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U\zDܠ}%t^Md`@@W@@M(eq_trans @2@6Equivalence_TransitiveW+D=.@@@*Transitive++-@0i@/倐ِ̐TL@(W\ĐD=-&ΠǠohYB1۠@@@@*eq_sym_iff @VF?rנ[J@F@!xZ@ui!ya@|i%Logic$Init@#iffС)r@倠BAz@倠@@@@AA@@A@'neq_sym @)"ʠà@A!n@i!m@i@C@#notШ@> @倠EC@@@@>(eq_stepl @Ӡàd]T>-ؠ@B}@i{@i!z@i@@倠0u@@倠D}@ 倠>@@@[Aۓό^)Morphisms-@a@AB@@(succ_inj @IB3q @C"n1@6i"n2"@=i@(@>倠.@?8߀4@E8߀:@PS@@@AviAE@AAۀߠKH@L@@ABC@@ɠ+succ_inj_wd @^NGzߠɠcR@NDG`@{iEe@il@倠r@8߀Dw@8߀C|@ @@@A@A8ӠA4Relation_Definitions)Relations@AB@@ /succ_inj_wd_neg @0)Ѡʠ  @E@i@iE@倠@8߀@8߀@O@@@@@F1central_induction @۠ˠĠle \F5@F!A@@iA$A_wd@&ProperL@@ i @*respectful%WO?@i@'@!i@PԶ@t@)iEީ@*8߀@9iF@@@@@'add_0_r @:*#ˠĠleV@@A&NZBase%NZMul&NatInt'Numbers#Coq@><@A@%NZMul&NatInt'Numbers#Coq@)NZMulPropMK@A@@@@"NZBA@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@,@$zero,筷T@@A@ABCDE@A@E@@i@倠@Q@U@@@@@0@ABC@@AA@@A@*add_succ_r @٠ҠzsjTC~p@{AU@ iS@i@倠@6@8߀ѩ @8߀@G@@G@@D-add_succ_comm @*\UFՠР@B0@Ki5@Pi:@P倠@@wF@W8߀ҩK@٩Q@b8߀@@@@(add_comm @k[T֠Š1$@Cךq@i՚v@i{@倠@@ @@@@'add_1_l @1*Ҡˠ! fYKF5'@2D @i@倠@@( B@8߀@@@@'add_1_r @ՠŠf_V@/{j\@gEA@i@倠@s@( @8߀@@'@@$)add_assoc @ <5&udРà@Fv@+it@0i!p@7i!@7倠'@^p-@d2@i8@oQĐ@@o@@l,add_cancel_l @RB;ܠ}nӠ @GX@si]@xiHb@}ii@倠o@u@z@@@@@,add_cancel_r @}%ƠZM?:)@&H@i@i@iC@倠@>@E@Ր@@@@,add_shuffle0 @֠Ơg`WA0|k]@hIB@i@@i̚@i@倠@(@.@3@9Q@@9@@6,add_shuffle1 @ NG8vՠǠ @J"@=i'@Bi,@Gi!q3@Ni8@N倠>@uD@{VJ@ةO@U@gZ@@@@@,add_shuffle2 @td]ߠΠ:- @Kz@iޚ@ij@iX@i@倠@@V@-@@7@ɐ@@@@䠠,add_shuffle3 @ʠ[TK5$up_Q@\L6@i4@i@i@倠@.@"y@'~@-E@@-@@*'sub_1_r @  B;,{j֠ɠ@M| @ 1i @ 1倠 !@A.1 &@J(  +@Aj@@b@@_'mul_0_r @ E 5 .֠ϠwpaƠ @ @ H@ ci M@ c倠 S@}%ŀ X@d \@h@@@@AA@@A@*mul_succ_r @ y i b  Ӡ?2$@=A⚠ |@ i @ i @ 倠 @%ŀ @8߀^ @ @%ŀ+*@@@@A(mul_comm @    H AӠ8"}pb@{B  @ i @ i @ 倠 @%ŀX @%ŀU@@@@s/mul_add_distr_r @  ٠ Ҡ z s   jTC@CR @ iP @ iܚ @ i @ 倠 @+%ŀ @>  @C @<%ŀa @A%ŀ@@N@@/mul_add_distr_l @ 1 !     c \ M@D 4@ Oi 9@ Ti$ >@ Yi C@ Y倠 I@s%ŀ O@ݩ T@ Z@%ŀs _@%ŀ@@@@)mul_assoc @ y i b     Ӡ?2$@=E⚠ |@ i @ il @ i @ 倠 @%ŀک @%ŀ% @%ŀ @%ŀ.@@@@F'mul_1_l @    M F   ؠ = ' ug@F% @ i @ 倠 @%ŀ @( [[@@@@s'mul_1_r @  ٠ Ҡ z s    j T C@GR @ i @ 倠 @ !%ŀ @ ( @@3@@,mul_shuffle0 @      H A 2   pܠϠ@H @ 4i} @ 9i  #@ >i (@ >倠 .@ X%ŀ 4@ ^%ŀM 9@ c%ŀ ?@ i%ŀ̐@@v@@㠠,mul_shuffle1 @ Y I B     u ڠ Ġ @Iš \@ wi a@ |iL f@ i: k@ i p@ 倠 v@ %ŀ |@ %ŀ8 @ %ŀ @ %ŀ @ %ŀ @ %ŀ@@@@5,mul_shuffle2 @    < 5 ݠ ֠ Ǡ ,  qdV@oJ @ i @ i @ i @ i @ 倠 @ %ŀ @ %ŀ @ %ŀa @ %ŀ @ %ŀ4 @ %ŀ@@@@,mul_shuffle3 @      / (  ~ h Wà@Kf @ id @ i𚠐 @ %i @ %倠 @ ?%ŀ^ @ E%ŀ @ J%ŀ &@ P%ŀu@@]@@ʠ%le_wd @ @ 0 ) Ѡ ʠ r k \   @@A&NZBase'NZOrder&NatInt'Numbers#Coq@ D B@A@'NZOrder&NatInt'Numbers#Coq@+NZOrderProp S Q@A@@@@"NZ H G@A@@y@ e@ i@ j@ ix p@ i@ u@ i y@ 倩 @ i @  ! @ ^@@@@AA@@A@*lt_le_incl @    5 . ֠ Ϡ  %  dWI@bA  @ i  @ i@ @"ltm A @ ^ѐ@@@@@1'le_refl @ Ҡ    c \    S = ,w@B ; @ i @ ^ g g@@ @@@T.lt_succ_diag_r @   ޠ   '   v ` O@C ^ @ i @Km  @ 8߀ϐ@@@@@|.le_succ_diag_r @      O H 9   wݠР@D  @ ;i %@ f^  +@ <8߀@@@@@/neq_succ_diag_l @ E 5 . ֠ Ϡ w p a Ơ  @E  H@ ci  O@ e倠 U@ f8߀ ! @@ @@Π/neq_succ_diag_r @ o _ X       ڠ ɠ/"@-F ؚ r@ i ΐ y@ 倠  @ 8߀ K@@@@@/nlt_succ_diag_l @    * # ˠ Ġ    YL>@WG  @ i  @m @ 8߀ u 5@@@@@"/nle_succ_diag_l @ à   T M   ߠ D . vh@H , @ i " @ ^ @ 8߀  _@@@@@L)le_succ_l @  ݠ ֠ ~ w    n X G@I V @ i T @i  @ =^@ 8߀  @Um @@ @@+le_gt_cases @!    S L =   {Ԡ@J $@?i )@Di @"or @6@ w^ ĩ;@m @@r@@-lt_trichotomy @UE>  ߠ   q ֠  @K X@si ]@xi4d@m ;k@ p@m @@@@@頠(lt_asymm @zs      J=/@HL 󚠐@i 񚠐@i@@m & @m @@ e@@(lt_trans @JC   ՠ : $ yl^@wM "@i @i@i@@m @@ m 멚@%m  @@ @@Q(le_trans @۠|$ s ] L@N [@i Y@i嚠@i@@ F^ @ @ L^ $@ Q^9@@ G@@+lt_strorder @*\UF   ݠ@O_@+StrictOrder.[@4@Oi8@m@@ @@+le_preorder @RB;ܠ}n Ӡ  @P@(PreOrder@\@wi`@ ^@@ @@٠/le_partialorder @zjc    Ԡ:-@8Q@,PartialOrder5͛К@i@倚@)$Ѐ@ ^@V. @@@ Y@A @ @ AhC@ABC@@)lt_compat @@%lt_wdJC@ ̰@@i@@i ݩ @i@@i 蚠@倩 @i 򚠐@ t@(m@@@%lt_wd']ɠߠ(!waP@@@@@l(lt_total @{#,@3v䀠F?0@ w@,i u@1i@km $@: )@wm O@aǠ@0)Ѡʠrk\@@@@@'le_lteq @o@+lt_eq_cases*Ȁ\LEx@ Y@ti ^@yi e@^ <l@m q@ @+lt_eq_cases&ywpG:,@@@@@0Private_OrderTacA'IsTotal A @=6ޠנ@@+Equivalence~@@i@倐@(eq_equiv7Lʠ[TK5$}o@@@@@@ @@;{Ԡ͠un@@i@4m@,Q^-&|fU@@@@@q^ @@fe G@1@ (@@/i@@4i 9 '@:i@$@?i D(@>倩 6.@Ii N2@H К6@m\I92ڠӠ{teʠ @@@@@S @R@[/C0cSL@ ƚ`@{i Ěe@i<l@m Cs@x@m O{t K>0@@@@@K @@SY6/נР@@i@iF@^<@mC@HI͠^WN8'r@@@@@C@@@@Ɛ@#TacMA @ݠ֠~w nXG@@D!P)OrdersTac*Structures#Coq@rk@@@A@@C!O@@&@@!t&J@@A@A&a@@"eq&a^@@A@&@ @"le&ڠh@@A@AB&@@"lt&h@@A@&̌@@#one&̌T@@A@A&̔I@@#two&̔IT@@A@&H@@$pred&Hd@@A@&؝[@"@$succ&؝[r@@A@&C@'@$zero&CT@@A@ABCDE@A@G,MakeOrderTacBi?@A@@@BONR@@@A@@AEUH@@@N@!tJ@@A@A$:@@"eq$:^@@A@$@ @"le$h@@A@$@@"lt$ h@@A@ABC@A@+*OrderFacts.#@A@@s@h@im@@@@@AA@@A@ @yr I<.WPJ/@A:@i@^@@@@*)lt_irrefl @B;ܠ͠2 qdVĠxrW@Bb@i@m@@&@@T @ڠʠàkd [E4ݠ@C@i@i@@|@ <@@X@@*le_antisym @ ?8)xg͠ ۠ԠΠ@!D@1i@6i@!@b^@'@h^p,@B @@@@ @E5.֠ϠwpaƠXH @YEN@iiS@ni@[@qb@x@@@@@*interp_ord @!o)OrdersTac@#ordȠ@ @@@@@@@@@@@D@@ @t1@t"+$̠Š@&$@@i@@i +k= R'6'6'6''+k+7T'+7T'6'ࠒA@@"lt\ @f@"eq8L@hlР@PȠ@@MLK@@A@B@A@xA܀Ѡʠrk bL;@@@@@w%trans @0)iX̠Š@F"o' @&i@+i6@0i@9L@! f@BU@! I\@!@)trans_ord-0uG%~@@@@ɠD @higjHsz@b МA۠_OH{@\@wia@|if@i@l@@r@w@ L+k() LL7#'#@$t~&ǠUH:c\V;@@@@@+ @NʷH˷̩bМ8C65JC@^@i\@iᚠ@i@@^߶@@ ^婚@^ L+k() NN7#'Zt۠Ԡ|ulVE@@@@@O @ !"МBGA:+@ @'i@,i7@1i@@jm5@"@pm;'@umP L+k() MM7#'tA1*Ҡˠsl]  TD@@@@@נ+le_lt_trans @wxvyWߠcSL@`@{ie@ij@i@p@^@v@m{@m L+k() MN7#'t~&Ǡ U H :c\V;@@@@@++lt_le_trans @O˷I̷ͩc43HA@\@iZ@iߚ@i@@ mݶ@@ ^㩚@ m L+k() NM7#'Xt٠ҠzsjTC   @@@@@%eq_lt @ !C =6'@@#i @(i3@-i@@.1@@ lm7#@ qmL L+k() ML7#'t=-&ΠǠohY   P@ @@@@@Ӡ%lt_eq @stru S۠_OH{@\@wia@|if@i@l@ m@r@w@ m L+k() LM7#'tz"à Q D 6_XR7@@@@@'%eq_le @KǷEȷɩ_W0/D=ޠ@X@iV@iۚ@i@@ٶ@@^ߩ@ ^ L+k() NL7#'TtՠΠvofP?   @@@@@{%le_eq @Q?92#@@i @$i/@)i@@U^-@@03@`^H L+k() LN7#'t9)"ʠàkdU   ޠL<@@@@@Ϡ&eq_neq @VF?rנ   iY$@jG_@zid@ii@i@o@@̐w@Ӑ~@@@@@@&neq_eq @)"ʠà X K =f_Y>@HI@iG@i̚@i@@̶@@ҩ@@@@@@S*not_neq_eq @ڠʠàkd [E4   ݠ@I@i@i@EG@@ @@@@)not_ge_lt @C<-|k Ѡ Ġ $ߠؠҠ@%Jš@5i@:i@|'@h^,@ zmE@@@@)not_gt_le @F6/נРxqbǠ   YI @ZKO@jiT@oi@\@ m⩚a@^z@@@@@)le_neq_lt @{kd ՠ ; . ~IB<!@L,@i*@i@@^@쐩@@ mX@@@@@/@@@@@@@@@@<_neq @KD֠;% z m _@ xR#@i!@i@@ mV$@@@@@@ H&le_neq @٠ҠzsjTC   @ SR@iP@ i@ Fm@#andЖw@@F^a @"@@@@@ *eq_le_incl @&XQB  ٠ @ T)@Di.@Ii@4@J©9@z^R@@@@@ (lt_stepl @SC<ݠ~oԠ   @ UV@qi[@vi`@{i@f@ m@l@~q@ m|@@@@@ ꠠ(lt_stepr @{t  K > 0@ IV6@i4@i@i@@ m@@@ m Ґ@@@@@ "(le_stepl @àTMߠD.  v h@ Wn@il@i񚠐@i@@^@@@"^@@@@@ Z(le_stepr @-&|fU   @ X@i@i)@#i@@O^'@@*-@Z^ B@@@@@ Ր @2"àd]N   @ Y5@Pi:@Ui%?@Zi@E@^^@K@ mdP@ m y@@@@@ ɠ @iYRԠà)@'ZҚl@iКq@i\v@i@|@ m@@^@ m @@@@@+le_antisymm @2+Ӡ̠" aTF@_[ @i@i@@^=@@^@v@@@@@3)le_succ_r @ԠĠe^U?.y@\=@i;@i{@$^q@8߀ @1^~@ 倠@ 8߀ǐ@@@@@t)lt_succ_l @G@1oՠȠ@]~@3i|@8i@#@qm)@:8߀.@|mG@@@@@,le_le_succ_r @H81٠Ҡzsdɠ@^K@fiP@ki@V@^䩚[@^a@r8߀3@@@@@ڠ,lt_lt_succ_r @{kd ՠ;. @9_䚠~@i⚠@i@@m@mש@8߀f@@@@@ ,succ_lt_mono @?8٠ʠ/naS@l`@i@iU@ mK@m@8߀@8߀@@@@@F,succ_le_mono @נРxqhRA@aP@iN@ i@7^@<^@8߀ө@8߀Ґ@@@@@<_0_1 @  RK<zӠ@b#@qm(@4,@P( @@@@&le_0_1 @F6/נРxqbǠ@cI@^N@ZR@v( @@@@@ˠ<_1_2 @l\UנƠ,@*do@mt@( x@/̀@@>@@<_0_2 @{#ĠRE7@Pe@m@@/̀@@@@&le_0_2 @IBԠ9#xk]@vf@^@@/̀@@@@@=<_1_l @ޠΠǠoh _I8@gG@iE@i@@:m@~@@Em@Jm@%( I@@@@@h@ABC@@}+lt_ge_cases @PI:xޠѠ@h!@<i&@Ai-@{m2@s^@@@@@+le_ge_cases @L<5ݠ֠~wh͠ @ iO@jiT@oi+[@^驚`@^@@@@@٠+lt_gt_cases @zjc Ԡ:-@8j㚠}@iᚠ@i!@b@m @m@@@@@,eq_decidable @B;ܠ͠2 qdV@ok@i@i)DecidableZ@)decidable r@W@@@@@B&eq_dne @Ӡ̠tmdN=@lL@iJ@iIK@ @@@@@@t&le_ngt @G@1oՠȠ@m~@3i|@8i$@e^+@ym@@@@@&nlt_ge @E5.֠ϠwpaƠ@nH@ciM@hi젩V@m䩚[@^@@@@@Ԡ,lt_decidable @ue^Ϡ5(@3oޚx@iܚ}@iĐ@m@@@@@<_dne @/(Рɠ ^QC@\p@i@iE@m?@mD@@@@@/&nle_gt @РaZQ;*u@q9@i7@iw6@"^o@4ml@@@@@_<_nge @2+kZ@ri@ig@#i@]mk@W^@@@@@,le_decidable @0 b[L@s3@Ni8@Si?@^͐@@@@@&le_dne @YIBuڠĠ @tš\@wia@|il@^q@^@@@@@ꠠ*nlt_succ_r @{t K>0@Iu@i򚠐@i2񐩚@m)@8߀t@m5@@@@@ 5lt_exists_pred_strong @RKݠB,tf@v嚠@i/@i-@i@@"m#@@^#z@"ex @@ i!k@ i@ 倠@ 8߀Ʃ@@^@@@@A@ @Ah@iAРD.Morphisms_Prop ;@oA1@D@ABCD@@.lt_exists_pred @ (  ZSD۠@wL +@ Fi 0@ Ki@ 6@mĩ\ <@ WiV A@ \iI H@ ^倠 N@_8߀ S@^@@@I@A@@͠,lt_succ_pred @ n ^ W٠Ƞ.!@,x q@ iܚ v@ i@ |@m  @ 倠 @8߀ @j_@@@S@AP@@@@ABCD@@ &rs_rs' @    = 6ޠנȠ-l_Q@jyҶ@ @ iжΩ̰@ @ iة @ iޚ @ `N蚠 @ i@@4 @ i@ @^#@  @8߀&I @ i@ @*^?@V @! iT @!i@ @<^ @!@PmKK'֩H)@@y@@(rs'_rs'' @! !!    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