*B@*@@ަ@@@ަH@ަ @@@@@G&neq_eq @ՠŠaZˠ Ԡ Ǡ Ӡv@H@*@*@*@Đ@ަ@@ަѐ@ަ K@@@@@*not_neq_eq @ A:ݠ֠Ǡ%   '۠ՠ@(IK$@?*I)@D*@3@Iަ@8@Nަ̐@@w@@Ġ)not_ge_lt @RB;ޠנzs^H7 Q D 6`P@aJ]@x*b@}*@<j@4oo@ .@@]@@)not_gt_le @yrMF7n  { mRKE*@K@*@*@s@ *.@p:@@@@@2)le_neq_lt @LE}n̠   Π|a@L򚠐@*𚠐@*@@@@ަr@ l.@@@@@o@@@@ߐ@@@@@@<_neq @)"Š    @ R隠@!*皠 @&*@@ .ꐩ@.ަ@@@@@ Π&le_neq @2"ZS>( 1 $ @ /S7@R*<@W*ZC@ .Pb@#andЖw@P@])W@mަd@@@@@ *eq_le_incl @qaZ5.}gV p c U@ nTYv@*W{@*@@ަ@P@@@@@ <(lt_stepl @,%Ƞd]N   @ U̚@*ʚ@*K@*@@ >.I@@ަH@ I.F@@@@@ v(lt_stepr @ڠʠàf_Р ٠ ̠ @ V@*@*@*@@ x.@@ ަ@ . F@@@@@ (le_stepl @<5ؠѠ    @W@@4*>@9*#@>*@)@@/@Eަ4@@@@@@ ꠠ(le_stepr @N>7ڠӠvo ZD3M@2@KXzS@n*xX@s*]@x*@c@-@i@ަn@8 @@@@@$ @wp KD5}lyk@Yo@*m@*ᚠ@*@@f0@@+.6@0. @@@@@] @LE}n̠@Z@*@*@*@@^.i@@o@i. ,@@@@@+le_antisymm @"ߠ@[⚠@*@*@ @@@֩@+ަ.@@@@@ˠ)le_succ_r @/WPݠ;%.!@,\4@O*9@T*W@@ MF@ь FM@ZR@hަ_X@ь@@@@@)lt_succ_l @rb[6/ ~hWqdV@o]Zw@*X|@*@@ .@"ь@.!@@@@@C,le_le_succ_r @3,ϠȠkdU@^@*@*@@ĩ@@\ь @@@@@x,lt_lt_succ_r @ܠ̠ŠhaҠ۠Π@_Ě@*š@*@@u.@z.@ьB@@@@@,succ_lt_mono @92ՠΠ@`@1*@6*9"@./'@.-@ьx2@ьw@@@@@蠠,succ_le_mono @L<5ؠѠtm XB1K>0@Ia4Q@l*2V@q*t]@'jb@,h@ьm@ь@@@@@#<_0_1 @wp KD5}lyk@b@.@&2ƀ@B{̀@@@@K&le_0_1 @;4נРsl]@c@~@N2ƀ@j{̀@@@@@s<_1_2 @נǠc\֠͠ɠ@d@e.@{̀@Ń@@$@@<_0_2 @' à @e@. @2ƀ @Ń@@@@à&le_0_2 @'OHՠ3 & @$f,@1@2ƀ5@Ń@@@@@렠<_1_l @O?8۠Ԡwp [E4NA3@Lg7T@o*5Y@t*@_@.d@2ƀp@j@.o@.t@!{̀9@@@@@d@ABC@@-+lt_ge_cases @zUN?vu@hy@*w@*@+.@q@@@@@]+le_ge_cases @MF~o͠@i@*@*ˠ@ߩ@ܐ@@@@@+lt_gt_cases @ڠ}v֠@jٚ@*ך@*֐@ަ @.@.@@@@@Ơ,eq_decidable @*RKؠ6 )@'k/@J*4@O*)DecidableTi@)decidable rD@ZަQ@@@@@&eq_dne @^NG" jTC]PB@[lFc@~*Dh@*CEs@ަx@ަ@@@@@.&le_ngt @{VO@wv@mz@*x@*@m|@3.@@@@@`&nlt_ge @ĠPIrРà@n@*@*젩@`.䩚@@@@@@,lt_decidable @ߠ{۠@oޚ@ *ܚ @ *̐ @.@@@@@<_dne @ !  IBޠϠ- @p  &@ A* +@ F*I 6@.C ;@.H@@@@@&nle_gt @ U E >ڠ}vaK:TG9@Rq= Z@ u*; _@ z*}: h@2u m@.r@@@@@#<_nge @  w p  KD5}lyk@ro @ *m @ * @!.q @i@@@@@U,le_decidable @    E >ڠ}vgŠ@s @ * @ * @א@@@@@&le_dne @  Ԡ ͠ p i  ڠɠ֠@t̚ @!*ʚ @! * ɐː @ @ @@@@@*nlt_succ_r @!!!   @ 9ܠՠƠ$ @u!@!8*!"@!=*@!+@.7!1@ь|!6@.C@@@@@젠5lt_exists_pred_strong @!P!@!9 ܠ ՠ x q  \F5OB4@Mv񚠐!U@!p*=!Z@!u*;!_@!z*@!e@.@!k@51@"ex @!w@!*!k!~@!*;!@!ަ!@%ьЩ!@Z@@@@A@@A@AРD.Morphisms_Prop!@A1@D@ABCD@@U.lt_exists_pred @!!!!E!>  ڠ } v gŠ@wZ!@!*!@!*@!@R.֩^!@!*X!@!*!@!ަ!@{ь&!@@@@K@A@@,lt_succ_pred @"!!!!!)!" Š   @x"@"!*" @"&*@"@."@",ަ"@ь""@8+m.@@@a@A^@@@@ABCD@@ݠ&rs_rs' @"A"1"*!͠!Ơ!i!b!   M 7 &@3%@>yܶ@"G@"b*ڶةְ@"O@"j*"U@"p*蚠"Y@"oަvd"^@"y*@˶@H"e@"*@"k@ 5@ ""t@ ь:]"z@"*@"@ J1@j"@"*h"@"*@"@ \ G@"@"._K'橐H)@@{@@S(rs'_rs'' @""""C"<!ߠ!ؠ!{!t!e à  @zR@"@"*PNL@"@"*XF"@"*^"@"ަڶp"@"*@"@"*@"@ @ʚ"@#*Ț"@#*@"@ U@"@.IFP䚠#@#*욠# @#$*ꚠ#@#)*@#@ "@#@.!_# @#;*#%@#@*@#+@ 9@#1@.8#7@ ь|@@v@@%rbase @#Q#A#:"ݠ"֠"y"r""!!]!G!6PC5@N{@#W@#r*#\@#w*E#b@#}*C#g@#*@#m@!7@#s@.9x@@@@@))A'A_right @##}#v##"""Q"J";!!!rq@|(@#@#*&4#@#*@#@#*#@#*#@#*@#@!y`@#@>.{jB#@#*@#@!q$ @@@@@x6strong_right_induction @#ܠ#̠#Š#h#a#""""!!Ҡ!۠Π@}w@#@#*usq@#@$*}k#@$ *#@$ ަ #@$*@⚠#@$*@$@! @$ @$'*횠$@$,*@$@!%@$@.$h" $"@$=*@$(@!@@@g@A@!@@ABC@@⠠/right_induction @$F$6$/#Ҡ#ˠ#n#g# #""R"<"+E8*@C~@$L@$g*߶ݩ۰@$T@$o*$Z@$u*횠$^@$tަ { i$c@$~*@ж@ M$j@$*@$p@": @&$x@"ь `$}@$*@$@"M48ϐ@@@@@:0right_induction' @$$$$*$##Ơ##b#[#L"""@9@$@$*753@$@$*?-$@$*E$@$ަ   W$@$*@ $@$*@$@" @ $@$*@$@"@h$@"uьg Ú$@$*5@@!@@7strong_right_induction' @$$$$$$"$####""ߠ@@@%@%*@%@%#*%@%)*%@%(ަ!/! %@%2*@!%@%8*@%#@"!(b@! %)@%D*@%/@"߶@!%6@%Q*!%;@%V*@%A@# @%G@.! J !1%N@%i*@@@@@&ls_ls' @%h%X%Q$$$$$,$%$#t#^#MgZL@eA @%n@%* @%v@%* %|@%* %@%ަ!!!!%@%*@@!o%@%*@%@.@萩%@#4ь !%@%*@%@#o!!2@!%@%*!%@%*@%@#!m@%@#!S&%@#_ь!*@@Q@@{(ls'_ls'' @%ߠ%Ϡ%Ƞ%k%d%%$$$##ՠ#ĠޠѠ@B z@%@&* x v t@%@&*  n%@&* %@& ަ""!%@&*@!嚠&@&*@&@#" @!򚠐&@&**!𚠐&@&/*@&@#"& }@&!@#!(&'@#ь!r,"&,@&G*"J"&4@&O*"&9@&T*@&?@$ %@&E@$!L!".&K@&f*",&P@&k*@&V@$ <@&\@$&!c&b@#ь!@@@@%lbase @&|&l&e&&%%%@%9%*$$r$a{n`@yC!@&@&*!"#&@&*"p&@&*"n&@&*@&@$b@&@$h"2&@$>ь!@@@@@Z(A'A_left @&&&&J&C%%ߠ%%{%l$ʠ$$@D!Y@&@&*!W"e&@&*@"&@&*"&@&*"&@&*@&@$;@&@$"z1"+"Κ&@'*@&@$0@@#@@5strong_left_induction @' &&&&&3&,%Ϡ%Ƞ%%%$ @E!@'@',*!!!@'@'4*!!'@':*!'#@'9ަ#@#."Ě'(@'C*@#'.@'I*@'4@$#9@#';@'V*#'@@'[*@'F@%,@'L@%"S'R@$ь"W#:'W@'r*@']@%'Q@@@"@$L@@ABC@@.left_induction @'z'j'c'&&&&>&7&(%%p%_yl^@wF"@'@'*"""@'@'*"" '@'*"!'@'ަ###3'@'*@"@#'@'*@'@-.@!'@%Fь"#'@'*@'@%4@@@@@m/left_induction' @'Ѡ''']'V&&&&&%ݠ%Ǡ%Рà@G"l@'@'*"j"h"f@'@'*"r"`'@(*"x'@'ަ$##'@( *@#ך'@(*@'@%$9@#㚠(@(*@(@.@"\(@%ь#YY#(@(.*"h@@ @@ɠ6strong_left_induction' @(-(('''U'N&&&۠&9&#&,@*H"ȶ@(3@(N*"ƶ"ĩ"°@(;@(V*"Ω"(A@(\*"Ԛ(E@([ަ$b$P#暠(J@(e*@$3(P@(k*@(V@& $c@$?(\@(w*@(b@&, V@$L(i@(*$J(n@(*@(t@&>$"ɶ@({@&E$4(@&ь#̶$i(@(*"ې@@ @@@,Y*(,C@,^*$,H@,c*(f,O@.'㩚,T@.,Z@*gO$N,_@*gO(d@@&@@AA@@A@-add_lt_mono_r @,|,l,e,,+++@+9+***r*a@*]A(a,~@,*(_,@,*$Ӛ,@,*(,@.(#,@.,@*ZgO$J,@*_gO(@@&G@@@+add_lt_mono @,,,,E,>++ڠ+}+v+g*Š**@*B(,@,*(,@,*%,@,*#,@,*@,@Y.#@,@_.(j,@d.,@*gO'6(n,@*gO3@@@@@-add_le_mono_l @-,,,,,),"+Š+++ **@*C(暠-@-*(䚠-@-#*%X- @-(*)+-@*(-@*-@*gO%-$@*gO))@@&@@Š-add_le_mono_r @->-.-',ʠ,à,f,_,+++J+4+#@+D)#-@@-[*)!-E@-`*%-J@-e*)h-Q@+(婚-V@+ -\@+gO% -a@+!gO)n@@' @@+add_le_mono @-{-k-d--,,,?,8,)++q+`@+\E)`-}@-*)^-@-*%Қ-@-*$-@-*@-@+\$@-@+b),-@+g-@+cgO©-@+hgO@@@@@I.add_lt_le_mono @- ---N-G,,,,,p+Π++@+F)-@-*)-@-*&-@-*$-@-*@-@ b.$@-@+)s-@ m.-@+gO -@+gO;@@@@@.add_le_lt_mono @. -----1-*,͠,Ơ,,++@+G). @.&*)욠.@.+*&`.@.0*%F.@.5*@. @+%9@.&@ .).+@ ..1@+gOP.6@+gO@@@@@נ+add_pos_pos @.P.@.9-ܠ-ՠ-x-q-- ,,\,F,5@,1H*5.R@.m**3.W@.r*@.]@ ..b@+2ƀ*n@.h@ ..m@,2ƀ*y.r@ ..w@, 2ƀ.|@,<gO%@@@@.add_pos_nonneg @....".---Z-S-D,,,{@,wI*{.@.**y.@.*@.@!,..@,=2ƀ*@.@,x.@,H2ƀ*.@!A..@,R2ƀ.@,gO%ې@@N@@c.add_nonneg_pos @.ܠ.̠.Š.h.a.----,,Ҡ,@,J*.@.**.@.*@.@,.@,2ƀ*@.@!}..@,2ƀ+.@!./@,2ƀ/@,gO&!@@@@1add_nonneg_nonneg @/"// ...J.C--ߠ-Р-.--@-K+/$@/?*+/)@/D*@//@,/4@,2ƀ+@@/:@-/?@,2ƀ+K/D@-/I@,2ƀ/N@-gO&g@@@@,lt_add_pos_l @/h/X/Q.....,.%.-t-^-M@-IL+M/j@/*+K/o@/*@/u@!./z@-2ƀ+/@".+/@-EgO+@@@*@,t@AB@@(,lt_add_pos_r @////-/&.ɠ. .e.^.O---@-M+/@/*+/@/*@/@"7./@-H2ƀ+/@"A.+ũ/@-~gO+@@*@@_,le_lt_add_lt @/ؠ/Ƞ//d/]/....--Π-@-N+/@/*+/@/*(//@/*'/@0*@/@-'@/@"~./@-gO+0@-gO, *c0@".'@@@@@,lt_le_add_lt @0 00 ///H/A..ݠ.Π.,..@.O,0"@0=*,0'@0B*(w0,@0G*']01@0L*@07@".'P@0=@.0C@.gO+ɩ0H@.gOH0M@".'f@@@@@,le_le_add_le @0g0W0P/////+/$/.s.].L@.HP,L0i@0*,J0n@0*(0s@0*'0x@0*@0~@.H'@0@.N0@.JgO,0@.OgO0@.^'@@@@@5,add_lt_cases @0000:03/֠/Ϡ/r/k/\...@.Q,0@0*,0@0*)0@0*'뚠0@0*@0@#N.0@.gO'䩚0@.gO,ݩ!Р0@#`.!#0@#e.,i@@@@@},add_le_cases @000ߠ00{00////..@.R,ۚ0@1*,ٚ0@1*)M1@1*(31@1"*@1 @.1@.gO(,1@.gO-%"1@.!k1$@.,@@@@@Š-add_neg_cases @1>1.1'0ʠ0à0f0_0///J/4/#@/S-#1@@1[*-!1E@1`*@1K@#.1Q@/gO-^1U@.2ƀ"U1\@#.-"1a@.2ƀ1f@#.-s1k@/2ƀ@@,@@ -add_pos_cases @11u1n11 000I0B03//{/j@/fT-j1@1*-h1@1*@1@$.1@/,2ƀ1@/\gO-"1@$,.1@/=2ƀ-m1@$6.1@/G2ƀ-@@>@@S0add_nonpos_cases @1̠111X1Q00000z/ؠ/ /@/U-1@1*-1@1*@1@/1@/gO-욠1@/x2ƀ"㠩1@/-1@/2ƀ1@/.1@/2ƀ@@-8@@0add_nonneg_cases @221111;140נ0Р000 /@/V-2@20*-2@25*@2 @/2%@/2ƀ2*@/gO.7#*21@/26@/2ƀ-2;@02@@/2ƀ.L@@@@ᠠ-le_exists_sub @2Z2J2C11ߠ11{1110f0P0?@0;W.?2\@2w*.=2a@2|*@2g@01.t2m@2**2r@2*/2y@2ަ.?2@0?gO ک2@0N2@02ƀ.@@@-@A.@/z@A/zAA@@-@ABCD@@3+mul_lt_pred @22228211Ԡ1͠1p1i1Z0@1X@*2@2*)ݚ2@2*.2@2*.2@2*@2@2ަ2@0aь.T..栩2@%X.2@0y#!2@0y-=.穚2@%i.2@0gO2@0y*.2@0gO2@0y..@@,@@AA@@A@1mul_lt_mono_pos_l @332222<251ؠ1Ѡ1 1 @1A+_3@3/*.3@34*.3@39*@3$@%.3)@02ƀ./G30@%..ĩ35@%.3;@0y*T3@@0y.͐@@@.@A.|@.@01@@ABCD@@N1mul_lt_mono_pos_r @3_3O3H22222#22 1k@2 B+3_@3z*/G3d@3*/E3i@3*@3o@%.3t@1 2ƀ/9/3{@&./3@& .3@19y/ 3@1>y//@@-4@@1mul_lt_mono_neg_l @333323+2Π2Ǡ2j2c2T1@2RC+񚠐3@3*/3@3*/3@3*@3@&?./|3@1P2ƀ/٠3@&K./V3@&P.3@1y/Z3@1y*@@@@۠1mul_lt_mono_neg_r @33ܠ3ՠ3x3q33 2221@2D,73@4*/Ԛ3@4 */Қ3@4*@3@&./š4@12ƀ04@&./4 @&.4@1y4@1y/@@-@@!4mul_le_mono_nonneg_l @424"4333Z3S2222>@2E042@4M*047@4R*,4<@4W*@4B@2 4G@12ƀ0R@4M@2+f4R@24X@2 y0.4^@2y/@@/@@g4mul_le_mono_nonpos_l @4x4h4a43333<353&2@3$F0[4x@4*0Y4}@4*,͚4@4*@4@2R04@2"2ƀ@4@2]+4@2b4@2Qy0$4@2VyK@@/@@4mul_le_mono_nonneg_r @4444I4B33ޠ33z3k2@3iG04@4*04@4*-4@4*@4@24@2g2ƀ0ݶ@4@2+񩚠4@24@2y%/4@2y,@@0'@@4mul_le_mono_nonpos_r @544444*4#3Ơ333@3H0嚠5@5*0㚠5@5"*-W5 @5'*@5@215@22ƀ@5@2,65"@25(@2y,A5-@2y%y@@0l@@6,mul_cancel_l @5G57504Ӡ4̠4o4h4 433S@3I1*5G@5b*1(5L@5g*-5Q@5l*@1+5Y@5oަ1e5^@22ƀ1|5e@5{ަ5k@3yਗ਼5p@3#y165u@5ަ,@@@0@2d@0q@ABC@@,mul_cancel_r @555{55444V4O4@3@4>J1u5@5*1s5@5*-皠5@5*@1v5@5ަ15@3>2ƀ1Ǡ5@5ަ5@3iy1C5@3ny1O5@5ަ,ِ@@/h@@ɠ(mul_id_l @5ڠ5ʠ5à5f5_544443@4K15@5*15@5*@15@5ަ1󚠐5@32ƀ2 5@6 ަ5@3y125@6ަ1Ě6@3{̀@@1@@ (mul_id_r @66 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