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U\DҠàkb LEנiXok@@b@@X(eq_trans @]@6Equivalence_TransitiveWV@U%Z@T%^@S# 0|@p@*Transitive++-i@h%m@g%RET;LW@-W\=D7'_XB;,֠XGĠ@@@@*eq_sym_iff @VF7ߠ֠~waZKݠ̠wf@b@!x@%!y@%%Logic$Init@#iffС)@%BA@%@@@@AA@@A@'neq_sym @'Ơg`=%@A!n@%!m @%@I@#notШ@%D @%KC@@@@@D(eq_stepl @ߠϠh_IBԠ~fU@BA@@%F@E%!zM@L%@S@M%6@Z@T%Da@[%D@@@Aۓό^)Morphismsu@@AB@@(succ_inj @) QJ4-ȠJ9@5C"n1@%"n2@%@@%@m2DΩ@s2Dө@%Y@@@AviAK@AAۀߠKN@R@@ABC@@۠+succ_inj_wd @vfW?8٠zk@DM@%K@%@%@2DJ@2DI@%"@@@G@A8ӠA4Relation_Definitions)Relations@AB@@"/succ_inj_wd_neg @F=ޠ' Ƞ\D3ޠ@E@%$@#%c-@'%3@2D8@2D.?@9%m@@@@@d1central_induction @' Ƞib u @ F!A@d@c%A$A_wd@&ProperL@v@u% @*respectful%WO?@%@%?@%@n@@%ѠE@j2D@%F@@@@@ɠ'add_0_r @dTE-&ΠǠohYڠ@@A&NZBase%NZMul&NatInt'Numbers#Coq@@A@%NZMul&NatInt'Numbers#Coq@)NZMulProp@A@@@@"NZ@A@%NZAdd&NatInt'Numbers#Coq@)NZAddProp/@@@A@@@A&NZBase%NZAdd&NatInt'Numbers#Coq@A@A@@@"NZ+@@,R@2@!t,RJ@@A@A,Y@@"eq,Y^@@A@,@@ @#add,@d@@A@AB,w@@#mul,wd@@A@,@@#one,T@@A@A,@@#sub,d@@A@,@@#two,T@@A@A,@"@$pred,d@@A@,@'@$succ,r@@A@,筷@,@$zero,筷T@@A@ABCDE@A@E@CC@B%H@B%N@A&:{S@2걀@@@@@v@ABC@@AA@@A@*add_succ_r @ A:۠|$v@A@~%}@%@%@&:@b2D署@g2D@&:ΐ@@M@@J-add_succ_comm @`PA)"ʠàkdU֠ܠˠ@Bƚ@%Ě@%@%@&:@2D<@&:@2DA@@@@(add_comm @0'ϠȠpi F.C6(#@C  @ % @%@%@&:K"@&:H@@@@̠'add_1_l @Ҡàkb LEנiX~qc^M?@JDHH@G%M@G%S@F&:X@8&3]@*2D@@ @@'add_1_r @ E>ߠ(!z@E@%@%@&:@s&3@e2D@@E@@B)add_assoc @XH9ؠy! c\MߠΠ٠Ԡà@F@%@%!p@%@%@&:@&: @&:@&:@@@@,add_cancel_l @/&ΠǠoh E-B5'"@G  @ %  @ %N @ %U @ % #@&:P )@&:O .@ (%ې@@@@ؠ,add_cancel_r @ޠϠwnXQud}ojYK@VHT T@ S%R Y@ X% ^@ ]% e@ _% k@^&:N q@d&: v@ p%#@@#@@ ,add_shuffle0 @ 6 & ^WA:+ՠҠŠ@I @ % @ %ޚ @ % @ % @&: @&:d㩚 @&: @&:W@@o@@l,add_shuffle1 @  r c  KDw! @J蚠 @ %暠 @ %* @ %!q @ % @ % @&: @&: @&:> @&: @&: @&:@@@@ʠ,add_shuffle2 @  Р  i `  JCՠgV|oa\K=@HKF F@ E%D K@ J% P@ O%^ U@ T% Z@ T% `@S&: f@Y&:\ k@^&: p@c&: v@i&: |@o&:)@@)@@&,add_shuffle3 @ < ,  Š  d ] G@1۠àؠˠ@L @ % @ %䚠 @ % @ % @&: @&:멚 @&:𩚠 @&:]@@u@@r'sub_1_r @  x i     Q J}'$ @M @ % @ % @&9݀& @&3 @1]@@@@'mul_0_r @ à   L C     - &ΠǠbJ9_RD@]@& &@ %% +@ %% 1@&1q^ 6@2걀 :@2걀@@@@AA@@A@*mul_succ_r @   ޠ  } %  Ơ  g `  s~@A` `@ _%^ e@ d% j@ d% p@V&1q v@C2DЩ {@n&: @g&1q@@.@@G(mul_comm @ A 1 " ʠ  i b     L E 6ȠݠР@B @ % @ % @ % @&1q⩚ @&1qߐ@@f@@/mul_add_distr_r @ y i Z     B ;  ܠ  } n  @Cܚ @ %ښ @ % @ % @ % @&1q @&:# @&: @&1q @&1q5@@@@͠/mul_add_distr_l @ Ǡ   P G     1 * Ҡ ˠ  f N =cVH@aD* *@ )%( /@ .%l 4@ 3% 9@ 3% ?@ %&1q" E@ 8&:s J@ =&: P@ 6&1q U@ ;&1q@@@@)mul_assoc @      = 6 ޠ נ  x      @Ex x@ w%v }@ |% @ % @ % @ s&1qp @ y&1q @ ~&1q @ &1qKʐ@@K@@d'mul_1_l @ ^ N ?  ޠ   '  Ƞ  i b S   Ԡ@F @ % @ % @ &1q @ &3@@~@@'mul_1_r @   r     Z S      0  - @+G @ % @ % @ &1q, @ &30@@@@ʠ,mul_shuffle0 @ Ġ   M D     . ' Ϡ Ƞ  c K :`SE@^H' '@ &%% ,@ +%i 1@ 0% 6@ 0% <@ "&1q B@ (&1qn G@ -&1q M@ 3&1qz@@@@,mul_shuffle1 @      5 . ֠ Ϡ w p      @Ip p@ o%n u@ t% z@ y% @ ~% @ ~% @ p&1q @ v&1q @ {&1qé @ &1q @ &1qF @ &1q:@@R@@k,mul_shuffle2 @ e U F     . ' Ϡ Ƞ p i Z   ۠@JȚ @ %ƚ @ %  @ % @ % @ % @ &1q @ &1qީ @ &1q  @ &1q @ &1q @ &1q@@@@à,mul_shuffle3 @    F =  ޠ   '  Ƞ   \ D 3YL>@WK  @% %@$%b*@)%/@)%5@ &1q ;@ !&1q i@@ &&1q nF@ ,&1qې@@@@ %le_wd @     . ' Ϡ Ƞ p i      |@@A&NZBase'NZOrder&NatInt'Numbers#Coq@ & $@A@'NZOrder&NatInt'Numbers#Coq@+NZOrderProp 5 3@A@@@@"NZ * )@A@@@@%@@%,@%@@%7@%)@%A@% 皠@ % @@Z@@AA@@A@*lt_le_incl @p`Q     9 2 ڠ Ӡ { t e   j]O@hA Ӛ@% њ@%@@"lt% @ %  @@@@@7'le_refl @-$ ̠ Š m f      C + @B @% @ %  9 9@@ .@@`.lt_succ_diag_r @͠VM     7 0 ؠ Ѡ   l T CǠ@C 00@/%5@W% b;@ 2D @@@@@.le_succ_diag_r @ܠ{# Ġ  e ^      q@D ^^@]%c@ `%  i@ 62D Ð@@@@@/neq_succ_diag_l @) QJ     4 -  Ƞ  #@!E @% @%@ f2D  Ő@@ @@젠/neq_succ_diag_r @YI:٠z" à  d ] N   ϠSF8@QF @% @% 𩚠@ 2D #@@@@@/nlt_succ_diag_l @yj RK     ~ (  vh@G 욠@% ␩@%@ 2D S %@@@@@L/nle_succ_diag_l @B9ڠ{# Ġ   X @ /@H @% #@ % )@ 2D  U@@@@@|)le_succ_l @٠ʠri SL   ޠ  p _֠@I LL@K% JQ@P% X@ U% ^@ +2D  c@% @@ @@+le_gt_cases @#KD.'    @J @% @% @"or @@ %  Ʃ@% Ð@@ J@@-lt_trichotomy @]M>ݠ~&ǠhaR   ӠWJ<@UK @% @%:@% A@% @% @@@@@+(lt_asymm @y!aZ7w@L @% @%@@(% 4  @/% @@ a@@`(lt_trans @͠VM70ؠѠ lTCǠ@M 00@/% .5@4%r:@9%@@@b% @F@h% 󩚠K@m%  .@@ @@(le_trans @ 4-ՠΠvo@N oo@n% mt@s%y@x%@@|%  ,@@%  2@% ?@@ 7@@ݠ+lt_strorder @J:+Ӡʠrk UN?ѠD7)@BO@+StrictOrder.[@@%@%@@ @@ +le_preorder @xhYA:۠|mreW@pP@(PreOrder@@%@% @@@@9/le_partialorder @/&ΠǠoh E-@Q@,PartialOrder5͛К@%@ %@ # 0|@% :@b3@@@ s@A @B@CAhC5@ABC@@y)lt_compat @4@%lt_wd @ ̰@<@;%@A@@% ݩ G@F%@L@K% 蚠P@J% V@U% 򚠐Z@T% ^@%@@@%lt_wd']ɠ HA+$ @@@@@Р(lt_total @@̀МA76@W@%U@%Κ@%@@%̶@%@%ҩ*@$%ߐ L+k() LL7#'S@$tޠϠwnXQud۠͠ޠΠq@@@@@a @_@^%d@c%i@h%d"@">̀МbCV@y@x%~@}%8@%@@% 6@@% <@% I L+k() NN7#'jtTD5ݠԠ|u_XI۠ʠ N A 3D4@@@@@Ǡ @ @% @%@%@B">̀МB@'@%%@%@%@@ %@@ %@ % L+k() MM7#'АtC:۠|$ŠYA0   e^X=@@@@@-+le_lt_trans @t,@+%r1@0%뚠6@5%X1@">̀g kj@C@B%H@G%M@L%@S@P% @Y@ {%^@ % L+k() MN7#'4tF?)"   ɠ @@@@@+lt_le_trans @ؚ@%֚@%O@%S@ ">̀1@@%횠@%f@%@@ %d@@% j@ %w L+k() NM7#'trc KDw!  | o arb-& @@@@@%eq_lt @<@%:@%@% @q">̀/32@S @ %Q@%ʚ@%@@%ȶ@!@ C%Ω&@ H%ې L+k() ML7#'t֠ǠofPI۠m\  Ӡ Š֠Ơi@@@@@Y%lt_eq @X@W%]@\%b@a%]@">̀cL@o@n%t@s%.y@x%@@ %,@@%2@ %? L+k() LM7#'`tJ:+Ӡʠrk UN?Ѡ D 7 ):*@@@@@%eq_le @@%@%{@%@9">̀]@@%@%@%@@%@@% @%  L+k() NL7#'Đt7.֠ϠwpM5$   YRL1@@@@@!%le_eq @h @%f%@$%ߚ*@)%L%@">̀+_^@7@6%}<@;%A@@%@G@D% @M@G%R@O%   L+k() LN7#'(t:3۠Ԡ|u   @@@@@&eq_neq @5%]V@9*Ԡ / " %٠Ӡ@&G暠@%䚠@%]@%@@%[@@%c@% r@@@@@͠&neq_eq @}m^F?r w j \m](!@nH.@%,@%@%@琩@%@@%@% @@@@@*not_neq_eq @ŠNE/(РɠdL;   picH@Iv.@-%t3@2%@*,=@7%kB@<%@@@@R)not_ge_lt @*#ˠĠle x   @Jk@j%p@o%@gx@u% }@ %*@@@@)not_gt_le @=-Ơe^HA2ܠĠ 7 * -۠@.K@%욠@%@@ %٩@% e@@@@@Ƞ)le_neq_lt @xhYA:۠|m r e WhX#@iL)@%'@%@@% @㐩@%@ %@@@@@ @@@@'@@@@@'@<_neq @F=ޠ' Ƞ\D3   @ R  @%%@$%@+@ M%Y!2@,%ߐ@@@@@ &le_neq @Ӡ{r\Uyh  ߠ @ SUU@T%SZ@Y%a@ %@#andЖw@n@k% du@o%@@@@@ Ƞ*eq_le_incl @5%]V@9*Ԡ/"@-T@%@%@@%ѩ@% U@@@@@ (lt_stepl @hXI1*Ҡˠsl]ޠbUG@`U@%@%@%@@ %@@%@%@@@@@9(lt_stepr @/&ΠǠoh E-@VQ @%O@ %Ț@%@@;%ƶ@@%̩$@F% ِ@@@@@w(le_stepl @ԠŠmd NG٠kZޠѠ@WG@F%L@K%Q@P%@W@T% @]@W%b@_% @@@@@(le_stepr @"JC-&@X͚@%˚@%D@%@@% B@@%H@%  U@@@@@ @_O@ߠ(!ɠ jcTՠYL>@WYš@%@%@%@@% @@%@% @@@@@0\ @}%Ġe^;#{@Z@%@%A @%@@1%@@% ©@<% ϐ@@@@@m+le_antisymm @ڠʠcZD=ޠϠyaPԠǠ@[==@<%;B@A%@H@E% v@N@K% 1S@M%I@@@@@)le_succ_r @;4ܠՠ}v @ \vv@u%t{@z%@% @U2D @% @%©@g2D@@@@@)lt_succ_l @ZJ;ڠ{#Ġe^OРTG9@R]@%@%@@%@2D.@%@@@@@&,le_le_succ_r @t\U2 r@^@%@%@@% /@% 驚 @2Dl@@@@@_,lt_lt_succ_r @̠UL6/נРkSBƠ@_//@.%-4@3%@:@\%h?@a%"E@2D@@@@@,succ_lt_mono @-&ΠǠoh {@`hh@g%fm@l%t@%y@%@L2Dߩ@Q2Dސ@@@@@נ,succ_le_mono @D4%͠Ġle OH9ˠ>1#@ߠРzbQՠȠ@d >@`% C@#&3 G@&;y@@@@<_0_2 @ /(Рɠqj }@e j@% o@72걀 s@I&;y@@@@Ơ&le_0_2 @ 3 # [T>7(Ҡ- @+f @%  @c2걀 @u&;y@@@@@<_1_l @ _ O @ߠ(!ɠ jcTՠYL>@Wgš @ % @ %@ @% @2걀@ @% @% @&3Đ@@@5@3@@ABC@@8+lt_ge_cases @    . %͠ƠngD,@h!@!%! @! %!@6%B!@% ?@@@@@l+le_ge_cases @ ٠ ɠ  b Y C<ݠΠx`OӠƠ@i<!<@!;%:!A@!@%!H@E% v!M@J% s@@@@@+lt_gt_cases @!      5 .֠Ϡwp@jp!p@!o%n!u@!t%m!~@!x%󠩚!@%!@%@@@@@ݠ,eq_decidable @!J!:!+ Ӡ ʠ r k  UN?ѠD7)@Bk!@!%!@!%)Decidable!@)decidable r!@!%@@@@@&eq_dne @!!r!c! !   K Dw! |oa@zl嚠!@!%㚠!@!%)␩䐩!@!%#!@!%(@@@@@M&le_ngt @!!!!C!:  ۠  | $ ŠYA0@m"@"%""@"!%a")@&% W"0@R%V@@@@@&nlt_ge @!!!Ѡ!y!p!!   Z Swfݠ@nS"S@"R%Q"X@"W%P"a@%"f@c% @@@@@,lt_decidable @"&""!!!N!G     1 * Š @o"@"%"@"%ܐ"@%Ð@@@@@蠠<_dne @"U"E"6!ޠ!ՠ!}!v!!   ` Y JܠˠOB4@Mp"@"%"@"%"@%"@%@@@@@ &nle_gt @""}"n"" !!!V!O      ,  zl@q𚠐"@"%"@"%4퐩"@% ,#@%%)@@@@@V<_nge @"à"""L"C!!!!!-!& Π Ǡ  b J 9@r&#&@#%%$#+@#*%j#2@T%`(#9@ 6% _@@@@@,le_decidable @"""ڠ""y"!"! !!c!\!     o@s\#\@#[%Z#a@#`%#h@ e% @@@@@&le_dne @#(## """P"I!!!!!3!,! 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