Transcript
Page 1: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

HAL Id: inria-00072274https://hal.inria.fr/inria-00072274

Submitted on 23 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Mathematics and Proof Presentation in PcoqAhmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau

To cite this version:Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau. Mathematics and Proof Presentationin Pcoq. RR-4313, INRIA. 2001. �inria-00072274�

Page 2: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--43

13--

FR

+E

NG

ap por t de r ech er ch e

THÈME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Mathematics and Proof Presentation in Pcoq

Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau

N° 4313

Novembre 2001

Page 3: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité
Page 4: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

������������� ��������������������������� ������������!���"�!�����#��$

%'& (*),+-%'(*),.0/213+54�6�73)98�:;)9.!<>=3<94@?=BADCFEG=B<0<>HI)9.,4@?13J .0),K C9)MLNHO+ )913J

P5QSRUT;V�W�XZY\[U]S^_Va`cbed,^cf!^_VU`V!ghf!ij`kf�l2`nmpoqT�r3be`c^ks,l2VtvuDbjwxV!g yzV!T{T{V

|ij}2}Bb,upg~�V�uDVUf�QSVUuDf�QSV�]3���,�2�U�MX��b��,V!T�rSu�VaWj�,�S�hX��0W�}2iedeVUm

�����U�!���S�e�e�G�MV}2u�V0mpVU],g��\�3���M�,i�lBmpVUup��^_]9gDV!u��Iief�V#� beu@g�QSVG¡#b�s�m�o�mxgDV!T�¢ QS^cf�Q{}SuDb��q^c~SVUm�V!`kijr3beu�i�gDVT�ijg�QSVUT{ijg�^kf!ie`z� b,u�T�lS`ci,mh`ki>o9�£b,l�gU¤a�¥V�¦2uDmpgampQSb�¢�QSb�¢§g�QSV{b,u�d9ij]S^c¨Uijg�^cbe]©bj��g�QSV;deu�ij}SQ2^cfUij`�l2mpVUup�^c]9g�VUup�Ii,f�V�ijuDbelS]B~Fmpg�uDl2fªgDlSu�V0~M~2i�gDi'T�ij«,VUm�^_g�}Bb9m�m�^crS`_V;gDb©ber�g�ij^c]FgDQS^cmU¤�¬­]®}2iju�g�^kf�l2`cieuU�n¢�V{^_]Bmp^kmxgb,]¥g�QSV¯g�QSuDV!V!�£`cV!�,V!`vV�°qgDV!]2m�^_b,]±T{VUf�Q2ie]S^cm�T�m�g�QBi�g�m�lS}S}3beu�gDm�gDQSV�f!l2mpg�beT{^ki�gD^_b,]¥be�5g�QSVN`ki>oeb,l�g�� beum�}3VUf�^_¦Bf�ij}2}S`_^kf!ijg�^cbe]2mU¤ ¬­]¯gDQSVa]SV!°9gm�VUf�g�^cbe]n��¢�Va~�VUmDf�uD^_r3VGg�QSV�~SijgDij��mpg�uDl2fªgDlSuDVhgDQ2i�g ^km5l2mpV0~¯^c]'g�QSV`cbed,^cfUij`�V!]Sd,^_]2V�g�b'uDVUf�b,uD~©²B³�´0´xµD¶ª·­¸º¹2·x»I¼q½!·­¾�¿>À>¸º¹9Á�ie]2~¢�V�m�QSb�¢�gDQS^cm�~SijgDij��mpg�uDl2fªgDlSuDV�f!ie]Âr3V;lBmpV0~i,m{iMrBiem�^cm�g�b®}SuDb�~�l2f!V¯gDV�°qgN^_]Ãi¥u�V0mxgDu�^kfªgDVU~ÄmplSrBmpV!g{be�G]2ijg�lSu�ij`5`kij]Sd,l2ijd,Ve¤ÆÅhV!uDVe��g�QSV©]2ijg�lSu�ij`gDV�°qg ^cm mpg�^c`c`Çber�g�ij^c]SVU~¯r9oNi;`ki>oeb,l�g�mpg�u�i�gDV!deo�b��,V!u�i;mpg�uDl2f�g�lSuDVU~¯}S^_V0f�V�bj��~Si�g�iS�Sij]B~N¢�V�m�QSb�¢ÈgDQ2i�g^c]9g�VUuDi,fªg�^c�eV�T{ie]S^c}SlS`ki�g�^cbe]Fb��eV!ug�QS^km\}S^_V0f�V�bj��~SijgDiN^km\mxgD^_`c` }Bb9m�m�^_r2`_V,�2gDQ9lBmG}2u�b��q^k~�^_]2dNiNT{VUie]2mhbe�~SV!�eVU`_b,}S^c]Sd�}SuDbqbj�x��¢ QS^c`cVG`cbqbe«q^_]2d;~�^cuDVUfªgD`_oNijg5g�V�°qg r3V!^c]Sd{}Su�b�~�lBf�VU~n¤��¥Vaf�b,]2f�`cl2~�V\rqo¯~�VUmDf�uD^crS^_]2die`_`zg�QSV�]SV!¢És9lSV0mxgD^_b,]2m5g�QBi�ghgDQS^kmhV!°q}3V!uD^cT;VU]9ghuDie^cm�VUm¢ ^ÊgDQ©u�V0mp}3VUf�gg�b�}Su�bqbe��V!°q}2`cie]2i�gD^_b,]Âij]2~Ë^_gDm^c]9g�VUdeu�i�g�^cbe]�^_]Ë}SuDbqbj� ~�V!�,V!`cbe}ST{V!]9g V!]q�q^cu�b,]ST{V!]9gDmU¤Ì©Í9ÎÐÏDÑ�Ò2��Ó��e�Ô^c]9g�VUup�Ii,f�Ve��}2u�bqbj�x��T�ijg�QSVUT{ijg�^kf!ie`Ç}Su�^c]9g�^c]Sd2�2}Su�bqbe��iemDmp^kmpgDij]9g

Page 5: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

���������������!$���� � ��������������� � ������� ������$

�q���� �®���\ij]2mf!VauDie}S}3beu�g]Sbel2m}Su�[0mpVU]9g�be]Bm�G�Ð���@�SlS]2Va^_]9g�VUup�Ii,f�Va}3belSuh`_V�m�o�mxgDR!T{V�¡#b�s¯s9lS^}3V!uDT{V�g�l2]±u�VU]2~�l±[!`kijr3beuD[N~�V0ma� beuDT�l2`_V0maT�i�g�Q2[!T�i�gD^cs9lSV0m!¤©�b,l2m�T{b,],gDu�b,]2m�~�� ijr3beu�~®f�b,T;T{VU],g`�� beuDd,ij]2^cmDi�gD^_b,]{~SVh`�� ^c]9g�V!u��Iief!Vhl�gD^_`c^km�ijg�V!l2u5iel�g�b,lSu#~SV\~�b,]S]S[UVUm�mxgDu�l2f�g�lSuD[!V0m@uDV!]2~�f�VU`cia}3b,mDm�^_rS`cVe¤��v]}Biju�g�^kf�lS`c^_VUua]SbelBm�^_]2m�^kmxgDbe]2mam�lSua`cV�T;[0f!ie]S^cm�T{V��'g�uDbe^km�]S^_�,VUiel�°Ms9lS^v}3V!uDT;V!g�~�� ie~Sie}�g�VUua`_V�uDV!]2~Sl~SVUm � b,u�T�lS`cVUmiel�°'ie}S}S`c^cfUi�g�^cbe]Bm5g�u�ij^_g�[UVUmU¤��@]2mpl2^ÊgDV�]SbelBmh~�[0f�uD^_�,be]2m5`kiNmpg�uDl2fªgDlSu�V�~SV�~�be]2]S[!V0ml�gD^Ê�`c^kmp[UVh}2ieu�`cV\T;beg�VUlSu#`cbed,^cs9lSVG}Bb,lSu5V!]SuDV!d,^cmpg�uDV!u�`_V0m#}2u�VUlS�eV0m�VU]�f�b,lSu�m�V!g�]Sb,l2m�T{be]9g�uDbe]2m#f�b,T;T{VU],gf!V�g�g�V�mpg�uDl2f�g�lSuDV¯~�V'~�be]S]2[!VUm�}3V!l�g���g�uDV¯lSg�^c`_^kmp[UV¯}3belSu�}SuDbq~SlS^_uDV�~SVUmagDV�°qg�V0m�~SV¯}SuDV!lS�,V¯~Sie]2m�l2]m�belBmx��V!]2m�V!T�rS`cV�uDVUmpg�uDV!^c],g�~�l¥`cie]Sd,iedeV�]Bi�g�l2u�VU`O�nm�lSu�`_V0m�s9lSVU`cm\`cVUm\T�ij]S^c}SlS`ki�gD^_b,]2m\^_]9gDV!u�iefªgD^_�,VUmGu�V0mx�gDV!]9gN}3b,mDmp^crS`cVUm��'b,]Ã}BVUl�g¯~�[!�,V!`cbe}S}3V!u�l2]SV©}SuDV!lS�,V'VU]Ã]SV©u�VUd,ieuD~Sie]9g;s9lSVÂm�be]Äg�V�°qgDV©V!]Ã`cie]Sdel2V]Bi�g�l2u�VU`_`cVe¤��v]�¦2]�]SbelBm�f�b,]2f�`clSb,]2mvV!]�~�[Uf!u�^c��ij]9g�`cVUm�s9lSV0mxgD^_b,]2m@]2belS�,V!`c`_V0m@s9lSVh]Sb9m@V!°q}3[!uD^cV!]2f!VUmvb,],gm�bel2`_VU�e[!V0mhf!be]2f!V!uD]2ij]9gh`�� V!°�}S`_^kf!ijg�^cbe]F~SVUmG}Su�VUlS�eV0mV!g\`cV!lSu\^c],gD[!d,uDijg�^cbe]Â~Sie]2mG~SVUmGV!]q�q^_uDbe]S]2V!T{V!]9gDm~SV�~S[!�eVU`_b,}S}3V!T{V!]9gU¤��Ò2�0��Ï��! "q�'� ^c],gDV!u��Iief!VUmU�9}2u�VUlS�eV0m!�Si$#�f�QBijdeV\T�i�gDQS[!T�i�gD^cs9lSV,�SiemDmp^kmpgDij]9g~�V�}SuDV!lS�,V

Page 6: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�F¿e» ¼���¾�¿j»O¸����2½�¶�½U»Ç²B³D½�����½ª¶�,¿�¹S¶� ��´�� �

� � ��������� �¯���������

�\�Ð���*^cmhg�Q2V�`ki�gDVUmpg\}SuDb�~�l2f�gG^c]Mi¯~SVUf!i,~�V���`cbe]SdNV��3b,upgGg�b'}SuDbq~Sl2f�V�deu�ij}SQS^kf!ie`zl2m�V!u���^_]9g�VUup�Ii,f�V0m5� beu}2u�bqbj��m�o�mxgDV!T�m!¤�¬�g�^_]SQ2V!uD^Êg�m¯T�ij]qoÃf�Q2iju�ief�g�VUu�^kmxgD^cfUm;� u�b,T gDQSVF}2u�VU�9^cbelBm������\���Émpo�mpg�V!TË� gDQ2i�g}2u�b��q^k~�VU~®m�l2f�Q*i©deu�ij}2QS^cfUij`vl2m�V!u���^_]9g�VUup�Ii,f�V�r2i,mpV0~®be]®gDQSVN}SuDbed,uDieT;T{^c]Sd'g�bqbe`km�i>��ie^_`kijrS`cV�^c]±g�QSVd,V!]SVUu�^kf'^_]9gDV!u�iefªgD^_�,VË}Su�b,deu�ijT{T{^_]2dMVU]9�q^cu�b,]ST{V!]9g�deVU]SV!u�i�gDbeu¯¡#V!]9gDielSu�� ���£¤-P5QSV©]SVU¢ d,uDie}SQS^kf!ie`^c]9g�VUup�Ii,f�V^km�}Su�b,deu�ijT{T{VU~;^c]�i�~S^��ÐV!uDV!]9g5`cie]Sdel2iedeV,���,i>�>i2�,ie]2~NT{b,mpg#be�Çg�QSV�~�V0mp^cde]¯~SVUf�^km�^_b,]2m@gDQ2i�gQBie~�rBVUV!]ËT�ie~�V\� beu ���!�\���MQ2i>�,V\r3V!VU]Ëu�V!��f!be]2m�^c~SV!uDVU~Ǥ

P5QSV"�����\���-mpo�mpg�VUT ie~��,b�f!i�gDVU~�iM� V!¢ rBiem�^cf'}SuD^_]Bf�^c}S`_V0m!��T�ij^c]S`coÆ~�VUmDf�uD^crBV0~Æ^c]#�_��$q�v�2��W%���gDQSV'l2mpVUup��^c],gDV!u��Iief!V¯^km;i¥m�V!}2ieuDijg�V¯}2u�b�f�V0m�m�� uDbeT g�Q2V'`cbed,^cfUij`#V!]Sd,^_]SV,�5¡#b�s&�c�'�£��g�QSVË`cbed,^cfUij`#~2i�gDi^c]ÈgDQSVMlBmpVUup��^_]9gDV!u��Iief�V©^km�T{ie]S^c}SlS`ki�g�V0~ iem'mpg�uDl2fªgDlSu�V0~Ç� gDu�VUV���`_^c«eVÂ~SijgDi2�hij]2~Èg�QSV¥l2m�V!u��£^c]9g�V!u��Iief!VT�ie]S^_}2lS`cijg�V0m5g�QSV�mpV0s,l2V!]2f!Vabj�@f�b,T;T�ie]2~Smm�V!]9g g�bNg�QSV�`cbed,^cfUij`ÇVU]Sde^c]SV�i,mi�f�b,T;}2`_V!g�V�~�b�f!lST{V!]9gU�gDQSV�f�b,]2m�^cmpg�VU]2f�o�be��¢ QS^kf�Q'^ÊgQSVU`_}Bm5g�b{V!]BmplSuDVe¤( ~��,b�f!i�gD^_]2d�g�QSV;l2m�V!u��£^c]9g�VUup�Ii,f�V�i,mGi�mpVU}2iju�i�gDVa}Su�b�f!VUmDmhT�ij«,VUmg�Q2ijgGgDQSV�¢ QSb,`_V*)x}2u�bqbj��~SV!�eVU`Ê�

b,}ST{V!]9gaV!]q�q^_uDbe]2T;VU]9g,+;uDV!`c^cVUm�be]®f�b,T{T�lS]S^kf!ijg�^cbe]¥}SuDbjg�b�f!be`kmh� b,uaT�i�g�Q2V!T�i�gD^cfUij`�~SijgDi�g�QBi�g�¢�V!uDV}2u�V0f�lSu�mpb,uDm�g�b;gDQSVa]Sb�¢ ¢�V!`c`_�£«q]Sb�¢ ]�}Su�b,}Bb9m�ie`cm5ieu�b,lS]2~.-0/©y �21436567!8:9;�=<¥ie]2~�8:9;�=<�8?>5¤/©ie]S^c}SlS`ki�g�^c]SdT�ijg�QSVUT{ijg�^kf!ie`j� beuDT�l2`ci,mziemzmxgDu�l2f�g�lSuDVU~a~2i�gDi2�>¢#V@QBi>�eVvr3V!V!]�ijrS`cV@gDb^_]BmxgDu�lST{VU],g

gDQSV�}2u�bqbj�@~�VU�eVU`_b,}ST{V!]9ghV!]q�q^cu�b,]ST{V!]9g¢ ^ÊgDQFf!ie}2ijr2^_`c^ÊgD^_V0m gDQ2i�g\^_]2f!u�V0iem�V�g�QSV�r2ij]B~�¢ ^c~�g�QÂr3V�gx¢�V!VU]gDQSV�l2m�V!u ij]2~¯g�Q2V\`cbed,^cfUij`ÐV!]2de^c]SVe¤ ( ]'V�°SieT;}2`_V�^km�g�Q2ijg5be�@�³D´U´pµ�·BA'C�·k²S´�¸º¹3»I¸º¹qÁD� �E�£��¢ Q2V!uDV\f!beT{}S`cV�°f!beT{T�ij]2~2maf!ij]®r3VNf�b,]2mxgDu�lBfªg�V0~F^c]±be]2V{f!`_^kf�«Frqo©gDQSVNl2m�V!u0�ngDQSuDbelSd,Q¥ie]®ie]2ij`co�mp^km\bj��g�QSV�T{b,l2mpV}3b,m�^_g�^cbe]'^_]'g�Q2V�� beuDT�lS`kiS¤( ]SbjgDQSV!u;f!ij}BijrS^c`_^_gxo©gDQ2i�g�^km�^c]2mpg�uDlST{V!]9gDie`@^c]¥VU]SQ2ie]2f�^c]SdËg�Q2VNr2ij]B~�¢ ^c~�g�Q±r3V�gx¢�V!V!]®gDQSVNl2m�V!u

ie]2~�gDQSV#`cbed,^cfUij`,V!]Sd,^_]SV#^cmzg�Q2V#}3b,mDmp^crS^c`_^_gxo\g�bG`ci>o,bel�gngDQSV�T{ijg�QSVUT�i�g�^kf!ie`e� b,u�T�lS`kiemn^c]�ihT�ij]S]SVUuzgDQ2i�g^km5�eVUu�oNf!`_b9mpV\gDb�gDQSV�gxoq}BV0mpV!gpgD^_]Sd�}2uDi,fªg�^kf�V\bj� T�i�gDQSV!T�i�gD^cfUij`Ç`c^ÊgDV!u�i�gDlSu�VF�c�U����¤G�jlBmxg^c]2f�uDVUi,mp^c]Sd�g�QSVuDVUi,~Sijr2^_`c^Êgxo�bj�nT�i�gDQSV!T�ijg�^kf!ij`B� beuDT�l2`ci,m�m�}BVUVU~Sm#lS}�gDQSVh}SuDb��q^_]2d�}SuDbqf!VUmDmv^c]�i�m�^_d,]S^_¦Bf!ie],g�T�ij]S]SVUuU�V0mp}3VUf!^cie`_`coN� b,ugDQSV�«q^_]B~Ëbj�v`_b,de^kf!ie`ÇV!]Sd,^_]SV�g�Q2ijgh¢#V�l2mpV,�2¢ QSVUu�Va^c]9g�VUuDi,fªg�^c�eVa}2u�bqbj� ^km}SuDV�� VUu�uDVU~�g�biel�g�b,T�i�g�^kfG}Su�bqbe��mpV0iju�f�Qn¤@H9g�^c`_`Çb,]SV�Q2iem#gDb;T�ie«eV�mplSuDVGg�Q2ijg ~2i�gDi;~�^kmp}2`ci>o,VU~Nb,]NgDQSVam�f!u�VUV!]¯^km mpg�^c`_`i,fªgD^_�,Ve�j^c]{g�QSVm�V!]BmpV5gDQ2i�g@^_g#f!ie]{rBVm�V!`cVUf�g�VU~;¢ ^_g�Q{gDQSV}3be^c],gD^_]2d�~�V!�q^kf�Ve¤ P5QS^kmvT�ij«,VUm�g�Q2V}SuDberS`cV!T�,V!uDoN~�^I�3VUu�VU],g gDQ2ij]'}2i,m�m�^_�,V�T{ijg�QSVUT�i�g�^kf!ie`Ç`ci>o,bel�g iem5^c]©P � -4�_�0���£¤

P5QSV.�����\���Æ}SuDbqbj�#~SV!�eVU`_b,}ST{V!]9gGVU]q�9^cuDbe]ST{V!]9g\Q2iemGrBVUV!]¥l2m�VU~Â^c]9g�VU]2mp^c�eVU`_o'� b,u\i���ijuD^_V!gxo'be�ieT�rS^_g�^cbelBm!��`cieu�d,V��­m�fUij`cV�}2u�bqbj� ~�VU�eV!`cbe}2T;VU]9gDmJ�c�!�S�#�e�'��¤©P5QSV��\�3����VU]9�q^cu�b,]ST{V!]9g�^km�}S`kij]S]2VU~¥g�buDV!}2`ci,f�V\^Êg � uDbeT gDQSVa]SV!°9g�,V!u�mp^cbe]�bj�K�\���Mb,]n¤v¬�g^km i>�>ie^_`kijr2`_V\be]Ë^c],gDV!uD]SV�gijghie~S~�uDVUmDm

L�M;MON PRQ;QES�S;S2T�UWVEN XZY�[�\=Y%]GX_^O\`QOa�b�c�c=b;QENedWV�fie]2~�^ÊgQ2i,m ij`cu�V0ie~�o�r3V!VU]'g�VUmpg�V0~�� beu5g�V0ief�QS^c]Sd�f�b,T{}Sl�g�VUup�­ij^k~�V0~N}SuDbqbj�zgDbNmxgDl2~�VU],g�m ij]2~�� b,u ^_]9g�VUup�T{V0~�^cieu�oNmDf!ie`_V�T�i�gDQSV!T�ijg�^kf!ij`n~SV!�eVU`_b,}ST{V!]9gDm ieu�b,lS]2~�f�beT{}SlSg�V!uGijuD^ÊgDQST{V�g�^kf!mU¤

P5QSVUu�VÂieu�V©m�V!�,V!u�ij`5«q^_]2~2m�bj�aij}S}2`_^kf!ijg�^cbe]2m{� b,u�¢ QS^kf�Q�b,]SV©T�i>o*¢5ij]9g�g�b±^c]9g�V!d,uDijg�V©i®}SuDbqbj�}2u�V0mpVU],g�i�gD^_b,]Mf!ie}2ijr2^_`c^Êgxo©^c]±i¯}2u�bqbj�5~�V!�,V!`cbe}ST{VU],gaVU]9�q^cu�b,]ST{V!]9gU¤ ( ¦2u�mxgau�ij]Sd,V�be��ij}2}S`_^kf!ijg�^cbe]2mf!b��eVUuDmÇ~�^k~Si,fªg�^kf@]2V!VU~2m!�>V!^_g�Q2V!u�g�bQSVU`_}�g�VUi,f�Qag�Q2V@� bel2]2~Si�gD^_b,]2mnbe�S`_b,de^kfvbeung�bG~SVUmDf�uD^_r3Vvg�Q2V�mpg�uDl2fªgDlSuDVbe� g�QSV�}Su�bqbe�Imb�f!f!lSu�uD^c]Sd�^_]F¢#VU`_`_�£«q]Sb�¢ ]ËT�i�gDQSV!T�ijg�^kf!ij`�u�V0mpl2`Êg�m!¤ ( mpV0f�be]B~ËuDie]SdeVabe��ij}S}S`c^kf!i�gD^_b,]f!b��eVUuDm�^_]B~�l2mpg�uD^cie` ]SV!V0~Sm!¤ÃP5QS^km{T�l2mpg�rBVÂi,~S~�uDVUmDmpV0~±gDb¥T�ie«eV'`_b,de^kfª��r2iem�VU~®� b,u�T�ie` T;V!g�QSb�~Sm{be�m�bj�ºgx¢5ijuDVN~�VU�eV!`cbe}2T;VU]9g�iefUf�V!}Sg�VU~n¤MÅVUu�V�}SuDb9be� }Su�V0mpVU]9gDi�gD^_b,]±^cm�]SVUVU~�V0~±rBV0f!iel2mpV¯� beuDT�ij`�}2u�bqbj�Imbe�@mpbe�ºgx¢�ieu�V�ieu�Va]SVUVU~�V0~©^c]Âf�be]9gDuDi,fªg�lBij`n}2ieupg�mbe��gDQSV�^c]2~�lBmxgDu�^kij`�mpbe�ºgx¢�ieu�Va~�VU�eV!`cbe}2T;VU]9gh}SuDbqf!VUmDm �

g�gih%j6k�l�mRl

Page 7: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

� �G¾�½�³�Àj¿W�����G½�³�»�´j»�� ´j»£»O¸I½�³����h¸��,½D¿E�

¢ Q2V!]�}SlSu�f�Q2i,mp^c]Sd�ia}S^cVUf!Vbj�nm�bj�ºgx¢5ijuDVe��gDQSVGs9l2ie`_^_gxo;bj�Ç¢ QS^kf�Q�u�VU`_^cVUm�be];� beuDT�ij`2}2u�bqbj�ImU�e}2lSuDf�QBiem�V!u�m]2V!VU~'gDbNr3V�f�b,]9�q^c]2f�V0~�g�Q2ijgg�QSV;mxg�i�gDV!T{V!]9gDmrBVU^_]Sd¯}Su�b��,VU~Ëief�g�l2ie`_`co¯uDV�BVUfªghg�QSVU^_u\mp}3VUf!^ʦBfUi�gD^_b,]2m!¤P5Q2^_u�~{}2iju�gxo;^_]2m�}3VUfªgD^_b,]�bj�ÇgDQSV� b,u�T�ij`B~�V!�,V!`cbe}ST{V!]9g�T�l2mpg#r3VhT�ie~�Vh}Bb9m�m�^_r2`_V,�e� b,u�^c]2mxg�ij]2f!V¢ QSVU]f!be]2m�^k~�V!uD^_]2d�f�V!u�g�^_¦BfUi�g�^cbe]{¢ ^_g�QNuDVUm�}3VUfªg�g�b�f!V!u�gDij^c]{`_VU�eVU`cm�bj�ns9l2ij`c^Êgxo�`_^c«eVg�QSV\f�beT{T{be]�f!u�^_g�VUu�^ki\� beum�VUf!lSuD^Êgxo©VU�>ie`_lBi�g�^cbe]z¤�GlSu�¢�beuD«©^cm�T{b,u�V;beuD^_VU]9g�VU~Âg�b�¢5iju�~Smhg�QS^km�m�VUf�b,]2~Âu�ij]2deV;bj��ie}S}S`c^cfUi�gD^_b,]2m!¤�2beu�gDQSVUm�V¯ij}2}S`_^kf!ijg�^cbe]2mU�nV0iem�V�bj�5l2mpV{� b,u�r3V!de^c]S]SVUuDm�^kma`cVUmDma^cT;}3beu�gDie]9gag�Q2ie]¥}2u�b��q^k~�^_]2dËmDf!ie`cierS`cVgDbqbe`km!�qf!ie}2ijrS`cVbe�nQ2ij]2~S`_^c]Sd�gDQSV\}Su�bqbe�Imvg�QBi�g�ie]NV!]2de^c]SV!VUu�T�i>o{}Su�b�~�lBf�Vhij�ºg�V!u5ia� VU¢ ~Si>o�m@be�n¢#b,u�«b,]'i{m�^_]2de`cV\}2u�bqbj�x��}3b,mDm�^_rS`coNl2m�^c]Sd�V!`kijr3beu�i�gDVG}SuDbqbj� f�beT{T�ij]B~Sm!¤

¬­]�g�QSV#u�V0mxgzbe��g�QS^km�}2ij}3V!u0�>¢#V�¦2u�mxg�m�QSb�¢±QSb�¢¥gDQSV�b,u�d9ij]S^c¨Uijg�^cbe]�bj��gDQSV�d,uDie}SQS^kf!ij`,l2m�V!u��£^c]9g�V!u��Iief!Vieu�b,lS]2~Âmpg�uDl2f�g�lSuDVU~M~Si�g�i¯T�ij«,VUm\^_g�}3b,mDm�^_rS`cV�g�bËber�g�ij^c]FVU`cierBb,uDijg�V!`coË`cie^c~��£b,l�g\T�ijg�QSVUT{ijg�^kf!ie`�� b,up�T�lS`kiemU¤z¬­]{}2iju�g�^kf�l2`cieuU��¢�V5^_]2m�^kmxgvbe]�g�Q2V5g�QSuDV!V!�£`cV!�,V!`qV�°qg�VU]2m�^_b,]�T{VUf�QBij]S^kmpT�m�g�QBi�g@T�ij«,V�^_gv}Bb9m�m�^crS`_VgDb�f�l2mpg�b,T;^c¨!V\gDQSV�`ci>o,bel�g�g�b�ij}2}S`_^kf!ijg�^cbe]2mU¤ ¬­]¯gDQSVa]SV!°9gm�VUf�g�^cbe]n��¢�Va~�VUmDf�uD^_r3V\g�QSVa~SijgDij��mpg�uDl2fªgDlSuDVgDQ2i�g�^km{l2m�VU~Ä^_]�g�Q2V'`cbed,^cfUij`5V!]2de^c]SV¯gDb®u�V0f�beu�~±²3³�´0´xµD¶ª·­¸º¹2·x»I¼q½!·­¾{¿�À0¸º¹qÁÂie]2~Æ¢�VËmpQSb�¢ g�QS^kmN~SijgDi��mpg�uDl2f�g�lSuDV�f!ie]'r3Val2m�VU~©i,m i{r2iem�^cm � beu gDQSV�}SuDb�~�l2fªgD^_b,]Ëbj��g�V�°qgh^c]©i{u�V0mxgDu�^kfªgDVU~Ëm�lSr2m�V�gbj�v]2ijg�lSu�ij``kij]2del2iedeVe¤�ÅV!uDVe�eg�QSV ]Bi�g�l2uDie`,gDV�°qg�^kmvmxgD^_`c`Sb,r�gDie^_]SV0~�rqo;iG`ki>oeb,l�gvmxgDuDijg�V!d,o�b��eVUuvi\mpg�uDl2f�g�lSuDVU~�}S^cVUf�Vbe�@~Si�g�iS�Ðij]2~©¢�V�m�QSb�¢-g�Q2ijgG^c]9g�V!u�ief�g�^c�eVaT�ij]2^_}Sl2`cijg�^cbe]©b��eVUu g�Q2^cmG}S^cVUf�V�bj��~SijgDi�^kmhmpg�^c`c`�}3b,mDmp^crS`cVe�gDQql2m�}2u�b��q^k~�VU~;T{VUie]2m�be�n~�V!�,V!`cbe}S^c]Sd�}SuDbqbj�x�9¢ QS^c`_VG`_bqbe«q^c]Sd�~�^cuDVUfªgD`_o{i�g#g�V!°9g�rBVU^_]2d�}SuDb�~�l2f�V0~Ǥ �MVf!be]2f!`_lB~�V�rqoÆ~SVUmDf�uD^_rS^c]Sd¥ij`c`5g�QSVË]SVU¢ s9lSVUmpg�^cbe]Bm�gDQ2i�g{gDQS^cm{V!°�}BVUu�^cT{V!]9g;u�ij^km�VUm�¢ ^ÊgDQ�uDVUm�}3VUfªg;g�b}2u�bqbj��V�°�}S`kij]2ijg�^cbe]'ij]2~'^_gDm5^c],gDV!d,uDijg�^cbe]'^_]'}SuDb9be��~�VU�eVU`_b,}ST{V!]9g V!]q�q^_uDbe]ST{VU],g�m!¤

��� �����������������������N� ����� ��a���¯� �����������F�Ç�����¯��������� �

¬­]FbelSuGV�°�}BVUu�^cT{V!]9gDmU�3¢#V�Q2i>�eV�lBmpV0~Ëg�QSV �\���Æm�oqmpg�VUTÔiem gDQSV�`cbed,^cfUij`�V!]Sd,^_]SV,¤GP5QS^km\mpo�mpg�V!T l2m�VUmi'gxoq}3VU~��S��fUij`kf�lS`cl2m�i,m�g�Q2VNlS]2~�VUu�`coq^_]2d©`cie]Sdel2iedeV�g�bÂuDV!}2u�V0mpVU],g�ie`_`@T�ijg�QSVUT{ijg�^kf!ie`@]SbjgD^_b,]2mU¤ ( `c`T�ijg�QSVUT{ijg�^kf!ie`�be}3V!u�i�g�^cbe]BmvijuDV�uDV!}2u�V0mpVU],gDVU~;iem�gxoq}3VU~;� lS]2f�g�^cbe]2m�ij]2~;T{b,mpg@T�i�g�Q2V!T�i�gD^cfUij`Sm�V�gDm�ijuDVuDV!}2u�V0mpVU],gDVU~Ggxoq}BV0m!¤ /ËbeuDV!b��,V!u0�ªg�Q2V�`cbede^kf!ie`j� b,lS]2~Sijg�^cbe]auDV!`c^_V0mnbe]�ihf�beuDuDVUm�}Bb,]2~�V!]Bf�Vvr3V�gx¢�V!V!]�gxoq}3VUmie]2~±`cbed,^cfUij`@� b,u�T�lS`kiemU��«q]Sb�¢ ]*i,m�gDQSV�� ��³ª³'C�·���´���¿�³Z�N^kmpb,T{beuD}SQS^kmpTˤF¬­]9gDlS^ÊgD^_�,V!`coe� gDQSVN� lS]Bfªg�^cbe]gxoq}3VG^km5ij`kmpb�l2m�VU~�� b,u5u�VU}Su�V0mpVU]9g�^c]Sd�^cT;}2`_^kf!ijg�^cbe] �@i�}Su�bqbe� �Ii�� lS]2f�g�^cbe]"!�bj�$#&%�'(� T{VUie]S^_]2d ��g�QSV� l2]2fªgD^_b,]*Q2iem�gxoq}BV)#+*,'-!afUij]±r3V¯l2m�VU~¥gDbMf!be]2mpg�uDl2fªg;i©}SuDb9be� bj��' ¢ Q2V!]SVU�eV!u�^Êg;^cm�de^c�eVU]®i}2u�bqbj�nbe�$#.� ^c]¯gDQS^km�m�V!]BmpVG� lS]2f�g�^cbe]Ëij}S}S`c^kf!i�gD^_b,]N^km�~S^_uDVUf�g�`co�l2mpV0~NgDb�uDV!}2u�V0mpVU],g#g�QSV\`cbede^kf!ie`BuDlS`cVGbe�¾�´��%�q¶ª·k²�´j¹3½!¹S¶/!�¤@P5Q2V\¢ Q2be`cV\`cbed,^cfUij`ÇVU]Sde^c]SV�deVU]SV!u�ij`c^_¨UVUm#g�Q2^cm ^k~�V0iS¤

P5QSV-���­f!ie`cf!lS`_lBm^km «q]Sb�¢ ]Ë� beu^_gDmhV�°qg�uDV!T{V�m�^_T{}S`c^kf�^_gxo �#g�Q2V!uDV�ijuDV�be]S`co¯gDQSuDV!V�«q^c]2~Sm be��gDV!uDT�m ���ieu�^kijrS`cVUmU��ij}S}2`_^kf!ijg�^cbe]2mU��ij]2~����­ijrBmxgDuDi,fªg�^cbe]Bm!¤��v�eVU]�^_��¢�V\gDie«eV\^_]9g�b�i,f!f�b,lS]9g#gDQSVabe}3V!u�i�gDbeu�m@gDQ2i�gQBi>�eVËrBVUV!] i,~S~�VU~�gDb±Q2ij]B~�`_VÂgxo9}2^_]Sd*ie]2~Ã^c]2~�l2f�g�^c�eVFmpg�uDl2fªgDlSuDVUmU��gDQSVF~�^I�3VUu�VU]9g�f�b,]2mxgDu�lBfªgDm¯ijuDVmpg�^c`c`��,V!uDo¯� VU¢a¤ ( m\i¯u�V0mpl2`Êg0�2gDQSV���ieu�^cbel2mhT�i�g�Q2V!T�i�gD^cfUij`�be}3V!u�i�g�^cbe]Bm be]SV�T�i>o'VU]2f�b,lS]9g�VUuGijuDV�ie`_`VU]2f�b�~�V0~'l2mp^c]Sd;� lS]2f�g�^cbe]Âie}S}S`c^cfUi�g�^cbe]z�Sij`c`z~Si�g�i{`_bqb,«qm�gDQSV�mDijT{V�ij]2~'^_g^cm�eV!uDoNl2m�V�� l2`Çg�b�}SuDb��q^c~SVie]Nij}S}2u�b9ief�Q�g�b�m�¢ ^Êg�f�QNi�g�`cVUi,mxg�� u�b,T�gDQSVhl2m�l2ie`2}SuDV�¦S°;]Sbjg�i�gD^_b,]�bj�Ð� lS]2f�g�^cbe]¯ie}S}S`c^cfUi�g�^cbe]{gDb�^c]�¦S°]2bjgDijg�^cbe]2mU¤�0�l�gal2m�l2ij` T�i�gDQSV!T�ijg�^kf!ij` ]SbegDijg�^cbe]2m�ief�g�l2ie`_`co©deb'r3V!oeb,]2~©gDQS^km�`_VU�eV!` be����ijuD^_V!gxoe�nij]B~m�lSr��­mDf�uD^_}�g�m!��mplS}3V!u��­m�f!u�^c}�gDmU�zie]2~¥�>ieu�^cbelBmG«q^c]2~Smabe��d,V!b,T;V!g�uD^cf�`ki>oebelSgaieu�V�ie`cm�bËV!]2f!belS]9g�VUu�V0~Ǥ�Pzbf!be}3V�¢ ^_g�QMg�Q2^cm���ijuD^cV�gxoËbj��]SV!V0~Sm!�Ç¢�V;Q2i>�eV;~�VUm�^_d,]SVU~Mi¯`ki>oeb,l�g�T;V0f�Q2ij]2^cm�T g�Q2ijg�}2u�b��q^k~�VUmhg�QSuDV!V`cV!�,V!`km�bj�5V�°qg�VU]2m�^_rS^c`c^Êgxo,��uDVUs9lS^cu�^c]Sd�g�QSuDV!VN~S^��ÐV!uDV!]9g�`cV!�eVU`cm�bj�5V�°�}BVUupgD^cm�Ve¤�P5QS^cm�`ci>o,bel�gaT{VUf�Q2ie]S^kmpTT�ie«eVUm�^_g}Bb9m�m�^crS`_V\gDbN~�^cm�}S`ki>oNT�i�gDQSV!T�i�gD^cfUm iem m�QSb�¢ ]'^_]�¦2del2u�V{�e¤

132 g4165

Page 8: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�F¿e» ¼���¾�¿j»O¸����2½�¶�½U»Ç²B³D½�����½ª¶�,¿�¹S¶� ��´�� $

��^cdelSuDV;�!� � °SijT{}S`cVabj��T�i�g�Q2V!T�i�gD^cfUij`Ç]SbegDi�gD^_b,]2m���`c^c]SVUieuij`cdeV!r2uDi2¤

����� ����� ����������������������������� �!�"�$#%#&��#

P5Q2V©¦2u�mxg'`_VU�eV!`Gbj�aV�°qgDV!]2m�^_r2^_`c^Êgxoâ�beuD«�mNmp^cT{}S`coÄrqoÈiemDmpb�f!^cijg�^c]Sd±}SuDVU~�V!¦2]SVU~Ã`ki>oeb,l�g�� beuDT�i�g�m�g�b^k~�VU],gD^ʦBV!u�m!¤zP5Q2V#i>��ij^c`cierS`cV � beuDT{ijgDmnf!be]2m�^kmxgz^c]�g�Q2V�l2m�l2ij`,^_]S¦S°\]2bjgDijg�^cbe]n�>be��f!belSu�m�Ve�0rSl�gzgDQSV�m�o�mxgDV!Tie`cm�b\T{ie«eV0m ^Êg@}3b,mDmp^crS`cV�g�bal2m�V5`ci>o,bel�g@mpg�u�i�g�VUde^cVUm�g�QBi�g@^cT;}2`_o�T;b,u�V�V!`kijr3beu�i�gDV ~�^cm�}S`ki>o�f!ie}2ijr2^_`c^ÊgD^_V0m �m�lSr2mDf�uD^c}�gDmU�jV�°�}3be]SVU]9gDmU�j� u�iefªgD^_b,]2mU�empl2}BVUup��}Bb9mp^_g�^cbe]Bm!�>gDQSVu�ie~�^kf!ie`qm�^cde]n�('!)+*�u�bqbegvm�^cde]2mU�jb��eVUup��`c^_]S^c]SdB�l2]2~�V!uD`c^_]S^c]SdB�hV�g�fj¤ P5Q2V¥f!beuDu�V0mp}3be]2~SV!]2f!VÂ^cm'T�ie~SV¥i,f!f�V0m�m�^crS`_VM^_] g�QSV¥� b,u�T�be��i*g�ijrS`cVe�G¢ QS^kf�QfUij]¥VUi,mp^c`_o©r3V�VU~S^ÊgDVU~���i,~S~�^c]Sd'i'`c^_]2V{� beu�i�]SV!¢ be}3V!u�i�gDbeu0�Ðf�Q2b9b9mp^c]Sd�g�QSVNf!beuDu�V0mp}3be]B~�^_]2d¯`ki>oebelSgmpg�u�i�gDV!d,oe�z}2u�V0f�VU~SV!]2f!VUm\g�QBi�g�¢ ^c`_`#QSV!`c}*~�V0f�^k~�V{� beu�iel�g�b,T�i�g�^kf{}2ieu�VU],gDQSVUm�^kma^_]BmpVUupgD^_b,]n�n� b,],g�m!� ij]B~VU�eVU]©g�QSV{}3b,mDmp^crS^c`_^_gxoËg�bËf�QSbqb,m�V�m�oqT�r3be`kmhb,l�gDm�^k~�V;g�QSV;uDie]SdeV�l2m�l2ij`c`_oÂf!b��eVUu�V0~Ër9oËgDQSV{«eVUo9r3b,ieuD~n¤�2beu�^c]2mpgDij]Bf�Ve�#g�Q2V'`ki>oeb,l�g{� beu{� lS]Bfªg�^cbe]Bm-, N`a/.=U �0, c�.`aWM ij]B~1,(2 Y43 u�VU}SuDVUm�V!]9g�^c]Sd®ie~S~�^_g�^cbe]Ãij]B~~S^_�q^kmp^cbe]'bj��uDijg�^cbe]2ie`Ç]9l2T�r3V!u�m5^cm ~�V0f�`kijuDVU~�¢ ^ÊgDQ'g�QSV�� b,`_`cb�¢ ^_]2d�`c^_]2VUm be�zg�Q2V�gDijr2`_V �

G}3V!u�i�gDbeu �BijT{^_`co ynV!�ºg | ^cdeQ9g �Sbe]9g �2be]9g �Sb,],g PzV!°qg]2ieT{V }SuDVUf!VU~�VU]2f�V }SuDVUf!VU~�VU]2f�V ]2ijT{V mpgxoq`_V mp^c¨!V5 }S`cl2m ^c]�¦S° Wj� We� �h^kij`cbed t�y ( ¬x� �>W 65 T�lS`Êg ^c]�¦S° WW$ WO$ �h^kij`cbed t�y ( ¬x� �>W 75 ~�^c� � u�ief�g�^cbe] We�e� Wj�,� �h^kij`cbed t�y ( ¬x� �>W

y�i>oeb,l�g�be}3V!u�i�g�b,uDm{fUij]ÈuDVUf�VU^_�,V©i®�>ieu�^cV�gxoÄbj�aijuDdelST{VU],g�m�rSl�g¯g�QSVUoÄT�i>o*]Sbeg¯l2m�VÂij`c`hbe�GgDQSV!Tˤ�2beu'^_]Bmxg�ij]2f!VÂ� u�iefªgD^_b,] ~�bqVUm�]Sbeg'l2m�VMie]qoÄgDV�°qg©ijuDdelST{V!]9g0¤.��^cdelSuDVÆ�,�Gm�QSb�¢m¯i*� V!¢ be��g�QSV0mpVm�^cT;}2`_V{be}3V!u�i�gDbeu�m�~�^cm�}S`ki>oÂmxgDuDijg�VUde^cVUm��\i,~S~�^_g�^cbe]2mU�nT�l2`ÊgD^_}S`c^kf!i�gD^_b,]2mU��m�lSr2mpg�u�iefªgD^_b,]2mU�nij]2~®mDs,lBijuDVUmU¤¬­]Ëgxoq}3V��£g�QSVUbeuDo9�£r2i,mpV0~¯uDV!}2u�V0mpVU],g�i�gD^_b,]2m5bj�v`cbede^kf!ie`Ç� beuDT�l2`ci,m!�ST�ie]9o�be}3V!u�i�gDbeu�m�g�Q2ijgGl2m�l2ij`c`co¯Q2i>�,V

g�gih%j6k�l�mRl

Page 9: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

� �G¾�½�³�Àj¿W�����G½�³�»�´j»�� ´j»£»O¸I½�³����h¸��,½D¿E�

]WN;N=d���Y 2 bW[�M���N`]eY�\����^�VO\%c�.ea�]��%[`b��;a`YOU%M� Y 2 bE[�M��� ���

]EN;N=d�� Y 2 bW[�M���N�\6V 2 ���^�VO\%c�.ea�]��%[`b��;a`YOU%M� Y 2 bE[�M������� Y 2 bW[�M���� �������

��� �]EN;N=d�� Y 2 bW[�M���N`]eY�\ ���

^�VO\%c�.ea�]��%[`b��;a`YOU%M� Y 2 bE[�M������� Y 2 bW[�M���� ������ � ��� �������

T���! EL�3�"#�%$&�%'��(�! EL'��)����� ��� �����*�+�! EL,'�� ��� �������-�! EL'�� ��� ��� �*�.�0/

��^_d,lSuDVaW � ( ��32143;u�lS`cV\� beu~�^km�}S`ci>oq^c]Sd;g�uD^_}2`_V!gDmU¤

gx¢�baijuDdelST{VU],g�mvief�g�l2ie`_`co�uDVUf�VU^_�,V5V�°qg�u�iaijuDdelST{VU],g�m�g�baT�ij«,V5gxo9}3V ^c]�� beuDT�i�g�^cbe]{T{beuDV5}SuDVUf�^km�Ve¤��Sbeu^c]2mpgDie]2f�V,�jVUs9l2ie`_^_gxo�^km l2mplBij`c`_o��q^cV!¢�VU~;iemvi\uDV!`ki�g�^cbe]�rBV!gx¢#VUV!];gx¢�bGb,r�wxVUf�gDmU�jrSl�gvgDQSVUm�V ber�wxV0fªg�m ]SV!V0~gDb�Q2i>�,V#gDQSVGmDijT{V5gxoq}3Ve¤ �Sb,uvg�QS^km@uDVUi,mpb,]n��gDQSVgxoq}3V��£g�QSVUbeuDo,��r2i,mpV0~�uDV!}SuDVUm�V!]9gDijg�^cbe];bj�ÇVUs9l2ie`_^_gxo;^cm@igDQSuDV!V��­ijuDdel2T;VU]9g�uDV!`ki�gD^_b,]n��¢ QSV!uDV\g�QSV�¦2uDmpgijuDdelST{VU],g ^km5g�Q2V�gxo9}3Ve�2^_]Ë¢ QS^kf�Q'g�QSV�gx¢�b�bjgDQSV!u T�l2mxgr3V!`cbe]2d2¤�P�b¯Q2ie]2~�`cV�gDQS^cmU�Ç^c]�¦S°F]SbegDijg�^cbe]2m\ij}S}2`_o©VU�eV!]Âg�b�� lS]2fªgD^_b,]2m\¢ ^ÊgDQFgDQSuDV!V{ijuDdelST{V!]9g�m!�3rSl�gb,]S`co�g�QSVa`kiempg65;ijuDV�~S^cm�}S`ki>oeVU~n¤

��� � 70���98 � �28 ��� � ��%�����9: ��� ���� �;:=< �&:4>-?��A@ ���B: #(���98 � �����B:¬­]MdeVU]SV!u�ij`£�2gDQSV;}Su�V!� V!uDu�V0~©]SbegDi�gD^_b,]©� beu�i�de^c�eV!]FT{ijg�QSVUT�i�g�^kf!ie`z� l2]2fªgD^_b,]Fbeu\uDV!`ki�g�^cbe]Mf!ij]2]Sbjg�rBV~SVUmDf�uD^_r3VU~�rqo¯i�mp^cT{}S`cV\T�ie}S}S^c]Sd;g�b{g�Q2Va}Su�V0~�V�¦B]SVU~�`ci>o,bel�gmpg�u�i�gDV!de^cVUm5i>��ij^c`kijrS`cVG^c]'g�QSVab,}BVUuDijg�b,ug�ijrS`cVe¤ �2beu\g�QSV0mpV�be}3V!u�i�gD^_b,]2mGie]2~ÂuDV!`ki�gD^_b,]2mU�Bl2m�V!u�m\f!ij]M~�VUmDf�uD^_r3V�g�QSVU^_u�]SbegDi�gD^_b,]Âl2m�^_]2d�i'mp^cT{}S`_V}Bi�gpgDV!uD]���T{ijgDf�QS^c]SdGr2iem�VU~�`cie]Sdel2iedeV,�jf!ij`c`cVU~���321632¤�P5Q2^cm `kij]Sd,l2ijd,V�^km ^_]2QSV!uD^ÊgDVU~�� u�b,T�g�QSVh¡#VU],g�ijlSum�o�mxgDV!Tˤ

�2beu;^_]2mpgDie]2f�V,�@^_�g�QSV'T�i�gDQSV!T�i�gD^cfUij` mpg�l2~�o®be�ºg�V!]ÄT�ij«eV0m�lBmpV'bj� gDu�^c}S`cV�gDmU��f!be]2mpg�uDl2fªgDVU~±¢ ^_g�Qgxoq}3VU~'}2ie^_u�m!��¢ Q2V!uDVGgDQSV�gxoq}BV0m r3V!^c]Sd�l2m�VU~ËijuDVaij`c¢�i>o�m#g�QSV�mDijT{V #�� 'N�Bie]2~DC;�2l2m�V!u�m5T�i>oN¢�ie]9ggDb{u�VU}S`ci,f�V\g�Q2V�~�V��IielS`Êg]2bjgDijg�^cbe] �

E #GFIH�'KJLCGMON�HQP�F E 'RFSCTN�HVU�F;W�M;M¢ ^_g�Q©i;T{beuDVaf�be]Bf�^kmpV�ij]2~n�SQSV!]Bf�Ve�2T;b,u�V\uDVUie~2ijrS`cV�]Sbjg�i�g�^cbe] �

X P�F%U�F%WSY[ZP5QS^km¯fUij]Èr3VFV0iem�^_`coÆV!°q}2u�V0m�m�VU~�¢ ^_g�Q iÆ��32163*u�l2`_VËgDQ2i�g'Q2iemNg�QSV©� b,u�T de^c�eVU]Ã^_]ȦBdelSuDVFW�¤

P5Q2^cm `ki>oeb,l�guDlS`_V�f!beT�rS^_]2VUm5gx¢�b{}2iju�gDmU¤�P5Q2V�¦2uDmpgh^kmi{}2ijgpg�VUu�]ËgDQ2i�g¢ ^c`c`zr3V�f�b,T;}BijuDVU~'¢ ^ÊgDQËg�QSVV!°�}Su�V0m�m�^cbe]2mGrBVU^_]2dË~S^cm�}S`ki>oeVU~n¤�¬­]Mg�QS^km�}2i�g�g�VUu�]n�ng�QSV{]2ieT;V0m\`_^c«eV ]WN;N d � ^�VO\%c�.ea�]��%[`b��;a`YOU%M �Sij]B~Y 2 bW[�M f!beuDu�V0mp}3be]B~Ëg�b'g�uDV!V{b,}BVUuDijg�beu�m^c]¥g�QSV�~SijgDij��mpg�uDl2fªgDlSuDV�uDV!}SuDVUm�V!]9gD^_]Sd¯g�QSV{`cbed,^cfUij`�� beuDT�lS`kiS¤H�beT{Vbe}3V!u�i�gDbeu�mv`c^_«,V Y 2 bE[�M T�i>o{f!ieu�uDo�i�mxgDu�^c]Sd���ij`clSV,¤�¬­]Ng�QSVG}2i�g�g�V!uD]n�qmpb,T{Vhmpl2r��OgDV!uDT�m�ijuDV`cV��ºgl2]S«q]Sb�¢ ]n�>g�Q2V!o�ieu�V�uDV!}SuDVUm�V!]9g�V0~a¢ ^_g�Q{��ijuD^cierS`_V0m!�jie]2~agDQSV!uDV5^cmvi\`cV�°�^kf!ij`�f!be]q�eVU],gD^_b,]ag�QBi�g@T�ij«,VUm�^Êg

132 g4165

Page 10: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�F¿e» ¼���¾�¿j»O¸����2½�¶�½U»Ç²B³D½�����½ª¶�,¿�¹S¶� ��´�� �

}3b,mDm�^_rS`cVGg�b{u�V0f�b,de]S^c¨!VG�>ieu�^kijr2`_V0m ��g�QSVUo¯ij`c¢5i>oqm�mpgDiju�g5¢ ^ÊgDQ'i;mxg�iju0¤ ¬­]�b,lSu5V�°SijT{}S`cVe�9g�Q2V\��ieu�^kijrS`cVUmieu�V ��� � ��� �Sie]2~ � � ¤

P5QSV�m�VUf!be]2~F}2iju�g\be�@g�Q2V;uDlS`cV;f!be]9gDie^_]Bm\i�`ki>oebelSg\m�}BV0f�^_¦Bf!ijg�^cbe]Fb,u�d9ij]S^c¨!V0~©^c],gDb�r3b>°�VUmU¤��@i,f�Qr3b>°�^cm ie]S]SbegDi�gDVU~�¢ ^_g�QËi�`ki>oeb,l�g�~S^_uDVUf�g�^c�eVGg�Q2ijg�^c]2~�^kf!ijg�VUm�QSb�¢ g�QSV\��ijuD^_b,l2m#f!beT{}3be]SVU],g�m�}Su�V0mpVU]9g^c]¥g�QSV�r3b>°F¢ ^c`_`�rBVNieu�u�ij]Sd,VU~F¢ ^ÊgDQ¥uDVUm�}3VUfªgagDbËVUief�Q¥bjgDQSV!u0¤�¬­]¥g�QS^kmaV!°�ieT{}S`_V,�z¢�V�mpVUV;b,]S`coÂgx¢�b«q^c]2~Smzbj�S~�^cu�V0fªg�^c�eV0m � EL�3 "#�L$ � '�� ^_]2~S^cfUi�g�V0mÇgDQ2i�gngDQSV�VU`_VUT{V!]9gDm�mpQSb,lS`k~Gr3V�`kij^k~�belSgzQSbeuD^c¨!be]9g�ij`c`_o,�m�V!}Biju�i�g�V0~arqoabe]SV�lS]S^_gvbj�Bm�}2i,f�Ve¤�Åb�¢�V!�,V!u0�0^_�2gDQSV5��ijuD^_b,l2m�f!beT{}3be]SVU],g�m�~Sb\]2bjg�¦Sgvbe]�gDQSV5`_^c]SV,��g�QSV`ki>oeb,l�g�m�QSb,lS`c~�mxg�iju�g5ied,ie^_]'i�g�g�QSV�]SV!°9g `c^_]2Ve��¢ ^_g�QËij]¯^c]2~�VU]9gDi�gD^_b,]¯be� W�lS]S^_gDm5bj��mp}2i,f�V,�qie]2~N¢ ^_g�Q�;V!°9gDuDi{`c^c]SVUm be� ~S^cmpgDie]2f�V,¤�P5QSVabjgDQSV!u~S^_uDVUf�g�^c�eVe� %L�'�� ^_]B~�^cfUi�gDVUm5gDQ2i�g gDQSV���ijuD^_b,l2m f�b,T{}Bb,]SV!]9gDm^c]�gDQSVhr3b>°{m�QSb,lS`c~NrBV\ijuDuDie]SdeV0~�Q2beuD^_¨Ube]9gDie`_`coe�j¢ ^_g�Q�]Sb�m�}2ief!Vr3V�gx¢�V!V!]Ng�QSVUT'¤ �È^_g�Q¯g�QS^km�uDlS`cVe�9¢#VfUij]'}SuDVU~�^kfªg5g�QBi�ghi�g�lS}2`_V �,] ��� �'d�� ¢ ^_`c`nr3Va`cie^c~�belSg^_]Ëbe]2V\be��gDQSV\� be`c`_b�¢ ^c]Sd{T�ij]S]2V!u�m �

�,] ���B��d�� b,u �_] �� � d�� beu �,] ��� �

d�� beu�,] �� �d��

P5Q2Va`_V!�ºg�Q2ie]2~©mp^k~�Vabe��gDQSV�`ki>oebelSg5uDlS`cV�^cmhuDijg�QSVUuf�beT{}S`c^kf!i�gDVU~¯g�b�¢ uD^ÊgDVe�BrSl�g^_gh^cm}Bb9m�m�^_r2`_V�gDb�l2mpVgDQSVhd,uDie}SQS^kf!ie`�l2mpVUup��^c],gDV!u��Iief!V5g�b�}SuD^c],g#bel�g#g�QSVG}2i�g�g�V!uD]�`kij]Sd,l2ijd,V�uDV!}2u�V0mpVU],g�i�gD^_b,]�� b,u#iad,^_�,V!];g�uDV!Vmpg�uDl2f�g�lSuDVe¤P5Qql2mU�B`ki>oeb,l�guDlS`_V0mhieu�V�iN`_^_gpgD`_V�VUiem�^cV!u gDb�f�b,]2mpg�uDl2fªg0¤yzi>oeb,l�guDlS`_V0mhfUij]©r3V�ijuDu�ij]Sd,VU~^c]9g�b®}2i,f�«>iedeV0m;ie]2~Äg�QSV©`ki>oeb,l�g�mxgDuDijg�V!d,o±g�QBi�gN^km�iefªgDl2ij`c`co*l2m�VU~Äg�b±~�^kmp}2`ci>o±gDQSV©T�i�gDQSV!T�i�gD^cfUij`� b,u�T�lS`kiem�^km;ij]*b,uD~�VUu�V0~±f�b,T�rS^c]2i�gD^_b,]±bj�hm�V!�,V!u�ij`�}Bief�«�ijd,VUmU¤ �Sb,u�VU�eV!uDo¥}2i�g�g�V!uD]n� gDQSVN¦2u�mpg�uDlS`_VgDQ2i�g�ie}S}S`c^_V0m�^cm�l2mpV0~Mg�b©f�b,T;}2l�g�V{gDQSV�`ci>o,bel�g0¤�P5QS^cm�`ci>o,bel�gaT{VUf�Q2ie]S^kmpT }2u�b��q^k~�VUm�i'debqb�~F`cV!�,V!`be��V!°qg�V!]Bmp^crS^c`_^_gxoe�BrSl�ggDQSV�}SuD^_T{^_g�^c�eV0m }SuDb��9^k~�V0~'mpg�^c`c`z]SVUVU~�g�bNr3V�VU]SQ2ij]Bf�VU~�gDbNf�be]9gDu�b,`n`cieu�d,Vam�fUij`cVs9lSV0mxgD^_b,]2m m�l2f�Q'i,m5`_^c]SVarSuDVUie«q^_]SdB¤

����� � ��� �B8 � ��������� �%� � �#@ ����: #&�!? ����� ��P5Q2V���32163'`kij]2del2iedeV;be]S`coÂT�ij«,VUm�^_g�}Bb9m�m�^_r2`_V;g�bËT�ij}¥� b,u�T�lS`ci�}2ijgpg�VUu�]Bm\g�b©V�°�^kmxgD^_]2d'd,uDie}SQS^kf!ie`f!beT�rS^_]Bi�g�b,uDmU¤ ( ]SbjgDQSV!u�`cV!�,V!`�bj��V!°9gDV!]2m�^crS^_`c^_gxoËg�Q2ijg\¢�V�}SuDb��q^c~SV�^c]Fbel2uGd,uDie}SQS^kf!ij`�l2m�V!u��£^c]9g�V!u��Iief!V^km�g�QSVG}Bb9m�m�^_r2^_`c^Êgxo;gDb�ie~2~�]SV!¢Èd,uDie}SQS^kf!ie`Bf�b,T�rS^c]2ijg�beu�mU¤�P5QSV0mpV\f�beT�rS^c]2i�gDbeu�m@ijuDV¢ uD^Êg�g�V!]�^_] �ei>��iS�ie]2~�gDQSV!o¯be]S`coN]SVUVU~¯g�b{f�be]S� beuDT g�b�i�}2u�V0~�V�¦2]2VU~�^_]9g�VUup�Ii,f�V,¤���ief�QËf�beT�rS^c]2i�gDbeu�]SV!V0~Sm#g�b{rBVaierS`_VgDb®V�°�}SuDVUmDm�¢ QSVUu�VË`c^_]2VUmU�#m�o9T�rBb,`cmU��ie]2~Äm�lSr���V�°�}SuDVUmDm�^_b,]2m�ieu�VË~�u�i>¢ ]Æ¢ QSVU]ÆgDQS^km�f�beT�rS^c]2i�gDbeu;^cmr3V!^c]Sd�l2mpV0~ǤGP5Q2V;f!beT�r2^_]2ijg�b,uT�l2mpg\ie`cm�bNr3V;ierS`cV�g�b�u�V0ief�gg�b�T;V0m�mDijd,VUm ^c]2~�^kf!ijg�^c]SdNgDQ2i�gaiNm�lSr��V!°�}Su�V0m�m�^cbe]ËQ2iemGf�Q2ij]2deVU~©m�^c¨!VU~ �O~�lSV�g�b¯V0~�^_g�^c]Sd ! beugDQ2i�gGg�QSV;mp^c¨!V�ij`c`_b�f!ijg�V0~'g�bNgDQS^kmGf!beT�r2^_]2ijg�b,uQBiem{f�Q2ij]2deVU~n�#~�lSVËgDb±iMf�Q2ie]SdeV'bj�am�^_¨UV'^c]Ãij]Sbeg�QSVUu{f!beT�rS^_]Bi�g�b,uU¤ÃP5QSVËT�ij^c]ÄT{V!g�QSb�~Sm�ieu�V©iem� b,`_`cb�¢m��

��� ������������ ¤qP5QS^km5T;V!g�QSb�~�^cm fUij`c`_V0~�gDb�uDVUs9lS^cu�VGg�QBi�g5g�QSVaf!beT�r2^_]2ijg�b,u�f!beT{}Sl�gDVUm#g�QSV�uDV!`ki�gD^_�,V}Bb9mp^_g�^cbe]Bm�be�zij`c`B^_gDm�f�b,T;}3be]2V!]9gDmU¤ ( ]�ieu�d,lST{V!]9g@bj�ÇgDQSVGT{V!g�QSb�~�^km@gDQSVG~S^cmpgDie]2f�VG`_V!�ºg�b,]�g�QSVu�^cdeQ9g��£Q2ie]2~'mp^k~�Ve¤�P5QS^km~�^kmxg�ij]2f!Va^cm lBmpV0~¯gDbNf�QSbqb,m�V�¢ QSV!]'g�b¯deb;g�b{gDQSV�]SV!°9g`c^c]SVe�Sb,u¢ QSVU]g�b�l2m�Vai;�eV!u�g�^kf!ie`Ç`ci>o,bel�g5^c]2mxgDVUi,~¯be� ie]�QSb,u�^c¨!b,]9gDij`Ðb,]SVe¤

������ �!���#"%$ &���' ¤ÆP5QS^km�T{V!g�QSb�~*^km{f!ie`_`cVU~Æ¢ QSV!]Äg�QSVËf�b,T{}Bb,]SV!]9g{^cm{ief�g�l2ie`_`co*~�u�i>¢ ]*be]Æg�QSVm�f!u�VUV!]n¤ ( T{be]SdÄg�QSV®ijuDdel2T;VU]9gDmU�GijuDVUi,mN¢ QSVUu�VMg�Q2V¥~�u�i>¢ ^c]SdÆ^km'u�V0ij`c`_oÃ]SVUVU~�V0~ T�i>oÈrBV}SuDb��9^k~�V0~Ǥ

g�gih%j6k�l�mRl

Page 11: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

� �G¾�½�³�Àj¿W�����G½�³�»�´j»�� ´j»£»O¸I½�³����h¸��,½D¿E�

������ �!��� ¤SP5Q2^cm T{V!g�QSb�~Ë^kmf!ie`_`cVU~¯g�b�uDVUs9lS^cu�V\gDQ2i�g gDQSV�f�b,T;}3be]2V!]9g^cmh~�uDi>¢ ]'be]�g�Q2V�m�f!u�VUV!]n�rSl�gN^_�\b,]SV©~�V�gDVUf�gDm{gDQ2i�g{g�Q2VÂf�beT{}3be]SVU]9g{QBiem;g�Q2VÂm�ieT;V©m�^_¨UV©iem{^c]�i¥}SuDV!�q^cbel2m{~�u�i>¢ ^_]2dbe}3V!u�i�gD^_b,]n�qg�QSV�~SuDi>¢ ^c]Sd;¢ ^_`c`nbe]2`_oNr3V�g�u�ij]Bmp`ki�gDVU~Ǥ

�� �'�� ��� ��& � ¤vP5QS^cm#T{V�gDQSb�~N^km#fUij`c`_V0~;g�b�^c]2~�^kf!ijg�Vhg�Q2ijg5i�m�lSr���V�°�}SuDVUmDm�^_b,]�Q2iem#f�Q2ij]Sd,VU~�m�^_¨UVe�g�Qql2ma}3b,mDm�^_rS`coÂuDVUs9lS^cu�^c]Sd�g�Q2ijgag�QSV�beg�QSVUu�mpl2r��£V!°�}Su�V0m�m�^cbe]2m\bj�#g�QS^km�f!beT�r2^_]2ijg�b,u\¢ ^c`c`@rBV�uDV��� beuDT�i�g�V0~Mie]2~¥u�V!��~SuDi>¢ ]n¤�¬��#gDQSV�¢ QSb,`_V�f�b,T�rS^c]2ijg�beuafUij]¥rBV�uDV��­~�u�i>¢ ]F¢ ^_g�QSb,l�g�f�Q2ij]Sd,^_]2dg�QSV'mp^c¨!V,�zgDQSV!]®gDQSV � �'�� ��� ��&�� T{VUmDm�iedeV{T�i>oF]Sbeg�r3V�g�u�ij]2m�T{^Êg�g�V0~¥g�b©gDQSV¯f�b,T�rS^c]2ijg�beu�� m}2ijuDV!]9g0¤P5QS^cmhQSV!`c}2mhQ2i>�q^_]Sd�ij]©^c]2f�uDV!T{V!]9g�ij`n`ki>oeb,l�ghij`cdeb,u�^_g�QSTË�B¢ ^ÊgDQÂT{^_]S^cT�ij`�u�V!��~�u�i>¢ ^c]Sdief�g�^cbe]2m5¢ QSVU]©~Si�g�i�^km5T{b�~�^ʦBVU~Ç��� b,ui{T{beuDVGl2m�V!u��£� u�^cV!]2~S`_oNuDVUm�lS`_gU¤

� � ����� ��������� ��� �'���������� � � ������� �!� �' ����

����� �$� ����� ������� #(���98 � �28 ���P5Q2V¯l2m�l2ij`#T{bq~SV¯bj�}2u�bqbj�~�VU�eVU`_b,}ST{V!]9g�^c]±g�QSV'^c],gDV!u�ief�g�^c�eV�}SuDbqbj�mpo�mpg�VUT{m�bj� gDQSV 3q��F�IijT{^_`couDV!`c^cVUm�be]�Á9´0¿ � ��¸º³�½ �U»£½ ��²3³�´U´pµ�¤ÄP5QSV'l2m�V!u;V!]9g�VUuDm;iMdeb,ie`�`cbed,^cfUij`#� beuDT�lS`kiS� gDQSV!]Ãf�b,T;T�ie]2~Sm;ijuDVie}S}S`c^_V0~'g�b�gDQS^km`cbede^kf!ie`Ç� beuDT�l2`ci2�2rSuDVUie«9^c]Sd{^_g\~�b�¢ ]©^c]9g�b¯m�^_T{}S`cV!u� beuDT�lS`kiemU�Bm�b{g�Q2ijgG¢ QSVU]Ëg�QSV0mpVm�^cT;}2`_VUu�� b,u�T�lS`kiem ijuDV#«q]Sb�¢ ]�g�b�r3V�g�uDlSVe��gDQSV!];gDQSV5}Su�VU�q^_b,l2m�deb,ie`q^cmvm�lSuDV�gDb\r3V�g�uDlSVe¤ P5QSV m�^_T{}S`cV!u� b,u�T�lS`kiem�f!ij]¯g�QSVU]�r3V�u�V0~�l2f!VU~N^c]�g�QSVamDijT{VGT�ie]S]SV!u0¤ ¬­]9g�lS^_g�^c�eVU`_o,�,gDQSV�¢ QSbe`cV�}SuDb9be�zf!ie]¯gDQSV!]'rBVuDV!}2u�V0mpVU],gDVU~�¢ ^ÊgDQ{iGgDu�VUV5mpg�uDl2f�g�lSuDVe¤ ( fªgDl2ij`c`_o,��g�Q2^cmv^kmvQSb�¢Äi�}Su�bqbe�2rBVU^_]2d�f�be]BmxgDu�l2f�g�V0~�^km u�V0f�b,uD~�V0~^c]2m�^c~SVGgDQSVD�\���¥`_b,de^kf!ie`3VU]Sde^c]SV,¤

¬­]Mg�QSV;}SuDbqbj�º�£g�uDV!V�mpg�uDl2f�g�lSuDVe�nie`_` ]Sb�~�V0m\ieu�V;`kijr3V!`cVU~¢ ^_g�QMg�Q2V�`cbed,^cfUij`�� beuDT�lS`kiNgDQ2i�g�¢5iemGg�QSVd,b,ie`9¢ Q2V!]{g�Q2^cmv}2u�bqbj�ÐmxgDV!}{¢5iemvf�b,]2mpg�uDl2fªgDVU~Ç�,rSl�gvg�Q2V!uDVmxgD^_`c`2ijuDV�gx¢#b�«q^c]2~Smvbe�3]Sb�~�V0m � ´�²S½!¹a]Sb�~�VUmQBi>�eV¯]2bjg{oeV�g{r3V!V!]�ie~S~Su�V0m�m�VU~±rqo*iF}Su�bqbe�Gf�b,T;T�ie]2~Æie]2~®g�Q2V!oÆ~�b¥]Sbeg;Q2i>�,V�f�QS^c`k~�u�VU]*]Sb�~�V0m!�¢ Q2^_`cV:���_´�¶!½Z��]Sb�~�V0m�ijuDV�ie`cm�bM`kijr3V!`cVU~Æ¢ ^ÊgDQÈiF}SuDbqbj�Gf!beT{T�ij]2~Äij]2~*g�QSVUo*Q2i>�,V�f�QS^c`c~Su�VU]n¤ÄP5QSV}2u�bqbj� f�b,T;T�ie]2~Sm ijuDV�ij`kmpb�f!ie`_`cVU~¥»�¿W�U»I¸���¶ª¤

�¥V�Q2i>�,Va^_T{}S`cV!T{VU],gDVU~©i�}2u�b,deu�ijT gDQ2i�gGg�u�i>�eV!u�m�VUm#g�QSV�}SuDb9be�º�OgDu�VUVamxgDu�l2f�g�lSuDV�ie]2~©f�b,]2mxgDu�lBfªgDmi¯mxgDu�l2f�g�lSuDVU~©g�V�°qgGg�Q2ijgG^kmG~�^km�}S`ci>o,VU~Ël2m�^c]SdNbel2uGd,V!]SVUuDie`n`ci>o,bel�gGT;V0f�Q2ij]2^cm�Tˤ¬­]©g�Q2^cm\mxgDu�lBfªg�l2u�V0~gDV�°qgU�qV0ief�QNf�`cb,m�VU~N]Sb�~�Vh^km�uDV!}2u�V0mpVU],gDVU~{rqo{i�m�V!]9g�VU]2f�V,�,¢ QSVUu�Vhg�QSVG�eVUu�rN^km�f�QSb,m�V!]{gDb�~�V0m�f!u�^crBVhg�QSVg�ief�g�^kf�r3V!^c]SdËl2m�^_]Sd'be]Fg�QS^km�]Sb�~�V,¤ �Sb,uGb,}BVU]M]2bq~SVUmU�Ðg�QSV�m�V!]9g�VU]2f�V�g�Q2ijga^cm�f�QSb9mpVU]F^km���¾{¿UÁj¸º¹Ð½¯¿²3³�´0´xµ�´pµ��������2� be`c`_b�¢�VU~¯rqo�g�QSVad,b,ie`B� b,u�T�lS`kiS¤

¬­]¥deb9ij`zb,u�^cV!]9gDVU~©}SuDbqbj�ImU�Ðdeb,ie`cm\ijuDV�m�VUs9lSVU]9gDm�� g�QSVUo©}2ij^cuaiN`c^cmpg�bj�#i,m�m�lST{}�g�^cbe]BmG¢ ^_g�Q¥i�f!be]��f!`_lBmp^cbe]ËgDQ2i�g\]SV!V0~SmgDb¯r3V�}SuDb��eV0~Ǥ ¬­]©gDQSV�gDV�°qg�l2ie`�}2u�V0mpVU],g�i�gD^_b,]©bj�@}SuDbqbj�ImU�2be]2`_o'g�Q2V�f�b,]2f�`cl2m�^_b,]be�ng�Q2VGd,b,ij`3^km#~�^km�}S`ci>o,VU~Nij�ºg�VUu#gDQSV\¢#b,uD~Sm��ª¾�¿!Áe¸º¹3½�¿\²B³�´0´xµa´pµ��������qgDQSV�Q9oq}3bjgDQSVUm�VUm�ijuDVGmDf!ijgpg�VUu�V0~^c]'g�QSV�gDV�°qgU¤P5Q2V!uDV�ieu�V\gx¢�b�«q^_]2~2m bj��gDi,fªg�^kf!mU¤ Hqb,T{V�gDief�g�^kf!mieu�V AD¿>¶ª¸��!¤#P5QSVUo�f!beuDu�V0mp}3be]2~¯g�b�}2u�^cT{^ÊgD^_�,V�� beuDT{mbe��uDVUi,mpb,]S^c]Sd � g�QSV�^c]�� VUu�VU]2f�VauDlS`cVUm5bj� ]2ijg�lSu�ij`_��~SVU~�l2f�g�^cbe]Ëmxgxoq`cVampV0s,l2V!]9gf!ij`kf�l2`_l2m !�¤ hg�QSVUu5gDief�g�^kf!mieu�V �D´j¾h²S´>¶ª¸ »�½�¤*P5Q2V!o±f�b,u�uDVUm�}Bb,]2~¥g�bFg�QSV'uDVUiem�be]2^_]Sd©}3V!u�� beuDT{VU~±rqo±��ijuD^cbel2m�«9^c]2~Sm�bj�\~�VUf!^cm�^_b,]}2u�b�f�V0~�lSuDVUm b,uhijlSg�beT�ijg�^kfa}SuDb9be�@mpV0iju�f�Q'f!beT{T�ij]2~SmU¤ �Sb,uh}Su�ief�g�^kf!ie`n]SV!V0~SmU�B^_gh^kmhu�ijuDV!`coNl2m�V�� lS`�g�bm�V!Vhg�QSV�r2i,mp^kf^c]�� VUu�VU]2f�V�mxgDV!}2m�QS^k~S~�VU]¯r3V!QS^c]2~'ijl�gDbeT�i�gD^cfh}SuDbqf!VU~�l2u�V0m!¤@H9g�^c`c`O��^_�zgDQSVG}2u�bqbj�n}2u�b�f�V0m�m]2V!VU~2m�gDbMr3V�g�QSb,u�b,lSdeQS`co®V�°SijT{^c]SVU~n�@^Êg{T�i>o±rBV']SV0f�VUmDmDijuDoMgDb¥m�V!V¯g�QS^km�~SijgDiS¤*P�bMi,f!f!beT{b�~Si�gDVgDQS^km!�@¢�V'Q2i>�eV�V!]SQ2ie]2f�V0~±gDQSVË}Su�bqbe�º�OgDu�VUV¯~Si�g�iMmpg�uDl2f�g�lSuDV!��]Sb�~�VUm;f!ie]Äij`km�bFr3VËie]S]SbegDi�gDVU~±¢ ^_g�Q

132 g4165

Page 12: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�F¿e» ¼���¾�¿j»O¸����2½�¶�½U»Ç²B³D½�����½ª¶�,¿�¹S¶� ��´�� �

ie]SbjgDQSV!uv}2u�bqbj�º�£g�uDV!V,�>g�Q2ijg#~�V!gDij^c`km�g�QSVuDVUi,mpb,]S^c]SdGmpg�VU}2mv}3V!u�� beuDT;V0~;r9o�g�QSV g�ief�g�^kfj�e¢ Q2V!];g�QS^km g�iefªgD^cf^km f�b,T{}Bb9mp^_g�V,¤( ]SbjgDQSV!u�� lS]2f�g�^cbe]2ie`_^_gxo{g�QBi�g5^cm#}SuDb��9^k~�V0~�rqo�belSu5mxgDu�lBfªg�l2u�V\~�u�^c�eVU]N`ki>oebelSg#T{VUf�Q2ie]S^kmpT ^km�gDQ2i�g

]2bq~SVUmN^c]ÈgDQSVFgDu�VUVFmpg�uDl2fªgDlSu�VMf!ij] rBV¥ij]S]SbegDijg�VU~â ^_g�Q V�°qg�u�i±^c]�� beuDT�i�g�^cbe]z¤ ( ]S]2bjgDijg�^cbe]2m�f!ie]r3V�l2m�VU~Mg�bËgDV!`c`�¢ QSV!g�QSVUuag�Q2VNf�beT{}3b,m�^ÊgDV{gDi,fªg�^kf�ij`cbe]2V;^km�~�^km�}S`ci>o,VU~Mbeu�¢ Q2V�g�Q2V!uagDQSV�}SuDb9be�#gDQ2i�g^_g©f�be]BmxgDu�l2f�g�V0~È^cmË~�^km�}S`ci>o,VU~È^c]2mpg�V0ie~Ǥ P5QS^kmËf!ij}BijrS^c`_^_gxoÃ^cm�}SuDb��q^c~�V0~Ãg�b�l2m�V!u�m¯l2]2~�V!uËij] VUi,mpoV!°�}2ij]2m�^cbe]��­mpQ2u�^c]S«q^_]2d¥T{b�~�VM¢ QSV!uDVÂgDQSV!oÈf!ie]ÈT�ij]ql2ij`c`co�m�V!`cVUf�g�V�°�}SuDVUmDm�^_b,]2mNij]B~È}SuDbedeuDVUmDm�^_�,V!`couDVUs9lSV0mxg5� b,u5g�QSVU^_u V!°q}Bij]2m�^_b,]n¤

��� � 7 �����!��� � �&:4> ���!:=<����-����� ���"���!:=<t@u�bqbj�Im�g�QBi�g�uDV!`co®QSVUi>�q^c`_o®b,]*VUs9l2ie`_^_gxo±ij]2~±uDV!¢ uD^_g�^c]SdFijuDV¯V0mp}3VUf!^cie`_`co®~�^ #Nf!lS`Êg{gDbFuDV!]2~�VUu�^c]Æie]V #Nf�^cV!]9ghT�ij]S]2V!u0¤��¥V�Q2i>�,Va^_]9gDu�b�~�l2f!VU~©i�m�}BV0f�^_¦Bf�gDu�V0i�g�T{VU],gh� beugDQS^km«q^_]B~Ëbj�v}SuDb9be�v¢ QSV!uDVabe]SVuDVUf!bed,]S^_¨UVUm�gDQ2i�gh}2u�bqbj�Imbe]ËVUs9l2ie`_^_g�^cVUmieu�V�~Sbe]SV�b,]S`co¯¢ ^_g�Q©uDV!¢ uD^ÊgD^_]Sd{g�ief�g�^kf!mU¤#¬­]ËgDQS^cmGf!i,mpV,�2VUi,f�Qmpg�VU}Âl2m�l2ij`c`co'f!u�V0i�g�V0m be]2`_o'be]SV;mpl2r��£d,b,ie`O�S¢ Q2V!uDV�be]S`co©i�`c^Êg�g�`cV�^c]�� beuDT�i�g�^cbe]©QBiemhf�QBij]Sd,VU~ǤhP5Q2V!uDVT�ijgGr3V�i�� VU¢�V�°qg�u�i¯m�lSr���deb9ij`km!�Sg�Q2ijg�f!beuDu�V0mp}3be]2~�gDb�m�^c~�V;f�b,]2~�^_g�^cbe]2mh� beu\mpb,T;V�bj�vgDQSV�uDV!¢ uD^ÊgD^_]2db,}BVUuDijg�^cbe]2mU¤ �2beu gDQSVUm�V�«q^c]2~Smbe�v}Su�bqbe�Im!�S¢�VauDV��­ijuDuDie]SdeVGg�QSV�g�u�ij]2mp� beuDT�i�g�^cbe]Bm5bj� g�V!°9ghg�bNT�ij«,Va^Êg`cbqbe«N`c^c«eVai;}Su�b,deuDVUmDmp^c�eVhg�u�ij]2mp� beuDT�i�gD^_b,]Nbj��g�QSVa`cV��ºgDQ2ij]B~'m�^k~�V�^_]9g�b{gDQSVauD^_d,Q,gDQ2ij]B~¯m�^c~�V,¤

�2beu^c]2mpgDij]Bf�Ve��^_� g�QSVaT{b�~�^_¦Bf!ijg�^cbe]2m ie}S}3VUiju b,]S`coN^_]ËgDQSVa`cV��ºg�QBij]2~Ëm�^c~�V,��g�Q2V!]'g�QSV�mpV0s,l2V!]2f!V\be�d,b,ie`cm�Q2i,m�g�QSV�� b,`_`cb�¢ ^_]2d;m�Q2ij}3V!�

tvuDb��eV # � C� rqo¯m�beT{V�wxl2mpg�^_¦Bf!ijg�^cbe]n� #§^km5uDV!¢ uD^Êg�g�V!]Ë^c])' !tvuDb��eV ' � C

�2beu5gDQS^cmV�°SijT{}S`cVe��¢�V�uDijg�QSVUu�}2u�b�~�l2f!V\g�QSV�� b,`_`cb�¢ ^_]2d�gDV�°qg���MVaQBi>�eV # � '(�Ir9o'mpb,T{V#wxlBmxgD^ʦBfUi�gD^_b,]"!

� C��^cdelSuDV�ad,^_�,VUmvij]�V!°SijT{}S`cV bj�Ç}SuDb9be�3}2u�V0mpVU],g�i�gD^_b,]{i,f!f!beu�~�^_]2dGgDbag�Q2^cm#mxgDuDijg�V!d,o�� b,u�i�m�^_T{}S`cV}SuDbqbj�ierBb,l�gNu�i�g�^cbe]Bij` ]qlST�r3V!u�mU¤ �beg�VËg�QBi�g¯b,]SV©bj�\g�QSV{wxlBmxgD^ʦBfUi�gD^_b,]2m�uDVUs9lS^cu�V0m�ij]ÃV�°qg�u�i®}SuDbqbj� �Ëg�buDV!}2`ci,f�V�¢ ^_g�Q�

�9^Êg�^cm�]SVUf!VUmDm�ieu�o�g�b�f�Q2VUf�«�gDQ2i�g ��^km@]Sbeg��2¤��¥Vhl2m�Vhi�m�`c^_d,Q,gD`_o;T{b�~�^ʦBVU~�mpg�u�i�g�VUdeo

¢ Q2V!]{g�Q2Vu�VU¢ u�^_g�^c]Sd�mpg�VU}2m@ief�g�be]{gDQSVuD^_d,Q9gp��Q2ij]2~;m�^c~�Vhbj�Ðg�QSVhVUs9l2ij`c^_gxoe�,ie]2~{oeV!g#ij]2bjg�Q2V!u@mpg�u�i�g�VUdeo^_� be]SV�uDV!¢ uD^_g�^c]SdÂmxgDV!}*i,fªgDm�be]®rBbeg�QÆm�^k~�VUmabe�5g�QSVNV0s9l2ij`c^Êgxo,¤�Grq�q^_b,l2m�`_oFm�VUs9lSVU]2f�V0mabj� uDV!¢ uD^ÊgD^_]2df!belS`k~¯gDij«,V�}S`ci,f�V�^c]Ëi;�,V!uDoN`_b�fUij`c^_¨UVU~'mplSr2}2iju�g bj�zgDV!uDT�mij]2~�^_g¢#b,lS`c~�r3Val2mpV!� lS`ÇgDbN~�^cm�}S`ki>o�be]S`cogDQS^km }2iju�g ^_]'gDQSV�}Su�bqbe��uDV!}SuDVUm�V!]9gDijg�^cbe]n�qrSlSg g�QS^km Q2i,m5]Sbjgr3V!VU]Ëmpg�lB~�^_V0~'oeV�g0¤

����� � : �%�����$� �����B:�� � �� � ��� �����I?!��� ��� �!: ���!:=<P5Q2V;}2u�bqbj��g�V�°qga^kmamxgDu�l2f�g�lSuDVU~¥~SijgDi¯g�Q2ijgafUij]¥rBV{T�ij]2^_}Sl2`cijg�VU~Ml2m�^_]2d¯gDQSV�T{bel2m�Ve¤;P5QS^cmafUij]¥rBVlBmpV0~Â� b,u�f!be]2mpg�uDl2f�g�^c]SdË}Su�bqbe�#f�b,T;T�ie]2~Sm��\m�^_]Bf�V{¢#V{«q]Sb�¢ÉQ2b�¢Ég�b©~�^km�}S`ci>o©}2u�bqbj�ImGgDQ2i�g�ieu�V;]Sbjgo,V�g5¦2]S^km�QSVU~Ç�Sg�QS^km ^c],gDV!u�ief�g�^cbe]�fUij]Ër3Val2mpV0~NgDb�f�beT{}S`cV�gDVa}SuDb9be�Im!¤

¬­]Âb,lSuV!°q}3V!uD^cT;VU]9gU�3¢#V�f!beT�r2^_]SV�g�V�°qgDl2ij`�}Su�bqbe� }2u�V0mpVU],g�i�gD^_b,]'¢ ^_g�Q²B³�´0´xµ�·BA�C�·k²�´j¸º¹B»O¸º¹9Á;iemh~�V��mDf�uD^crBV0~F^c] � ����¤{¬�g�^km�^c]2mxgDu�l2T;VU]9gDij` gDQ2i�ga`cbed,^cfUij`�� beuDT�lS`kiem\ij}2}BV0ijuD^_]Sd'^c]FgDQSV{g�V!°9g�ieu�V{~�^kmp}S`ki>oeV0~

g�gih%j6k�l�mRl

Page 13: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�0� �G¾�½�³�Àj¿W�����G½�³�»�´j»�� ´j»£»O¸I½�³����h¸��,½D¿E�

��^cdel2u�Va� � ( l�g�b,T{ijg�^kf!ie`_`coNdeVU]SV!u�i�gDVU~¯}Su�bqbe�zg�V!°qg¢ ^ÊgDQËie]ËVUs9l2ij`c^_gxo¯mpV0s9lSV!]2f!Ve¤

132 g4165

Page 14: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�F¿e» ¼���¾�¿j»O¸����2½�¶�½U»Ç²B³D½�����½ª¶�,¿�¹S¶� ��´�� �,�

i,mGmpg�uDl2f�g�lSuDVU~Â~2i�gDi�V�°Si,fªg�`coË^_]Fg�QSV{mDijT{V�T�ij]2]SV!uaiemhg�Q2V!oÂijuDV�¢ Q2V!]FlBmp^c]Sd¯}2`cie^_]Mdeb,ie`Ê�­~�^cu�V0fªgDVU~}2u�bqbj�x¤��MV5T�l2mxg@ie`cm�bg�ij«eV�^c],gDbaief!f!belS]9g�g�QSV��Iief�g g�Q2ijgvm�V!�,V!u�ij`9deb9ij`kmzT�i>oab�f!f!lSuvi�g�g�QSV mDijT{V�gD^_T{V^c]�gDQSV}SuDbqbj�Ðg�V!°9g0¤��@ief�Q¯m�V!]9gDV!]2f!Vbj�Ðg�Q2V� beuDT �ª¾�¿!Áe¸º¹3½�¿h²B³D´U´pµ\´pµ ������T�l2mxg#g�QSVU]Nr3VGij]2]Sbjg�i�g�V0~¢ ^_g�QÂg�QSV�u�VU`_VU�>ie]9g^_]�� b,u�T�ijg�^cbe]Ëg�b¯^_]2~S^cfUi�g�V�¢ QS^kf�Q©deb,ie`Çg�Q2V!o©f�b,u�uDVUm�}Bb,]2~¯g�b2¤ h�vf!belSu�m�Ve�2}2u�bqbj�º�rqo9��}Bb,^_]9g�^c]Sd;^cm5}3b,mDm�^_rS`cV\]Sbjg be]2`_o�rqoN}3be^c]9g�^c]Sd�i�gmpl2r��£V!°�}Su�V0m�m�^cbe]2m�bj��d,b,ie`Ðf!be]2f!`_lBmp^cbe]2mU�qrSl�gij`km�brqo'}Bb,^_]9gD^_]Sd�i�g\m�lSr���V�°�}SuDVUmDm�^_b,]2m5bj�@iemDm�lST{}�g�^cbe]2mU�SgDQ2i�g\T�i>o�b�f!f!lSuGie]qo9¢ Q2V!uDVa^_]ËgDQSV�gDV�°qgU¤hP5Q2^cm^c]q�eb,`_�,VUmnm�V!�,V!u�ij`e}2u�b,rS`_VUT�m!�>¢ QS^kf�Q�¢�V�Q2i>�,V@]Sbeg�ij`c`�mpb,`_�,VU~Ǥ �Sbeu�^c]2mxg�ij]2f!Ve��^c]�uDV!del2`cieuzdeb9ij`9~�^_uDVUf�g�V0~}2u�bqbj�x�>}SuDbqbj�º��rqo,��}3be^c],gD^_]2dhij`cdeb,u�^_g�T§mpb,T{V�g�^cT{VUm ���c½D¿j³�¶nQqo9}3bjgDQSVUm�VUmzg�bGu�V0~�l2f!V�gDQSV#m�^_¨UV#be�2~�^kmp}S`ki>oeV0~d,b,ie`cmU¤ �ÈQSV!]'g�Q2V�¢ QSbe`cV�}SuDb9be�zgDV�°qgh^km ~�^cm�}S`ki>oeV0~Ç�qg�QS^km T�ij«,VUm5`c^Êg�g�`cV�m�V!]2m�Ve�Sm�^_]Bf�VaQqoq}Bbeg�QSV0mpV0m ijuDVb,]S`co©T;VU]9g�^cbe]SV0~MijgGgDQSV{`_b�f!ijg�^cbe]M¢ QSV!uDV;g�QSVUoFijuDV�^c]9g�uDbq~Sl2f�V0~F^c]FgDQSV�~�^km�f!belSu�mpV �huDV!T{b��q^_]2dNg�Q2^cmT{VU],gD^_b,]Âbj��ie]ÂQqoq}Bbeg�QSV0mp^kmhT�i>oËuDV!]2~�VUugDQSV�}2u�bqbj�º�£g�V!°qg\f�b,T{}S`_V!g�VU`_oËlS]SuDVUi,~Sijr2`_V,¤GP5QS^kmGm�lSd,deVUmpgDmgDQ2i�gvgDQSV}SuDbqbj�º��rqo,��}3be^c],gD^_]2d\ie`_d,beuD^ÊgDQST�m�QSbelS`k~�rBVhf!be]�¦BdelSuDVU~;~�^I�3VUu�VU],gD`_o;~�V!}3V!]B~�^_]2dabe];g�QSVmpgxoq`_Vbe��~�^km�}S`ci>o,¤

� � ����� ���¯�������

P5Q2V�^k~�V0iGbe�Bb,u�d9ij]S^c¨!^c]SdgDQSV5^_]9gDV!u��Iief�V�� b,u iGgDQSV!b,u�VUT�}Su�b��,V!u ijuDbelS]B~ag�Q2VmxgDu�l2f�g�lSuDV#g�Q2ijg�~�VUmDf�uD^_r3VUmgDQSV;}Su�bqbe�@^cm\]Sbeg\]SVU¢a¤aP5QS^cm�mpg�uDl2fªgDlSuDV�T�i>oËrBV{i¯]Bi�g�l2uDie`�~�V0~�l2fªgD^_b,]ÂgDu�VUVe�niemG^c] �,ij}3V � $����3b,u\ig�ief�g�^kf;g�uDV!V�iem�^c]®be`k~�V!uaV!°q}3V!uD^cT;VU]9gDm�rqoMP�¤�Y\uD^ #{]&� � ��¤)hg�QSVUu�mpo�mpg�VUT{m�rBiem�VU~Mbe]¥gxoq}BV{gDQSV!b,u�oie`cm�b�m�^_T{}S`co;~S^cm�}S`ki>o�g�Q2Vgxoq}BV!�OgDQSV!b,u�V!g�^kf ���£g�VUu�T�g�QBi�g#~�V0m�f!u�^cr3VUmvgDQSV}SuDbqbj�@�c�0W���¤vP�oq}3V��£g�QSVUbeuDo�f!ie]VU�eVU]¥r3VNl2m�VU~¥gDbÂ`cVUie~®f!beT{T�ij]2~®}2ieuDm�^c]Sd©ij]2~±~�^km�}S`ci>oq^c]SdË� beu�g�QSV¯m�lSr2m�V�g�bj� ]2i�gDlSu�ij`@`kij]Sd,l2ijd,VlBmpV0~{^_]N}2u�bqbj�ImU�,i,m@^c] � �%��¤�¬­]¯belSu�fUiem�Ve�,g�QSV\~Si�g�i��­mxgDu�lBfªg�l2u�V l2m�VU~�i,m�iarBiem�^cm@g�b�}2u�b�~�l2f!V g�Q2Vh}SuDbqbj�}2u�V0mpVU],g�i�gD^_b,];^km�f!`_b9mpVUu g�b�iagDief�g�^kf5g�uDV!V,¤ Hqg�^c`_`£�q¢#Vie`cm�barBVU]SV�¦2g�� uDbeT ^_]2m�^cdeQ9gDm@f!beT{^c]Sda� uDbeT�V0ijuD`_^cV!u¢�beuD«;^_]�belSu#g�V0ijT � ���Ç¢ QSV!uDVgDQSV\}Su�^cT{^ÊgD^_�,V\~2i�gDij��mpg�uDl2f�g�lSuDV¢5iem�~�^_uDVUf�g�`co;g�QSV ���£g�VUu�Tˤ@P5QS^km#¢#b,u�«ie`cm�b�m�QSb�¢�VU~Ng�Q2ijg5^_g ¢�i,m�u�VU`_VU�>ie]9g�g�b{l2m�V�^_]S� beuDT{ijg�^cbe]¯� u�b,T g�QSV\gDi,fªgD^cfGg�uDV!VGg�b;deV!g T;b,u�V�f�b,]2f�^km�V}2u�bqbj��}SuDVUm�V!]9gDijg�^cbe]2mU¤

¬­]±b,lSu�fUiem�Ve�ngDQSV¯f!be]2f!^cm�V!]SV0m�m�u�VU`_^cVUmab,]®g�QSV�~�^cmpg�^c]2f�g�^cbe]±r3V�gx¢�V!V!]*f!beT{}3b,m�^ÊgDV¯ij]2~¥}SuD^cT;^_g�^c�eVg�ief�g�^kf!m@ie]2~�g�QSVhm�}3VUf�^_¦Bf5gDu�V0i�gDV!T{V!]9g@de^c�eVU]�gDb�}SuDbqbj�Im bj�ÇV0s,lBij`c^ÊgD^_V0m r9o;uDV!¢ uD^ÊgD^_]SdB¤ ¡#VUupg�ij^c]S`_o�T{beuDV¢�beuD«�m�QSb,lS`c~Ër3V{~�be]SV�� beuh}2u�bqbj�Img�Q2ijgGl2m�V�uDV!¢ uD^ÊgD^_]2d,m���i�uDV!¢ uD^_g�^c]Sd¯mpg�V!}FT{i>oËf�Q2ie]SdeV�i¯T;^c]ql�g�V~SV�gDie^_`�^c]Mi�`cieu�d,V�deb9ij`£�3ie]2~©^_g\^km�f�bel2],gDV!u��£}2u�b�~�l2f�g�^c�eVagDb¯uDV���}SuDbq~Sl2f�VagDQSV{¢ QSbe`cV�d,b,ie`O¤ �Sb,uG]Sb�¢a�¢�V�Q2i>�eV\]Sbeg^_T{}S`cV!T{V!]9gDVU~Ëij]qo¯m�be`cl�g�^cbe]�g�b;gDQS^cm}Su�b,rS`cV!Tˤ

� ��� : � � ����>=<��� �A: ��# ��¥V ¢�belS`k~;`_^c«eV5gDb�g�Q2ie]S«�t iemDf!ie`SynV0s,lBij]Sda¢ QSb�¢�beuD«eV0~�b,]{i�}2u�VU�9^cbelBm �eV!u�m�^_b,]�bj�z�\�3���@�eyzielSu�VU]9gÅGiemDf�b���g0� 0�uDlS]Sb�¡#be]B~�l2fªgD^_VUu�ij]B~�yzijl2u�VU],g�P5Q2[!uDo�¢ QSb�~�VU�eVU`_b,}BV0~�gDQSVh`ki>oeb,l�gv`_^crSu�ijuDo ��^cdelSV,�,ij]B~ÅGij]2ie]SV �hi,f�^cu�^S¢ Q2baV�°qg�VU]2~�VU~�gDQS^cm@`_^crSu�ijuDoag�b�i�`kijuDdeV�]qlST�r3V!u@be�3T�ijg�QSVUT{ijg�^kf!ie`Sbe}3V!u�i�gDbeu�m!¤zP5Q2^cm¢�beuD«�¢5iem5mpl2}S}Bb,upgDVU~'rqo �\iemDm�ielS`Êg�� ( �q^cijg�^cbe]�lS]2~�VUuf�be`c`kijr3beu�i�g�^cbe]ËY\[!]S^cV�ij]B~¯rqoN¬x�G| ¬ ( ¤

g�gih%j6k�l�mRl

Page 15: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

�>W �G¾�½�³�Àj¿W�����G½�³�»�´j»�� ´j»£»O¸I½�³����h¸��,½D¿E�

� � ��� ����¯��� �

������������ � ����������������������! "�# ��� ��!$����&%&�! ��'�& ��'�(�)�&%&� "�&*+��,�,������� �$'��-.�&/0��132��!$�4��657�650$� �8�$�,������#/9�65"�!$��)� "��:;�7�!�'4��6 "��:;�& ���<(=(�' "��>?�� �$��(�)�@ "�'>?�'�!,����A��B7����'CD8!�E�'�'�!FG�"�IHJ$K ��!$� )�L�<&$. ��6MN>O ��(��PQB"�������R�����6�S $�PT$� �$��#UK$V:W�'�6�#��,6XZY[$K ������'�GY0\�����!L]$)U�<!�&�^U��', S $�_��' ��a`@b�`�c6�(dfe S dfB7�IB�!g��&�R�.�)h'h@i6�& �$�j6����$)F9j@$K ������'�F���R�� ���8&�&��$)F�kV�l��<m ��6M#�

� bI�on[j@$��G��$� ������)�pL�<&$m [�K ��6Mq�R=�R��$�1WrtsZ$�����g��p�'�#Fu�� �U�<!�l��$)U����& �$��pv]w'x�y3z'{�|0}D~&������}�wR���aw�y0~#������(�'��'��r b�b'�)X6b��@c6�?�)h'h�h�

� c��on[j@$��N�[$� ����'�)�a/0��,�,�$��7�N�'<!���?���#FJ�A�'�& �$��6�7L�<&2� �=(�t>? ��(�'P8(=�_��'���6�����&g&�tdf�����!��w'x������D��z�{O|Z}D~&������}twR��aw�y0~#����Kx���wf�����[z'x��K�!j@��,��!13$�i'��ho��Pa�+�������6x��N��w����K}Z���q�aw�y0~#����KxN�������K�#���K�#_#��g�$������&�KX!�)\'`�?�)h'h'��

� �'�on[j@$��Z��$� ����'�9���#F��A�'� �$��(�7L�<!2� �=(��B�g�$��&$� ���U��'_!_ ��@�'U�<����W8!�&��,�F���&g;�&��$� 7���(��$� �P���U�$��NPT�' 7��<!$��� �$�1_ ��Ij@$� ����V�6w'�6x���z�{]wR�Z�!��y3��w�{ �D���aw'y0~#��Dz����Ew'�!��b��6r���\�KX!�)h��&�O�)h'h���

� �I� S ��U�<!�� �F����' ��!���9�'�!F���$� ��!�� �Fq�6�&P^ �������%@��_�$�r9�;,�����$K ��',E�?,���g�<(�fk�$���g�<(�)�?���(��$� ���UK����j@$3_ ��(�'PV���������R���'�(�)�LA$)U�<&�!��U���, S $�_��' ��\��&�'�Q�0�!$�$���:W�I �=W���#FtH�$��R�� #$�,^FW ���,�,�$�g�$��#¡0�!��j@$K ������¢=��'Pa�����#F��������h�h��&�

� \���>?�I�� ���U�£G���' � ������6sZ��13���!��M(�!$ �,�2�13$��6�)�@L�<!��$� � �=GsZ$���_�$�=6 ����¤Q�6%@�'�&$��[df�#UK$� �_!�E�/0��,�,�$����N��<!�A�(��$� ��#�I �F�A�'�&g&�'�'�!FNY[�',�2� ���$�>?�'��U��!�',E� �$��(���'� )rA��<&$[�R=�R��$�1W�'df�����6�Tx�¥0�!��y0~w�}��T�y¦w'�o��wf�����[z'x��[§9�K¨'��{�wK~!y��K�#�© �!¨)�Tx�w'�&y��K�#��}��A�)h��'��"ªDB",����3��_!_�$)�I ��"���"d¢e S dRB S $�_��' ���!���i'i�i�«��

� iI�on9�[ �����U��)=(�./9�[�N��<!���.�'�!F¬�A�'� �$��(��L�<!2� �=(�u-?¤6�� ���UK�����&g���$�¤6�oP^ ���1­_ ��(�'P����pdf�p:��asZ$�4)���!�V�'�!F/9�&>O,��'��£6���A�&$)F&�l���' ����Q®�xRw)������¥'�T�6�'}7wR���^�!�����#���Kx���z����Ew'��z�{]��w'�)�)�Kx��K�����;�Q�.�]|¯w'�m����~&��¥�{�z'yo��¥�z�°f��z�{����{ �z���¥tzK~@~�{ �D��z����Ew'�!}�± © ¥'�Ty3��w'�6xD�'�&j@��,��&13$�h�`�bG�'P��'~#xK�T�6�(�KxK°�²!�KxK{�z)�3�+�G�?�!�!B"_ ���,?��h�h@�(�

� ����L���13����<6=;/" ��l³3���0�Nw��Dz'���Dw'��z'{?¥(��´a�&�E���Dw'��z'�#¥µ�DwK~�°¢¥@w'�O��x���´a���Ky��K�!�Q��w'xN���#���KxRz@������¨'�"~#xRw)wR�G¥@�K¨'��{�wK~�°y3�K�#�a}���}����KyG}���>a<&s���<!$��������# ��' ��&$�,�,]¡0�&��j@$� ����l�f=(�Q�)h'���6�

� h���L�<&��1����[50�',�,�g' �$������#FoB"�� ��!$ S �'�(�����B"�o$�¤6��$��&����8!,�$._& ��(��P���$K¤�a$)F�l���' )�Qdf�3�]w��'�D�+��w�x.®�x�w��'xRz'yGy9���6�z���¥G|�&�Dw�y3z�����¥3¶V��z'}Kw'�!�T�6�'�#j���,��!13$7�)h@�'�o�'Pa�]�o�?�&�!_!�'g�$��0i�`IX(�'�&�Q�_& ����&g�$� Ya$� �,��'g&�#b'`'`�`6�

����`��3s0���#��,�F;-.�#���6�&��<������!�t� ©6· ��w)wI¸���BVF!F�����'�&¹DHJ$���,�$K=(���)h�h'c����'����>O��$� � �$Z��$������&4�$�=o�'�#F��A�'�& �$��6��L�<!2� �=(��*&�' �1��',���4����!gG�6��º�,�1��� �U�£Q» �[�',�g��� ��l��<!1¼���µ ��6M#�Ad¢��%&�65Z�I � ��������

���#F�:��AB0��g@���I �F��A$)F&�l���� ����Z���!��w'x��Ky½®�x�w'¨)���(�;����¾0���I�!�Kx;¿�x�¥(�Kx��]w��'�D��}�À�Á'Â@�^�mÃ��#���Kx���z����Ew'��z�{.�aw'�!°���Kx��K�����K±o��®[¾G¿a�O}GÄÅ@Å@Å��Oj���,��!13$;����\'hm��P"�+�������6x�����w����K}3���Æ�aw'y"~#����Kxt�#�����K�����K�O_!�'g�$��Gc'�@i)X@�@`��&��6_& ����!g'$� �¹�Y[$K �,^��g&�#b'`'`�`�

���)bI����$��#�9:W�'g'�6�!�������µ�'�!Fµ�[$��!g��.e0�� �F&�R�� �Ç�1W�AL�<&$ZB0�A*�_& ��(��P+$)F�l���' V���#F3�l����_& ��(��P+$��&g����!$��]df�È�&��~&�K}��w'xW®�xRw)wR�K}�z'�#¥É®�xRw��'x�z'y9}���j@��,��&13$��'`�\É�'Po�+�������6x����Nw'���K}u�aw�y0~#����Kx��������K�#�K�K�Z_#�'g'$��mb(�)cIX6b'c�i6�e"� ÊR13$�g'$��A�Q�)h'h'����_ ����&g�$� �¹�Ya$� �,��'g�

����c��357��e0��U��l ������#F3�O� S �^F$)�'���QB³3U�<!�'g�$V���(��$� ���U�����P¢�68!��F&��13$��&�������!�&$�,$������!UK �2�13$��(���',!F&$�PT�' �1G�&,�$��[1��I��<&¹2�1�������M(�!$����?d¢�¬�]|¶VÃ&Ë ÄÅ@Å@ÅI±?|�!�Dz'��z�xK��¨�wmÌ�Í�z@¥@z)�@z�}���z'x¢Î)�Ab'`'`�`�a����*& �$��#U�<A�

�����'���A�'� �$��(�?L�<!2K �=(�(BÉU�$� ����l #$�F7j@$K ������'�9��P#�[�#U�<8�$K �g�$� )» �?��,�g'�' ��l��<!1W�@df�G|��Dw'y3z�����¥0§9��¥'�&�����Dw'��Ì��]|§ © °Á'Ï�Î)�!j@�',��!13$G���(b6�N��Pa�+�������6x��N��w����K}����;|xK��� ´����Ez�{�Ã��!����{�{ ���(�K�#�K�K���_ ����&g�$� �¹�Ya$� �,��'g�!%'�!,l=���h�h'��

���)�I���A�'� �$��(�tL�<!2� �=Q��n[j@$�����$K ����'�)��'�!FÐ/0��,�,�$��;�N��<!�A� S $)��,0L�<&$��' �$�1È>? ��Ij@$� ���sZ$���$K �j@$ S $)�',¡0��$� �¹d¢�(��$� �PE�'U�$������#wR�����[z'x�� © �6�'�������Kx����(���Nw'���K}��.�)iª���«��.�)h�h�b6�m>? ��6U�$�$�F&���!g'�G�'P.��<&$t�I��<É�(=13_��������!1Ñ�'��6�'P^�fk.�I �$�sZ$�j@$�,��'_!13$��(�V-O�6j6�� ��'�!13$��(�����

132 g4165

Page 16: Mathematics and Proof Presentation in Pcoq · Mathematics and Proof Presentation in Pcoq Ahmed Amerkad, Yves Bertot, Loïc Pottier, Laurence Rideau N° 4313 Novembre 2001. Unité

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)� � ����� ��� � � ��� � '� � �� � �

ISSN 0249-6399


Top Related