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U\D{VOܠiXok@@b@@X(eq_trans @{@6Equivalence_TransitiveWt@s={x@r=|@qB@@*Transitive++-@={@=pcTYLu@-W\[DUE6٠РslG@1֠XGĠ@@@@*eq_sym_iff @tdU.'ʠàf_Pݠ̠wf@b@!x@={!y@={%Logic$Init@#iffС)@=BA@=@@@@AA@@A@'neq_sym @@7ڠӠvo =%@A!n @={!m'@&={@I@#notШ4@.=D ;@5=KC@@@@@D(eq_stepl @ޠxSL٠~fU@B_@^={d@c={!zk@j={@q@k=CB@z@t=D@{=D@@@Aۓό^)Morphisms@@AB@@(succ_inj @J:+ΠŠha<5&ˠM<@8C"n1@={"n2@={@@=@p=Vѩ@v=V֩@=\@@@AviAK@AAۀߠKN@R@@ABC@@ޠ+succ_inj_wd @xQJs@DM@={K@={@= @=VJ@=VI@=%@@@G@A8ӠA4Relation_Definitions)Relations)@AB@@%/succ_inj_wd_neg @ޠΠbY4-Рɠ_G6@E"@@?={ E@D={fN@H=T@=VY@=V1`@Z=p@@@@@g1central_induction @ >7ڠӠvo x#@F!A@@={A$A_wd@&ProperL@@={ @*respectful%WO?@={@=B@={@q@@={ԠE@m=V@={F@@@@@̠'add_0_r @uf ?8۠Ԡwpaݠ@@A&NZBase%NZMul&NatInt'Numbers#Coq@@A@%NZMul&NatInt'Numbers#Coq@)NZMulProp@A@@@@"NZ@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@,@$zero,筷T@@A@ABCDE@A@E@Fd@c={i@c=o@D=1~t@=@@@@@y@ABC@@AA@@A@*add_succ_r @:*XQ,%v@A@={@={@=@=1@e=V署@j=V񀐩@=1ѐ@@M@@J-add_succ_comm @qb;4נРsl]٠ܠˠ@Bɚ@={ǚ@={@=@=1@=V< @=1@=VA@@@@(add_comm @ȠLCߠ{I1 C6(#@C.@-={3@2={8@2=>@=1NC@=1K@@@@̠'add_1_l @~!YRߠl[~qc^M?@JDKi@h={n@h=t@I=1y@;=F~@-=V@@ @@'add_1_r @>. \U0)z@E@={@=@=1@v=F@h=V@@E@@B)add_assoc @yiZ3,ϠȠkdUѠ٠Ԡà@F@={@={!p@={@=@=1@=1  @=1 @=1@@@@,add_cancel_l @ǠKBޠzH0B5'"@G -@ ,={  2@ 1={N 7@ 6={X >@ 8= D@=1S  J@=1R O@ I=ސ@@@@ؠ,add_cancel_r @ -&ɠ e^xg}ojYK@VHW u@ t={U z@ y={ @ ~={ @ = @a=1Q @g=1 @ =&@@#@@ ,add_shuffle0 @ W G 8۠Ҡun IB3ؠҠŠ@I @ ={ @ ={ޚ @ ={ @ = @=1 @=1g橚 @=1 @=1W@@o@@l,add_shuffle1 @    ' ]V$ @J뚠 @ ={隠 @ ={* @ ={!q @ ={ @ = %@=1 +@=1𩚠 1@=1A 6@ =1 <@=1© A@=1@@@@ʠ,add_shuffle2 @     |  WPݠjY|oa\K=@HKI g@ f={G l@ k={ q@ p={^ v@ u={ {@ u= @V=1 @\=1\ @a=1 @f=1 @l=1 @r=1,@@)@@&,add_shuffle3 @ ] M >  ؠ { t  OH9ޠƠؠˠ@L @ ={ @ ={䚠 @ ={ @ = @=1 @=1 @=1󩚠 @=1]@@u@@r'sub_1_r @    - $ Ǡ  c \*$ @M񚠐 @ ={ @ = @=LC) @=F $@=|]@@@@'mul_0_r @  Ԡ Š h _     : 3֠ϠeM<_RD@]@) G@ F={ L@ F= R@=C׀a W@= [@=@@@@AA@@A@*mul_succ_r @      < 5 ؠ Ѡ t m  v~@Ac @ ={a @ ={ @ = @Y=C׀ @F=VЩ @q=1 @j=C׀@@.@@G(mul_comm @ b R C  ݠ  y     T M >ˠݠР@B @ ={ @ ={ @ = @=C׀婚 @=C׀@@f@@/mul_add_distr_r @   {     T M     v  @Cߚ @ ={ݚ @ ={ @ ={ @ = @=C׀ @=1& @=1 #@=C׀ (@=C׀8@@@@͠/mul_add_distr_l @  ؠ ɠ l c     > 7 ڠ Ӡ Ġ i Q @cVH@aD- K@ J={+ P@ O={l U@ T={ Z@ T= `@ (=C׀% f@ ;=1v k@ @=1 q@ 9=C׀ v@ >=C׀@@@@)mul_assoc @ 6 &    T M     ( !    @E{ @ ={y @ ={ @ ={ @ = @ v=C׀s @ |=C׀ĩ @ =C׀ @ =C׀N͐@@K@@d'mul_1_l @  o `     9 2 ՠ Π q j [   נ@FĚ @ ={ @ = @ =C׀ @ =F@@~@@'mul_1_r @    6 - Р ɠ l e      3  - @+G @ ={ @ = @ =C׀/ %@ =F3@@@@ʠ,mul_shuffle0 @  ՠ Ơ i `     ; 4 נ Р  f N =`SE@^H* H@ G={( M@ L={i R@ Q={ W@ Q= ]@ %=C׀ c@ +=C׀q h@ 0=C׀ n@ 6=C׀}@@@@,mul_shuffle1 @ .     L E    }      @Is @ ={q @ ={ @ ={ @ ={ @ = @ s=C׀ @ y=C׀ @ ~=C׀Ʃ @ =C׀ @ =C׀G @ =C׀:@@R@@k,mul_shuffle2 @  v g     @ 9 ܠ ՠ x q b   ޠ@J˚ @ ={ɚ @ ={  @ ={ @ ={ @ =@ =C׀ @ =C׀ީ@ =C׀ @ =C׀@ =C׀@ =C׀@@@@à,mul_shuffle3 @ ޠ Π  b Y     4 - Р ɠ  _ G 6YL>@WK #A@@={ !F@E={bK@J={P@J=V@ =C׀ \@ $=C׀ la@ )=C׀ qg@ /=C׀ې@@@@ %le_wd @'   E >  ڠ } v      @@A&NZBase'NZOrder&NatInt'Numbers#Coq@ ) '@A@'NZOrder&NatInt'Numbers#Coq@+NZOrderProp 8 6@A@@@@"NZ - ,@A@@@@={@@={,@={@@={7@=)@={A@= ꚠ@ =p@@Z@@AA@@A@*lt_le_incl @r    K D    | m   j]O@hA ֚@={ Ԛ@={@@"lt= @ =p @@@@@7'le_refl @ŠI@  ܠ  x      F . @B (@'={-@ =p < <@@ 1@@`.lt_succ_diag_r @ޠϠri    D =  ٠ ʠ o W FǠ@C 3Q@P={V@W= e\@ =V @@@@@.le_succ_diag_r @ :3 ֠ Ϡ r k      t@D a@~={@ c=p @ 9=V Ð@@@@@/neq_succ_diag_l @J:+ΠŠha    < 5 & ˠ  #@!E @={ @=@ i=V  Ȑ@@ @@젠/neq_succ_diag_r @zj[4- Р ɠ l e V   ҠSF8@QF @={ @= 󩚠@ =V #@@@@@/nlt_succ_diag_l @.%Ƞd]     +  vh@G  @ ={ 吩@=@ =V S (@@@@@L/nle_succ_diag_l @ڠʠ^U0) ̠ Š  [ C 2@H =@<={ D@ #=pJ@ =V  X@@@@@|)le_succ_l @ (!Ġ`Y     s b֠@I Om@l={ Mr@q={ y@ X=p@ .=V  @= @@ @@+le_gt_cases @D4%Ƞb[6/  Š  @J @={ @={ @"or @@ =p ɩ@= Ɛ@@ J@@-lt_trichotomy @~n_81Ԡ͠piZ   ֠WJ<@UK Ú@={ @={:@= A@= @= @@@@@+(lt_asymm @=4נРsl:"w@L @={ !@ ={@'@(= 7 .@/= @@ a@@`(lt_trans @ޠϠri D=٠ʠoWFǠ@M 3Q@P={ 1V@U={r[@Z={@a@b= @g@h= l@m=  1@@ @@(le_trans @-KD| @N r@={ p@={@={@@=p /@@=p 5@=p?@@ 7@@ݠ+lt_strorder @k[L%]VGԠàD7)@BO@+StrictOrder.[@@={@=@@ @@ +le_preorder @zSLureW@pP @(PreOrder@@={@=p@@ @@9/le_partialorder @ǠKBޠzH0@Q7@,PartialOrder5͛К0@/={4@.=8@-B‚<@=p[@bPN@@@ s@A @E@FAhCV@ABC@@y)lt_compat @U@%lt_wd>_@ ̰@]@\={@b@a={ ݩ h@g={@m@l={ 蚠q@k= w@v={ 򚠐{@u= @=@@@%lt_wd']ɠA1"Š_X3,  @@@@@Р(lt_total @@>@ @={ @={ @= ͩ@= ԩ@= ѐJ3@ aǠvg @9ܠՠxqbޠ_RD@@@@@'le_lteq @@+lt_eq_cases=]`@ ٚ@={ ך@={@=pW @ =@ =@+lt_eq_cases&yˠOF~!L4#@@@@@Z0Private_OrderTacLA'IsTotalT A1 @D@9B@W@+Equivalence~@P@O={T@N=@(eq_equiv7L.'ʠàf_yhܠ@@@@@ @@3Հ@@={@=@,Q^?/ à]V1* @@@@@ΠU @@] Af@ @@={@@={ 0 @={@@={ ;@= -@={ E@=@=S{lE>ڠ}vg dWI@@@@@J @@R b@ܚ@={ښ@={S@=Z @=@=IˠOF~!L4#@@@@@ZE @P@M=5@:@9={?@>={`F@%=pVM@N=]R@L=bC )"ŠaZtcנ@@@@@@@@@"A@#TacGA @1!OH#  @@D!P)OrdersTac*Structures#Coq@rk@@@A@@C!O@@&@ @!t&J@@A@A&a@@"eq&a^@@A@&@ @"le&ڠh@@A@AB&@@"lt&h@@A@&̌@@#one&̌T@@A@A&̔I@@#two&̔IT@@A@&H@@$pred&Hd@@A@&؝[@"@$succ&؝[r@@A@&C@'@$zero&CT@@A@ABCDE@A@G,MakeOrderTacBo?@A@@@BONR@@@A@@AEUH@@@N@!tJ@@A@A$:@@"eq$:^@@A@$@ @"le$h@@A@$@@"lt$ h@@A@ABC@A@+*OrderFacts.#@A@@s@6 @ ={@ =@@@@AA@@A@ @ӠàWN)"ŠT<+]VP5@Af<@;={A@ =p@@D@@0)lt_irrefl @|WPݠjYڠ͠Р~c@Bj@i={Bq@r=D@@t@@`g @0 NG"  @CÚ@={@={@@=@=n@@@@*le_antisym @iYJ#[TEҠB5'8(@9D@={@={@@=p@@=p@= @@@@נ @+"ŠaZ(sevf1*$ @wE:@={8@={@@=-$@=@@@@@*interp_ord @!o)OrdersTac3@#ordȠ@ @@@@@@@@@@@D@F@E={@K@J={YP@J=T@U=X@7=p@*(@^@]={@c@b={ސ +k= R'6'6'6''+k+7T'+7T'6'ࠒA̠ܠ۠@@"lt\ @@"eq8L@lР@TȠ@@QPO@@A@B@A@xA܀M=.ѠȠkd?8)Π&  נРʠ@@@@@%trans @p`Q*#Ơb[L٠ȠI<.?/@@F"o'@={@={~@={@ @a5:@@a5 @a5@)trans_ord-0uGDR̐@@;@@ @@@={>@={ @={B@r=gМA:9@Z0@/={X5@4={њ:@9={@@@:=϶@F@@=թK@E=ߐ L+k() LL7#'S@$t-&ɠ e^xg۠͠ޠΠq@@@@@a @@={@={!@={d"@=gМbCY@Ě@={š@={;@={@@=p9@@=p?@=pI L+k() NN7#'jtueV/(ˠĠg`Qޠ͠ N A 3D4@@@@@Ǡ @@={@={@={@B=gМB  @*@={(@={ @ ={@@ =@@ =@ = L+k() MM7#'Аt۠ˠ_V1*͠Ơ\D3   e^X=@@@@@-+le_lt_trans @wM@L={uR@Q={W@V={y1@=gg#nm@d@c={i@h={n@m={@t@S=p@z@ {= @ = L+k() MN7#'4t?/ à]V1*   ɠ @@@@@+lt_le_trans @ۚ@={ٚ@={R@={S@ =g1@򚠐@={𚠐@={i@={@@ =g@@=pm@ =w L+k() NM7#'t']V$  | o arb-& @@@@@%eq_lt @?@={=@={@={A@q=g/65@V,@+={T1@0={͚6@5={@<@6=˶@B@ C=ѩG@ H=ې L+k() ML7#'t%]Vp_  Ӡ Š֠Ơi@@@@@Y%lt_eq @y@x={~@}={@={]@=gcO@@={@={1@={@@ =/@@=5@ =? L+k() LM7#'`tk[L%]VGԠà D 7 ):*@@@@@%eq_le @@={@={~@={ @9=g]@@={@={@={@@=@ @=p@=p L+k() NL7#'ĐtϠSJ%P8'   YRL1@@@@@!%le_eq @kA@@={iF@E={⚠K@J={m%@=g+ba@X@W={]@\={b@a={@h@G=p@n@h=s@R=p  L+k() LN7#'(t3#QJ%   @@@@@&eq_neq @VF7ڠѠtm HA2נ / " %٠Ӡ@&G隠@={皠@={`@={@@=^@@=f@= r@@@@@͠&neq_eq @"XQz w j \m](!@nH1@={/ @ ={@={@ꐩ@=@@=&@ = @@@@@*not_neq_eq @֠Ǡja<5ؠѠ gO>   picH@IyO@N={wT@S={@-/^@X=nc@]=@@@@R)not_ge_lt @#A:ݠ֠yr{   @J@={@={@j@x=p@ =-@@@@)not_gt_le @^N?٠|uPI:ߠǠ 7 * -۠@.K񚠐@={@={@@ =ܩ@=ph@@@@@Ƞ)le_neq_lt @zSLu r e WhX#@iL,@={*@={@ @=p@搩@=@ =@@@@@ @@@@)@@@@@)@<_neq @ޠΠbY4-Рɠ_G6   @ R#A@@={!F@E={@L@ M=\$S@M=@@@@@ &le_neq @1*͠Ơib|k  ߠ @ SXv@u={V{@z={@ =@#andЖw@@n=pg@=@@@@@ Ƞ*eq_le_incl @VF7ڠѠtm HA2נ/"@-T@={@={@@=ԩ@=pX@@@@@ (lt_stepl @yj C<ߠؠ{te bUG@`U@={@={@={@@ =@@=@=@@@@@9(lt_stepr @ǠKBޠzH0@VT*@)={R/@.={˚4@3={@:@;=ɶ@@@:=ϩE@F= ِ@@@@@w(le_stepl @#[Tn]ޠѠ@Wh@g={m@l={ r@q={@x@W=p@~@x=@b=p@@@@@(le_stepr @C3$ǠaZ5.Ġ@XК@={Κ@={G@={@@=pE@@=K@=p U@@@@@ @pa:3֠Ϡrk\ؠYL>@WYŚ@={Ú@={@={@@=p@@=@= @@@@@0\ @A8۠Ԡwp >&{@Z @={%@$={A*@)={@0@1=@6@=pũ;@<= ϐ@@@@@m+le_antisymm @ܠvQJנ|dSԠǠ@[@^@]={>c@b={@i@H=py@o@N=p4t@n=I@@@@@)le_succ_r @4$RK& @ \y@={w@={@=p@X=V @=p@=ũ@j=V@@@@@)lt_succ_l @{k\5.ѠʠmfWӠTG9@R]@={@={@@=@=V.@=@@@@@&,le_le_succ_r @8/Ҡˠng 5 r@^@={@={@"@=p2'@=p쩚-@=Vl@@@@@_,lt_lt_succ_r @ݠΠqh C<ߠؠɠnVEƠ@_2P@O={0U@T={@[@\=k`@a=%f@=V@@@@@,succ_lt_mono @&D=٠|u~@`k@={i@={@=@=@O=Vߩ@T=Vސ@@@@@נ,succ_le_mono @eUF|WPAΠ>1#@@WgŚ @ ={Ú @ ={@ @= @=@ @= @=!@=Fǐ@@@5@3@@ABC@@8+lt_ge_cases @ Ơ   J AݠyG/@h !)@!(={ !.@!-={!5@6=E!:@=pB@@@@@l+le_ge_cases @   ۠ ~ u  PI֠{cRӠƠ@i?!]@!\={=!b@!a={!i@H=py!n@M=pv@@@@@+lt_gt_cases @!.!!   L E}  @js!@!={q!@!={p!@!=󠩚!@=!@=@@@@@ݠ,eq_decidable @!k![!L     % ]VGԠàD7)@Bk!@!={!@!={)Decidable!@)decidable r!@!=@@@@@&eq_dne @!!!!'!   ] V$ |oa@zl蚠"@"={暠" @" ={,吩琩"@"=&"@"=+@@@@@M&le_ngt @!۠!ˠ!!_!V     1 *͠Ơ\D3@m ">@"=={"C@"B={d"J@)=pZ""Q@R=Y@@@@@&nlt_ge @""!!!!/!( ˠ Ġ g ` ziݠ@nV"t@"s={T"y@"x={S"@="@f=p@@@@@,lt_decidable @"G"7"(!ˠ! !e!^!    9 2 #Ƞ @o"@"={"@"={ܐ"@=Ɛ@@@@@蠠<_dne @"v"f"W!!!!!0!) ̠ Š h a RߠΠOB4@Mp"@"={"@"={"@="@=@@@@@ &nle_gt @""""2")!̠!Š!h!a!     /  zl@q󚠐#@#={񚠐#@#={7𐩚#@=p/#$@%=,@@@@@V<_nge @""Ԡ"Š"h"_"!!!!:!3 ֠ Ϡ  e M <@r)#G@#F={'#L@#K={m#S@T=c+#Z@ 9=pb@@@@@,le_decidable @## """"8"1!Ԡ!͠!p!i! !    r@s_#}@#|={]#@#={#@ h=p@@@@@&le_dne @#I#9#*"͠"Ġ"g"`"!!!!;!4!% ʠ  "@ t#@#={#@#={Ҡ#@ =p̩#@ =pѐ@@@@@*nlt_succ_r @##q#b#"""";"4!נ!Р!s!l!]!  ٠ZM?@Xuƚ#@#={Ě#@#={ Ð#@=#@ =V7#@= @@@@@/5lt_exists_pred_strong @####A#8"۠"Ԡ"w"p"" !!!!>!&!{@v$ @$={$%@$$={$*@$)={@$0@1=@$6@!=pX@"ex @$B@$A={!k$I@$H={Ǡ$P@$J=֩$V@!=V$[@!:=pj@@@@A@@A b@ cAРD.Morphisms_Prop$s@ iA1@D@ABCD@@.lt_exists_pred @$*$$ ###H#A""ݠ""y"""!!!@w$$@$={t$@$={@$@=b$@$={\$@$={!$@$=o$@!_=V驚$@!=p@@@O@A@@蠠,lt_succ_pred @$v$f$W#####0#)"̠"Š"h"a"R!!ߠ!ΠOB4@Mxp$@$={$@$={@$@=$@$=$@!=V񀐩$@!=|4 @@@(@A%@t@ @@ABCD@@,&rs_rs' @$$$$>$5#ؠ#Ѡ#t#m## """";"#"x@y@%@%={@%&@%%={%,@%+={%0@%*= P >̚%5@%4={@@ %<@%;={@%B@"!=p@>%K@!=V  3%Q@%P={@%W@"6=p@ @%^@%]={ >%c@%b={@%i@"H=pc x@%p@q= 5K'H)@@@@(rs'_rs'' @%4%$%$$$R$K#####&##""" @ z@%@%={@%@%={ %@%={!%@%=   F%@%={@ %@%={@%@"=p ˶@ %@%={ %@%={@%@"=p ܶ@%@= I  %@%={  ƚ%@%={ Ě%@%={@%@"=p"@%@= ! . ݚ%@%={ ۚ&@%={@&@"=p9@& @ = 8&@"=V K@@ E@@D%rbase @%Ҡ% %%V%M$$$$$($!#Ġ###S#;#*@{@&6@&5={ Қ&;@&:={!#&A@&@={!!&F@&E={@&L@#+=p@&R@S=!| @@@@@)A'A_right @&&%%%%0%)$̠$Š$h$a$####{#jޠ@|@&v@&u={!&{@&z={@!c&@&={!i&@&={!g&@&={@&@#q=p@&@=!]N Ҷ!&@&={@&@#=p @@@@@נ6strong_right_induction @&e&U&F%%%%|%%$$$W$P$A##Π#>1#@<} F@&@&={ D B @@&@&={ L :&@&={ R&@&=!!!w&@&={@!Ț&@&={@&@#=p!@!՚&@&={!Ӛ&@&={@&@#=p1@'@=!0!C.!뚠' @'={@'@#=p̐@@@!B@A!@#@@ABC@@E/right_induction @&Ӡ&à&&W&N%%%%%)%"$Š$$$T$<$+@~ @'7@'6={   @'?@'>={  'E@'D={ 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