"`?# )ZMulOrder(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@$ZMul(Abstract'Integer'Numbers#Coq@#ZLt(Abstract'Integer'Numbers#Coq@)ZAddOrder(Abstract'Integer'Numbers#Coq@)Notations$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"H#ZLt(Abstract'Integer'Numbers#Coq@0xK|j2[&=}۠+0w:}H#NK}{K69Aqw:w#~&Р)ZMulOrder(Abstract'Integer'Numbers#Coq@ A-ZMulOrderProp@@!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@ }@ @@@@@$pred @d@@& @* @@@@@)'succ_wd @@)Morphisms'Classes#Coq@@&ProperL@@F @J @*respectful%WO?@U @Y @"eq @ @] "f@@@AviA4'Classes#Coq@@A@@n'pred_wd @@E@@ @ 9@ @ @4 @8 @} "@@@A3@A@@)pred_succ @@!n@ @S @ ""@ "fA@@@@@,bi_induction @@!A@5@ A@@?@  E@ I@ %Logic$Init@#iffС)0@B[@ # @Ra@ DCl@ "fJbq@  EQ@@@kA@A@@#one @T@ @@@@@#two @T@+ @@@@@*(one_succ @@@ @@@1 U @A "f@$zero # @@@@@O(two_succ @@@ @A ]s@_ "f@Y U@@@@@l"lt @h@@!t @@ A@@@@@AA@@A@"le @h@@ @@ @@@@@%lt_wd @@q@@ @@ f@ @!@ 욠%@b u+@ /@l 暠3@"lt @@@,Ah@A@@Ϡ+lt_eq_cases @@5D@ !mK@ R@` 0 @"or @`@- e@ @@@@@)lt_irrefl @@ds@  +@#notШ@L ]]@@@@@)lt_succ_r @@@( J@- K@g F@1 "f~@ S@@@@@>#add @d@@N @@S @W @@@@@V#sub @d@@f @@k @o @D@@m#mul @d@@} @@ @ @[@@&add_wd @@[@@ @@ @ T @ @@ @ @T g@ !@ %@b )@f -@ A4@@@%Aa@A@@Ƞ&sub_wd @@@?@ @D@ H@ N@ @S@ W@ [@ a@ e@ i@ m@ q@ [׀@@@iA@A@@ &mul_wd @@㰶@@ @@" @& @, @@1 @5 @ @? @C @ @ @ Sk@@@A@A@@P'add_0_l @@@_ @ @& A4@x # @@@@@n*add_succ_l @@Ԛ@} @ @* @I A4@ "fש@ "f@[ A4@@@@@'sub_0_r @@@ @U @\ [׀#@ # @@@@@*sub_succ_r @@"1@ 횠6@ ;@x A@ [׀G@ "f%L@ "R@ [׀@@@@@렠'mul_0_l @@Q`@ e@ k@ Skp@ # Mt@ # @@@@@ *mul_succ_l @@s@ >@! @ @ Sk@) "fu@ A4@ SkQ@@@@@<#max @d@@!t @@ @  @@@@@AA@@A@%max_l @@!x@ !y@$ @@"le AB@"eq @D QC@@@@@2%max_r @@/@J -@O @@+ )* @( @k Q'4@@@@@W#min @d@"@p @'@u +@y @@@@@o%min_l @@l9@ j>@ @D@h =I@d O@7 Rcb@@@@@%min_r @@]@ b@ @h@ m@ s@[ R@@@@@)succ_pred @@!n@ @"eq @  "f@ "A@@@@@AA@@A@#opp @d@@C @G @@@@@&opp_wd @@)Morphisms'Classes#Coq@@&ProperL@@c @g @*respectful%WO?@r @v @X @\ @I U‐@@@ %'A*Equalities*StructuresPAviA9'Classes#Coq@@A@@`%opp_0 @@@}  @l U‐@ # @ # @@@@@}(opp_succ @@"@ '@ -@ U‐3@ "f8@ ">@ U@@@@@@'ZAxioms(Abstract'Integer'Numbers#Coq@/ZAxiomsMiniSig'@@@@ @Y@U J@@A@A @\@= ^@@A@ @`@n h@@A@AB @c@ h@@A@ A4@h@ A4d@@A@A Q@k@ Qd@@A@ R@o@W Rd@@A@ABC Sk@r@ Skd@@A@ U@x@ UT@@A@A U@{@ Ud@@A@ [@@ [נd@@A@AB ]s@@ ]sT@@A@ "@@ "d@@A@ "f@@ "fr@@A@ # @@1 # T@@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@@$zero]lT@@A@ABCD@A@@L<.+*@A@4*BackportEq@@@A@@@@ %'A*Equalities*StructuresA݋X_A@A@@AA@@A@&eq_sym @@5Equivalence_Symmetric. U3,ϠȠkd@@)SymmetricFgq@ @ TL@,. U\D֠Ơb[6/ |fUlh@@_@@U(eq_trans @@6Equivalence_TransitiveW~!YRC@+@*Transitive++-@ @ ܐTҐL@,W\ԐD(PIrΠRA@@@@*eq_sym_iff @E5.Ѡʠmf ՠĠo^@Z@!xM@h !yT@o %Logic$Init@#iffС)e@{ BAm@ @@@@AA@@A@'neq_sym @{tOHՠ1 @A!n@ !m@ @G@#notШ@ B @ IC@@6@@B(eq_stepl @ʠVO*#pZI@B@ @ !z@ @@ CB@@ D@ D@@@zAۓό^)Morphisms+@@AB@@(succ_inj @=6٠Ҡun_?.@*C"n1@8 "n2$@? @*@@ 0@b "f˩6@h "fЩ<@R Z@@@AviAI@AAۀߠKL@P@@ABC@@ؠ+succ_inj_wd @`PI$ߠy@uDKf@ Ik@ r@ x@ "fH}@ "fG@ @@@E@ A8ӠA4Relation_Definitions)Relations@AB@@/succ_inj_wd_neg @1*͠ƠibK5$Ϡ@E@ @ \@ @ "f@ "f)@ f@@@@@]1central_induction @ՠΠqj E>/ud@F!A@@  A$A_wd@&ProperL@@  @*respectful%WO? @& @% :@/ @g@@7 ȠE'@Y "f,@G F@@@@@'add_0_r @H81Ԡ͠pi ؠǠ@@A&NZBase%NZMul&NatInt'Numbers#Coq@qo@A@%NZMul&NatInt'Numbers#Coq@)NZMulProp~@A@@@@"NZut@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@:@ @ @. A4p@ # t@@@@@c@ABC@@AA@@A@*add_succ_r @#[TEzt@At@  r @% @% @m A4@M "f婚 @R "f&@~ A4@@K@@H-add_succ_comm @@0)̠ŠhaРݠؠǠ@BJ@e O@j T@j Z@ A4`@ "f0e@ A4k@ "f5@@@@(add_comm @un IBޠϠ+=0" @ C@ @ @ @ A4:@ A47@@@@Ơ'add_1_l @JCߠ{dN=vi[VE7@BD7@ @ @+ A4@ Uq@ "f@@@@'add_1_r @|WPAv~p@{Ep@ @  @d A4@V U@H "f@@;@@8)add_assoc @0 XQz֠۠͠Ƞ@F:@U ?@Z !pF@a K@a Q@ A4W@ A4򩚠\@ A4b@ A4@@@@,add_cancel_l @|le@9ܠՠƠ" 4'@G@ 󚠐@ L@ <@ @ A47署@ A46@ Ɛ@@@@ʠ,add_cancel_r @ NG" hRAzm_ZI;@FH;@ 9@ @ @ @; A45}@A A4@   @@@@,add_shuffle0 @ 0)̠ŠhaR@I @ -  @ 2 ؚ @ 7  !@ 7  '@ A4 -@ A4KƩ 2@ A4 8@ A4UҐ@@]@@Z,add_shuffle1 @ R B ;ޠנzsѠ ٠@J˚ \@ w ɚ a@ | " f@  !q m@   r@   x@ A4 ~@ A4Щ @ A4 @ A4 @ A4 @ A4@@@@,add_shuffle2 @    : 3֠ϠrkT>-fYKF5'@2K' @  % @  ~ @  \ @   @   @* A4 @0 A4Z @5 A4x @: A4 @@ A4 @F A4 @@@@,add_shuffle3 @      0 )̠ŠhaR@L @ -  @ 2 ؚ @ 7  !@ 7  '@ A4y -@ A4ȩ 2@ A4ͩ 8@ A4U@@]@@Z'sub_1_r @ R B ; ޠ נ z s  Ѡ ٠@M˚ \@ w  a@ w  g@ [׀ l@ U q@ ";@@@@'mul_0_r @  { t     O Hՠ1 C6(@A@ @   @   @ Sk7 @ #  @ # @@@@AA@@A@*mul_succ_r @ à   O H     #  iSB{n`@yA9 @  7 @   @   @% Sku @ "f @= A4 @6 Sk@@@@E(mul_comm @      - & ɠ   e ^ O@B{ @ ' y @ ,  @ ,  @g Sk !@l Sk@@F@@{/mul_add_distr_r @ ; + $ Ǡ  c \     ˠ@C B@ ]  G@ b  L@ g  Q@ g  W@ Sk ]@ A4{ b@ A4 h@ Sk m@ Sk@@@@Ǡ/mul_add_distr_l @  w p     K D   Ѡ -  ?2$@=D @   @  T @   @   @ Sk @  A4D @  A4 @ Skҩ @  Sk֐@@@@)mul_assoc @ Ӡ à  _ X     3 ,  y c R~p@EI @  G @   @   @   @ : SkA @ @ Sk @ E Sk @ K Sk@@%@@Z'mul_1_l @      B ; ޠ נ z s d   ҠŠ@F !@ <  &@ <  ,@ w Sk 1@ v Uʐ@@V@@'mul_1_r @ K ; 4 נ Р s l       ۠ ʠ@G R@ m  W@ m  ]@  Sk b@  U@@@@,mul_shuffle0 @ | l e     @ 9 ܠ ՠ Ơ "  4'@2H򚠐 @  𚠐 @  I @   @   @  Sk @  Sk7 @  Sk @  SkC@@@@,mul_shuffle1 @ à   O H     #   i S B{n`@yI9 @  7 @   @  n @   @   @ / Sk @ 5 Skl @ : Sk @ ? Sk @ E Sk @ J Sk@@$@@Y,mul_shuffle2 @      A : ݠ ֠ y r c   ѠĠ@J @ ;  %@ @ 暠 *@ E Ě /@ J  4@ J  :@  Sk @@  Sk© E@  Skਗ਼ J@  Sk P@  Skh U@  Sks@@z@@,mul_shuffle3 @ o _ X     3 , Ϡ Ƞ    ' @%K嚠 v@  㚠 {@  < @   @   @  Skݩ @  Sk , @  Sk 1 @  Sk@@@@%le_wd @    B ; ޠ נ z s    \ F 5@@A&NZBase'NZOrder&NatInt'Numbers#Coq@  @A@'NZOrder&NatInt'Numbers#Coq@+NZOrderProp  @A@@@@"NZ  @A@@尶@ @  @ @   @ @ @   @   @   @  @ c @@&@@AA@@A@*lt_le_incl @   F ?  ۠ ~ w h Ġ  h[M@fA %@@  *@E @0@"lt  ̩6@   T@@@@@5'le_refl @P@9 ܠ ՠ x q        Ϡ@B ƚW@r \@    @@ @@\.lt_succ_diag_r @wg`    ; 4 נ Р    @C 횠~@ @S  @  "f S@@@@@.le_succ_diag_r @/( ˠ Ġ g `    I 3 "@D @ @   I@  "f @@@@@/neq_succ_diag_l @Ϡ[T     / (  u _ N @E E@  ;@ @  "f  |@@ @@⠠/neq_succ_diag_r @%   ] V G   |G:,@EF s@  i @!  @ C "f ې@@@@@/nlt_succ_diag_l @+SL     u Ѡ  uhZ@sG 2@M  9@  ?@ q "f  ؐ@@@@@>/nle_succ_diag_l @YIBޠz      ؠ@H Ϛ`@{  Őg@  m@  "f 7 @@@@@l)le_succ_l @wp KD   Ѡ -  ѠĠ@I @  @  ?@  @  "f p 9@u  @@@ i@@+le_gt_cases @KD|  e O > @J 5@  3@  x@"or @@ :  s@  p@@ @@ܠ-lt_trichotomy @|WPA   vA4&@?K m@  k@ 8 @  ?@'  @  @@@@@(lt_asymm @0 XQz ֠  zm_@xL 7@R  <@W @B@  ݩ I@  @@ @@H(lt_trans @cSL' à   @M ٚj@  ךo@ 0t@ @z@J  @@P  @U  g א@@ J@@(le_trans @,%Ƞd]F0ݠ@N @  @ m@ @@  ն@@  ۩@$ =@@ @@+lt_strorder @ܠ̠Šha<5&l[& @$O@+StrictOrder.[@@ @ @@ t@@+le_preorder @0)̠ŠhaRRE7@PPA@(PreOrder@@1 @| @@ @@/le_partialorder @4$\U~ڠĠ~qc@|Ql@,PartialOrder5͛КA@\ E@[ I@ZRvM@ #@^)mAE@@@ @A ^@ @ AhC@ABC@@W)lt_compat @@%lt_wdwg`;4נР@ @z@ @@   @ @@  @  @  @  C@l @͠@@%lt_wd']ɠD=٠|u^H7@@@@@(lt_total @b@3v䀠ڠʠàf_:3$@ J@  H@ @  @  @  W*@aǠ 2+ΠǠjcTTG9@@@@@ 'le_lteq @=@+lt_eq_cases*Ȁ*RKt@ +@F  0@K  ܠ7@  ҩl>@  ٩C@Y  ސ@+lt_eq_cases&yYIBޠzؠ@@@@@Y0Private_OrderTac5A'IsTotal= A @rkF?۠@@+Equivalence~@@ @ @(eq_equiv7L0)̠ŠhaJ4#@@@@@ @@;{ NG" @ࠚ@ @ @,Q^ݠ͠Ơib=6'm\' @@@@@ݠ @$@e #[TE@ @@ @@    @$ @@)  @(  @3  *@2 ǚ @ 3#[T}٠à}pb@@@@@3{ @z@/C0QA:ݠ֠yr@R@m W@r ^@. e@{ j@: w}mf A:ݠ֠Ǡ# Ǡ@@@@@}s @@{Y' à_X@ @  @ M@  Cݠ@ J@ OqǠSL' mWF@@@@@@@@@Đ@#TacuA @۠Ԡwp KD5{j5(@@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@j@ o@ @@@@AA@@A@< @{tOHՠ1 ՠȠ[TN3@AM@ @ A@@#@@.)lt_irrefl @C<ߠؠ{t]G6̠z_@By@ )@ o@@Q@@\D @Ԡ͠pi D=.tc.!@C@  @ @@ @ S@@@@*le_antisym @ C<ߠؠ{teeXJ0 ޠ@1Dݚ(@C ۚ-@H @3@ ζ@9@ >@T  @@@@Ϡ͐ @WG@ܠx֠l\' @mEd@ i@ @ϐq@  ֐x@ ʐ@@@@@ *interp_ord @!o)OrdersTac @#ordȠ@ @@@@@@@@@@@D@@ @@ 8@ @x @ @*(@@ @@ ɐ +k= R'6'6'6''+k+7T'+7T'6'ࠒA ҠѠ@@"lt\ @~@"eq8L@lР@TȠ@@QPO@@A@B@A@xA܀#[TEzE8*ˠĠ@@@@@%trans @ D=٠|uf fYK1!ߠ@2F"o'⚠-@H 2@M ]7@R @p@F!ڶ@y@F !@F@)trans_ord-0uG+9@@@@頠 @@f МA  qj E>ڠ@7@ 5@ @ @@ @@ @  L+k() LL7#'G@$tG@ܠxaK: Р~c@@@@@S< @jМLC$mlӠ̠oh C<-@@ @ @  @@V @@\ @a z L+k() NN7#'bt A:ݠ֠yrc c V H.ܠ@@@@@א @RSVTМBʠA1*͠Ơib@B@] G@b rL@g @R@ " p@X@ ( v]@ - ؐ L+k() MM7#'twg`;4נР   |G@:@@@@@+le_lt_trans @B<'_'&)"ŠaZ@S@ Q@ Κ@ @@ ̶@@  ҩ@  4 L+k() MN7#'tӠà_X3,ycR   ؠ{@@@@@k+lt_le_trans @ :~!YRC@@ @ *@ @ @  (@@r .@   L+k() NM7#'xt/WPyՠ y l ^D4@@@@@Ǡ%eq_lt @ijmkޠUE>ڠ}v@ V@q  [@v `@{ @f@| @l@ < q@ A 쐐 L+k() ML7#'Ԑt{tOHՠ1  ՠ Ƞ [TN3@@@@@#%lt_eq @VŷPƷǩ;s3;:=6٠Ҡun @g@ e@ ⚠@ @@  @@ 橚@  H L+k() LM7#'0tנРslG@1wf 1 $ @@@@@%eq_le @!"%#-N 5.ѠʠmfW@Ú@) @. >@3 @@4 <@$@ B)@  L+k() NL7#'tC3,ϠȠkdӠ    rXH @@@@@۠%le_eq @}~iYR-&ɠ @j@ o@ t@ @z@ @@ @  L+k() LN7#'t+$Ǡc\E/  ܠ ΠohbG@@@@@7&eq_neq @LE}  fP?   ՠŠh@G@ @ @ @@ @C@ J@  g@@@@@}&neq_eq @.'ʠàf_P P C 5 ֠Ϡɠ@HȚ@. ƚ@3 C@8 @%@; C@+@A I2@H  @@@@@à*not_neq_eq @L<5ؠѠtm ܠˠ   {aQ@bIY@t  ^@y @ĐƐh@~ m@ @@1@@)not_ge_lt @wp KDѠ- Ѡ Ġ WPJ/@JI@ G@ @@ 4@ v Đ@@,@@7)not_gt_le @LE}  fP?   ՠŠh@K@ @ @8@  m@A @@@@@p)le_neq_lt @~!YRCx C 6 (ɠ @L@!  @& @@s @w@/ 7@  @@@@@@@@@@@@@@@<_neq @<,%Ƞd]̠  y k@ RC@^ H@c @N@  U@k s@@@@@ T&le_neq @o_X3,ϠȠ   @ S嚠v@ 㚠{@ '@ R /@#andЖw@@ *@ 1@@@@@ *eq_le_incl @<5ؠѠtm V@/   @ T&@ $@ @@ ]@) @@@@@ Ơ(lt_stepl @Ѡʠmf A:+q`+@)U@ @ @  @@  @@ @  @@@@@(lt_stepr @ E>ڠ}vgàgZL@eVٚ$@? ך)@D T.@I @4@ R@:@P X?@  @@@@@>(le_stepl @YIBޠzؠ@W`@{ e@ j@ @p@ @v@ {@ @@@@@z(le_stepr @~!YRߠ;%ߠҠ@XQ@ O@ ̚@ @@ ʶ@@ Щ@  2@@@@@9 @Р\U0)v`O @YF@ D@ @ @@I @@  @  m@@@@@ @ 3,ϠȠkdUUH:@SZ@- @2 ؚ@7 @"@ @@(@ F-@  @@@@@,+le_antisymm @G70Ӡ̠oh נƠv@[N@i S@n @Y@ @_@ d@z @@@@@c)le_succ_r @~ng B;ޠנȠ$Ƞ@\@ 򚠐@ 6@ ,@ "fa ̠@ 9@ >@ "fs@@@@@)lt_succ_l @àOH# iSB @ ]9@ 7@ @@ @  "ft@ @@@@@ߠ,le_le_succ_r @"ZSDyD7)@B^p@ n@! @ @n @s c@I "f@@@@@,lt_lt_succ_r @1!YR{נ{n`@y_8@S =@X @C@ ީH@ N@ "f@@@@@M,succ_lt_mono @hXQ,%Ƞ@`ޚo@ ܚt@  {@K @P @ "fV@ "fU@@@@@,succ_le_mono @1*͠ƠibK5$@a@ @ ]@ S@ @ "f@ "f@@@@@Ǡ<_0_1 @Ҡˠng B;,ra,@*b@ @ # @7 U@@@@&le_0_1 @ 4-РɠleVVI;@Tc@u @E # @a U@@@@@<_1_2 @6& ^WܠƠse@~d=@  B@ UF@ ]s@@ @@E<_0_2 @`PI$ߠ@eg@7 l@ # p@ ]s@@@@o&le_0_2 @zsNGԠ0 ԠǠ@f@ @ # @ ]s@@@@@<_1_l @@9ܠՠxq ZD3@g*@ (@ @@ @ # e@@ 署@ @  U,@@@@@c@ABC@@ݠ+lt_ge_cases @} XQBwB5'@@hn@  l @  9 @  @r @@@@@+le_ge_cases @ *  RKtРtgY@ri 1@ L  6@ Q k =@ ة B@ Ր@@@@@A+lt_gt_cases @ \ L E} ۠@jҚ c@ ~ К h@  ϐ q@    x@H  }@M @@@@@|,eq_decidable @    # [T='Ԡ@k  @    @  )DecidableQ @)decidable r @  N@@@@@&eq_dne @ ͠   Y R-&s]L @lC @  A @  @B @   @  @@@@@蠠&le_ngt @!     + $Ǡc\MM@2@Kmy! @!% w!@!* !@x {!@ @@@@@&nlt_ge @!7!'!  à  _ XݠǠtf@n!>@!Y !C@!^ !L@ 穚!Q@ @@@@@P,lt_decidable @!k![!T     / (ˠĠ@oᚠ!r@! ߚ!w@! Ԑ!~@N @@@@@}<_dne @!!!!$!   \ U>(ՠ@p!@!  !@! P  !@ J!@ O@@@@@&nle_gt @!Π!!!Z!S     . ' t^M @qD!@! B!@! A!@E ~!@ {@@@@@砠<_nge @"!!!!!*!# Ơ  b [ LL?1@Jrx" @"$ v"@") "@ z"@~ @@@@@,le_decidable @"6"&"! !!^!W     ܠƠse@~s"=@"X "B@"] "I@ @@@@@H&le_dne @"c"S"L!!!!!'!  à   @tٚ"j@" ך"o@" ֐ؐ"z@ "@ @@@@@~*nlt_succ_r @""""%"!!!]!V    ? ) ֠@u"@"  "@" Q "@~ H"@ "f"@ T@@@@@5lt_exists_pred_strong @"Ӡ"à""_"X!!!!!3!,! y c R@v"@" N"@" L"@" @"@ @"@ R B@"ex @"@# !k#@# # @#  #@ B "fک#@ w @@@@A@%@A@AРD.Morphisms_Prop#Q@A1@D@ABCD@@#.lt_exists_pred @#>#.#'"ʠ"à"f"_"!!!!  Π {m@wk#E@#` #J@#e @#P@  `#V@#q Z#[@#v ٠#b@#x #h@  "f2#m@  @@@M@A@@m,lt_succ_pred @##x#q## """L"E!!!Ҡ!.!!ҠŠ@x#@# #@# @#@j 5#@# #@  "f#@  "{E@@@o@Al@@ 4@@ABCD@@&rs_rs' @#ʠ###V#O"""""*"#"!p!Z!I@y@#@# శ@#@# #@# 򚠐#@# }#@$ @ն@_#@$  @#@!X @ȩ #@!1 "fQt$@$  @$ @!m #@$@$- $@$2 @$@!  @$$@ vK'H)@@}@@'(rs'_rs'' @$B$2$+#Π#Ǡ#j#c#""""!!Ҡ!q@z^@$J@$e \ZX@$R@$m dR$X@$s j$\@$r  $a@$| @֚$g@$ @$m@!  @㚠$t@$ ᚠ$y@$ @$@! a @$@V ةIX\$@$  : $@$  $@$ @$@" "@$@w !q $@$  $@$ @$@" 9@$@  8$@! "f@@@@à%rbase @$ޠ$Π$Ǡ$j$c$####>#7#(""n"]( @&{@$@%  $@%  `$@%   ^$@% @$@"^ @%@  Tz̐@@@@@)A'A_right @%% %$$$D$=##٠#|#u#f" ""fYK@d|8@%$@%? 6 O%)@%D @ %/@%J  %5@%P  %:@%U @%@@" X@%F@  H  %L@%g @%R@" o4 #@@@@@R6strong_right_induction @%m%]%V$$$$$1$*#͠#Ơ##""@}@%u@% @%}@% }%@% %@% !2!  %@% @!%@% @%@" !3@!%@% ! %@% @%@#  +@%@ !* (!$%@% @%@# Đ@@@ @A @"C@@ABC@@/right_induction @%٠%ɠ% %e%^%$$$$9$2$###i#X#@!~@%@% ﰶ@%@& %@&   %@&  !!!%@& @@!n%@& @&@#g "@& @#? "f!&@&- @&@#z 0 @@@@@0right_induction' @&3&#&%%%[%T$$$$$}#٠#à#}pb@{ O@&;@&V  M K I@&C@&^  U C&I@&d  [&M@&c !!!x&R@&m @!ǚ&X@&s @&^@# ! @!Ӛ&d@& @&j@# @t&r@# "fs!暠&w@&  K@@"@@v7strong_right_induction' @&&&z&&%%%U%N$$$۠$7$!$۠Π@@ @&@&    @&@&   &@&  &@& "V"D!֚&@& @"%&@& @&@$ "Oj@"1&@& @&@$* @">&@& "<&@& @&@$< Ŷ@&@ "2J!"V&@'  @@@@@栠&ls_ls' @'&&&&&)&"%Š%%a%Z%K$$$K>0@IA!@' @'$ !!!@'@', !#!'@'2 !)'@'1 """F' @'; @! @"''@'B @'-@ @!'5@$g "f""':@'U @'@@$ ""U@"'H@'c "'M@'h @'S@$ "@'Z@$ "x6'`@$ "f"0:@@@@_(ls'_ls'' @'z'j'c'&&&&>&7%ڠ%Ӡ%Ġ% % $Ġ@B!@'@' !!!@'@' !!'@' !'@' #?#-"'@' @#'@' @'@% #8@#'@' #'@' @'@% #Q!@'@%  "8'@$ "f"<#8'@' #u#@'@' #>'@' @'@%> %@'@%D #\"#W'@( #U'@( @'@%U <@'@%[ #s'@%1 "f"ɐ@@$@@%lbase @(( ('''A':&ݠ&֠&y&r&c%%%cVH@aC"5@(!@(< "3#L(&@(A #(,@(G #(1@(L @(7@% @(=@% #[(C@%u "f# @@@@@B(A'A_left @(](M(F''''~'!'&&&&%%ܠ@D"y@(e@( "w#(j@( @#ߚ(p@( #嚠(v@( #㚠({@( @(@% A@(@% #A#Q#(@( @(@% @@@%@@5strong_left_induction @((((8(1'Ԡ'͠'p'i' '&&R&<&+@E"ȶ@(@( "ƶ"ĩ"°@(@( "Ω"(@( "Ԛ(@( $q$_#񚠐(@( @$@(@( @(@&9 $j@$M(@( $K(@( @(@&K 2@(@&Q $ i(@&' "f#m$i(@) @)@&b  c @@@#@%@@ABC@@.left_induction @)) )(((E(>''ڠ'}'v'g&à&&gZL@eF#9@)%@)@ #7#5#3@)-@)H #?#-)3@)N #E)7@)M $$ж$b)<@)W @#(@$)C@)^ @)I@  @#)Q@& "f$! $Ś)V@)q @)\@& D@@@@@[/left_induction' @)v)f)_)((((:(3'֠'Ϡ'''&@G#@)~@) ###@)@) ##)@) #)@) %;%)$)@) @% )@) @)@' %<O@%)@) @)@} !@#)@& "f$o%))@) #@@ @@6strong_left_induction' @)Ԡ)Ġ))`)Y(((((4(-('z'd'S@H#@)@) ###갶@)@) ##)@* #)@* %%%)@* @%h)@* @)@'a %@%t*@*  @* @'m !n@%*@*- %*@*2 @*@' %#@*$@' %BD**@'\ "f$%*/@*J $@@ T@@./order_induction @*I*9*2)ՠ)Π)q)j) )(((''٠'Ƞx@I$e@*Q@*l $c$a$_@*Y@*t $k$Y*_@*z $q*c@*y &%%*h@* @$T@%ޚ*o@* @*u@' !@ *}@' "f~@%򚠐*@* @*@Y I@4*@' "f%ak&*@* @%a@@@%Z@%X@AB@@0order_induction' @****?*8)۠)Ԡ)w)p)) ((Y(C(2@J$϶@*@* $Ͷ$˩$ɰ@*@* $թ$*@* $ۚ*@* &x&f%*@* @$@&H*@* @*@(A !@*@( "f@&\*@+ @*@(U @ܩΐ*@(2 "&o+@+ j@@i@@1order_induction_0 @++ +***B*;)ޠ)נ)z)s)d(((dWI@bK%6@+"@+= %4%2%0@+*@+E %<%*+0@+K %B+4@+J &&Ͷ@&А+;@(h # @&+A@+\ @+G@( +L@(y # &@%!+T@( "fU@&ɚ+Z@+u @+`@0 &+e@( # @%:+m@( "f&='&ᚠ+r@+ %F@@@@@q2order_induction'_0 @++|+u++***P*I)))֠)2)) ֠ɠ@L%@+@+ %%%@+@+ %%+@+ %+@+ 'Q'?@'B+@( # @'"+@+ @+@) +@( # 'W@m%+@( "fǶ@';+@+ @+@)4 'l+@) # @+@) "'S+@+ %@@@@@㠠<_ind @++++++&+* **^*W*H)))}H;-@FM&@,@,! &&&@,@,) & &,@,/ &&,@,. ''',@,8 @'p,%@)W "f&@',+@,F @,1@ #N@;ܐ,9@)k "f:',>@,Y @,D@ "\,@@@@@C&le_ind @,^,N,G+++++"+****))ݠ@N&z@,f@, &x&v&t@,n@, &&n,t@, &,x@, (#('욠,}@, @&i@'욠,@, @,@) #@5,@) "f',@, @,@) "@@@@@&Rlt_wd @,,,,C,<+ߠ+ؠ+{+t+++*]*G*6@O'䚠,@, &˰@,@, @,@, &ܩ&,@, @,@, &皠,@, &,@, &񚠐,@, ((W,@- (U,@- k,@*V (,@ (@@&@@&Rgt_wd @--,,,,;,4+נ+Р+s+l+]***]PB@[P(@-@-5 ''@-!@-< @-&@-A '8'&-,@-G @-1@-L 'C-5@-K '5-;@-V 'M-?@-U ((-D@-_ (-I@-d Ǡ-P@  (㩚-U@* (@@&z@@T%lt_wf @-o-_-X,,,,,3,,+Ϡ+Ƞ++**@Q(-v@- "Wf)"@,well_founded8E-@- (-@- (-@-  -@* (-@j )5@@@* A𱍠G @*$A7iA)@AB@@%gt_wf @----H-A,,ݠ,,y,,,+b+L+; @ R(隠-@- M-@- )=-@- );-@- Q-@ )m-@+A )1@@E@@ޠ-add_lt_mono_l @-----~-!-,,,Y,R,C+++x@+t@)l-@. )j.@. %Ú.@." ).@ ),.@ .@+q A4%|.@+v A4)@@'C@@AA@@A@-add_lt_mono_r @.;.+.$-Ǡ--c-\,,,,,++ˠ+@+A).?@.Z ).D@._ &.I@.d ).P@  )n.U@ % .[@+ A4%x.`@+ A4)@@'@@B+add_lt_mono @.z.j.c.---->-7,ڠ,Ӡ,Ġ, , +@+B)횠.~@. )뚠.@. &D.@. %".@. @.@ c %@.@ i ).@ n .@+ A4(x).@, A4%@@@@@-add_le_mono_l @.Ġ...P.I-----$--,j,T,C@,?C*7.@. *5.@. &.@. *~.@,; ).@,@ .@,< A4&G.@,A A4*|@@(@@ˠ-add_le_mono_r @/.....+.$-Ǡ--c-\-M,,,@,~D*v/@/" *t/ @/' &͚/@/, */@,z *6/@, /#@,{ A4&@/(@, A4*Ð@@(M@@ +add_le_mono @/B/2/+.Π.Ǡ.j.c.----,,Ҡ,@,E*/F@/a */K@/f ' /P@/k %ꚠ/U@/p @/[@, %ݶ@/a@, */f@, /l@, A4ȩ/q@, A4@@@@@S.add_lt_le_mono @//{/t//...O.H---ՠ-1-- @-F*/@/ */@/ 'U/@/ &3/@/ @/@!t &&@/@-  *ȩ/@! /@-  A4/@- A45@@@@@.add_le_lt_mono @/Ԡ/Ġ//`/Y.....4.-.-z-d-S@-OG+G/@/ +E/@/ '/@/ &|/@0 @/@-O &o@/@! +/@! /@-V A4Z0@-[ A4~@@@@@堠+add_pos_pos @00 0///E/>..ڠ.}.v.g-à--@-H+0!@0< +0&@0A @0,@! 01@-^ # +˶@07@" 0<@-i # +֩0A@" 0F@-s # 0K@- A4&͐@@p@@-.add_pos_nonneg @0e0U0N/////)/".Š... --@-I+ؚ0i@0 +֚0n@0 @0t@"D 0y@- # ,@0@- 0@- # ,0@"Y 0@- # 0@- A4'@@@@u.add_nonneg_pos @0000902/ՠ/Π/q/j/ /..S.=.,@.(J, 0@0 ,0@0 @0@. 0@- # ,[@0@" 0@- # ,f0@" 0@. # 0@.3 A4']@@@@1add_nonneg_nonneg @000ޠ00z00///U/N/?...t@.pK,h0@1 ,f0@1 @1@.f 1 @.6 # ,@1@.q 1@.A # ,1@.{ 1@.K # 1#@.{ A4'@@H@@,lt_add_pos_l @1=1-1&0ɠ0 0e0^0////..͠.@.L,1A@1\ ,1F@1a @1L@# 1Q@.~ # ,멚1V@#& ,񩚠1\@. A4,z@@@,@-@AB@@@,lt_add_pos_r @1x1h1a10000<05/ؠ/Ѡ/ //.@.M,뚠1|@1 ,隠1@1 @1@#W 1@. # -&1@#a -,1@. A4,@@,[@@y,le_lt_add_lt @1111=160٠0Ҡ0u0n00 //W/A/0@/,N-$1@1 -"1@1 ){1@1 (Y1@1 @1@/, (L@1@# 1@/. A4,䩚1@/3 A4-v+1@# (c@@@@@à,lt_le_add_lt @111111#1000[0T0E///z@/vO-n1@2 -l2@2 )Ś2 @2$ (2@2) @2@# (@2@/| 2 @/x A4-.2%@/} A4J2*@# (@@@@@ ,le_le_add_le @2D242-1Р1ɠ1l1e11000//Ԡ/@/P-2H@2c -2M@2h *2R@2m (욠2W@2r @2]@/ (߶@2c@/ 2i@/ A4-w2n@/ A42s@/ (@@@@@U,add_lt_cases @22}2v22111Q1J000נ0300 @0Q.2@2 -2@2 *W2@2 )52@2 @2@$v 2@0 A4).2@0  A4.L"栩2@$ "32@$ -Ґ@@@@@,add_le_cases @2נ2Ǡ22c2\111117101!0}0g0V@0RR.J2@2 .H2@2 *2@3 )2@3 @2@0R 2@0N A4)x2@0S A4.#03@0d "}3@0i .@@@@@頠-add_neg_cases @3!33 222I2B11ޠ11z1k0Ǡ00@0S.3%@3@ .3*@3E @30@% 36@0 A4.њ3:@0g # #o3A@% .3F@0s # 3K@% .暠3P@0} # @@.@@2-add_pos_cases @3j3Z3S22222.2'1ʠ1à1100@0T.ݚ3n@3 .ۚ3s@3 @3y@%I 3~@0 # 3@0 A4/#3@%Z 3@0 # .ਗ਼3@%d 3@0 # /3@@@@{0add_nonpos_cases @3333?382۠2Ԡ2w2p22 11Y1C12@1.U/&3@3 /$3@3 @3@1$ 3@1  A4/c3@0 # $3@15 /%3@1 # 3@1? /x3@1 # @@.@@Ġ0add_nonneg_cases @333333$3222\2U2F111{@1wV/o4@4 /m4@4  @4 @1m 4@1= # 4@1m A4/$J4@1~ 4!@1N # /r4&@1 4+@1X # /Ő@@P@@ -le_exists_sub @4E454.3Ѡ3ʠ3m3f3 322211ՠ1@1W/4I@4d /4N@4i @4T@1 /d4Z@4u ,4_@4z ݠ4f@4| /4l@1 A4 ,4q@1 4v@1 # 0@@@/:@A/@0@A0AA_@]@.@ABCD@@a+mul_lt_pred @4444%4333]3V2222?@2@,W4@4 +54@4 04@4 04@4 @4@4 4@1 "f/00c4@& 4@2 Sk$?4@2 Sk.0d4@& 4@2- A44@2& Sk+]0u4@28 A44@21 Sk/07@@. @@AA@@A@1mul_lt_mono_pos_l @544444+4$3Ǡ33c3\3M2@3KA,5@5  0y5 @5% 0w5@5* @5@& 5@2G # 0k0Ơ5!@& 0?5&@& 5,@2w Sk+51@2| Sk0F@@@/@A/@0?@1@@ABCD@@P1mul_lt_mono_pos_r @5P5@594ܠ4ՠ4x4q44 3332@3B-5R@5m 0ƚ5W@5r 0Ě5\@5w @5b@'2 5g@2 # 015n@'> 05s@'C 5y@2 Sk05~@2 Sk00@@.@@1mul_lt_mono_neg_l @5555%5444]4V3333?@3C-W5@5 15@5 1 5@5 @5@'{ 05@2 # 1\5@' 0թ5@' 5@3  Sk0ש5@3 Sk,I@@@@ᠠ1mul_lt_mono_neg_r @55Ѡ5ʠ5m5f5 5444A4:4+3@4)D-5@5 1W5@6 1U5@6 @5@' 1;1;$@"lt 1;,@#mul Sk1@@/*@@4mul_le_mono_nonneg_l @666555G5@44ܠ44x4i3@4gE16!@6< 16&@6A -皠6+@6F @61@3 66@3c # 1϶@6<@3 ,6A@3 6G@3 Sk10)6M@3 Sk1[@@1@@g4mul_le_mono_nonpos_l @6g6W6P55555+5$4Ǡ444 @4F1ؚ6i@6 1֚6n@6 ./6s@6 @6y@3 26~@3 # @6@3 -6@3 6@3 Sk16@3 SkM@@1X@@4mul_le_mono_nonneg_r @6666:635֠5Ϡ5r5k5544T@4G26@6 26@6 .v6@6 @6@4" 6@3 # 2^@6@4- -M6@42 6@4! Sk&Q6@4& Sk-]@@1@@4mul_le_mono_nonpos_r @666ޠ66z66555U5N5?4@5=H2f6@7 2d6@7 .7@7 @7@4i 27 @49 # @7@4t -7@4y 7@4h Sk-7"@4m Sk&@@1@@<,mul_cancel_l @7<7,7%6Ƞ66d6]655554@5I27>@7Y 27C@7^ /7H@7c @27P@7f 2ꚠ7U@4 # 37\@7r 7b@4 Sk䩚7g@4 Sk27l@7 -@@@2/@3@1@ABC@@,mul_cancel_r @77y7r77666M6F555Ӡ5/@5J27@7 27@7 /Q7@7 @27@7 377@4 # 3N7@7 7@4 Sk2ĩ7@4 Sk2ҩ7@7 .;@@0@@Ӡ(mul_id_l @7Ӡ7à77_7X6666636,65y@6K3D7@7 3B7@7 @3@7@7 3|7@5 # 37@8 7@5? Sk337@8 3K7@5C U@@3 @@(mul_id_r @888777@796ܠ6ՠ6x6q6b5@6`L38@85 38@8: @38'@8= 3š8,@5Y # 3ؠ83@8I 89@5 Sk3W38>@8T 3ٚ8C@5 U@@1h@@]1mul_le_mono_pos_l @8]8M8F7777~7!76666@6M3Κ8_@8z 3̚8d@8 0%8i@8 @8o@*? 8t@5 # 4 4 8{@5 .8@5 8@5 Sk8@5 Sk3ݐ@@3O@@1mul_le_mono_pos_r @888818*7͠7Ơ7i7b7666K@6N48@8 48@8 0m8@8 @8@* 8@5 # 4U4h8@6% /E8@6* 8@6 Sk3㩚8@6 Sk3@@1@@1mul_le_mono_neg_l @88ݠ8֠8y8r88777M7F776@75O4^8@9  4\8@9 08@9 @8@* 49@61 # 49 @6m /9@6r 9@6a Sk4h9@6f Sk@@@@@51mul_le_mono_neg_r @959%9888]8V777776@7}P497@9R 49<@9W 09A@9\ @9G@+ 4ᚠ9L@6y # 49S@6 /թ9X@6 9^@6 Sk4|9c@6 Sk4x@@2@@}2mul_lt_mono_nonneg @9}9m9f9 9888A8:7ݠ7֠7Ǡ7#@7Q49@9 4욠9@9 1E9@9 0#9@9 @9@6 9@6 # 4@9@+o 34@9@7 9@6 # 4@9@+ 9@+ 9@7 Sk39@7  Sk#l3@@@@@ݠ2mul_le_mono_nonneg @9ݠ9͠9Ơ9i9b98888=868'7@8%R5N9@9 5L9@9 19@: 09@:  @9@7V 9@7& # 5 @9@7a `@:@7g : @77 # 5@:@7r q:@7w :@7f Sk_: @7k Sk^@@4@@:+mul_pos_pos @:::*:#9Ơ99b9[888887@8S5:<@:W 5:A@:\ @:G@, :L@7y # 5@:R@," :W@7 # 5񩚠:\@,, :a@7 # :f@7 Sk0@@@@+mul_neg_neg @::p:i: :999D9=88٠8ʠ8&@8T5񚠐:@: 5:@: @:@,] 6(:@7 # @:@,h 63:@7 # :@,r :@7 # :@7 Sk1.@@@@Ơ+mul_pos_neg @:Ơ:::R:K99999&998l@9U67:@: 65:@: @:@, :@8 # 6r@:@, 6y:@8 # :@, :@89 Sk1p:@8 # @@5@@ +mul_neg_pos @; :::::4:-9Р9ɠ9l9e9V8@9TV6};@;) 6{;@;. @;@, 6;@8K # @;$@, ;)@8V # 6é;.@, ;4@8 Sk1;8@8e # @@]@@R1mul_nonneg_nonneg @;R;B;;:ޠ:נ:z:s::9998@9W6Ú;T@;o 6;Y@;t @;_@8 ;d@8 # 6@;j@8 ;o@8 # 7 ;t@8 ;y@8 # ;~@8 Sk2@@@@0mul_pos_cancel_l @;;;;$;:::\:U9999>@9X7 ;@; 7;@; @;@-u ;@8 # 7D7V;@- ;@8 # ;@9 Sk6٩;@- ;@8 # 7_@@ i@@ߠ0mul_pos_cancel_r @;ߠ;Ϡ;Ƞ;k;d;;:::?:8:)9@:'Y7P;@; 7N;@< @;@- ;@9 # 77;@- ;@9* # <@9M Sk7 <@- < @99 # 7]@@ @@&3mul_nonneg_cancel_l @<&<<;;;N;G:::::p9@:nZ7<(@@@,lt_1_mul_pos @<<<<@<9;ܠ;ՠ;x;q;; ::Z@:\8%<@< 8#<@< @<@. <@:  U8`@<@. <@9 # 8k<@. <@:  U<@:+ Sk3b@@@@(eq_mul_0 @<<<<<<"<;;;Z;S;D:@;B]8k<@= 8i=@= 8=@= =@:Y Sk8=@:? # -G=@=/ 8=@:K # =#@=9 8=(@:U # @@6M@@B)neq_mul_0 @=B=2=+<Π<Ǡ=@= =@: Sk/C=@: # =@= 9B=@: # @@@@@Ǡ*eq_mul_0_l @=Ǡ===S=L<<<<<'< <;m@<`98=@= 96=@= @=@= =@;% Sk9u=@;  # @9D=@= 9=@; # =@> 9=@;" # @@@@@ *eq_mul_0_r @>=====7=0<Ӡ<̠@>, 9~>@>1 @>@>2 >"@;m Sk9>&@;S # @9>.@>D 9>3@;` # >8@>N 9>=@;j # @@@@@ W(mul_eq_0 @,@fĈ>[>K>D====|==<<<@9˚>\@>w 9ɚ>a@>| : >h@>~ >n@; Sk: >r@; # .>y@> :>~@; # >@> :>@; # <ڠ<٠<ؠ<@<0@^\>>>>/>(=ˠ=Ġ=g=`=<<>>>I>B==ޠ==z===@:->@> :+>@> @>@> >@< Sk:j>@< # @:9>@> :v>@<  # >@> 9>@< # b@=24pe>>>>>>'> =à==_=X=I<@@@@@ *mul_eq_0_r @-B@0*?? ?>>>A>:=ݠ=֠=y=r=c@:?@?5 :?@?: @?%@?; ?+@>>>|>>====@@@@@ q(lt_0_mul @?q?a?Z>>>>>5>.=Ѡ=ʠ==@=b:⚠?s@? :?x@? ;$?@1O ?@< # ?@< Sk;$/$ ?@1b ?@< # ;1?@1l ?@< # ;:$?@1x ;B?@< # ?@1 ;M?@< # @@:{@@ Ѡ5square_lt_mono_nonneg @?Ѡ???]?V>>>>>1>*>=w@>c;B?@? ;@?@? @?@=@ ?@= # ;}@?@1 ;?@1 ?@=? Sk; ; ?@=E Sk;L;L@@@@@ 5square_le_mono_nonneg @@@????=?6>٠>Ҡ>u>n>_=@>]d;@@@2 ;@@@7 @@"@= @'@=T # ;@@-@= ;K@2@= @8@= SkD@=@= SkC@@@@@ W6square_lt_simpl_nonneg @@W@G@@??ܠ??x??>>>=@>e;Ț@Y@@t ;ƚ@^@@y @@d@= @i@= # <@@o@2? @u@= Sk{@z@= Sk<<@@2P 7@@@@@ 6square_le_simpl_nonneg @@@@@&@? ??^?W>>>>@@>f< @@@ < @@@ @@@>  @@= #  @@> Sk@@> SkC@@>$ 7D@@@@@ ܠ,mul_2_mono_l @@ܠ@̠@Š@h@a@????<?5?&>@?$gL A4@@>> U@@>I SkA@>> ]sS SkA @>H ]s<@@:2@@ '*add_le_mul @A'AA@@@O@H?????q>@?oh!aA+@AF !bA2@AM @A8@3 A=@> U<׶@AC@3 AH@> U<⩚AM@> AS@> A47թAX@> Sk7ڐ@@<@@ r-square_nonneg @ArAbA[@@@@@6@/?Ҡ?ˠ??@?iKAt@A Ay@> A~@> # A@> Sk3'@@!@@ 5crossmul_le_addsquare @AAAA)A"@Š@@a@Z????C@?jvA@A tA@A @A@?  A@> # =I@A@? A@> # =TA@?! A@? A4A@? Sk<٩A@? Sk8RA@?- A4A@?& Sk穚A@?+ Sk@@5@@ -add_square_le @AAAAAA"A@@@Z@S@D?@@BkӚA@B њB@B @B@?i B @?9 # =@B@?t B@?D # =B@?~ B"@?z A4B(@?s Sk4B-@?x Sk3B2@?} SkB8@? A48B=@? A48@@=@@ W-square_add_le @BWBGB@AAܠAAxAA@@@?@@l0BY@Bt .B^@By @Bd@? Bi@? # >@Bo@? Bt@? # >By@? B@? SkB@? A49B@? A49 B@? SkB@? ]sB@? A4B@? SkB@? Sk@@=h@@ 4quadmul_le_squareadd @BBBBJBCAAߠAA{AAA@d@AmB@B B@B @B@@- B@? # >j@B@@8 B@@ # >uB@@B B@@1 SkB@@7 SkB@@= SkB@@2 ]sB@@6 ]s>>LC@@K SkC@@^ A49C @@c A49@@=@@%(pred_inj @C%CCBBBMBFAAA@A@!nC&@!t !mC.@ @C4@"eq C;@$pred "BCC@ "ACJ@ CB@@@ %'A*Equalities*Structures#Coq@AviA)Morphisms'Classes#Coq@@AAۀߠK @Aۓό^ @A݋X_A/RelationClasses'Classes#Coq@@ABC@@AA@@A@+pred_inj_wd @CC~CwCCBBBRBKA@BHA"n1C@i "n2C@p %Logic$Init#Coq@@#iffС)C@u C@t "BC@{ "AC@ BA@@@rAj@tA8ӠA4Relation_Definitions)Relations#Coq@@AB@@['succ_m1 @CC֠CϠCrCkCCBBBF@BBC@ C@$succ "fC@B U‐C@B  UC@$zero # @@@A@A@A@ABC@@*add_pred_l @DD DCCCDC=B@C:@!nD@!t !mD#@ D(@"eq D/@#add A4D6@$pred "BAD?@  "DE@ A4BA@@@ %'A*Equalities*Structures#Coq@AviA)Morphisms'Classes#Coq@@Aۓό^ @A݋X_A/RelationClasses'Classes#Coq@@ABC@@AA@@A@*add_pred_r @DDuDnDD CCCI@CAiD@h gD@n D@f D@e A4BD@e "AD@l "D@y A4BA@@@cA[@eAS@gAQ@ABC@@I)add_opp_r @DˠDDDWDPCCC@CBD@ D@ D@ D@ A4BD@Cb U‐AD@#sub [׀BA@@@A@A@A@ABC@@'sub_0_l @E DDDDD4D-C@D*C𚠐E @ E@ E@- [׀E@$zero # AE!@C U‐A@@@A@A@A@ABC@@*sub_succ_l @ECE3E,DϠDȠDkDdD@DaD'EA@& %EG@, EL@$ ER@j [׀EX@$succ "fBAEa@  "fEg@ [׀BA@@@"A@$A@&A@ABC@@*sub_pred_l @EEzEsEEDDDN@DEnE@m lE@s E@k E@ [׀E@i "BAE@q "E@ [׀BA@@@hA`@jAX@lAV@ABC@@N*sub_pred_r @EРEEE\EUDDD@DFE@ E@ E@ E@ [׀BE@ "AE@ "fE@  [׀BA@@@A@A@A@ABC@@(opp_pred @FFEEEE>E7D@E4GF@ F@ F@D U‐F%@ "AF,@ "fF2@D U‐A@@@A@A@AB@@Р(sub_diag @FRFBF;EޠEנEzEsE@EpH6FP@5 FU@- F[@s [׀AAFb@H # @@@A@A @A@ABC@@.add_opp_diag_l @FFrFkFFEEEF@EIfF@e F@] F@\ A4F@E U‐AAF@~ # @@@PAH@RA@@TA>@ABC@@6.add_opp_diag_r @FFFFDF=EE٠E|@EJF@ F@ F@ A4AF@EI U‐AF@ # @@@A~@Av@At@ABC@@l)add_opp_l @FFޠFנFzFsFFE@F KҚF@ КF@ F@ F@ A4G@E U‐ABG @# [׀BA@@@A@A@A@ABC@@-add_sub_assoc @G.GGFFFVFOE@FLLG,@ G2@ !pG9@ G>@ GD@ A4CGK@c [׀BAGS@k [׀GY@* A4CBA@@@A @A@A@ABC@@.opp_involutive @G}GmGfG GFFFA@FMaG{@` G@X G@F U‐G@F  U‐AA@@@GA?@IA7@KA5@ABC@@--opp_add_distr @GGGG;G4FנFРFs@FNG@ G@ G@ G@F? 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