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U\DNG1*|fUlh@@_@@U(eq_trans @@6Equivalence_TransitiveWРɠqj TM>@@*Transitive++-@# @# !ܐTҐL@,W\ԐDA:۠|mΠRA@@@@*eq_sym_iff @,^WՠĠo^@Z@!x4@O# !y;@V# %Logic$Initq@#iffС)L@b# !BAT@j# !@@@@AA@@A@'neq_sym @rb[E>ߠР1 @A!nz@# !m@# @G@#notШ@# !B @# !IC@@6@@B(eq_stepl @B;ܠ}%pZI@B@# @# !z@# @@# !4}@@# !D@# !B@@@wAۓό^)Morphisms@}@AB@@(succ_inj @+$̠ŠmfW<+@'C"n1@# "n2@## @@$# !@_#ȩ@e#ͩ @6# !W@@@AviAI@AAۀߠKL@P@@ABC@@ՠ+succ_inj_wd @D4-ՠΠvoܠv@rDKJ@e# IO@j# V@l# !\@#Ha@#Gf@|# !@@@E@A8ӠA4Relation_Definitions)Relations@AB@@/succ_inj_wd_neg @yr\UH2!̠@E@# @# Y@# !@#@#&@# !c@@@@@Z1central_induction @ɠZS=6'ra @F!A@@# A$A_wd@&ProperL@@#  @*respectful%WO?@ # @ # !7@# @d@@# ŠE @V#@+# F@@@@@'add_0_r @,^WՠĠ@@A&NZBase%NZMul&NatInt'Numbers#Coq@nl@A@%NZMul&NatInt'Numbers#Coq@)NZMulProp}{@A@@@@"NZrq@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@,@$zero,筷T@@A@ABCDE@A@E@7@# @# !@+# m@#u(q@@@@@`@ABC@@AA@@A@*add_succ_r @ߠϠȠpi SL=wt@Aq@# o@ # @ # !@j# @J#婚@O# @{# @@K@@H-add_succ_comm @$ VO͠ݠؠǠ@B.@I# 3@N# 8@N# !>@# D@#0I@# O@#5@@@@(add_comm @iYR<5ݠ֠Ǡ(=0" @ Cs@# x@# }@# !@# 7@# 4@@@@Ơ'add_1_l @3,Ԡ͠unaK:vi[VE7@BD4@# @# !@(# @# .n@ #@@@@'add_1_r @۠ˠĠle OH9s~p@{Em@# @# !@a# @S# .@E#@@;@@8)add_assoc @F?rӠ۠͠Ƞ@F@9# #@># !p*@E# /@E# !5@# ;@# 署@@# F@# }@@@@,add_cancel_l @`PI3,Ԡ͠ 4'@G򚠐j@# 𚠐o@# Lt@# 9{@# !@# 4쩚@# 3@# !Ð@@@@ʠ,add_cancel_r @70ؠѠyreO>zm_ZI;@FH8@# 6@# @# @# !@8# 2z@># @# ! @@@@,add_shuffle0 @ܠՠ}v`YJ@I~@ # |@ # ؚ @ #  @ # ! @|#  @# Hé @#  @# Uϐ@@]@@Z,add_shuffle1 @ 6 & Ǡha ߠΠ ٠@JȚ @@ [# ƚ E@ `# " J@ e# !q Q@ l#  V@ l# ! \@#  b@# ͩ h@#  m@#  s@#  x@# @@@@,add_shuffle2 @   { # Ġe^Q;*fYKF5'@2K$ @ # " @ # ~ @ # \ @ #  @ # ! @'#  @-# Z @2# u @7#  @=#  @C#  @@@@,add_shuffle3 @  ܠ ՠ } v  `YJ@L~ @ # | @ # ؚ @ #  @ # ! @|# v @# ũ @# ʩ @# U@@]@@Z'sub_1_r @ 6 &  Ǡ  h a  ߠΠ ٠@MȚ @@ [#  E@ [# ! K@# T P@# . U@#z;@@@@'mul_0_r @ o _ X     B ;ܠ͠.C6(@A@ v@ #  {@ # ! @# 耠4 @#u( @#u(@@@@AA@@A@*mul_succ_r @    8 1 ٠ Ҡ z s   fP?{n`@yA6 @ # 4 @ #  @ # ! @"# 耠r @# @:#  @3# @@@@E(mul_comm @  ٠ Ҡ z s     ] V G@Bx @ # v @ #  @ # ! @d#  @i# @@F@@{/mul_add_distr_r @      Q J     }ޠȠ@C &@ A#  +@ F#  0@ K#  5@ K# ! ;@# 耠 A@# x󩚠 F@#  L@#  Q@# @@@@Ǡ/mul_add_distr_l @ k [ T     > 7 ߠ ؠ ɠ *  ?2$@=D r@ #  w@ # T |@ #  @ # ! @# 耠򩚠 @# A @ #  @# ϩ @ # ֐@@@@)mul_assoc @    H A     + $  v ` O~p@EF @ # D @ #  @ #  @ # ! @ 7# 耠> @ =#  @ B# 耠 @ H# @@%@@Z'mul_1_l @      0 ) Ѡ ʠ r k \   ҠŠ@F @ #  @ # ! @ t# 耠 @ s# .ǐ@@V@@'mul_1_r @ /     a Z       ؠ Ǡ@G 6@ Q#  ;@ Q# ! A@ # 耠 F@ # .@@@@,mul_shuffle0 @ ` P I     3 , Ԡ ͠    4'@2H g@ # 횠 l@ # I q@ #  v@ # ! |@ # 耠 @ # 4 @ # 耠 @ # @@@@@,mul_shuffle1 @    8 1 ٠ Ҡ z s    f P ?{n`@yI6 @ # 4 @ #  @ # n @ #  @ # ! @ ,# 耠 @ 2# l @ 7#  @ <# 耠 @ B#  @ G# @@$@@Y,mul_shuffle2 @      / ( Р ɠ q j [   ѠĠ@J @ #  @ $# 暠 @ )# Ě @ .#  @ .# ! @ # 耠 $@ # © )@ # ݩ .@ # 耠 4@ # h 9@ # p@@z@@,mul_shuffle3 @ S C <  ݠ  ~ &  Ǡ     ' @%K⚠ Z@ u#  _@ z# < d@ #  i@ # ! o@ # 耠ک u@ #  ) z@ # 耠 . @ # @@@@%le_wd @    + $ ̠ Š m f    Y C 2@@A&NZBase'NZOrder&NatInt'Numbers#Coq@  @A@'NZOrder&NatInt'Numbers#Coq@+NZOrderProp  @A@@@@"NZ  @A@@尶@ @ # @ @ #  @ # @ @ #  @ # ! @ #   @ # !  @ `# !@@&@@AA@@A@*lt_le_incl @     4 - ՠ Π v o `   h[M@fA  @$#  @)# @@"lt# ! ɩ@ # ! Q@@@@@5'le_refl @4$ Š  f _       ݠ ̠@B Ú;@V# @@ # !  @@ @@\.lt_succ_diag_r @[KD     . ' Ϡ Ƞ    @C ꚠb@}# g@S# ! m@ # S@@@@@.le_succ_diag_r @wp   Z S    F 0 @D @# @ # ! F@ # @@@@@/neq_succ_diag_l @D=  ޠ   '   r \ K @E B@#  8@# !@ #  y@@ @@⠠/neq_succ_diag_r @Ѡʠrk    U N ?   yG:,@EF p@#  f@# ! @ @# ې@@@@@/nlt_succ_diag_l @A:  ۠  | m Π  uhZ@sG @1#  @ # !#@ n#  Ր@@@@@>/nle_succ_diag_l @=-&ΠǠoh       ՠ@H ̚D@_#  K@ # !Q@ # 7 @@@@@l)le_succ_l @k[T>7 ߠ ؠ ɠ *  ѠĠ@I r@#  w@#  <~@ # !@ # p 6@u# ! =@@ i@@+le_gt_cases @4-ՠΠvo b L ; @J 2@#  0@#  u@"or @@ 7# ! p@# ! m@@ @@ܠ-lt_trichotomy @۠ˠĠle OH9   sA4&@?K j@#  h@# 8@# ! ?@ # ! @# ! @@@@@(lt_asymm @F?r Ӡ  zm_@xL @6#   @;# @&@# ! ک -@# ! @@ @@H(lt_trans @G70ؠѠyr  ߠ@M ֚N@i#  ԚS@n# 0X@s# @^@J# ! @d@P# ! i@U# ! g Ԑ@@ J@@(le_trans @tmWPC-ݠ@N @#  @# m@# @@# ! Ҷ@@# ! ة@!# !=@@ @@+lt_strorder @QJ4-iX& @$O@+StrictOrder.[@@# @# !@@ q@@+le_preorder @ܠՠ}v`YJRE7@PP%@(PreOrder@@# @y# !@@ @@/le_partialorder @JCvנ~qc@|QP@,PartialOrder5͛К%@@# )@?# !-@>3󀚠1@# !U@^@@@ @A ^@ @ AhCo@ABC@@W)lt_compat @n@%lt_wd[KD.'ϠȠ@ @^@y# @c@~#   i@# @n@#  r@# ! x@#  |@# ! @@l# !@͠@@%lt_wd']ɠ-&ΠǠoh [E4@@@@@(lt_total @_@3v䀠OH2+@ G@#  E@# @# ! @# ! @# ! W*@aǠޠנx b[LTG9@@@@@ 'le_lteq @!@+lt_eq_cases*Ȁ@9ڠ{l@ @*#  @/#  ٠@# ! ϩl"@# ! ֩'@=# ! ې@+lt_eq_cases&y=-&ΠǠoh ՠ@@@@@Y0Private_OrderTacgA'IsTotalo Ac @kfVO92ڠӠ@@+Equivalence~@n@# r@# !@(eq_equiv7Lxq[TG1 @@@@@ @@;{70ؠѠyr@ࠚ@# @# !@,Q^RK5.jY' @@@@@ݠ @$@e ߠϠȠpi SL=@ @@# @@#   @# @@ #  @ # ! @#  *@# !Ě@# !IBu֠}pb@@@@@3{ @z@/C05%Ơg`@6@Q# ;@V# B@.# !I@_# !N@:# !waQJ4-ՠΠ  Ǡ@@@@@}s @@{Yoh RK@@# @# J@# !@ݠ@# !G@# !Lq<5ݠ֠~w jTC@@@@@@@@@@#TacuA @Ϡ`YC<-xg5(@@D!P)OrdersTac*Structures#Coq@C-@@@A@@C!O0@@&@7@!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@'@$zero&CT@@A@ABCDE@A@G,MakeOrderTacBm?@A@@@BONR@@@A@@AEUH@@@N@!tJ@@A@A$:@@"eq$:^@@A@$@ @"le$h@@A@$@@"lt$ h@@A@ABC@A@+*OrderFacts.#@A@@s@N@i# S@i# !@@@@AA@@A@< @o_XB;ܠ͠.ՠȠ[TN3@AJ|@# @# !A@@ @@.)lt_irrefl @,%͠ƠngZD3̠z_@Bv@# &@# !o@@N@@\A @ȠYR<5&q`.!@C@# @# @@# !@# !P@@@@*le_antisym @1*Ҡˠsl]eXJ0 ޠ@1Dښ @'# ؚ@,# @@# !˶@@# !"@8# ! @@@@Ϡʐ @;+$̠ŠmfӠl\' @mEH@c# M@h# @̐U@k# ! Ӑ\@r# !ǐ@@@@@ *interp_ord @!o)OrdersTac@#ordȠ@ @@@@@@@@@@@D@~@# @@# 5@# !@x# !@ # !@*(@@# @@# ɐ +k= R'6'6'6''+k+7T'+7T'6'ࠒA ҠѠ@@"lt\ @~@"eq8L@lР@TȠ@@QPO@@A@B@A@xA܀ߠϠȠpi SL=wE8*ˠĠ@@@@@%trans @2+Ӡ̠tm^fYK1!ߠ@2F"o'ߚ@,# ݚ@1# Z@6# @Cp@:ٝ!׶@Ly@:ٝ S@:ٝ@)trans_ord-0uG+9@@@@頠| @nop@f МAeUN81٠Ҡ@4f@# 2k@# p@# @v@# !@|@# !@# ! L+k() LL7#'G@$t0)Ѡʠrk ^H7 Р~c@@@@@S< @ط|ٷکjМLC!jiǠXQ;4%@@# @# @# @@S# !@@Y# !@^# !z L+k() NN7#'bt/(Рɠqj[ c V H.ܠ@@@@@א @67S8МBǠ%WP@&@A# 򚠐+@F# o0@K# @6@ "# !m@<@ (# !sA@ -# !ؐ L+k() MM7#'t[KD.'ϠȠ   |G@:@@@@@+le_lt_trans @?9'_$#qj TM@P@# N@# ˚@# @@ # !ɶ@@ # !ϩ@ # !4 L+k() MN7#'tHA+$v`O   ؠ{@@@@@k+lt_le_trans @7ݠ͠ƠngQJ;@@# @# '@# @@ # !%@@o# !+@ # ! L+k() NM7#'xtE>ߠqҠ y l ^D4@@@@@Ǡ%eq_lt @MNjO۠9)"ʠàkd @:@U# ?@Z# D@_# @J@`# !@P@ <# !U@ A# !쐐 L+k() ML7#'Ԑto_XB;ܠ͠. ՠ Ƞ [TN3@@@@@#%lt_eq @SM;s387~&Ǡha @d@# b@# ߚ@# @@ # !ݶ@@# !㩚@ # !H L+k() LM7#'0tˠ\U?8)tc 1 $ @@@@@%eq_le @"-Kڠ{#Ġe^O@@ # @# ;@# @@# !9@@# !? @# ! L+k() NL7#'t'YRР   rXH @@@@@۠%le_eq @ ab~cM=6ޠנx @N@i# S@n# X@s# @^@# !@d@z# !i@# ! L+k() LN7#'tsl VOB,  ܠ ΠohbG@@@@@7&eq_neq @5.֠ϠwpcM<   ՠŠh@G@# }@# @# @@# !@@@# !G@# ! g@@@@@}&neq_eq @ڠӠ{t^WH P C 5 ֠Ϡɠ@HŚ@# Ú@# @@# @ @# !@@@%# !F@,# ! @@@@@à*not_neq_eq @0 b[٠Ƞ   {aQ@bI =@X#  B@]# @ÐL@b# !Q@g# !@@1@@)not_ge_lt @k[T>7ߠؠɠ* Ѡ Ġ WPJ/@JFx@# D}@# @@# !1@ v# !@@)@@7)not_gt_le @5.֠ϠwpcM<   ՠŠh@K@# }@# @5@ # !j@># !@@@@@p)le_neq_lt @ݠ͠ƠngQJ;u C 6 (ɠ @L@# @ # @@p# !@t@# !4@ # !@@@@@@@@@@@@@@@<_neq @  RK~ߠɠ  y k@ R'@B# ,@G# @2@ # !9@O# !p@@@@@ T&le_neq @SC<ݠ~&Ǡ   @ S⚠Z@u# _@z# $f@ R# !,@#andЖw@s@# !'񐩚z@# !.@@@@@ *eq_le_incl @}%Ơg`S=,   @ T#@# !@# @@# !Z@&# !@@@@@ Ơ(lt_stepl @ŠVO92#n]+@)U@# @# @# @@ # !@@# !@ # !@@@@@(lt_stepr @3,Ԡ͠un_gZL@eV֚@## Ԛ @(# Q@-# @@# !O@@4# !U#@# ! @@@@@>(le_stepl @=-&ΠǠoh ՠ@WD@_# I@d# N@i# @T@# !@Z@p# !_@# !@@@@@z(le_stepr @yib LEנ8"ߠҠ@XN@# L@# ɚ@# @@ # !Ƕ@@# !ͩ@# ! 2@@@@@9 @E>ߠ(!s]L @YC@# A@# @# @@F# !@@# !@# ! m@@@@@ @ߠؠy! c\MUH:@SZ~@# |@# ؚ@# @@# !=@ @# !C@# ! @@@@@,+le_antisymm @+]VԠàv@[2@M# 7@R# @=@# !@C@# !H@^# !@@@@@c)le_succ_r @bRK5.֠Ϡ! Ƞ@\񚠐i@# n@# 3u@# !){@#a ̠@# !6@# !;@#s@@@@@)lt_succ_l @81٠ҠzsfP? @ ]6@# 4@# @@# !@ #q@# !@@@@@ߠ,le_le_succ_r @ޠΠǠoh RK<vD7)@B^m@# k@# @@k# !@p# !`@F#@@@@@,lt_lt_succ_r @G@sԠ{n`@y_@7# !@<# @'@# !۩,@# !2@}#@@@@@M,succ_lt_mono @L<5ݠ֠~w @`ۚS@n# ٚX@s# _@K# !d@P# !j@#Vo@#U@@@@@,succ_le_mono @yr\UH2!@a@# @# Z@# !P@# !@#@#@@@@@Ǡ<_0_1 @ƠWP:3$o^,@*b@# !@#u(@4# .@@@@&le_0_1 @٠z"àd]NVI;@Tc@r# !@B#u(@^# .@@@@@<_1_2 @ LEx٠àse@~d!@ # !&@# .*@~# @@ @@E<_0_2 @D4-ՠΠvoܠ@eK@7# !P@#u(T@# @@@@o&le_0_2 @n^WA:̠۠-ԠǠ@fu@# !z@#u(~@# @@@@@<_1_l @)"ʠàkd WA0@g'@# %@# @@# !@#u(b@@# !쩚@# !@# .)@@@@@`@ABC@@ݠ+lt_ge_cases @ܠ̠ŠmfPI:tB5'@@hk@# i@ # 9@# !@o# !@@@@@+le_ge_cases @ @9ڠ{l͠tgY@ri @ 0#  @ 5# k !@# !թ &@# !Ґ@@@@@A+lt_gt_cases @ @ 0 )Ѡʠrk ؠ@jϚ G@ b# ͚ L@ g# ̐ U@ k# !  \@H# ! a@M# ! @@@@@|,eq_decidable @ { k d  NG٠:$Ԡ@k  @ #  @ # )DecidableN @)decidable r @ # !K@@@@@&eq_dne @    B ;ܠ}%pZI @l@ @ # > @ # =? @ # !| @ # !@@@@@蠠&le_ngt @  נ Р x q  [TEM@2@Kmv @! # t @!#  @u# !x!@# !@@@@@&nlt_ge @!! !   M FyڠĠtf@n!"@!=# !'@!B# 젩!0@# !䩚!5@# !@@@@@P,lt_decidable @!O!?!8  ٠  z " à@oޚ!V@!q# ܚ![@!v# Ԑ!b@N# !@@@@@}<_dne @!|!l!e! !   O Hڠ;%ՠ@p !@!#  !@!# M !@# !G!@# !L@@@@@&nle_gt @!!!!C!<  ݠ  ~ &  q[J @qA!@!# ?!@!# >!@B# !{!@# !x@@@@@砠<_nge @!!֠!Ϡ!w!p!!   Z S D~L?1@Jru!@"# s!@" # !@# !w"@{# !@@@@@,le_decidable @"" "!!!L!E     x٠àse@~s"!@"<# "&@"A# "-@# !@@@@@H&le_dne @"G"7"0!ؠ!Ѡ!y!r!!    ߠ@t֚"N@"i# Ԛ"S@"n# ӐՐ"^@# !"c@# !@@@@@~*nlt_succ_r @"}"m"f""!!!P!I   ۠ < & ֠@u "@"#  "@"# N "@~# !E"@#"@# !Q@@@@@5lt_exists_pred_strong @""""H"A!!!!!+!$! v ` O@v"@"# K"@"# I"@"# @"@# !@"@ O# !?@"ex @"@"# !k"@## "@## !"@ ?#ک"@ t# !@@@@A@%@A@AРD.Morphisms_Prop#5@A1@D@ABCD@@#.lt_exists_pred @#"## """T"M!!!!!  ˠ {m@wh#)@#D# #.@#I# @#4@ # !`#:@#U# Z#?@#Z# ٠#F@#\# !#L@ #2#Q@ # !@@@M@A@@m,lt_succ_pred @#l#\#U"""""?"8!!٠!ʠ!+!!ҠŠ@x#s@## #x@## @#~@j# !2#@## !#@ ##@ #z{B@@@o@Al@@ 1@@ABCD@@&rs_rs' @####?#8""٠""z"""" !m!W!F@y@#@## శ@#@## #@## 򚠐#@## !z #@## @ն@\#@## @#@!U# !@ȩ #@!.#Nq#@$# @#@!j# !#@~#@$# |#@$# @$@!|# ! @$@# !sK'H)@@}@@'(rs'_rs'' @$&$$###X#Q"""""!!Ϡ!q@z^@$.@$I# \ZX@$6@$Q# dR$<@$W# j$@@$V# ! $E@$`# @Ӛ$K@$f# @$Q@!# ! @$X@$s# ޚ$]@$x# @$c@!# !a @$j@V# !թIX\$r@$#  7 $z@$#  $@$# @$@"# !"@$@w# !!q $@$#  $@$# @$@"# !9@$@# ! 8$@!#@@@@à%rbase @$ $$$S$L#####6#/# ""k"Z( @&{@$@$#  $@$#  ]$@$#  [$@$# @$@"[# !@$@# ! 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!%@'@%A# !"\"#T'@'# #R'@'# @'@%R# !<@'@%X# !#s'@%.#"ɐ@@$@@%lbase @''''''/'(&Р&ɠ&q&j&[%%%cVH@aC"5@(@( # "3#I( @(%# #(@(+# #(@(0# @(@%# !@(!@%# !#X('@%r## @@@@@B(A'A_left @(A(1(*'Ҡ'ˠ's'l'' &&&&%%٠@D"y@(I@(d# "w#(N@(i# @#ܚ(T@(o# #⚠(Z@(u# #(_@(z# @(e@%# !A@(k@%# !#A#Q#(p@(# @(v@%# !@@@%@@5strong_left_induction @(((y(!(' ''c'\'&&&O&9&(@E"ȶ@(@(# "ƶ"ĩ"°@(@(# "Ω"(@(# "Ԛ(@(# !$n$\#(@(# @$=(@(# @(@&6# !$g@$J(@(# $H(@(# @(@&H# !2@(@&N# !$ i(@&$##m$f(@(# @(@&_# ! c @@@#@%@@ABC@@.left_induction @)(((((3(,'Ԡ'͠'u'n'_&&&gZL@eF#9@) @)$# #7#5#3@)@),# #?#-)@)2# #E)@)1# !$$Ͷ$_) @);# @#(@$)'@)B# @)-@# ! @#)5@&#$! $š):@)U# @)@@&# !D@@@@@[/left_induction' @)Z)J)C(((((-(&'Π'Ǡ'''&@G#@)b@)}# ###@)j@)# ##)p@)# #)t@)# !%8%&$)y@)# @%)@)# @)@'# !%9O@%)@)# @)@}# !!@#)@&#$o%&)@)# #@@ @@6strong_left_induction' @))))I)B(((((,(%('w'a'P@H#@)@)# ###갶@)@)# ##)@)# #)@)# !%%%)@)# @%e)@)# @)@'^# !%@%q)@*# @)@'j# !!n@%~)@*# %|)@*# @*@'|# !%#@*@'# !%?D*@'Y#$%*@*.# $@@ T@@./order_induction @*-**)))_)X)((((''֠'Šx@I$e@*5@*P# $c$a$_@*=@*X# $k$Y*C@*^# $q*G@*]# !& %%*L@*g# @$T@%ۚ*S@*n# @*Y@'# !!@ *a@'#~@%*g@*# @*m@Y# !I@4*u@'#%ak&*z@*# @%a@@@%Z@%X@AB@@0order_induction' @****(*!)ɠ) )j)c) )((V(@(/@J$϶@*@*# $Ͷ$˩$ɰ@*@*# $թ$*@*# $ۚ*@*# !&u&c%*@*# @$@&E*@*# @*@(># !!@*@(#@&Y*@*# @*@(R# !@ܩΐ*@(/#z&l*@*# j@@i@@1order_induction_0 @******0*))Ѡ)ʠ)r)k)\(((dWI@bK%6@+@+!# %4%2%0@+@+)# %<%*+@+/# %B+@+.# !&&ʶ@&͐+@(e#u(@&+%@+@# @++@(# !+0@(v#u(&@%!+8@(#U@&ƚ+>@+Y# @+D@0# !&+I@(#u(@%:+Q@(#&='&ޚ+V@+q# %F@@@@@q2order_induction'_0 @+p+`+Y+****C*<))ݠ)Π)/))֠ɠ@L%@+x@+# %%%@+@+# %%+@+# %+@+# !'N'<@'?+@(#u(@'+@+# @+@)# !+@(#u('T@m%+@(#Ƕ@'8+@+# @+@)1# !'i+@)#u(@+@)#z'P+@+# %@@@@@㠠<_ind @++Ҡ+ˠ+s+l++ ***V*O*@)))zH;-@FM&@+@,# &&&@+@, # & &+@,# &&+@,# !''',@,# @'m, @)T#&@',@,*# @,@# !#N@;ܐ,@)h#:',"@,=# @,(@# !"\,@@@@@C&le_ind @,B,2,++Ӡ+̠+t+m++****))ڠ@N&z@,J@,e# &x&v&t@,R@,m# &&n,X@,s# &,\@,r# !( ('隠,a@,|# @&i@'隠,h@,# @,n@)# !#@5,v@)#',{@,# @,@)# !"@@@@@&Rlt_wd @,,,,,,%+͠+Ơ+n+g++**Z*D*3@O'ᚠ,@,# &˰@,@,# @,@,# &ܩ&,@,# @,@,# &皠,@,# !&,@,# 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@@@@@S.add_lt_le_mono @/o/_/X/....B.;--ܠ-͠-.--@-F*/s@/# */x@/# 'U/}@/# &3/@/# @/@!t# !&&@/@- # !*ũ/@!# !/@- # /@-# 5@@@@@.add_le_lt_mono @////I/B.....,.%.-w-a-P@-LG+D/@/# +B/@/# '/@/# &|/@/# @/@-L# !&o@/@!# !+/@!# !/@-S# Z/@-X# ~@@@@@堠+add_pos_pos @0/////3/,.Ԡ.͠.u.n._---@-H+0@0 # +0 @0%# @0@!# !0@-[#u(+ȶ@0@"# !0 @-f#u(+ө0%@"# !0*@-p#u(0/@-# &͐@@p@@-.add_pos_nonneg @0I0902/ڠ/Ӡ/{/t//....--@-I+՚0M@0h# +Ӛ0R@0m# @0X@"D# !0]@-#u(,@0c@-# !0h@-#u(,0m@"Y# !0r@-#u(0w@-# '@@@@u.add_nonneg_pos @000z0"0/à//d/]/...P.:.)@.%J,0@0# ,0@0# @0@.# !0@-#u(,X@0@"# !0@-#u(,c0@"# !0@.#u(0@.0# ']@@@@1add_nonneg_nonneg @0٠0ɠ0 0j0c0 0///M/F/7...q@.mK,e0@0# ,c0@0# @0@.c# !0@.3#u(,@0@.n# !0@.>#u(,0@.x# !1@.H#u(1@.x# '@@H@@,lt_add_pos_l @1!11 000S0L/////..ʠ.@.L,1%@1@# ,1*@1E# @10@## !15@.{#u(,詚1:@#&# !,1@@.# ,w@@@,@-@AB@@@,lt_add_pos_r @1\1L1E00000/0(/Р/ɠ///.@.M,蚠1`@1{# ,暠1e@1# @1k@#W# !1p@.#u(-#1u@#a# !-)1{@.# ,@@,[@@y,le_lt_add_lt @111~1&10Ǡ00h0a0 0//T/>/-@/)N-!1@1# -1@1# ){1@1# (Y1@1# @1@/)# !(L@1@## !1@/+# 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@4)44333[3T2222211Ҡ1@1W/4-@4H# /42@4M# @48@1# !/d4>@4Y# ,4C@4^# ݠ4J@4`# !/4P@1#  ,4U@1# !4Z@1#u(0 @@@/:@A/@0@A0AA_@]@.@ABCD@@a+mul_lt_pred @4}4m4f44333P3I222۠2<@2@,W4@4# +54@4# 04@4# 04@4# @4@4# !4@1#/00`4@&# !4@2 # $?4@2# 耠.0a4@&# !4@2*# 4@2## +]0r4@25# 4@2.# /04@@. @@AA@@A@1mul_lt_mono_pos_l @44נ4Р4x4q44333[3T3E2@3CA,4@5# 0v4@5 # 0t4@5# @4@&# !4@2D#u(0h0à5@&# !0<5 @&# !5@2t# +5@2y# 0E@@@/@A/@0?@1@@ABCD@@P1mul_lt_mono_pos_r @545$54Š44f4_443332@3B-56@5Q# 0Ú5;@5V# 05@@5[# @5F@'2# !5K@2#u(015R@'># !05W@'C# !5]@2# 05b@2# 耠10@@.@@1mul_lt_mono_neg_l @5}5m5f55444P4I333۠3<@3C-W5@5# 1 5@5# 1 5@5# @5@'{# !05@2#u(1Y5@'# !0ҩ5@'# !5@3 # 0֩5@3# ,I@@@@ᠠ1mul_lt_mono_neg_r @5Š555V5O444449424#3@4!D-5@5# 1T5@5# 1R5@5# @5@'# !1B5@3"#u(15@'# !15@'# !5@3R# 5@3W# 1@@/4@@)4mul_le_mono_nonneg_l @6 55555?5844٠44z4k3@4iE16@6*# 16@6/# -񚠐6@64# @6@3# !6$@3j#u(1ֶ@6*@3# !,ȩ6/@3# !65@3# 耠1036;@3# 1d@@1@@q4mul_le_mono_nonpos_l @6U6E6>55ߠ555(5!4ɠ4 44@4F1ߚ6W@6r# 1ݚ6\@6w# .96a@6|# @6g@3# !26l@3#u(@6r@3# !-6w@3# !6}@3# 16@3# M@@1b@@4mul_le_mono_nonneg_r @6666-6&5Π5Ǡ5o5h55 44[@4G2&6@6# 2$6@6# .6@6# @6@4)# !6@3#u(2e@6@44# !-W6@49# !6@4(# &[6@4-# -g@@1@@4mul_le_mono_nonpos_r @66Ӡ6̠6t6m66555W5P5A4@5?H2m6@7# 2k6@7# .ǚ6@7 # @6@4p# !26@4@#u(@7@4{# !-7@4# !7 @4o# -7@4t# &@@1@@F,mul_cancel_l @7*77666\6U555554@5I27,@7G# 271@7L# /76@7Q# @27>@7T# !2񚠐7C@4#u(37J@7`# !7P@4# 7U@4# 27Z@7p# !-@@@29@3@1@ABC@@,mul_cancel_r @7w7g7`77666J6C555ՠ56@5J37y@7# 27~@7# /[7@7# @37@7# !3>7@4#u(3U7@7# !7@5# 2ͩ7@5# 2٩7@7# !.E@@0@@ݠ(mul_id_l @7777R7K6666656.65@6K3K7@7# 3I7@7# @3G7@7# !37@5#u(37@7# !7@5F# 337@7# !3R7@5J# .@@3@@"(mul_id_r @8777778716٠6Ҡ6z6s6d5@6bL38@8## 38 @8(# @38@8+# !3ɚ8@5`#u(3ߠ8!@87# !8'@5# 3^38,@8B# !381@5# .@@1r@@g1mul_le_mono_pos_l @8K8;847ܠ7ՠ7}7v776666 @6M3՚8M@8h# 3Ӛ8R@8m# 0/8W@8r# @8]@*I# !8b@5#u(44'8i@5# !/8n@5# !8t@5# 8y@5# 3@@3Y@@1mul_le_mono_pos_r @888|8$87Š77f7_7766R@6N48@8# 48@8# 0w8@8# @8@*# !8@5#u(4\4o8@6,# !/O8@61# !8@6 # 3쩚8@6%# 3@@2@@1mul_le_mono_neg_l @8۠8ˠ8Ġ8l8e8 8777O7H796@77O4e8@8# 4c8@8# 08@9# @8@*# !48@68#u(48@6t# !/8@6y# !9@6h# 4o9 @6m# @@J@@?1mul_le_mono_neg_r @9#99 888U8N777776@7P49%@9@# 49*@9E# 19/@9J# @95@+!# !4蚠9:@6#u(49A@6# !/ߩ9F@6# !9L@6# 49Q@6# 4@@2@@2mul_lt_mono_nonneg @9k9[9T88888>877ߠ7ؠ7ɠ7*@7Q49m@9# 4󚠐9r@9# 1O9w@9# 0-9|@9# @9@6# !9@6#u(4@9@+y# !34@9@7# !9@6#u(4ȶ@9@+# !9@+# !9@7# 耠39@7# 耠#v3@@@@@砠2mul_le_mono_nonneg @9ˠ999\9U88888?888)7@8'R5U9@9# 5S9@9# 19@9# 09@9# @9@7]# !9@7-#u(5@9@7h# !`@9@7n# !9@7>#u(5'@9@7y# !q:@7~# !: @7m# _:@7r# ^@@4@@D+mul_pos_pos @:(::999Z9S888887@8S5:*@:E# 5:/@:J# @:5@,!# !::@7#u(5@:@@,,# !:E@7#u(5:J@,6# !:O@7#u(:T@7# 0@@@@+mul_neg_neg @:n:^:W99999A9:88۠8̠8-@8T5:p@:# 5:u@:# @:{@,g# !6/:@7#u(@:@,r# !6::@7#u(:@,|# !:@7#u(:@7# 18@@@@Р+mul_pos_neg @::::E:>99ߠ999(9!98s@9U6>:@:# 6<:@:# @:@,# !:@8 #u(6y@:@,# !6:@8#u(:@,# !:@8@# 1z:@8&#u(@@5@@+mul_neg_pos @::::::,:%9͠9Ơ9n9g9X8@9VV6:@;# 6;@;# @;@,# !6; @8R#u(@;@,# !;@8]#u(6ʩ;@-# !;"@8# 1;&@8l#u(@@g@@\1mul_nonneg_nonneg @;@;0;):Ѡ:ʠ:r:k:: 9998@9W6ʚ;B@;]# 6Ț;G@;b# @;M@8# !;R@8#u(7@;X@8# !;]@8#u(7;b@8# !;g@8#u(;l@8# 2 @@@@0mul_pos_cancel_l @;;v;o;;:::Y:R9999E@9X7;@;# 7;@;# @;@-# !;@8#u(7K7];@-# !;@8#u(;@9 # 6ਗ਼;@-# !;@8#u(7f@@ s@@頠0mul_pos_cancel_r @;͠;;;^;W:::::A:::+9@:)Y7W;@;# 7U;@;# @;@-# !;@9%#u(77;@-# !;@91#u(;@9T# 7';@-# !;@9@#u(7d@@ @@03mul_nonneg_cancel_l @<<;;;;F;?:::::r9@:pZ7<@<1# 7<@<6# @# 9>@># @> @> # !>@;t# 9Ě>@;Z#u(@9>@>2# !9>!@;g#u(>&@><# !9>+@;q#u(@@@@@ a(mul_eq_0 @,@fĈ>I>9>2=ڠ=Ӡ={=t==<<<@9Қ>J@>e# 9К>O@>j# :>V@>l# !>\@;# :>`@;#u(.>g@>}# !:>l@;#u(>q@># !:$>v@;#u(<ܠ<۠<ڠ<@<7@^\>>>z>">=à==d=]=<<>>><>5=ݠ=֠=~=w=== @:4>@># :2>@># @>@># !>@@<#u(@:@>@># !:}>@<#u(>@># !:>@<#u(b@=24pe>>ݠ>֠>~>w>>===a=Z=K<@@@@@ *mul_eq_0_r @-L@0*?>>>>>9>2=ڠ=Ӡ={=t=e@:?@?## :? @?(# @?@?)# !?@<}# :͚?@ڠ>Ӡ>{>t>>====@@@@@ {(lt_0_mul @?_?O?H>>>>>2>+=Ӡ=̠==@=b:隠?a@?|# :皠?f@?# ;+?m@1Y# !?r@<#u(?w@<# ;+/Ƞ$?@1l# !?@<#u(;8?@1v# !?@<#u(;A$)?@1# !;I?@<#u(?@1# !;T?@<#u(@@:@@ ۠5square_lt_mono_nonneg @????P?I>>>>>3>,>=~@>c;I?@?# ;G?@?# @?@=G# !?@=#u(;@?@1# !;?@1# !?@=F# 耠;;?@=L# 耠;S;S@@@@@ 5square_le_mono_nonneg @@?????5?.>֠>Ϡ>w>p>a=@>_d;@@@ # ;@ @@%# @@@=# !@@=[#u(;ȶ@@@=# !;R@ @=# !@&@=# D@+@=# C@@@@@ a6square_lt_simpl_nonneg @@E@5@.?֠?Ϡ?w?p??>>>>@>e;Ϛ@G@@b# ;͚@L@@g# @@R@=# !@W@=#u(< @@]@2I# !@c@=# {@h@=# 耠<<@n@2Z# !7 @@@@@ 6square_le_simpl_nonneg @@@x@q@@???[?T>>>>G@>f<@@@# <@@@# @@@># !@@=#u(# !@@> # @@># C@@>+# !7N@@@@@ 栠,mul_2_mono_l @@ʠ@@@[@T?????>?7?(>@?&gS# @@>E# .@@>P# 耠@@>E# <[@@>Z# 耠@@>O# <@@:<@@ 1*add_le_mul @AA@@@@G@@?????s>@?qh!aA@A4# !bA @A;# @A&@3# !A+@># .<޶@A1@3# !A6@># .<驚A;@># !AA@># 7ߩAF@># 7@@<&@@ |-square_nonneg @A`APAI@@@@@3@,?Ԡ?͠??@?iKAb@A}# Ag@># !Al@>#u(Aq@># 31@@!@@ 5crossmul_le_addsquare @AA{AtAA@@@^@W????J@?jvA@A# tA@A# @A@?# !A@>#u(=P@A@?# !A@>#u(=[A@?(# !A@?$# A@?# <⩚A@?"# 8\A@?4# A@?-# 穚A@?2# @@?@@ -add_square_le @AAؠAѠAyArAA@@@\@U@F?@@DkӚA@B# њA@B # @A@?p# !A@?@#u(=@B@?{# !B@?K#u(=B @?# !B@?# B@?z# 4B@?# 3B @?# 耠B&@?# 8ĩB+@?# 8ɐ@@= @@ a-square_add_le @BEB5B.A֠AϠAwApAA@@@@@@l0BG@Bb# .BL@Bg# @BR@?# !BW@?#u(> @B]@?# !Bb@?#u(>Bg@?# !Bm@?# 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UO@*lt_le_incl1#rkҷР@#andЖw@BBB@@@@D@@נ@@p@)le_succ_l6'ٲ@@@c˰e6ᠩP-ĩ$(sSSSj U@*add_pred_r!Oˀ멚@&sub_gtE ItC@:䐑)'()멷b &0@#one# .e$7WQ3 WA!YC#saG5@'add_1_l#wux1J$xzX8P*v\<* `@:4O@-add_sub_assoc9)7nlbmPi3Mm7D5$xXRr;s_MBPPo@'le_reflrI$l1, o =*q,d_,, @(sub_diag _Щ{*]\$:m::@'add_0_r#voLNשr$թ TT5@(add_comm[KLa$M1eeFihhW$ >L\@@ttU 󩚠@'add_sub#Ƞt0+nȐ"H1'ҠyZ$-k{__ZɩVeޠ;7/z4 O hcY  STT] n@4iff_impl_subrelation5*B9VL4|o/-#.6/I@,add_sub_eq_lKҩoj4640e b A?5@HA"%4@0double_induction x1O2P@24M񵵷#Y!Z$,]*^r!(@C6HF_АII@LG./ Pi RA  $wWI ,yYR8#9G{[\u>`yCB$e4"GU##FF1@i]&kKAM(CA7& ^' ($t#'g UpVV.22wrW@8per_partial_app_morphismp.^yTRCVT@whm@pC.ggC87i̩Vϩ%Z74CB_@'sub_0_lQ߀\|@)False_induُ<֩SO4J!BY@N>\N;1  ,RP3ƠN^~$yQQˠU]D&)Ԡy,cUo@ XSjQ@$g00oj2@'add_0_l$ өSN>uܷK%.F*ؐQ744y.%` $ ^($4T,ҩf0j>@@*add_succ_lu}(#$K+k C@1g3.YWMX`YGз?]>9ȩYJ<y21g/h-+((LG.gHJY SN5nIQ`9${[ ,}]V<',_v-YA?M,aZhgAO686C:g8++C԰-J$mg, q jP,_kfΩyt{o{(~- l Hݩڐl$ϩp sG9,43,wb@*add_succ_r!hȀ!$d  in'qƠ $  ~{  MIXCѠ̠ΠĐ4ީ$ ڠܠywoj ⩷Ca=ө$5 ~^Y꩷Tr$$Dk(-+!,з00͵56(9;CE%C٩4JM-O/ Q11D S33F!ĠةA@*eq_le_incl(qꀰηQ\-4`'/;;#O@(le_trans(PЩV@)le_pred_l5==Co1vtju/[y\z,]QUKIGE"x0TRKH(><2=nabaBs'g'+$JH&LJ@@/iff_equivalenceZ 4$/r3)ss #,haL=bpTyyy \h(^\R A$`^ C$>B v  ecC-gegmC;JJCEDLũA@g@g@&le_0_l܀ ֩S$=@Z@\ $@*nle_succ_0/Tz,eݠѩG@-iff_Symmetric!x ՠ@,succ_le_monoō&F$$,y3C Ȑ!o@+le_ge_cases驛@"or @BAAAA@@@@@D@C;_<6U"LEL|2,H(GI$ BW`Y! ?TCUD7c+  X(H@(sub_0_le ؀/p,eEd?'y3c"GE2P3ة,sS&r+M5)MAa@'sub_addOI䐑u@oF@jJ*P}@*sub_add_le ZS婷ΩJ"|>&q@-add_le_mono_rBFkyj#s@#LE'94EDyup{   yl#۠ՠFΩ 㠩ݠ#DQ"͠e4qpH }Z0۠hzi1, JIk3ҠKǩ@/le_sub_le_add_raQ͐    "LT @+le_lt_trans?VUѠ @-add_lt_mono_r%q( 3 1' 2: 3k;,\?'*+;, > - >@%lt_wd#pr ? 1@/lt_sub_lt_add_r> O MC NV OP  @u@X9Y@\V? `<w@ c$pF '@@ 8 9, p P#o(hN s S)M A ݩ  ߩ4{B1- ~ ^ L | 3 X_Ȱ `L B @ ] el - R R    ϩ z:"4k ; >j s @ @ g  \ Z @/le_add_le_sub_rl  r M    @ǩ qT*c Y $ } { q\^ Q,    `_   o 5 b   $    $        } =   8!zȐ#GE' 驷 v  Uܠ*ߠة Ʃ v,  ;  :Z  à . 갩 ޠ ҩF   Ϡ ѩ>@<@ @@ @)lt_irrefl2Uҩ'ԩ M ٩4t e  `   $  "멚 @+add_le_mono!m   _  )fF @'nlt_0_r ߀  > < 2 =E > v $   235   6@/lt_add_lt_sub_r, T R H S8BE  D@(le_sub_l(xߩ]t/@9Sp ? FVO'  l  ]o H  S  A u@+lt_eq_cases"' H#LT' L"EQ o@-add_sub_eq_nzH cH O l  @@%Falsee@ w rr lY b<<  0Private_OrderTac#Tac@/"-0C z @#€O O r@%le_eq mW z {@(eq_trans0f+   C:   )@)not_ge_ltlԀ8 Ҡ<867@%lt_eq fRz֠||  àܠ ]4ih@ݩ   [   V ) ٩谩 ˠְ   C0       @   ^ wQ )  䠩 @,transitivity>4   @3PreOrder_Transitive4(ۀ &% @+le_preorder   D 1 / % 08 1 : j 栩6\ 9 3 ة~?] &B ߩ<  A  1  KJ, C   R P F QY R@t@_W@Z`]@^] L@-add_le_mono_lB *@@@ ) l j ` ks l ; H - . s q g r `@-case_analysis gPз ] { \ ~@#mul# 耠   o _  p # U  S  Q O L( B @ 6 p m   n o q x u  Ʃ v! q y 5$   0 $,    % @&mul_wd48  j   n    .,  )  (/   o H m {   o  b  d b C ,  f  d99C   ;     x$   ۩> X ĩ@ b A,  <  ;B [1     ] $  é  L W  A4 M L  $}    i A   i i? @'mul_0_r&   q c   h  v   з  ۩      $   e K cg 4 h,  c  biD N f   l , Q L  f ة & Ѡu  ߩo     Ѡ b b p  d d @*mul_succ_r5 쩷 < 砩  *      * ,  y yY  T ⠩  ð   t R >    : <  , "    >  $  @   = ^  "   .  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