une formalisation des faisceaux et des schémas affines en théorie des types avec Coq

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(1)une formalisation des faisceaux et des schémas affines en théorie des types avec Coq Laurent Chicli. To cite this version: Laurent Chicli. une formalisation des faisceaux et des schémas affines en théorie des types avec Coq. RR-4216, INRIA. 2001. �inria-00072403�. HAL Id: inria-00072403 https://hal.inria.fr/inria-00072403 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Une formalisation des faisceaux et des schémas affines en théorie des types avec Coq Laurent Chicli. N° 4216 Juin 2001. ISSN 0249-6399. ISRN INRIA/RR--4216--FR. ` THEME 2. apport de recherche.

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(59) ! " # %$ & ' ( . L. Let E be a set. Definition. union_part:℘(℘(E)) ⇒ ℘(℘(E)) 1: Intros P. 1: Apply (!Build_Predicate ? x:E , ∃ A:℘(E) , (A ∈ P) ∧ (x ∈ A)). Definition (union_in_top). given top of type ℘(℘(E)) , we define the proposition union_in_top(top) by ∀ open_family:℘(℘(E)) , open_family ⊂ top ⇒ union_part(open_family) ∈ top Definition (inter_in_top). given top of type ℘(℘(E)) , we define the proposition inter_in_top(top) by ∀ open1,open2:℘(E) , open1 ∈ top ⇒ open2 ∈ top ⇒ open1 ∩ open2 ∈ top. `bX,kreS|}eJdieJfgacXpn v Xjpkldce v p&I . Definition. a Topology is a structure with the following components: top: of type Predicate(℘(E)) (coercion) union_in_top_prf: a proof of union_in_top(top) inter_in_top_prf: a proof of inter_in_top(top) total_prf: a proof of full(E) ∈ top empty_prf: a proof of ∅ ∈ top. Xjkno`7Xjp|olghXkeJ|DegdcegfJactnbX/Xjpk!no`7XjòpXjYsbdcX~beJò`X^ lJXjhno`bX,keJ|DegdcegfgacXI Definition. a topological space is a structure with the following components: et_setoid: a set (coercion) open: of type Topology(et_setoid). v v v 20! #H |DpX egv dcleg fJno`,ac`t~nbX^XjX^ dhXH^ lgdmYl die XHXFv `Spnok

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(65) ¥. N . Yd.;- acXH`SòkrX hvX p|oXjhl krv aceJh`7eSò~bpX,k v hnbXHa pv X©X d.-v XjY òpXj^ Xjp&YI sbdiX. : (2 >?0 / T?E]?. ~bXHpµX v YXH^ pheJ`SkrXHòlg`Sk ³ |onbamp eJ`| v XHò~op. Let E be a topological space. Let A be of type ℘(E). Definition. closed_containing:℘(℘(E)) 1: Apply (!Build_Predicate ? F:℘(E) , A ⊂ F ∧ F is closed). Definition. let adh be intersection_part(closed_containing) (its type is ℘(E)).. h- XH`$prk Y;dceJX`S|bk dmv noXplg|Ddie Xv krp©ack diXHpX v diY XHYzX,^ YtnbXHaµp hheSdmlg`SpkrpracamXjtònb`bXjX p³ lam òpa·tnbXdml| v eJ| v aX^ kbX*^ hjl v lghkbX ^ v amprkactnbX~bX/d.- lJ~ U X ^ v XHòhXWI. Lemma (included_adh). A ⊂ adh. Lemma (adh_closed). ∀ A:℘(E) , adh(A) is closed. Lemma (closed_containing_adh). ∀ F,G:℘(E) , F is closed ⇒ G ⊂ F ⇒ adh(G) ⊂ F.. v v v v v v amòhdcOynop!- am`S~}kblgX ^ òacp Xjn³ ~9- no`bX|ol kracX ³ p- X¡o| acY;X~bX/Qlh eS`$pacY;acdclga XPI}h- XHprk!d.- n}`baieS`~bX,kreSnop!dcXjp eJnogX kp. Definition. open_included:℘(℘(E)) 1: Apply (!Build_Predicate ℘(E) U:℘(E) , U ⊂ A ∧ U is open). Definition. let int be union_part(open_included) (its type is ℘(E)).. 'k oheJYzY;X|}eSn v d- lg~ U X ^ v Xj`}hXWI M. sutvsxw.

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(73) H. ¥. N . Definition (infinite_inter_closed). given close of type ℘(℘(E)) , we define the proposition infinite_inter_closed(close) by ∀ closed_family:℘(℘(E)) , closed_family ⊂ close ⇒ intersection_part(closed_family) ∈ close Definition (finite_union_closed). given close of type ℘(℘(E)) , we define the proposition finite_union_closed(close) by ∀ closed1,closed2:℘(E) , closed1 ∈ close ⇒ closed2 ∈ close ⇒ (closed1 ∪ closed2) ∈ close Let F_empty be of type ∅ ∈ F. Let F_full be of type full(E) ∈ F. Let F_prop1 be of type infinite_inter_closed(F). Let F_prop2 be of type finite_union_closed(F).. e~ogX/achk Xj>p|}(l v: krac=Xjp9I?0 / ' dcleSòhkaieS`,tSnoa l!V no`/Xj`}prXHY sbdcX

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(83). . . Definition. a open_base is a structure with the following components: open_base_setoid: of type Predicate(℘(E)) (coercion) base_inter_in_top_prf: a proof of inter_in_top(open_base_setoid) base_total_prf: a proof of full(E) ∈ open_base_setoid base_empty_prf: a proof of ∅ ∈ open_base_setoid. 'k eJ`~X^ D`back9I M. Let B be of type open_base. Definition. generated_part_by_union:℘(℘(E)) 1: Apply (!Build_Predicate ? x:℘(E) , ∃ Y:℘(℘(E)) , Y ⊂ B ∧ (x = union_part(Y))).. . dLòX v XHprkX/|odcnop*tSnH- lzV ~XH^ Y;eJ`Sk v X v tnbX. >?. / 0 ( / ' 9?E>?. Lemma (full_in_top_for_bases). full(E) ∈ generated_part_by_union. Lemma (empty_set_in_top_for_bases). ∅ ∈ generated_part_by_union. Lemma (union_in_top_for_bases). union_in_top(generated_part_by_union). Lemma (inter_in_top_for_bases). inter_in_top(generated_part_by_union).. |DeJn v |}eSnbgeJa v ~X^ D`ba v dcl;keJ|DegdcegfgacX/Xj`bfJXjò~ v Xj^ XnI. n?B %. Xjpk!sbaiXH`no`bX,kreS|}eJdieJfgacXWI.

(84) Z. ¥. N . Definition. let generated_topology_o be (!Build_Topology E generated_part_by_union(B) union_in_top_for_bases inter_in_top_for_bases full_in_top_for_bases emptyset_in_top_for_bases) (its type is Topology(E))..

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(87). . . Definition (cont_in_pt). given f of type (Map E F) and a of type E , we define the proposition (cont_in_pt f a) by ∀ W:open of F , f(a) ∈ W ⇒ f−1(W) is open. '~Xk!^ Dnoò`baikX,aieSlg`9|oI |odiamhlgkraceJ`$Xjpk heJ`Skram`nbX,praWXjdidcX/Xjpk!heJ`Skac`nbX/Xj`7h U lJhno`7~bXjp!|Degam`Skp!~bX,peJ`7~beJYzlgac`bX,~bX M. Definition (cont). given f of type (Map E F) , we define the proposition cont(f) by ∀ W:open of F , f−1(W) is open Lemma (cont_then_conteverywhere). ∀ f:(Map E F) , f is continuous ↔ ∀ a:E , (cont_in_pt f a).. `8Qlgaik soaiXH` >p n v diX@dcacXj` lgXHh;diXHplgnbk v XHp`beJkraceJòpj /XH^ tSnoaildcXj`SkXjpj L~bXheS`Skram`SnoaikbX ^ ; ma Y;lgfgX¤am`SgX v pX H~ - no`$X v YXJ^ hjl v lghkbX ^ v amplgkraceJ`¤|ol v d- lg~ U X ^ v Xj`}hXD I. Lemma (cont_with_closed). ∀ f:(Map E F) , f is continuous ⇒ ∀ C:closed(F) , f−1(C) is closed. Lemma (im_of_adh). ∀ f:(Map E F) , f is continuous ⇒ ∀ C:℘(E) , f(adh(C)) ⊂ adh(f(C)). Lemma (cont_with_adh_inv). ∀ f:(Map E F) , (∀ C:℘(F) , f−1(adh(C)) = adh(f−1(C))) ⇒ f is continuous.. sutvsxw.

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(99) . / . !dgf. . v XjY;acX v v Xjpk!no`$V ac~Xj^ ld·| v eJ| v X©X ^ v a¨DlJ`Sk©dml| v eJ| v aX^ kbX*^ pnbaclJ`SkrX=I $ ` c a ~ j X ^ l W d | v V v p a lJ|o|ol kaiXH`Jk l lgdie p eJn lJ|o|ol kaiXH`Jk l Let R be a commutative ring. Definition. let is_prime be the function defined by , for all I:ideal(R) , ∀ x,y:R , x*y ∈ I ⇒ (x ∈ I) ∨ (y ∈ I). Definition. let propre be the function defined by , for all I:℘(R) , I ≠ full(R). Definition. a prime_ideal is a structure with the following components: prime_ideal_ideal: of type ideal(R) (coercion) prime_ideal_prop: a proof of prime_ideal_ideal is a prime ideal prime_ideal_propre: a proof of prime_ideal_ideal is a proper ideal. Lemma (prime_ideal_propre2). ∀ p:prime_ideal , 1R ∉ p. Lemma (is_prime_reverse). ∀ p:prime_ideal , ∀ x,y:R , y ∉ p ∧ x ∉ p ⇒ x*y ∉ p.. V -. lJnbk v XjpdiXHY;Y;XHp9I. Lemma (inter_included_prime). ∀ p:prime ideal of R , ∀ I,J:ideal(R) , (I ∩ J) ⊂ p ⇒ (I ⊂ p) ∨ (J ⊂ p).. ;. u v v v v v l;T>2kkrXj(`Snb!kacp@aieS=`diXHIJp¤am hdi@XHa5Yzdcl DY | XHv peS|}teSnbpra*ackrhjaceJl `lghkbX ^>2 amprXH/ `Sk(1dcXjp¤=-am~XH^lgn ¡ /0: | hXHe Y;vvaiXXjppz|DeJXHò` ~¤krpX acYzY;|bXjdcp@XjY;~HXH- lJ`Jòk `bl,V XjlJdml n¡ | v teJnb|DeJeJkrpacaiXjkai`SeSk` p. n?B %.

(100) j. ¥. N . Definition. let idomain_proposition be the function defined by , for all R:CRING , ∀ x,y:R , x ≠ 0R ⇒ y ≠ 0R ⇒ x*y ≠ 0R. Lemma (prime_integral). I is a prime ideal ⇒ R/I is an integral domain. Lemma (integral_prime). R/I is an integral domain ⇒ I is a prime ideal..

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(105). Definition. let is_maximal be the function defined by , for all I:ideal(R) , ∀ J:ideal(R) , I ⊂ J ⇒ I=J ∨ (J = full(R)). Definition. a maximal_ideal is a structure with the following components: maximal_ideal_ideal: of type ideal(R) (coercion) maximal_ideal_prop: of type maximal_ideal_ideal is a maximal ideal maximal_ideal_propre: of type maximal_ideal_ideal is a proper ideal. v|}l `bXX¤klgn}nb`k v X^XdXj^ QY;l h XjeJ`S`¶k tH~ nb- XX¡bdmh| eSv òamY;tnbX vX1t _9n ~b- n}X`¶ am~`9XH^ - llgd |o|} l Xjv pkrk/Xj`}Yzlg`Sl¡kam|}Yzlglp dFlV Xjp k,Xj~bpXzk!d.~b- lJa v òX¤`btXjlJnbnXd. - ac~krXjeJ^ lnodFk!XHXj`b`SfgkrXHacXòv ~ I v X ^ Lemma (maximal_prop1). I is a maximal ideal ⇒ ∀ x:R , x ∉ I ⇒ full(R) = I+<{x}>. Lemma (maximal_prop1_rev). (∀ x:R , x ∉ I ⇒ full(R) = I+<{x}>) ⇒ I is a maximal ideal.. v am`SgX v p`@acso~diXWX^ DI òaik

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(113) pX n}`bdmXl7QlgeJY;`}hacdikrdcacXeJ` 9~ - am~XH^ lgn¡ 9` - Xjpk¤|olgpzbn hteJnbYza'Y;heJXòpnok ` nbXjack òpwrXjnoYprksbXjY;diXXj~b`SXk. Let ideal_family be of type ℘(Set_Ideal).. Definition. ideals_containing:℘(Set_Ideal) 1: Apply (!Build_Predicate Set_Ideal a:ideal(R) , ∀ i:ideal_family , i ⊂ a). Definition. sum_ideals:ideal(R) 1: Vinyl2 x:R , ∀ a:ideals_containing , x ∈ a.. v XXj`}pnbackrX©diXHpL~bXjnb¡| v eJ| v aX^ kbXj^ p5hl v lJhkbX ^ v acpkramtnbXjp·~bXdml!preSYzYX

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(128). . Let R be a ring. Definition. a multiplicative_part is a structure with the following components: multi_part: of type Predicate(R) (coercion) multi_part_one: of type 1R ∈ multi_part multi_part_prop: of type ∀ x,y:R , x ∈ multi_part ⇒ y ∈ multi_part ⇒ x*y ∈ multi_part. v v v v v Xjp|D Xja h% kaiJXjXjY;pkXj`Sn}k `bX dcX/k| XvdcdiebX;~o|onbl ackkr~bacXXY~bXHnbndi¡¢kac|oXj^ didamXj^hY;lgkrXjac`SgkXgp 5~beJX`7 eSn Xk!`baid.k- n}di`bXHpaikbX,^ eShòeShY;kaieSY;òX/pkrtSX nov Y;a |DXX ~bY;X,Xkqbkrk|DXjX `Sk ~o X geJa Let M be of type multiplicative_part. Definition. let build_elt_mp_prod be the function defined by , for all m1,m2:M , (m1*m2[in M]). Definition. let multi_part_unit be Build_subtype(multi_part_one(M)).. v nback 9 : = ' k S e ` h J e o ` p k tnbXz M. ( / r]?. dml|}l v kracX/Ynbdckram|bdiamhlgkracgX/Y;ac`bamYzldcX=IohXjdidcX,tnbaW`bX,heJ`SkaiXH`Jk. Definition. multi_part_min:multiplicative_part 1: Apply !Build_multiplicative_part({1R}). 1: Abstract ( Auto with algebra ).. p¯ ± `@|DXjnokYzlam`SkrXHòlg`Sk'~X^ Dòa v dcX v lJ~bachjld}~H- noàc~Xj^ lgd v XdmlkracgXHY;Xj`Sk lV noòX|ol v kaiXY nbdckram|bdcachjlkracgX tnbXdmheSòtnbXPI

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(131) . ¥. N . Definition. radical:℘(R) 1: Apply (!Build_Predicate ? x:R , ∃ m:M , ∃ n:nat , m*x ^ n ∈ I).. `|DXjnbk~Xj^ Y;eJ`Sk v X v dcXjp dcXjYzY;Xjppnbaclg`Skp9I. Lemma (included_radical). I ⊂ radical. 1: Red. Lemma (radical_increasing). I ⊂ J ⇒ (radical I M) ⊂ (radical J M).. 'kpra { XjpkheSYzY nbklkraizI M. Lemma (radical_multiplicative). ∀ R:cring , ∀ M:multiplicative part of R , ∀ I:ideal(R) , ∀ x:(radical I M) , ∀ y:R , x*y ∈ (radical I M).. =@@. u eSn v dcX v lJ~bachjld v Xjdclgkr ai lV dml|}l v kracXY nodikac|bdcamhlkaiJXY;am`bamY;lgdiX XjktSnoXd.- eJ``begkX preSnopz¥P.°¦H§ oeS`$l¤lJnoppraI. ;< / 1 /0:. . ; 9 : = ( / A]?. Lemma (radical_propre). √I = full(R) ⇒ I = full(R). Lemma (radical_to_natural_def). ∀ x:R , x ∈ √I ⇒ ∃ n:nat , x ^ n ∈ I. Lemma (natural_def_to_radical). ∀ x:R , (∃ n:nat , x ^ n ∈ I) ⇒ x ∈ √I.. 4y=eWq j%ketjgyi. !d !d. lve6` q%y =e6`. + p Tp + ]o p. vi_i. ¬ S ¯ F¯ b] p¬ + d.h- eJU YzlgsbY;ac`kX/n}lgdc~bòl¤X!`b~bXjXHeS`;lgòn$Yz`diXjlêbX k hjU l; XjkrdL^ YzeSXHnprlwxkkeJacnon t`v nbp X!lJ~bòamX!l`bXjhdmlJl¤eJn$òYzptaclg~nbh Xa^ v v `be X Xv hu eS>eS`Jn kaiv XH `SY;D k~9acY;tS- nonHX `v- nohlJ`7Xjòham`b~abXjXHeJ^ lJl`;ndL|}Yz ;XHnbl }¡bk©~Ham~-Y;noX^ `7lgDdª`bam D~a tSv XH^ nolnoX,dL`bYzXd- eJlgl`7`}¡b`baml¤Y;XjlJprlgeSn d nb dcgebXHh~b`SlX k d ; JXk©9~ - no`bX| v XjnbJX9~ - n}`bachaikbX*^ ~bXHp

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(166) p~}dclgX$òp| v¢dXjY;ml/acX pv |Xjh^ hlga¨p DhjhlX$kaipeSX `v l~bX no` pam `bfg; dcXk eJ`WD µh~}e lgvv òXjp p|Dd -eJlgònb~ k vv lX d.- XjòpXjYsbdiX/am~bXg. Parameter (incl_dec). ∀ U,V:open of X , { U ⊂ V }+{ ¬U ⊂ V }. . . . . . .

(167). . . . . . Let U, V be open of X . Definition. a prf_included is a structure with the following components: prf_included_prf: of type U ⊂ V (coercion). Definition. let eq_triv be the function defined by , for all t1,t2:prf_included , True (its type is prf_included ⇒ prf_included ⇒ Prop). Lemma (eq_triv_equiv). eq_triv is an equivalence.. Definition. let HomTop be the set Build_Setoid(eq_triv_equiv).. n?B %.

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(170) `9- XjpkL`XH^ hXjppla v XJ XjkµpXjnbdmpWdcXjpLhjlgpWpaifS`ba¨5hlka¨QpµY;lgk U Xj^ YzlkactnbXHYXH`Sk p V h|DXjeJnp¡8ackraceeJn`@~b X/XH~bprXHk,nsb¡acXjY;`8e amv ò| hU dmamnoppY;~oXjlJp'òeJp` ~bµegac~okeghacgeJXjò`Spkk v Xnb k a vv XXnoXj^ k` noXj~b^ daXj^XH^ Y;p Xju `Sk©l v ~bX X¡b; XHY;!|o%diXJ L( |}eS n v @ ~X^ l;VD`b|oa l v v krdmla v h~HeS- Y'no` E Xj^ dXj^ Y;Xj>`SkzpreS~bnoX p ;diXHp !U %q>|D( egk UWXHV prXH@ p Xk¤ ~9- no`)XkX^ dXH^ Y XH`S k;~b DX hX,; tnbaL%Xjpk!( amY;YXH^@ ~b amlhk- Xj pk lV ~oa v X7n}`bX_| v XjnbJX7~bX . . . . . . . . !yi tzlA`fi@k &yiOt0ie _e_ijHePiOt?k l9t chi wl Je6j%k&lve6`. ed c. . U eJacpa5~oX*e v YzldcamprX v dcXjp| v X^ QlamphXjlJVn¡zXv k'Qv lamphXjlgnb¡ lV v ldcXjn v v ~olg`}p©dmlhjlkbX^ fJe v aiX*´ { , \ $ `. l h ~o| Xjv Xp¶^ QllJamòph`bXjXHlJlgnn¡ ¡ lV heSkY;eJnbYkX¶nbkhlglgkrkbaiQXj^ pfg e v acX ``bX |ol|}v XHhnbX¶k tl n9- |eJ` aie q a;nb|okrlgacdip¶ampX¶fXH^ d.`'-xXj^X ^ fJlgldidcacamkbprXX^ ~bXH~oplJòpp Xkre1¥02m¦H~b§ Xjpjdml d/`bpeJnkr¤aceJk`h~bXX E |DtnbXj`}Xjdi~odcX¤lg`Slgk*nb~bk v X Xzv hXHlgY;kbXj|o^ fgdclJe hv XacX v ~b~oeSlg`Jòk/p*dc`bXjegp/k veSX swx~oXXkgp,Xj|}dieSXH|onb|DgXjXHY;`Jk Xj`SXj k!k v dcX;X hY;eegX vk*h·hX^ dXg^ X ´ v p{ diXH, p/\ prXj|okre1l 02cv ~bhXHXjp/dcnb|}a eSn`bX v eJ`9sb- amkY;Xj`b|Da e v v krdclX òpk egv krnoaceJhk` n v ~bXjX_p!9~| v- lgX^ `}Ql`bamXjplJhXjnlJ¡namòh~oe nbvv ackrXjXHpp|D }eJtònb~oaLlJ~o`SkrXXg Fv eSy `Sl kXpr^ XHnbac~odiX XjYzv XjY;pk Xjv `Samhk krXaceJk v` X/hlgeS~oòlJh|bX kbv Xj^òpjX diXHp@p|Xj^ ha¨DhjlkaieSòpz~bXjp `| v X^ QlgacphXHlgn 9~ - lJò`bXHlgn¡¢heSY;YnbklgkraiQppn v n}`¢Xjp|olghXkreS|}eJdieJfgamtSnoX ]Xjpkno` eSòhkXjn v hkleJk`Sa¨kQpjv l l v aclJ`Sk~bX dml$hjlkbX^ fJe v aiX ; %>( / @ ~olJòpdcl$hlkbX^ fJe v aiX´ { , \>~bXHp/lgò`bXHlgn¡_heJYzY Fn E yXjp;eJ`}hkrXHn v p;heJ`Sk v ll v aclJ`SkpXH`Sk v X7~bXHn¡ hjlkbX^ fge v acXjp Xjk peJ`Skz~X^ D`backzheJYzY;X7~}lgòp 3 ³f 587 ·Xj`_pX~beSòòlJ`Jkno`bXzlg|o|odiamhlgkraceJ`Xj`Sk v XdiXHp*eSswxXkp*XjknoòXlgnok v X Xj`Sk v XdiXHpY;e v | U ampYXHp plkacprQlamplg`Sk~oXjn¡| v eS| v aXj^ kbXH^ @p I ; D oXjk ; HD ; HD ; D

(171) .

(172) . . . . .

(173). . . . . Definition. a functor is a structure with the following components: fctr_ob: a function of type Ob(C1) ⇒ Ob(C2) (coercion) fctr_morph: of type ∀ a,b:Ob(C1) , (a → b) ∼→ (fctr_ob(a) → fctr_ob(b)) im_of_id_prf: of type ∀ a:Ob(C1) , (fctr_morph a a)(Ida) = Idfctr_ob(a) distrib_prf: of type ∀ a,b,c:C1 , ∀ fa:a → b , ∀ fb:b → c , (fctr_morph a c)(fb o fa) = ((fctr_morph b c)(fb) o (fctr_morph a b)(fa)). 'k eJ`|}XHnbk DòlgdiXHY;Xj`SkXj^ h v a v XWI M. Definition. let CRpresheaf be (Cfunctor Top_cat(X) CRING).. v X^ QlgacphXHlgn v -7U lJppebhacX

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(191) nokr` XdLtXjnbhX5eJnb XjY; $ Xj`S=k@&eSnb gX k7~bX !krXjdcptnbX "!$# %& $'!$#¶ldce p@acd/X¡bacpkrX o ¯ p F¯ oT p¬ +µ _± o ¬ o "p + p ¯ p v v v v v v v ~oX.d - Xj`¤òphXjeJYYzsbY;diXXj~oòXjhpXeJ|onol gX ~v Xk^ Dp!`bta nbaLdcXj~op egack!Xjh©eJX ^ nbv a¨}XXjvY;dmXjl¤`Sk| pFv eJeJ| nbv JaXX^ kbkX*^ pp9~ nb- aino`blJX*`Jk|}=X l I kracX ³ ~b6X M* Jh - Xjpk©no`bX|ol kaiX . . . .

(192). .

(193). . Definition. let covering_prop be the function defined by , for all P:℘(open of E) , A:℘(E) , ∀ x:E , x ∈ A ⇒ ∃ Pi:open of E , (Pi ∈ P) ∧ (x ∈ Pi). Record Open_covering_of [A:℘(E)]: Type := { Open_covering_setoid: of type ℘(open of E) (coercion); Open_covering_prf: a proof of (covering_prop Open_covering_setoid A) }. Let U be a open of X. Let P be of type Open_covering_of(U). Let s, s’ be of type F(U) . Definition. let equal_over_a_covering be ∀ V:Open_covering_setoid(P) , s|U ∩. n?B %. V. = s’|U ∩ V..

(194) Z. ¥. N . v eS| v aXj^ kbX ^ ha>Ev ~bXHpvpnopXjpkno`|}V XHn~oa>/*X ^ v Xj`SkX;~bX¤V hXdcdiX¤~beSò`Xj^ X;~olJòp/v dml7~X^ DòaikaieS`¶~bXHpQlampgE y 7 l | h|}Xjl lJv nkrac¡WXz~}Mn `@v XXj/·hXeJk nbeS `v XjY;XjXjpk`Sk/Xam~b`JX kp)Xkacd©p`9- - l Xjpk,|olgpXe k v h·|oXj^ lgY;p Xjl `Sk ac`}pachYzdmno|bp,dcXj~oY;lg`}XH`Jp k |ol ; noh` Xtv XHnbhX/eJnoY Xj v Y;XjY;XXHp`Ja k/ |DQXjlgnbaikk v v v v v v v Vv v v lg- ~ac`}XH^ s}prae I ~oX ->D·´'XkkrXz| eJ| aX^ kbX^ ~'X^ 5`baiXJ DeS`¶|DXjnbke YzldcamprX dml| XjY;a X X | eJ| aX^ kbX^ ~bXjpQlgacphXHlgn¡ . . . Definition. let unicity_presheaf be the function defined by , for all U:open of X , P:Open_covering_of(U) , s,s’:F(U) , (!equal_over_a_covering U P s s’) ⇒ s = s’.. ¯ p F¯ oT p¬ +µ _± o ¬ o "p ¯ ¬ ±±b²+ v eJsod XHV Y;X v |}eSv n v dcl$prXHheSò~bX| v eS| v aXj^ kbX,^ XHprk/~bXzprX;~beJòòX v dcl7heJdidcXjhkraceJ`_~9-rXj^ dXH^ Y;Xj`Skp5 ` y ; X | v JeJno~ lgaik heSòpram~X ^ X dcX,kqb|}X,pnbaclJ`SkI . . o.

(195). .

(196). . Definition. a stick_elt is a structure with the following components: stick_elt_set: a open of X stick_elt_elt: of type F(stick_elt_set). XH~o`_X;QhlgXa v pX¤Xkrnoe1`²02m~bXgprXj kr e102c`_~oXgl7 L~b|oeJnbòamp,h~bs}a XHv X@preJtac`8nbX¤|}eSpX¤n v ~beJhXjòdcòl$X ~9v - non}`b`bX¤X kX^ XfJdclgdiX@diackbhX^ egdcpdin XHvhkhaieSX;`8kqbh|}- XHXJpr·k,yplX@ò~begeJkròacòeJ`8X v ~9no-xX`b^ fSX¤l|odcailkbX,v^ kdcailX |oòdcXznop |DXjònolk kn X vk Xjv X¤didcX¤hebprX~v X^ lgX;ai5k krX $dcdcX@ tS no'XdcprdiX¤a ;3|o( l / v hXz t^ nb@ Xz*dcXjp/;3(kqb/ |DXj p^~bX @ ;J Xk: ;J ^ : @ Xjk,^ ~b@MX * ;J ;J: : ^ ^@ @ v LXHYzp|DlXjamhp/kracXgdcXdiX E Y;~oeJXHò`Jh/k |o; lJ0 p!;Hlg( n7/ Y Xj ^Y;@X,@ XjpbXjk ^ kr1e ; 002c~o;HX ( D/ ^ @@ peJ`Skz~baG/*X ^ v XH`Skp ; ;J : ^ @ Xjk ;J : ^ @ `9- lg|}|ol v kracXjòòXj`Sk { XjYzl v t^ nbeJòp |ol v heS^`J k v X/tnbXpa ;H(^/ ^ @ *);H( / v ^ @ beS`$l;Xj`|ol v kachjnbdcaiX v d - ac`}hdmnopraceJ` ;3( / @ ;3( / @ Xjk ;< : @ |DXjnbk~beSòh X k X/bnheSY;Y;X,no`£Xj^ dXH^ Y;Xj`Sk~oXkqb|DX ; 0 ;H( / ^ @@ Saml;diXHpY;e v | U ampY;XHp!~bX v XHprk v amhkaieSòp `7~ba v lz~beSòh/tnbX. Lpa ;3( / ^ @*6;3( / ^ @ Xk ; ? rD ; ? D

(197) I Definition. let equ_stick_elt be the function defined by , for all st1,st2:stick_elt , ∃ proof_equal_sets:st1_SET = st2_SET , st1_ELT = st2_ELT|st1_SET.. sutvsxw.

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(200) ! " # %$ & ' ( . Lemma (equ_stick_equiv). equ_stick_elt is an equivalence. Proof:. Definition. let Stick_elt be the set Build_Setoid(equ_stick_equiv).. v ;32= : : 20@ Xkp !dcXkX v YX a eS`¶`beJkrX;preSnop¥P¥

(201) ¦j§ p dcX kX Y;X v ; ; = : XHp^ Xh vv l7a v X,dc`bldmXl@~oeJ|eJòacv p!eJ`|d.X^ - vX¤XjaòX^9~ pkbXj-X^noY`b~bsbX@X diX v|oXHl hv eJkdiaidcX XjY;2 Xj`S( kj/ ~o `bXjp7X;~bXjX ^~bdeJXj^Y;`}1`'XjX^`SXkp*~o X~b: Xv Xjdml;h µeJ~9die dc-XjnoY;Y`bXHXX `J|k v ; Xj 2no1g X¤t= nb]?X¤4 dD - XjD`}/ ~pr XXH^ / DY `bsbac? kdcX@ }XHeJ~b` `Xjp ¥|DXjeJ¦Hnb§nFkDE JX v kp¤!?lJ p p eb ha Xj^ p! e v Y; X_ sb acXj!`¢ no` v XjheJnb v XjY;Xj`Sk¤~bX ; ©Xk~9- no`bX8| v XHnbgX_tnbX/ ; D ; D 26#H1 : ;32= : 2 02 @. . . . . . . . . . . Let U be a open of X. Let P be of type Open_covering_of(U). Let s, s’ be of type F(U) . Definition. let equal_over_intersections be the function defined by , for all Vi,Vj:open of X , si:F(Vi) , sj:F(Vj) , si|Vi ∩. Vj. = sj|Vi ∩. Vj.. Record sticking_data_on [U:open of X]: Type := { st_part: of type Predicate(Stick_elt(F)) (coercion); st_covering_prf: of type ∀ x:U , ∃ st:st_part , x ∈ st_SET; st_prf: of type ∀ U,V:st_part , (equal_over_intersections U_ELT V_ELT) }.. y lz| v eS| v aXj^ kbX^ ~bX v XHheJdidcXjY;Xj`Sk!pX v llgdie v p&I. Definition. let sticking be ∀ U:open of X , ∀ st:sticking_data_on(U) , ∃ s:F(U) , ∀ Si:st , (equal_over_intersections. 'k eJ`|}XHnbk!Xj`bD`~'X^ 5`ba v dcXjp QlgacphXHlgn¡~H- lJò`bXjlJn¡$heJYzY nbklka¨QpBI M. n?B %. s Si_ELT)..

(202) Zg. ¥. N . Record CRsheaf [X:topological space]: Type := { pre: a presheaf of commutative rings over X (coercion); unicity_prf: of type ∀ U:open of X , ∀ P:Open_covering_of(U) , ∀ s,s’:U , (unicity_presheaf(P) s s’); sticking_prf: of type sticking(pre) }.. ed. vlvi@m]lrk _ohi?lrklHt m]yichj%kFm]lrk. . ²; v X ~H- no`¤| v X^ QlamphXjlJn Xj`zno`z|}eJac`Sk ; ¤~bX -XHprkd.- XHòpXjY v sodiXv ~bXHp

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