&KDSLWUH
/DGpGXFWLRQQDWXUHOOH
$SUqVDYRLUGRQQpOHVGp¿QLWLRQVGXODQJDJHL+HWGHVQRWLRQVFOHIVGHODVpPDQWLTXHGHVSUpGLFDWV QRXVpWXGLHURQVGDQVFHWWHOHoRQGHX[FDOFXOVV\QWD[LTXHVSRXUSURXYHUGHVWKpRUqPHVGHODORJLTXH GHVSUpGLFDWV /HSUHPLHUFDOFXOHVWXQFDOFXOD[LRPDWLTXH TXLFRQVLVWHHQTXHOTXHVD[LRPHVHWGHX[
UqJOHVG¶LQIpUHQFHVjO¶DLGHGHVTXHOOHRQHQGpGXLWGHVWKpRUqPHV /DGHX[LqPHPpWKRGHV\QWD[LTXH HVWXQHH[WHQVLRQGHODPpWKRGHGHVDUEUHVSRXUODORJLTXHSURSRVLWLRQQHOOH DYHFTXDWUHUqJOHVVXSĥ SOpPHQWDLUHVSRXUOHVSKUDVHVTXDQWL¿pHVHWOHXUVQpJDWLRQV
/HVLQGLYLGXVDUELWUDLUHV
3RXUSRXYRLUWUDLWHUGHVSKUDVHVTXDQWL¿pHVSDUOHVPpWKRGHVGHODGpGXFWLRQQDWXUHOOH QRXVDYRQV EHVRLQGHUqJOHVG¶LQWURGXFWLRQHWG¶pOLPLQDWLRQSRXUOHVTXDQWL¿FDWHXUV
&RPPHQRXVO¶DYRQVIDLWSRXUODPpWKRGHGHVDUEUHV QRXVGHYRQVSUHQGUHHQFRQVLGpUDWLRQOHIDLW TX¶XQHTXDQWL¿FDWLRQXQLYHUVHOOHVXUXQGRPDLQH¿QLHVWpTXLYDOHQWHjODFRQMRQFWLRQGHWRXWHVVHV LQVWDQFLDWLRQV HWTX¶XQHTXDQWL¿FDWLRQH[LVWHQWLHOOHVXUXQWHOGRPDLQH¿QLFRUUHVSRQGjXQHGLVĥ MRQFWLRQGHVHVLQVWDQFLDWLRQV eOLPLQHUXQTXDQWL¿FDWHXUXQLYHUVHOUHYLHQWGRQFjO¶LQVWDQFLHUSRXU WRXVOHVPHPEUHVGXGRPDLQH &RQVLGpURQVO¶LQIpUHQFHVXLYDQWH
Īī 7RXVOHVKRPPHVVRQWPRUWHOV 6RFUDWHHVWXQKRPPH
6RFUDWHHVWPRUWHO
/¶LQIpUHQFHYDOLGH Īī FRUUHVSRQGUDjFHWWHSUHXYHĥFLGDQVOHFDOFXOTXHQRXVGpYHORSSHURQVGDQVFHWWH OHoRQ
Ha,∀x(Hx→M x) # Ha SUpPLVVH
Ha,∀x(Hx→M x) # ∀x(Hx→M x) SUpPLVVH
Ha,∀x(Hx→M x) # Ha→M a GHĪīSDUĪalī
Ha,∀x(Hx→M x) # M a GHĪīHWĪīDYHFĪJSī
/DUqJOHĪalī DSSHOpH©VSpFLDOLVDWLRQ XQLYHUVHʃ˳ª QRXVSHUPHWGHSDVVHUG¶XQHTXDQWL¿FDWLRQXQLĥ YHUVHOOHjQ¶LPSRUWHTXHOOHLQVWDQFLDWLRQGHODSKUDVHRXYHUWHXQLYHUVHOOHPHQWTXDQWL¿pHSRXUXQH FRQVWDQWH /¶pOLPLQDWLRQGXTXDQWL¿FDWHXUXQLYHUVHOUHYLHQWGRQFjXQHLQVWDQFLDWLRQGHODSKUDVH RXYHUWHTX¶LOJRXYHUQH UHPSODoDQWWRXWHVOHVRFFXUUHQFHVOLEUHVG¶XQHYDULDEOHSDUGHVRFFXUUHQFHV G¶XQHFRQVWDQWH
&RPPHQWSRXYRQVQRXVLQWURGXLU˳XQTXDQWL¿FDWHXUXQLYHUVHOGDQVXQHIRUPXOH " /¶DQDORJLHDYHFOD
FRQMRQFWLRQSRXUUDLWQRXVIDLUHSHQVHUTX¶LOVXɷUDLW SRXUpWDEOLUODYpULWpGH©∀x(F x)ª SDUH[HPSOH GHSURXYHU©F(a1)ª ©F(a2)ªHW©F(a3)ª VLQRXVQRXVWURXYRQVGDQVXQHVWUXFWXUH¿QLHQHFRQWHQDQW TXHOHVWURLVREMHWVa1a2HWa3GDQVVRQGRPDLQH &HWWHPpWKRGH FHSHQGDQW QHV¶DYqUHDXFXQHPHQW YDOLGH 1RQVHXOHPHQWHOOHVHUDLWLQDSSOLFDEOHGDQVOHFDVG¶XQGRPDLQHLQ¿QLHWOHVHUDLWDXVVLGDQV OHFDVRQRXVQHGLVSRVRQVSDVGHQRPVSRXUWRXVOHVREMHWVGDQVOHGRPDLQH 6RQDSSOLFDWLRQVLJQLĥ
¿HUDLWpJDOHPHQWXWLOLVHU jO¶LQWpULHXUG¶XQHSUHXYH XQHLQIRUPDWLRQTXLQHQRXVHVWGRQQpHTXHGH O¶H[WpULHXUGXPRGqOH PrPHVLQRXVDUULYLRQVµDFFLGHQWHOOHPHQW¶jSURXYHU©F xªGHWRXVOHVPHPEUHV GXGRPDLQHĦHWDLQVLjSURXYHU SRXUWRXWHFRQVWDQWH©aª TXHF aĦ QRXVQ¶DXULRQVSDVHQFRUHSURXYp TXH ©F xªHVWYUDLGHWRXVOHVPHPEUHVGXGRPDLQHVĦSRXUFHOD QRXVDXULRQVEHVRLQG¶XQHJDUDQWLHTXH ODWRWDOLWpGHVLQGLYLGXVGRQWQRXVDYRQVSURXYp©F xªFRPSUHQGUpHOOHPHQWWRXVOHVLQGLYLGXVGX GRPDLQH
8QHSLVWHSOXVSURPHWWHXVHQRXVHVWGpVLJQpHSDUOHVSUHXYHVPDWKpPDWLTXHVHWSDUO¶LQWURGXFWLRQGH QRXYHOOHVFRQVWDQWHVGDQVODPpWKRGHGHVDUEUHV 6LXQJpRPqWUH SDUH[HPSOH YHXWSURXYHUTXHOD VRPPHGHVDQJOHVLQWpULHXUVGHQ¶LPSRUWHTXHOWULDQJOHHVWpJDOHjGHJUpV LOGHVVLQHUDDXWDEOHDX QRLUXQWULDQJOHSDUWLFXOLHU QRPPpABCG¶DSUqVVHVWURLVDQJOHV 6¶LOQHIDLWXVDJH GDQVVDSUHXYH GXWKpRUqPH G¶DXFXQHSURSULpWpGHFHWULDQJOHDXWUHTXHFHOOHVTXLOXLVRQWLPSRVpHVSDUODGp¿QLĥ WLRQPrPHG¶XQWULDQJOH OHWKpRUqPHYDXGUDSRXUWRXWWULDQJOH /HWULDQJOHSDUWLFXOLHUABCDXUD UHSUpVHQWpWRXVOHVWULDQJOHV 'DQVFHVHQV OHWULDQJOHABCSHXWrWUHDSSHOp© WULDQJOHDUELWUDLUH ª 1RXVVRPPHVSDVVpHVSDUXQHpWDSHDQDORJXHGDQVOHGpYHORSSHPHQWGHODPpWKRGHGHVDUEUHV QRXV DYRQVPRQWUpTX¶LOpWDLWSRVVLEOHGHVLPSOL¿HUOHVUqJOHVSRXUOHTXDQWL¿FDWHXUH[LVWHQWLHOHWODQpJDĥ WLRQGXTXDQWL¿FDWHXUXQLYHUVHO HQQRXVOLPLWDQWjXQHVHXOHFRQVWDQWH QRXYHOOH TXLUHSUpVHQWDLW pJDOHPHQWOHVDQFLHQQHVFRQVWDQWHVGpMjLQWURGXLWHVVXUODEUDQFKHFRUUHVSRQGDQWH $XOLHXGHGLUH TXHODFRQVWDQWHQRXYHOOHUHSUpVHQWDLWOHVLQGLYLGXVTXHQRXVQ¶DYLRQVSDVHQFRUHFRQVLGpUpVGDQV QRWUHSUHXYH QRXVDYRQVVLPSOHPHQWGLWTX¶HOOHUHSUpVHQWDLW Q·LPSRUWH TXHOLQGLYLGXGXGRPDLQH 1RXVQRXVDYRQVVHUYLGHODFRQVWDQWHSRXUGpVLJQHUXQLQGLYLGXDUELWUDLUH
3RXUSURXYHUXQHTXDQWL¿FDWLRQXQLYHUVHOOHjSDUWLUGHVSUpPLVVHVSDUWLFXOLqUHV QRXVH[LJHURQVGRQF TXHFHVSUpPLVVHVVRLHQWYUDLHVG¶XQLQGLYLGXDUELWUDLUH8QWHOLQGLYLGXQRXVHVWIRXUQLSDUH[HPSOH SDUODUqJOHGHVSpFLDOLVDWLRQXQLYHUVHOOH FRPPHF¶HVWOHFDVGDQVODSUHXYHVXLYDQWH
∀x(F x),∀x(F x→Gx) # ∀x(F x) SUpPLVVH ∀x(F x),∀x(F x→Gx) # ∀x(F x→Gx) SUpPLVVH
∀x(F x),∀x(F x→Gx) # F a GHĪīSDUĪalī
∀x(F x),∀x(F x→Gx) # F a→Ga GHĪīSDUĪalī
∀x(F x),∀x(F x→Gx) # Ga GHĪīHWĪīDYHFĪJSī
∀x(F x),∀x(F x→Gx) # ∀x(Gx) GHĪīDYHFĪ:lī 1RXVDSSHOOHURQVFHWWHUqJOH TXLQRXVSHUPHWG¶pWDEOLUXQHSKUDVHJpQpUDOHjSDUWLUG¶XQHSKUDVH VLQJXOLqUHSDUODQWG¶XQLQGLYLGXDUELWUDLUH© JpQpUDOLVDWLRQ XQLYHUVHʃ˳ ªĪ:lī
&HSHQGDQW LOHVWFODLUTXHQRXVGHYRQVUHVWUHLQGUHQRWUHXVDJHGHĪ:lī /HUDLVRQQHPHQWVXLYDQWHVW FODLUHPHQWIDOODFLHX[
F a # F a SUpPLVVH
F a # ∀x(F x) GHĪīSDUĪ:lī
1RXVQHSRXYRQVSDV GXIDLWTX¶XQFHUWDLQLQGLYLGXaVRLWF FRQFOXUHTXHWRXWHVOHVFKRVHVGDQV O¶XQLYHUVGHGLVFRXUVVRLHQWF ĦOHaHQTXHVWLRQGRLWrWUHDUELWUDLUH 0DLVTX¶HVWĥFHTXHFHODVLJQL¿H FRQFUqWHPHQW " 'DQVOHFDVGHODPpWKRGHGHVDUEUHV FHFLVLJQL¿DLWTXHODFRQVWDQWHpWDLW© QRXYHOOH ª
-HSUpIqUHFHWWHSUpVHQWDWLRQjFHOOHGH/HPPRQĪD īTXLSDUOHGHQRPVDUELWUDLUHVSRXUGHVUDLVRQVHVTXLVVpHVGDQV ODOHoRQWURLV 5()(5(1&( :+(5( (;$&7/< "3RXUXQHWKpRULHGpYHORSSpHGHVLQGLYLGXVDUELWUDLUHV YRLU)LQHĪī HW SRXUXQHUHYXHFULWLTXHGHGLɱpUHQWHVWKpRULHV1HIĪī
/HVLQGLYLGXVDUELWUDLUHV SRXUODEUDQFKH F¶HVWĥjĥGLUHQ¶DYDLWSDVG¶RFFXUUHQFHSUpFpGHQWH 'DQVOHFDVGXJpRPqWUHTXLSURXYH GHVWKpRUqPHVVXUWRXVOHVWULDQJOHV FHFLVLJQL¿HTXHODSUHXYHHQTXHVWLRQQHGpSHQGG¶DXFXQHDVĥ VRPSWLRQSDUWLFXOLqUHVXUOHWULDQJOHpWDQWFRQVLGpUpSDUDGLJPDWLTXHRXDUELWUDLUH 'DQVODGpGXFWLRQ QDWXUHOOH QRXVFRPELQRQVFHVGHX[H[LJHQFHV ODFRQVWDQWHHQTXHVWLRQ ©aª QHGRLWDSSDUDvWUHGDQV DXFXQHVXSSRVLWLRQRXSUpPLVVHGRQWGpSHQGODSUHXYHGHODSKUDVHVLQJXOLqUH©F aª
'HPrPHTXHOHVUqJOHVSRXUO¶LQWURGXFWLRQHWO¶pOLPLQDWLRQGHODGLVMRQFWLRQVRQWSOXVFRPSOH[HV TXHFHOOHVSRXUODFRQMRQFWLRQ GHPrPHOHVUqJOHVSRXUOHTXDQWL¿FDWHXUH[LVWHQWLHOVRQWXQSHXSOXV FRPSOLTXpHVTXHFHOOHVSRXUOHTXDQWL¿FDWHXUXQLYHUVHO 3RXULQWURGXLUHOHTXDQWL¿FDWHXUH[LVWHQWLHO QRXVJpQpUDOLVRQVXQHSKUDVHSDUWLFXOLqUH VL©F aªHVWSURXYpHSRXUXQFHUWDLQa DORUVQRXVSRXYRQV SURXYHU©∃x(F x)ª /DUqJOHG¶LQWURGXFWLRQGXTXDQWL¿FDWHXUH[LVWHQWLHOHVWGRQFDSSHOpH©JpQpUDĦ OLVDWLRQ H[LVWHQWLHʃ˳ªĪ:1ī (QYRLFLXQH[HPSOH
∀x(F x) # ∀x(F x) SUpPLVVH
∀x(F x) # F a GHĪīSDUĪalī
∀x(F x) # ∃x(F x) GHĪīSDUĪ:1ī
0rPHVLODFRQVWDQWHLQGLYLGXHOOH©aªTXHQRXVLQWURGXLVRQVGDQVO¶DSSOLFDWLRQGHĪalīGpVLJQHXQ LQGLYLGXDUELWUDLUH QRXVSRXYRQVWRXMRXUVFRQFOXUH GXIDLWTXHFHWLQGLYLGXDUELWUDLUHHVWF TX¶LO\
HQDDXPRLQVXQF Ħa ELHQTXHDUELWUDLUH H[LVWH
3RXUODUqJOHG¶pOLPLQDWLRQGXTXDQWL¿FDWHXUH[LVWHQWLHO QRXVQRXVUDSSHORQVGHODUqJOHĪ∨(ī FHWWH UqJOHpOLPLQDLWXQHGLVMRQFWLRQGDQVOHVHQVTX¶HOOHQRXVSHUPHWWDLWGHSURXYHUXQHIRUPXOHSURXYpH jSDUWLUGHVGHX[GLVMRLQWVGLUHFWHPHQWGHODGLVMRQFWLRQHOOHĥPrPH '¶XQHPDQLqUHDQDORJXH ODUqJOH G¶pOLPLQDWLRQGXTXDQWL¿FDWHXUH[LVWHQWLHOQRXVSHUPHWGHSDVVHUGHVSUHXYHVG¶XQHIRUPXOHφjSDUWLU GHWRXWHODVpULHGHVLQVWDQFLDWLRQV©F a1∨F a2∨. . .∨F an∨. . .ªjXQHSUHXYHGHφGLUHFWHPHQWj SDUWLUGH©∃x(F x)ª
$XOLHXGHPRQWUHUTXHφHVWXQHFRQVpTXHQFHGHWRXWHVOHVLQVWDQFLDWLRQVGHODTXDQWL¿FDWLRQXQLĥ YHUVHOOH LOVXɷWGHPRQWUHUTX¶HOOHV¶HQVXLWG¶XQHLQVWDQFLDWLRQTXHOFRQTX˳ĦG¶XQHLQVWDQFLDWLRQSDU XQLQGLYLGXDUELWUDLUH R© DUELWUDLUH ªYHXWGLUHODPrPHFKRVHTXHGDQVO¶DSSOLFDWLRQGHODUqJOH GHJpQpUDOLVDWLRQXQLYHUVHOOHĪĪ:lī O¶LQWURGXFWLRQGXTXDQWL¿FDWHXUXQLYHUVHOī 1RXVDSSHORQVFHWWH LQVWDQFLDWLRQOH© GLVMRLQW W\SLTX˳ ªTXLFRUUHVSRQGjODTXDQWL¿FDWLRQH[LVWHQWLHOOH 1RXVDSSHORQV
© Īa1ī ªRX© VSpFLDOLVDWLRQ H[LVWHQWLHʃ˳ ªODUqJOHTXLpOLPLQHODTXDQWL¿FDWLRQH[LVWHQWLHOOHHQIDYHXU GXGLVMRLQWW\SLTXH
/HVUqJOHVGHJpQpUDOLVDWLRQXQLYHUVHOOHĪ:līHWGHVSpFLDOLVDWLRQH[LVWHQWLHOOHĪa1īVRQWLQWLPHPHQW OLpHV QRQVHXOHPHQWHOOHVVRQWWRXWHVGHX[UHVWUHLQWHVSDUODFRQGLWLRQVHORQODTXHOOHO¶LQGLYLGXHQ TXHVWLRQHVWXQLQGLYLGXDUELWUDLUH PDLVGHSOXVODQRWLRQG¶DUELWUDLUHTX¶HOOHVXWLOLVHQWHVWODPrPH Ī:līHVWDSSOLFDEOHjXQHSKUDVHφVLHWVHXOHPHQWVLφDXUDLWSXrWUHREWHQXH SDUĪa1ī GHVDTXDQWL¿ĥ FDWLRQH[LVWHQWLHOOH /¶DSSOLFDWLRQGHODUqJOHGHVSpFLDOLVDWLRQH[LVWHQWLHOOHFRPSUHQGTXDWUHpWDSHV ODSUHXYHG¶XQHTXDQWL¿FDWLRQH[LVWHQWLHOOHVRXVFHUWDLQHVVXSSRVLWLRQVHWjSDUWLUGHFHUWDLQHV
SUpPLVVHV
ODVXSSRVLWLRQGXGLVMRLQWW\SLTXH
XQHSUHXYHG¶XQHDXWUHIRUPXOHVRXVODVXSSRVLWLRQGXGLVMRLQWW\SLTXH
O¶DSSOLFDWLRQGHODUqJOH D\DQWFRPPHFRQFOXVLRQFHWWHGHUQLqUHIRUPXOHĪGHO¶pWDSHĪīī HW FRPPHVXSSRVLWLRQVHWSUpPLVVHVFHOOHVGHODTXDQWL¿FDWLRQH[LVWHQWLHOOHĪGHO¶pWDSHĪīī
/DUDLVRQVpPDQWLTXHSRXUODYDOLGLWpGXVpTXHQW©∀x(F x)" ∃x(F x)ªHVWTXHQRXVDYRQVH[FOXOHVGRPDLQHVYLGHV GDQVQRWUHGp¿QLWLRQG¶XQHVWUXFWXUHSRXUODORJLTXHGHVSUpGLFDWV 1RXVDYRQVSDUOpG¶XQH© ORJLTXHOLEUH ªTXLQHIDLVDQWSDV FHWWHSUpVXSSRVLWLRQGDQVOHFKĪFI Sī
9RLFL FRPPHH[HPSOH XQHSUHXYHGXVpTXHQW©∀x(F x→Gx),∃x(F x)# ∃x(Gx)ª ∀x(F x→Gx),∃x(F x) # ∀x(F x→Gx) SUpPLVVH ∀x(F x→Gx),∃x(F x) # ∃x(F x) SUpPLVVH
∀x(F x→Gx),∃x(F x) F a #∗ F a VXSSRVLWLRQĪGXGLVMRLQWW\SLTXHī ∀x(F x→Gx),∃x(F x) F a #∗ F a→Ga GHĪīDYHFĪalī
∀x(F x→Gx),∃x(F x) F a #∗ Ga GHĪīHWĪīDYHFĪJSī ∀x(F x→Gx),∃x(F x) F a #∗ ∃x(Gx) GHĪīDYHFĪ:1ī
∀x(F x→Gx),∃x(F x) # ∃x(Gx) GHĪī ĪīHWĪīDYHFĪa1ī eWDQWGRQQpOHVSUpPLVVHVTXHWRXWFHTXLHVWF HVWpJDOHPHQWG HWTX¶LO\DTXHOTXHFKRVHTXL HVWF QRXVVXSSRVRQV jODOLJQHĪī TX¶XQLQGLYLGXDUELWUDLUHaHVWF 6RXVFHWWHVXSSRVLWLRQ QRXV GpPRQWURQVHQVXLWHTX¶LO\DTXHOTXHFKRVHTXLHVWGĪOLJQHī 'HFHVGHX[OLJQHVHWGHODOLJQHR
QRXVDYRQVSURXYpODTXDQWL¿FDWLRQH[LVWHQWLHOOH QRXVFRQFOXRQVTX¶LO\DTXHOTXHFKRVHTXLHVWG ĪOLJQHī 'DQVXQHDSSOLFDWLRQGHĪa1ī QRXVLQGLTXRQVWURLVOLJQHV FHOOHRQRXVDYRQVSURXYpOD TXDQWL¿FDWLRQH[LVWHQWLHOOH FHOOHRQRXVDYRQVIDLWODVXSSRVLWLRQGXGLVMRLQWW\SLTXHHWFHOOHR
QRXVDYRQVPRQWUpTXHODSKUDVHHQTXHVWLRQSHXWrWUHGpPRQWUpHVRXVFHWWHVXSSRVLWLRQ
&RPPHO¶LQGLTXHOHSDUDOOpOLVPHHQWUHĪa1īHWĪ:lī QRXVGHYRQVpJDOHPHQWDGRSWHUTXHOTXHVUHVWULFĥ WLRQVSRXUpYLWHUGHVUDLVRQQHPHQWVIDOODFLHX[FRPPHOHVXLYDQW
∃x(F x) # ∃x(F x) SUpPLVVH
∃x(F x) F a #∗ F a VXSSRVLWLRQ
∃x(F x) # F a GHĪī ĪīHWĪīDYHFĪa1ī
∃x(F x) # ∀x(F x) GHĪīDYHFĪ:lī
/¶DSSOLFDWLRQGHĪ:līHVWFRUUHFWH SXLVTXH©∃x(F x)ªQHFRQWLHQWSDV©aª 0DLVO¶DSSOLFDWLRQGHĪa1ī HVWLQFRUUHFWH SXLVTXHODFRQFOXVLRQSURXYpHVRXVODVXSSRVLWLRQGXGLVMRLQWW\SLTXHFRQWLHQWHOOHĥ PrPHXQHRFFXUUHQFHGHODFRQVWDQWHSRXUO¶LQGLYLGXDUELWUDLUHa 0rPHVL©F aªHVWXQHFRQVpTXHQFH G¶HOOHĥPrPH LOQHV¶HQVXLWSDVGH©∃x(F x)ªTX¶XQLQGLYLGX DUELWUDLUHPHQWFKRLVL HVWF 1RXVGHYRQV GRQFOLPLWHUOHVFRQFOXVLRQVREWHQXHVjSDUWLUGXGLVMRLQWW\SLTXHjGHVSKUDVHVTXLSRUWHQWVXUGHV LQGLYLGXVDXWUHVTXHFHOXLTXLQRXVDVHUYLSRXUO¶LQVWDQFLDWLRQ
&HWWHUHVWULFWLRQFRUUHVSRQGjO¶H[LJHQFHTXH GHWRXWHVOHVSKUDVHVTXHQRXVGpULYRQVGHODVXSSRVLĥ WLRQGXGLVMRLQWW\SLTXH VHXOHVFHOOHVTXLQHFRQFHUQHQWSDVO¶LQGLYLGXFKRLVLFRPPHDUELWUDLUHVRQW pJDOHPHQWGHVFRQVpTXHQFHVGHODTXDQWL¿FDWLRQH[LVWHQWLHOOH
&HWWHSUpFDXWLRQ PrPHVLHOOHHVWQpFHVVDLUH Q¶HVWSDVHQFRUHVXɷVDQWH 1RXVGHYRQVpJDOHPHQW VXSSRVHUTXH©aªQ¶DSDVG¶RFFXUUHQFHGDQVOHVVXSSRVLWLRQVVRXVOHVTXHOOHVHVWREWHQXHODFRQFOXVLRQ GpULYpHGXGLVMRLQWW\SLTXH &HFLHVWPRQWUpSDUOHUDLVRQQHPHQWIDOODFLHX[VXLYDQW
∃x(F x), Ga # ∃x(F x) SUpPLVVH
∃x(F x), Ga # Ga SUpPLVVH
∃x(F x), Ga F a #∗ F a VXSSRVLWLRQ
∃x(F x), Ga F a #∗ F a∧Ga GHĪīHWĪīDYHFĪ∧,ī ∃x(F x), Ga F a #∗ ∃x(F x∧Gx) GHĪīDYHFĪ:1ī
∃x(F x), Ga # ∃x(F x∧Gx) GHĪī ĪīHWĪīDYHFĪa1ī
&HUDLVRQQHPHQWHVWIDOODFLHX[ĦQRXVSURFpGRQVGHGHX[SUpPLVVHVSRUWDQWVXUGHX[LQGLYLGXVĦ TX¶XQLQGLYLGXDUELWUDLUHHVWF HWTXHTXHOTXHFKRVHHVWGĦjODFRQFOXVLRQTX¶LO\DTXHOTXHFKRVH TXLHVWHQPrPHWHPSVF HWG /HSUREOqPHQ¶HVWSDVTXHODFRQFOXVLRQREWHQXH jODOLJQHĪī GHOD VXSSRVLWLRQGXGLVMRLQWW\SLTXH FRQWLHQW©aª /HFDUDFWqUHIDOODFLHX[GXUDLVRQQHPHQWHVWSOXW{WG
DXIDLWTX¶HOOHUHVWHVXUXQHVXSSRVLWLRQ jVDYRLU©Gaª DXWUHTXHOHGLVMRLQWW\SLTXH FRQWHQDQW©aª
©∃x(F x∧Gx)ªDpWpGpULYpHHQIDLVDQWXQHDVVRPSWLRQVXUO¶LQGLYLGXDUELWUDLUHHQTXHVWLRQĦTX¶LO
/HVUqJOHVGHODGpGXFWLRQQDWXUHOOHSRXUODORJLTXHGHVSUpGLFDWV
Q¶pWDLWSDVVHXOHPHQWF PDLVpJDOHPHQWG
&HVTXDWUHUqJOHVĪ:lī Īalī Ī:1īHWĪa1īDLQVLTXHOHVUqJOHVSRXUOHVFRQQHFWHXUVIRUPHQWXQFDOFXO GHGpGXFWLRQQDWXUHOOHSRXUODORJLTXHGHVSUpGLFDWV /HVUqJOHVSRXUOHVFRQQHFWHXUVVRQWPDLQWHQDQW LQWHUSUpWpHVFRPPHV¶DSSOLTXDQWjGHVSKUDVHVGXODQJDJHL+ ,OHVWLPSRUWDQWGHQRWHUTXHO¶RQQH SHXWSDV SDUH[HPSOH VXSSRVHUXQHSKUDVHRXYHUWHĦXQHWHOOHVXSSRVLWLRQGRQQHUDLWIDFLOHPHQWOLHX jGHVUDLVRQQHPHQWVIDOODFLHX[HWVHUDLWVpPDQWLTXHPHQWQRQĥVHQVpH QRXVVXSSRVRQVTX¶XQHSKUDVH HVWYUDLHHWQHSRXYRQVSDVVXSSRVHUTX¶XQHSKUDVHRXYHUWHHVWVDWLVIDLWHSDUTXHOTXHVREMHWV
/HVUqJOHVGHODGpGXFWLRQQDWXUHOOHSRXUODORJLTXHGHV SUpGLFDWV
1RXVVRPPHVPDLQWHQDQWHQPHVXUHGHIRUPXOHUQRVUqJOHVGHODGpGXFWLRQQDWXUHOOHSRXUOHVTXDQĥ WL¿FDWHXUV UqJOHVTXLVHUDMRXWHQWjFHOOHVSRXUOHVFRQQHFWHXUV PDLQWHQDQWLQWHUSUpWpHVFRPPH JRXYHUQDQWQRQVHXOHPHQWGHVFRQQHFWHXUVSURSRVLWLRQQHOV PDLVDXVVLGHVFRQQHFWHXUVUHOLDQWGHV SKUDVHVRXYHUWHV
/·pOLPLQDWLRQGXTXDQWLÀFDWHXUXQLYHUVHOHWO·LQWURGXFWLRQGXTXDQWLÀFDĦ WHXUH[LVWHQWLHO
6RLWφXQHIRUPXOHGHL+ ©xªXQHYDULDEOHHW©tªXQWHUPHOLEUHSRXU©xªGDQVφ6RLW!φ(x/t)"OH UpVXOWDWGHODVXEVWLWXWLRQĪXQLIRUPHīGH©tªSRXU©xªGDQVφ /DUqJOHGH© VSpFLDOLVDWLRQ XQLYHUVHʃ˳ ª ĪalīQRXVGRQQHDORUVOHGURLWGHFRQFOXUH!φ(x/t)"jSDUWLUGH!∀x(φ)"
P #!∀x(φ)"
Q #!φ(x/t)" GHĪmīDYHFĪalī
/DUqJOHGH© JpQpUDOLVDWLRQ H[LVWHQWLHʃ˳ªĪ:1īHVWODFRQYHUVHGHĪalī HOOHQRXVGRQQHOHGURLWG¶LQIpUHU
!∃x(φ)"jSDUWLUGH!φ(x/t)"
P #!φ(x/t)"
Q #!∃x(φ)" GHĪmīDYHFĪ:1ī
'DQVOHVGHX[FDV G¶pYHQWXHOOHVVXSSRVLWLRQVRXSUpPLVVHVjODOLJQHĪPīVRQWFRQVHUYpHVjODOLJQH ĪQī
/¶DSSOLFDWLRQGHVDQFLHQQHVUqJOHVDX[QRXYHOOHVIRUPXOHVSHXWFUpHUGHQRXYHDX[W\SHVG¶HUUHXUV 3RXUO¶DSSOLFDWLRQGH ODUqJOHGHSUHXYHFRQGLWLRQQHOOH SDUH[HPSOH LOIDXWJDUGHUjO¶HVSULWTX¶HOOHQHQRXVSHUPHWG¶HQOHYHUTXHODVXSSRVLWLRQWUDQVĥ IRUPpHHQO¶DQWpFpGHQWGHO¶LPSOLFDWLRQ 1RXVQHSRXYRQVSDV SDUH[HPSOH VXSSRVHU©F aª SURXYHUVRXVFHWWHVXSSRVLWLRQ TXH∃x(Gx)HWFRQFOXUHHQVXLWH©∃x(F x)→ ∃x(Gx)ª
1RXVDYRQVGp¿QLFHWWHQRWLRQGDQVODOHoRQĪFI Sī 8QWHUPHQ¶HVWSDVOLEUHSRXUXQHYDULDEOHGDQVXQHIRUPXOH VLXQHpYHQWXHOOHVXEVWLWXWLRQGHODYDULDEOHSDUOHWHUPHDYDLWFRPPHFRQVpTXHQFHTX¶XQHRFFXUUHQFHOLEUHGHFHWWHYDULDEOH GDQVOHWHUPHGHYHQDLWJRXYHUQpHSDUXQTXDQWL¿FDWHXUGDQVODIRUPXOH 'DQVOHFDVROHWHUPHHQTXHVWLRQHVWXQHYDULDEOH FHODYHXWGLUHTX¶LOVHUDLWVXEVWLWXpjO¶LQWpULHXUG¶XQTXDQWL¿FDWHXUTXLOHJRXYHUQHUD
/·LQWURGXFWLRQGXTXDQWLÀFDWHXUXQLYHUVHOHWO·pOLPLQDWLRQGXTXDQWLÀFDĦ WHXUH[LVWHQWLHO
6RLW!φ(a)"XQHIRUPXOHTXLFRQWLHQWXQHFRQVWDQWHLQGLYLGXHOOH©aª 6¶LOQ¶HVWSDVOHFDVTXH©aªDXQH
RFFXUUHQFHGDQVXQHGHVSUpPLVVHVGRQWGpSHQGODSUHXYHGHφ ODUqJOHGHJpQpUDOLVDWLRQ XQLYHUVHʃ˳
Ī:līQRXVSHUPHWG¶pWHQGUHXQHSUHXYHGH!φ(a)"jXQHSUHXYHGH!∀x(φ(a/x))"
P #!φ(a)"
Q #!∀x(φ(a/x))" GHĪmīDYHFĪ:lī
/DUqJOHGHVSpFLDOLVDWLRQ H[LVWHQWLHʃ˳Īa1īQRXVSHUPHWGHSURXYHU jSDUWLUGH!∃x(φ(a/x))"WRXWH IRUPXOHψTXHQRXVSRXYRQVSURXYHUjSDUWLUGHODVXSSRVLWLRQ!φ(a)"ĦV¶LOQ¶HVWSDVOHFDVTXH©aªD XQHRFFXUUHQFHGDQVψRXGDQVXQHVXSSRVLWLRQRXXQHSUpPLVVHGRQWGpSHQGODSUHXYHGHψjSDUWLU
GH!φ(a)"
P #!∃x(φ(a/x))"
Q !φ(a)" #∗ !φ(a)" VXSSRVLWLRQ
R !φ(a)" #∗ ψ
S #ψ GHĪmī ĪnīHWĪoīDYHFĪa1ī
/DOLJQHĪSīFRQWLHQGUDWRXWHVOHVSUpPLVVHVRXVXSSRVLWLRQVGHODOLJQHĪPīHWWRXWHVOHVVXSSRVLWLRQV QpFHVVDLUHVSRXUODSUHXYHGHψjSDUWLUGH!φ(a)"ĪDXWUHVTXH!φ(a)"ī
4XHOTXHVH[HPSOHV
/HSUHPLHUH[HPSOHLOOXVWUHOHERQXVDJHGHODUqJOHGHVSpFLDOLVDWLRQH[LVWHQWLHOOHĪa1ī 3RXUSURXYHU XQHFRQFOXVLRQjSDUWLUG¶XQHTXDQWL¿FDWLRQH[LVWHQWLHOOH QRXVHVVD\RQVGHODGpULYHUGHVRQGLVMRLQW W\SLTXH 9RLFLXQHSUHXYHGXVpTXHQW©∀x(Gx→Hx),∃x(F x∧Gx)# ∃x(F x∧Hx)ª 1RXVFRPĥ PHQoRQVSDUODVXSSRVLWLRQGXGLVMRLQWW\SLTXH
∀x(Gx→Hx),∃x(F x∧Gx) # ∀x(Gx→Hx) SUpPLVVH ∀x(Gx→Hx),∃x(F x∧Gx) # ∃x(F x∧Gx) SUpPLVVH ∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ F a∧Ga VXSSRVLWLRQ
8QHLQVWDQFLDWLRQGHODTXDQWL¿FDWLRQXQLYHUVHOOHSRXUODFRQVWDQWH©aªTXLUHSUpVHQWHO¶LQGLYLGX DUELWUDLUHTXLpWDLWGLWrWUHF HWGQRXVSHUPHWDORUVG¶DSSOLTXHUOHVUqJOHVRUGLQDLUHVGHFRQQHFWHXUV SURXYDQWTXHaQ¶HVWSDVVHXOHPHQWF PDLVDXVVLH
∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ Ga→Ha GHĪīDYHFĪalī ∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ Ga GHĪīDYHFĪ∧(ī ∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ Ha GHĪīHWĪīDYHFĪJSī ∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ F a GHĪīDYHFĪ∧(ī ∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ F a∧Ha GHĪīHWĪīDYHFĪ∧,ī
4XHOTXHVH[HPSOHV 3RXUFRPSOpWHUODSUHXYHGH©∃x(F x∧Hx)ª LOQRXVUHVWHjIDLUHXQHJpQpUDOLVDWLRQH[LVWHQWLHOOHSDU UDSSRUWjO¶LQGLYLGXDUELWUDLUHHWG¶DSSOLTXHUODUqJOHGHVSpFLDOLVDWLRQH[LVWHQWLHOOHDXUpVXOWDW
∀x(Gx→Hx),∃x(F x∧Gx) F a∧Ga #∗ ∃x(F x∧Hx) GHĪīDYHFĪ:1ī
∀x(Gx→Hx),∃x(F x∧Gx) # ∃x(F x∧Hx) GHĪī ĪīHWĪīDYHFĪa1ī /¶DSSOLFDWLRQGHĪa1īHVWOpJLWLPHFDU©aªQ¶DSDVG¶RFFXUUHQFHGDQV©∃x(F x∧Hx)ªHWSDUFHTXHOD SUHXYHGHODGHUQLqUHSKUDVHQHGpSHQGDLWSDVG¶DXWUHVVXSSRVLWLRQVVXUaTXHFHOOHTXHOHGLVMRLQW W\SLTXHpWDLWYUDL
3URXYRQVODGLVWULEXWLYLWpGXTXDQWL¿FDWHXUXQLYHUVHOVXUODFRQMRQFWLRQ F¶HVWĥjĥGLUHOHVpTXHQW©∀x(F x)∧
∀x(Gx)# ∀x(F x∧Gx)ª /DSUHXYHFRQVLVWHHQXQHLQVWDQFLDWLRQGHVGHX[FRQMRLQWV VXLYLHG¶XQH JpQpUDOLVDWLRQGHODFRQMRQFWLRQGHVLQVWDQFLDWLRQV
∀x(F x)∧ ∀x(Gx) # ∀x(F x)∧ ∀x(Gx) SUpPLVVH
∀x(F x)∧ ∀x(Gx) # ∀x(F x) GHĪīDYHFĪ∧(ī
∀x(F x)∧ ∀x(Gx) # F a GHĪīDYHFĪalī
∀x(F x)∧ ∀x(Gx) # ∀x(Gx) GHĪīDYHFĪ∧(ī
∀x(F x)∧ ∀x(Gx) # Ga GHĪīDYHFĪalī
∀x(F x)∧ ∀x(Gx) # F a∧Ga GHĪīHWĪīDYHFĪ∧,ī
∀x(F x)∧ ∀x(Gx) # ∀x(F x∧Gx) GHĪīDYHFĪ:lī
'DQVODSUHXYHSUpFpGHQWH QRXVQ¶DYRQVGIDLUHDXFXQHVXSSRVLWLRQ 3DUFRQWUH SRXUSURXYHUOD GLVWULEXWLYLWpGXTXDQWL¿FDWHXUH[LVWHQWLHOVXUODGLVMRQFWLRQ QRXVGHYRQVHQIDLUHXQH 3URXYRQV DORUVOHVpTXHQW©∃x(F x∨Gx)# ∃x(F x)∨ ∃x(Gx)ª
∃x(F x∨Gx) # ∃x(F x∨Gx) SUpPLVVH
∃x(F x∨Gx) F a∨Ga #∗ F a∨Ga VXSSRVLWLRQ ∃x(F x∨Gx) F a∨Ga, F a #∗ F a VXSSRVLWLRQ ∃x(F x∨Gx) F a∨Ga, F a #∗ ∃x(F x) GHĪīDYHFĪ:1ī ∃x(F x∨Gx) F a∨Ga, F a #∗ ∃x(F x)∨ ∃x(Gx) GHĪīDYHFĪ∨,ī
∃x(F x∨Gx) F a∨Ga, Ga #∗ Ga VXSSRVLWLRQ
∃x(F x∨Gx) F a∨Ga, Ga #∗ ∃x(Gx) GHĪīDYHFĪ:1ī ∃x(F x∨Gx) F a∨Ga, Ga #∗ ∃x(F x)∨ ∃x(Gx) GHĪīDYHFĪ∨,ī
∃x(F x∨Gx) F a∨Ga #∗ ∃x(F x)∨ ∃x(Gx) GHĪ īDYHFĪ∨(ī ∃x(F x∨Gx) # ∃x(F x)∨ ∃x(Gx) GHĪī ĪīHWĪīDYHFĪa1ī 1RXVSURXYRQVODGLVWULEXWLYLWpGXTXDQWL¿FDWHXUH[LVWHQWLHOVXUODFRQMRQFWLRQVRXVIRUPHGHWKpRĥ
UqPHSOXW{WTXHGHVpTXHQW *UkFHDXPpWDĥWKpRUHPGHGpGXFWLRQĪFI SīHWODSUpVHQFHG¶XQH UqJOHG¶LQIpUHQFHGHSUHXYHFRQGLWLRQQHOOHĪ*Sī FHODUHYLHQWjODPrPHFKRVH 9RLFLGRQFXQHSUHXYH GH©# ∃x(F x∧Gx)→(∃x(F x)∧ ∃x(Gx))ª
∃x(F x∧Gx) #∗∃x(F x∧Gx) VXSSRVLWLRQ
∃x(F x∧Gx), F a∧Ga #∗F a∧Ga VXSSRVLWLRQ
∃x(F x∧Gx), F a∧Ga #∗F a GHĪīSDUĪ∧(ī
∃x(F x∧Gx), F a∧Ga #∗∃x(F x) GHĪīSDUĪ:1ī
∃x(F x∧Gx), F a∧Ga #∗Ga GHĪīSDUĪ∧(ī
∃x(F x∧Gx), F a∧Ga #∗∃x(Gx) GHĪīSDUĪ:1ī
∃x(F x∧Gx), F a∧Ga #∗∃x(F x)∧ ∃x(Gx) GHĪīHWĪīSDUĪ∧,ī ∃x(F x∧Gx) #∗∃x(F x)∧ ∃x(Gx) GHĪī Īī ĪīSDUĪa1ī # ∃x(F x∧Gx)→(∃x(F x)∧ ∃x(Gx)) GHĪīHWĪīSDUĪS*ī
&RPPHQRXVO¶DYRQVYXGDQVODGLVFXVVLRQVXUOHVUqJOHVGHSDVVDJHĪSGDQVDOHoRQ OHTXDQWL¿ĥ FDWHXUH[LVWHQWLHOGLVWULEXHDXVVLVXUO¶DQWpFpGHQWG¶XQHLPSOLFDWLRQTXLQHFRQWLHQWSDVG¶RFFXUUHQFH OLEUHGHODYDULDEOHTX¶LOTXDQWL¿H 1RXVSRXYRQVGRQFSURXYHU©∃x(F a→F x)#F a→ ∃x(F x)ª
∃x(F a→F x) # F a→ ∃x(F x) SUpPLVVH
∃x(F a→F x) F a #∗ F a VXSSRVLWLRQ
∃x(F a→F x) F a, F a→F b #∗ F a→F b VXSSRVLWLRQ
∃x(F a→F x) F a, F a→F b #∗ F b GHĪīHWĪīDYHFĪJSī ∃x(F a→F x) F a, F a→F b #∗ ∃x(F x) GHĪīDYHFĪ:1ī
∃x(F a→F x) F a #∗ ∃x(F x) GHĪī ĪīHWĪīDYHFĪa1ī
∃x(F a→F x) # F a→ ∃x(F x) GHĪīHWĪīDYHFĪS*ī
3XLVTXHQRXVDYRQVGpMjXQHRFFXUUHQFHGHODFRQVWDQWH©aª QRXVGHYRQVFKRLVLU jODOLJQHĪī OD FRQVWDQWH©bªSRXUGpVLJQHUO¶LQGLYLGXDUELWUDLUHTXL¿JXUHGDQVOHGLVMRLQWW\SLTXHGHODTXDQWL¿FDWLRQ H[LVWHQWLHOOH
1RXVSRXYRQVDXVVLSURXYHUXQVpTXHQW ©∀x∀y(Rxy)# ∀y∀x(Rxy)ª TXLFRUUHVSRQGjXQHREVHUYDĥ WLRQTXHQRXVDYRQVIDLWHDXSDUDYDQW TXHOHVYDULDEOHV SULVHVLQGLYLGXHOOHPHQW VRQWLQWHUFKDQJHDEOHV ĦWRXWFHTXLGLVWLQJXHXQHYDULDEOHG¶XQHDXWUHVRQWOHXUVSURSULpWpVUHODWLRQQHOOHV HQSDUWLFXOLHUVL HOOHVVRQWOLpHVSDUGHVTXDQWL¿FDWHXUVGHGLɱpUHQWHVW\SHV
∀x∀y(Rxy) # ∀x∀y(Rxy) SUpPLVVH
∀x∀y(Rxy) # ∀y(Ray) GHĪīDYHFĪalī
∀x∀y(Rxy) # Rab GHĪīDYHFĪalī
∀x∀y(Rxy) # ∀x(Rxb) GHĪīDYHFĪ:lī
∀x∀y(Rxy) # ∀y∀x(Rxy) GHĪīDYHFĪ:lī
&RPPHGDQVODGHUQLqUHSUHXYH QRXVGHYRQVFKRLVLUGHX[FRQVWDQWHVLQGLYLGXHOOHVGLɱpUHQWHV FDU VDQVOHIDLUH QRXVGpGXLULRQVOHVpTXHQWLQYDOLGH©∀x∀y(Rxy)# ∀x(Rxx)ª D\DQWSHUGXWRXWHSRVVLĥ ELOLWpGHGLVWLQJXHUOHVGHX[SODFHVDUJXPHQWDOHV
)LQDOHPHQW QRXVSURXYRQVXQVpTXHQWFRPSOLTXp jVDYRLUOHVXLYDQW
∃x(F x∧ ∀y(Gy→Rxy)),∀x(F x→ ∀y(By→ ¬Rxy))# ∀x(Gx→ ¬Bx)
(QDEUpJHDQW©∃x(F x∧ ∀y(Gy→Rxy))ªSDU©$ªHW©∀x(F x→ ∀y(By→ ¬Rxy))ªSDU©%ª QRXV
/DFULWLTXHUpIpUHQWLDOLVWHGH)UHJH
REWHQRQVODSUHXYHVXLYDQWH
$,% # ∃x(F x∧ ∀y(Gy→Rxy)) SUpPLVVH
$,% # ∀x(F x→ ∀y(By→ ¬Rxy)) SUpPLVVH
$,% F a∧ ∀y(Gy→Ray) #∗ F a∧ ∀y(Gy→Ray) VXSSRVLWLRQ
$,% F a∧ ∀y(Gy→Ray) #∗ F a GHĪīDYHFĪ∧ī
$,% F a∧ ∀y(Gy→Ray) #∗ ∀y(Gy→Ray) GHĪīDYHFĪ∧ī $,% F a∧ ∀y(Gy→Ray) #∗ F a→ ∀y(By→ ¬Ray) GHĪīDYHFĪalī $,% F a∧ ∀y(Gy→Ray) #∗ ∀y(By→ ¬Ray) GHĪīHWĪīDYHFĪJSī
$,% F a∧ ∀y(Gy→Ray), Gb #∗ Gb VXSSRVLWLRQ
$,% F a∧ ∀y(Gy→Ray), Gb #∗ Gb→Rab GHĪīDYHFĪalī $,% F a∧ ∀y(Gy→Ray), Gb #∗ Rab GHĪīHWĪīDYHFĪJSī $,% F a∧ ∀y(Gy→Ray), Gb #∗ Bb→ ¬Rab GHĪīDYHFĪalī $,% F a∧ ∀y(Gy→Ray), Gb #∗ ¬¬Rab GHĪīDYHFĪ.Lī $,% F a∧ ∀y(Gy→Ray), Gb #∗ ¬Bb GHĪīHWĪīDYHFĪJhī
$,% F a∧ ∀y(Gy→Ray) #∗ Gb→ ¬Bb GHĪīHWĪīDYHFĪS*ī
$,% F a∧ ∀y(Gy→Ray) #∗ ∀x(Gx→ ¬Bx) GHĪīDYHFĪ:lī
$,% # ∀x(Gx→ ¬Bx) GHĪī ĪīHWĪīDYHFĪa1ī
&RPPHWRXMRXUV QRXVFRPPHQoRQVSDUGHVLQVWDQFLDWLRQVGHVTXDQWL¿FDWLRQV FHTXLQRXVSHUPHW HQVXLWHG¶DSSOLTXHUOHVUqJOHVSRXUOHVFRQQHFWHXUV (QYXHG¶XQHDSSOLFDWLRQGHĪS*ī QRXVVXSSRVRQV jODOLJQHĪī O¶DQWpFpGHQWGHO¶LPSOLFDWLRQGRQWQRXVYRXORQVSURXYHUODTXDQWL¿FDWLRQXQLYHUVHOOH 1RXVDXULRQVpJDOHPHQWSXVXSSRVHU©GaªHWIDLUHOHVLQVWDQWLDWLRQGDQVOHVOLJQHVĪīHWĪīDYHF
©aªDXOLHXGH©bª 1RXVJpQpUDOLVRQVOHUpVXOWDWREWHQXjODOLJQHĪīHWHQOHYRQVODVXSSRVLWLRQGX GLVMRLQWW\SLTXH FHTXLHVWSHUPLVSDUFHTXHODFRQFOXVLRQREWHQXHQ¶HQGpSHQGSDV
/DFULWLTXHUpIpUHQWLDOLVWHGH)UHJH
)U)UHJHN|QQHQ,GHQWLWlWVDXVVDJHQLQIRUPDWLYVHLQ ZHQQGLHEHLGHQVLQJXOlUHQ7HUPHYHUVFKLHĥ GHQH6LQQHKDEHQ 'LHVH6LQQHĪ³$UWHQGHV*HJHEHQVHLQV´īEHVWLPPH˿GLH5HIHUHQ] 5HIHUHQ]REMHNW GHV7HUPVLVWGDVMHQLJH'LQJ DXIGDVGHULP6LQQµNRGL¿]LHUWH¶GHVNULSWLYH*HKDOW]XWULɱW (LJHQQDĥ PHQZHUGHQGDPLWlKQOLFKZLHVLQJXOlUH%HVFKUHLEXQJHQEHKDQGHOW
4XHOTXHVSURSULpWpVGXVHQV)UHJpHQ
/HVHQVGpWHUPLQHODUpIpUHQFH ,OHVWVLQJXODULVDQWGDQVOHVHQVR SRXUXQQRPSURSUH XQHW XQVHXOLQGLYLGXWRPEHVRXVOHFRQWHQXFRQFHSWXHOUDWWDFKpDXQRPSURSUHĦOHVHQVHVWGRQF FRQoXG¶DSUqVOHPRGqOHGHVGHVFULSWLRQVGp¿QLHV &HFLH[FOXW S H[ ODSRVVLELOLWpTXHOHVHQV GH© MH ªVRLWODVLJQL¿FDWLRQOLQJXLVWLTXHFRQYHQWLRQQHOOHGRQQpHSDU© ODSHUVRQQHTXLSDUOH ª
&HWWHGHUQLqUHQHGpWHUPLQHSDVODUpIpUHQFH /DVLJQL¿FDWLRQOLQJXLVWLTXHUHVWHODPrPH PDLV ODUpIpUHQFHYDULHVHORQOHVSHUVRQQHVTXLXWLOLVHQWO¶H[SUHVVLRQ© MH ª
/HVHQVHVWFHTXHO¶RQVDLVLWVLRQFRPSUHQGXQHH[SUHVVLRQ
/HVHQVHVWODVLJQL¿FDWLRQOLQJXLVWLTXHFRQYHQWLRQQHOOHGHVH[SUHVVLRQVĪDXPRLQVGDQVGHV ODQJDJHVSDUIDLWVRLOQ¶\DQLG¶DPELJXwWpQLGpSHQGDQFHFRQWH[WXHOOHī
/HVHQVHVWREMHFWLIHWXQLYHUVHOHWSHXWrWUHVDLVLSDUQ¶LPSRUWHTXLĪFHTXLOHGLVWLQJXHGHV UHSUpVHQWDWLRQV LH GHVLPDJHVPHQWDOHVHWVXEMHFWLYHVDVVRFLpHVjXQHH[SUHVVLRQī
/HVHQVHVWLQGpSHQGDQWGHODUpIpUHQFH 2QSHXWWRXWjIDLWFRPSUHQGUHXQHH[SUHVVLRQVDQV SRXUDXWDQWrWUHFDSDEOHG¶HQLGHQWL¿HUOHUpIpUHQWĪH[ © O¶KRPPHOHSOXVULFKHGXPRQGH ªī ,O HVWSRVVLEOHTX¶RQQ¶DLWDFFqVjGHVUpIpUHQWVTX¶jWUDYHUVOHVHQVG¶XQHH[SUHVVLRQ &¶HVWSRXU FHWWHUDLVRQTX¶RQSHXWFRPSUHQGUHGHVpQRQFpVFRQWHQDQWVGHVQRPVSURSUHVYLGHV
/HVHQVHVWO¶REMHWGHVDWWLWXGHV F¶HVWFHTX¶RQFURLW$ 92,5 F¶HVWFHTX¶RQSHQVH),1
S H[ VLRQFURLWTXHOTXHFKRVH 6LTXHOTX¶XQGRQQHVRQDVVHQWLPHQWjRXDɷUPHXQHSKUDVH ĪS H[ © 6XSHUPDQSHXWYROHU ªī FHTX¶RQDɷUPHHWFURLWHVWXQHSHQVpH LH OHVHQVGHODSKUDVH ĪTXLHVWGLɱpUHQWGXVHQVGHODSKUDVH© &ODUN.HQWSHXWYROHU ªī
/HVHQVHVWOHUpIpUHQWGHVPRWVTXLVHWURXYHQWGDQVXQFRQWH[WHOLQJXLVWLTXHLQGLUHFW S H[ XQ FRQWH[WHOLQJXLVWLTXHLQWURGXLWSDU© -HFURLVTXH. . .ª
&HVSURSULpWpVGXVHQV)UHJpHQFRUUHVSRQGHQWjGHVSUREOqPHVGLVFXWpVSOXVKDXW &KDFXQHGHFHV SURSULpWpVDpWpFULWLTXpH
)U.ULSNHĪīXQG.DSODQĪīKLQJHJHQVLQG(LJHQQDPHQHKHUDP0RGHOOYRQ9DULDEOHQĪXQWHU HLQHU:HUW]XRUGQXQJīDXI]XIDVVHQ 'LH5HIHUHQ]LVWGLUHN̄ GK ZLUGQLFKWGXUFKHLQHQ6LQQYHUPLWĥ WHOW
.ULSNHJLEWLPZHVHQWOLFKGUHL$UJXPHQWH
'DVPRGDOH$UJXPHQW ³a=b´LVW ZHQQZDKU QRWZHQGLJHUZHLVHZDKU
'DVVHPDQWLVFKH$UJXPHQW DXFKZHQQ³F´GHQµ6LQQ¶YRQ³a´DQJLEW NDQQ³F a´V\QWKHWLVFK VHLQ
'DVHUNHQQWQLVWKHRUHWLVFKH$UJXPHQW LFKNDQQ³a´YHUVWHKHQ DXFKZHQQLFKQLFKWVEHUa ZHLVV
3RXUOHVH[SRVDQWVGHODWKpRULHGHODUpIpUHQFHGLUHFWH HWQRWDPPHQWSRXU.ULSNHĪīHW.DSODQ Īī ODYDULDEOHHVWO¶H[SUHVVLRQUpIpUHQWLHOOHSDUH[FHOOHQFHĦGDQVOHFDVGHODYDULDEOHJRXYHUQpH SDUXQTXDQWL¿FDWHXU OH½ PRGH ¾ GHVDUpIpUHQFHQHSHXWrWUHTXHGLUHFW
$ ODEDVHGHODWKpRULHGH)UHJHFRPPHHOOHpWDLWLQWHUSUpWpSDUVHVDGYHUVDLUHVGDQVOHVDQQpHVFLQĥ TXDQWHHWVRL[DQWHVHWURXYHXQH FRQFHSWLRQ VDWLVIDFWLRQHʃH īGHVFULSWLYLVWHĬ GX VHQV /DFRQFHSWLRQVDWLVĥ IDFWLRQQHOOHGXVHQVHVWODSHQVpHTX¶jFKDTXHPRWVRQWDWWDFKpHVTXHOTXHVFRQGLWLRQVTXHQ¶LPSRUWH TXHOOHFKRVHGDQVOHPRQGHGRLWVDWLVIDLUHSRXUrWUHOHUpIpUHQWGXPRW /HUpIpUHQWGH© 6XSHUPDQ ªHVW O¶LQGLYLGXXQLTXHTXLVDWLVIDLWFHVFRQGLWLRQV 6¶LOQ¶\SDVGHWHOREMHW RXV¶LO\HQDSOXVLHXUV ULHQQ¶HVW GpQRWpSDU© 6XSHUPDQ ª 3RXUOHVSKUDVHV FHWWHFRQFHSWLRQVDWLVIDFWLRQQHOOHQRXVDPqQHjLGHQWL¿HU ODSHQVpH F¶HVWĥjĥGLUHOHVHQVG¶XQHSKUDVH DYHFOHVFRQGLWLRQVGHYpULWpGHFHWWHSKUDVHĪOHVFRQGLĥ WLRQVTXHQRWUHPRQGHGRLWVDWLVIDLUHSRXUTXHODSKUDVHHQTXHVWLRQVRLWYUDLHī 0rPHOHVSDUWLVDQV GX1HRĥ)UHJpHQLVPH FHSHQGDQW UHMHWWHQWXQGHVFULSWLYLVPHJpQpUDO
$ FHWWHFRQFHSWLRQVDWLVIDFWLRQHOOHV¶RSSRVHXQH WKpRULH UHODWLRQQHʃ˳ FRPPHHOOHDpWpGpIHQGXHSDU 'DYLG.DSODQ 3RXUOHVWHUPHVVLQJXOLHUV.DSODQĪīSURSRVHGHFDUDFWpULVHUODWKpRULHGH)UHJH SDUOHVFKpPDVXLYDQW
Īī WHUPHVLQJXOLHUH[SULPH−→ FRQFHSWVHQVVDWLVIDFWLRQ
−→ LQGLYLGXTXLWRPEHOHFRQFHSW
/DUHODWLRQGHUpIpUHQFH(WHUPHVLQJXOLHUGpQRWH−→ LQGLYLGX)HVWFRQoXFRPPHSURGXLWGHFHVGHX[UHODĥ WLRQV
GpQRWH
−→=H[SULPH→ ◦ HVWVDWLVIDLWSDU
→
$ FHWWHFRQFHSWLRQ .DSODQRSSRVHO¶LGpHGHODUpIpUHQFH GLUHFW˳ LH G¶XQHUHODWLRQGHUpIpUHQFHTXL Q¶HVWSDVPpGLWDWpHSDUXQVHQV
$ ODEDVHGHFHWWHTXHUHOOH RQWURXYHGHX[FRQFHSWLRQVGLɱpUHQWHVGXUDSSRUWHQWUHODQJDJHHWPRQGH 6HORQXQFDPSĪUHSUpVHQWpS H[ SDU&DUQDSī OHPRQGHHVWG¶XQF{WpHWOHODQJDJHGHO¶DXWUH HWF¶HVW OHODQJDJHTXLLPSRVHGHVFRQGLWLRQV OHPRQGHTXLOHVVDWLVIDLWRXSDV 6HORQO¶DXWUH RQHVWIRUFpGH SUHQGUHHQFRPSWHOHVOLHQVĪSUDJPDWLTXHVSDUH[ī jO¶LQWpULHXUGXQLYHDXGXODQJDJHGpMjSRXUpWDEOLU OHVOLHQVGHUpIpUHQFHTXLOLHQWFHQLYHDXDYHFOHPRQGH LOIDXWSUHQGUHHQFRPSWHOHIDLWTXHOHVPRWV HX[DXVVL VRQWGDQVOHPRQGHĦDXPRLQVOHVPRWVFRQoXVFRPPHRFFXUUHQFHVĪ© WRNHQV ª HWQRQSDV FRPPHW\SHVīTXLRQWXQHORFDWLRQVSDWLRĥWHPSRUHOOHĪLOVVHWURXYHQWVXUOHWDEOHDXQRLUS H[ī
/DFULWLTXHUpIpUHQWLDOLVWHGH)UHJH
&HVRFFXUUHQFHVHQWUHQW HQWDQWTX¶REMHWVGXPRQGH GDQVGHVUHODWLRQVDYHFG¶DXWUHVREMHWVGX PRQGHĪFRPPHOHORFXWHXU OHVSHUVRQQHVSDUOHVTXHOOHVM¶DLDSSULVFHPRWĪFHTX¶RQDSSHOOHXQHFKDvQH GHWUDQVPLVVLRQī O¶DXGLWHXUHWFī/DUHODWLRQGHGpQRWDWLRQHQWUHXQHRFFXUUHQFHFRQFUqWHGH© FHERXW GHFUDLH ªHWFHERXWGHFUDLHĪODFKRVHEODQFKH SRLQWXH GDQVPDPDLQīHVWXQHUHODWLRQHQWUHGHX[
FKRVHVTXLVHWURXYHQWGDQVOHĪPrPH SXLVTX¶LOQ¶\DTX¶XQVHXOīPRQGH
/HQRXYHDXSDUDGLJPHGLWTXHFHTXL¿[HODUpIpUHQFHGHVPRWV FHQ¶HVWSDVOHFRQWHQXFRQFHSWXHO DWWDFKpDX[PRWVPDLVOHVUHODWLRQVHQWUHGHVRFFXUUHQFHVGHVPRWVHWWHOVHWWHOVREMHWV /HUpIpUHQW G¶XQĪHRFFXUUHQFHG¶XQīPRWHVWFHWWHFKRVHTXLVHWURXYHGDQVXQHWHOOHUHODWLRQVSpFL¿TXHDYHFFHWWH RFFXUUHQFH &¶HVWSRXUFHWWHUDLVRQTXHOHVLQGH[LFDX[SHXYHQWSDUIDLWHPHQWVHUYLUFRPPHH[HPSOHV SRXUDUULYHUDXUpIpUHQW LOIDXWGpWHUPLQHUTXHOREMHWGDQVOHPRQGHVHWURXYHGDQVODERQQHUHODWLRQ DYHFFHWWHRFFXUUHQFHĪODUHODWLRQG¶DYRLU pWp pQRQFpH SDUTXHOTX¶XQ HQO¶RFFXUUHQFHī
/HVSURSULpWpVVpPDQWLTXHVG¶XQHSKRWR S H[ OHIDLWTXHFHWWHSKRWRHVWG˳0PH 7KDWFKHU YLHQQHQW GXIDLWTX¶HOOHDpWpSULVHHQODSUpVHQFHGH0PH 7KDWFKHUHWTXHODSKRWRYLHQWGHO¶LPSUHVVLRQGH ODSHOOLFXOHFDXVpHSDU0PH 7KDWFKHU 0rPHVLF¶HVWELHQSRVVLEOHTXH0PH 7KDWFKHUVDWLVIDLW OH© FRQWHQXFRQFHSWXHO ªGHODSKRWRĪDOHVPrPHVDSSDUHQFHVTXHODSHUVRQQHSKRWRJUDSKLpHī FHFL Q¶HVWSDVQpFHVVDLUH FRPPHRQYRLWFRPPHVXLYDQW VL0PH 7KDWFKHUDYDLWXQHVRHXUMXPHOOHTXLOXL UHVVHPEODLWpQRUPpPHQW RQQHGLUDLWSDVTXHODSKRWRHVWXQHSKRWRGHFHWWHVRHXUMXPHOOH PrPHVL ODSKRWRQHUHVVHPEOHSDVGXWRXWj0PH7KDWFKHUĪSDUFHTX¶LOIDLVDLWQRLUHWFī RQGLWWRXMRXUVTX¶HOOH HVWXQHSKRWRĪUDWpHīG¶HOOH (QFRQVpTXHQW PrPHV¶LODUULYDLWTXHVHXO0PH 7KDWFKHUVDWLVIDVVHOH FRQWHQXFRQFHSWXHO FHWWHVDWLVIDFWLRQQ¶HVWSDVFHTXLIDLWGHODSKRWRXQHSKRWRGH0PH 7KDWFKHU PDLVFHTXLHQHVWUHVSRQVDEOHHVWXQHUHODWLRQUpHOOHĪGDQVOHVFDVSDUDGLJPDWLTXHV LOV¶DJLWG¶XQH UHODWLRQFDXVDOHī
8QGHVH[HPSOHVOHVSOXVFODLUVGHFHQRXYHDXSDUDGLJPHHWHQPrPHWHPSVXQHFULWLTXHGH)UHJHHVW FRQVWLWXpSDUFHTXH'RQQHOODQĪīDSSHOOHO·HPSORLUpIpUHQWLHOGHVGHVFULSWLRQVGpÀQLHV 6LMHGLV GDQV XQEDU © OHW\SHOjĥEDVTXLERLWGX0DUWLQLDGUDJXpPD¿OOH ªMHSHX[GpVLJQHUTXHOTX¶XQPrPHVLFH TXHOTX¶XQQHERLWTXHGHO¶HDXĦHWPrPHVL OjĥEDV FDFKpGHUULqUHXQULGHDX LO\DTXHOTX¶XQTXL $ 92,5 HQHɱHW$ /$ 3/$&( '( ĪHQIDLWī),1 ERLWGX0DUWLQL /DWKpRULHGH)UHJHQRXVGLUDLW TXHM¶DLIDLWUpIpUHQFHjODSHUVRQQHFDFKpHĦFHTXL LQWXLWLYHPHQW Q¶HVWSDVOHUpVXOWDWTX¶RQDLPHUDLW DYRLU&¶HVWSOXVFODLUHQFRUHSRXUOHVGHVFULSWLRQVLQFRPSOqWHVFRPPH© FHW\SH ªRX© OXL ª &HTXL
¿[HODUpIpUHQFHHVWS H[ ODVDLOODQFHSHUFHSWXHOOHGHO¶LQGLYLGXHQTXHVWLRQGDQVOHFRQWH[WH 8QHRSWLRQSRVVLEOHSRXUOH)UHJpHQVHUDLWGHGLUHTXH GDQVFHVFDVĥOj ODUHODWLRQPrPHHVWUHSUpĥ VHQWpHGDQVOHFRQFHSWĦOHVHQVGH© FHW\SH ªVHUDLWDORUVGRQQpSDU© OHW\SHTXHMHYRLVV¶DJLWHU ª /¶REMHFWLRQjFHOD GRQQpHSDU.ULSNHĪī HVWTXHOHVFKDvQHVGHWUDQVPLVVLRQUHOLDQWQRWUHXVDJH GHVQRPVSURSUHVDYHFOHVEDSWrPHVRULJLQDX[ HWGRQF¿QDOHPHQWDYHFOHVFKRVHVEDSWLVpHV QRXV SHUPHWWHQWGHSDUOHUGHVFKRVHVGRQW RQ Q·D TXH OHV QRPVHWGRQWRQQHVDLWULHQG¶DXWUH RXHQFRUH SDUUDSSRUWDX[TXHOOHVRQHVWGDQVGHVLOOXVLRQVWHUULEOHV $XPRLQVGDQVFHVFDVĥFL ODUpIpUHQFHQH VHPEOHSDVrWUHPpGLDWLVpHSDUXQFRQWHQXFRQFHSWXHO 2QQ¶HVWSDVFRQVFLHQWGHVFKDvQHVFDXVDOHV HWRQQ¶DSDVEHVRLQGHOHVFRQFHSWXDOLVHUSRXUIDLUHUpIpUHQFH
+LODU\3XWQDPĪDīDJpQpUDOLVpFHWWHWKpRULHjXQHVRXVĥHVSqFHGHVPRWVFRPPXQV F¶HVWĥjĥGLUH DX[WHUPHVG¶HVSqFHQDWXUHOOHFRPPH© RU ª © HDX ªHW© WLJUH ª 'DQVFHWWHWKpRULH LO\DXQFHUWDLQ
,OHVWGLɷFLOHGHFDUDFWpULVHUFHVUHODWLRQVVDQVSUpVXSSRVHUTX¶HOOHVVRQWWRXWHVUpGXFWLEOHVjGHVUHODWLRQVFDXVDOHV &HWWH GHUQLqUHUpGXFWLRQHVWOHEXWG¶XQFHUWDLQSURJUDPPHGHUHFKHUFKHTXLYLVHjQDWXUDOLVHUWRXWHVOHVUHODWLRQVVpPDQWLTXHV 'H WRXWHIDoRQ XQHFDUDFWpULVDWLRQDGpTXDWHGHFHWWHUHODWLRQGHYUDLWpYLWHUGHWULYLDOLVHUODWKqVHTXL$ 92,5 VXUYLHQW$ /$
3/$&( '( ĪDUULYDLWī),1VLRQFRPSWDLWODUHODWLRQGHVDWLVIDFWLRQFRPPHUHODWLRQHQTXHVWLRQ
3HXWĥrWUHXQHGLVWLQFWLRQHQWUH© UpIpUHQWVpPDQWLTXH ªHW© UpIpUHQWSUDJPDWLTXH ª SRXUUDLWDLGHULFLDX1pRĥ)UHJpHQ /HUpIpUHQWVpPDQWLTXHVHUDLWO¶KRPPHGHUULqUHOHULGHDXĪHWOHUpIpUHQWVpPDQWLTXHGH© -HDQ ªHVW-HDQī PrPHVLOHUpIpUHQW SUDJPDWLTXHHVWO¶KRPPHDXEDUTXLERLWGHO¶HDXĪFRPPHOHUpIpUHQWSUDJPDWLTXHGH© -HDQ ªSHXWrWUH3LHUUH VLOHORFXWHXUD YRXOXIDLUHUpIpUHQFHj3LHUUHī
/DVWUDWpJLHGHQQHPDUFKHSDVGDQVFHVFDV SXLVTX¶LOQ¶\DSDVGHUpIpUHQWĪXQLTXHīVpPDQWLTXH
LQYHUVHPHQWGHODUHODWLRQHQWUHVHQVHWUpIpUHQFH /DUpIpUHQFHHVWSUHPLqUH /HVUpIpUHQWVQRXVVRQW GRQQpVHWOHVHQVF¶HVWODIDoRQGRQWOHUpIpUHQWQRXVHVWGRQQpRXHQFRUHODIDoRQGRQWQRXVQRXVOH UHSUpVHQWRQV
/DORJLTXHGHVSUpGLFDWVGHGHX[LqPHRUGUH
jYHQLU
/DTXDQWLÀFDWLRQSOXULHOOH
jYHQLU
/H1pRĦ)UpJpDQLVPH
jYHQLU
3RLQWVjUHWHQLU
/DGpGXFWLRQQDWXUHOOHSRXUODORJLTXHGHVSUpGLFDWVFRQVLVWHHQOHVUqJOHVGHGpGXFWLRQQDWXĥ UHOOHSRXUOHVFRQQHFWHXUVHWHQTXDWUHUqJOHVG¶LQWURGXFWLRQHWG¶pOLPLQDWLRQGHTXDQWL¿FDWHXUV /DUqJOHĪalī DSSHOpH© VSpFLDOLVDWLRQ XQLYHUVHʃ˳ª QRXVSHUPHWGHSDVVHUG¶XQHTXDQWL¿FDWLRQ
XQLYHUVHOOHjXQHLQVWDQFLDWLRQGHODSKUDVHRXYHUWHSRXUQ¶LPSRUWHTXHOWHUPH
8QTXDQWL¿FDWHXUH[LVWHQWLHOHVWLQWURGXLWSDUODUqJOHGHJpQpUDOLVDWLRQ H[LVWHQWLHʃ˳Ī:1ī 3RXUTX¶XQHDSSOLFDWLRQGHVUqJOHVĪalīHWĪ:1īVRLWFRUUHFWH LOHVWUHTXLVTXHOHWHUPHUHPSODFp
SDUODYDULDEOHVRLWOLEUHSRXUFHWWHYDULDEOHGDQVODIRUPXOHTXLOHFRQWLHQW F¶HVWĥjĥGLUHTX¶LOQH VHWURXYHSDVGDQVODSRUWpHG¶XQTXDQWL¿FDWHXUTXLOLHFHWWHYDULDEOH
3RXUSURXYHUXQHTXDQWL¿FDWLRQXQLYHUVHOOHjSDUWLUG¶XQHSUpPLVVHSDUWLFXOLqUHjO¶DLGHGHOD UqJOHGHJpQpUDOLVDWLRQ XQLYHUVHʃ˳Ī:lī QRXVH[LJHRQVTXHFHWWHSUpPLVVHVRLWYUDLHG¶XQLQGLYLGX DUELWUDLUH F¶HVWĥjĥGLUHG¶XQLQGLYLGXTXLQ¶DSSDUDvWGDQVDXFXQHVXSSRVLWLRQRXSUpPLVVHGRQW GpSHQGODSUHXYHGHFHWWHSKUDVHSDUWLFXOLqUH
3DUĪ:lī QRXVSRXYRQVHQVXLWHLQIpUHUXQHTXDQWL¿FDWLRQXQLYHUVHOOHGDQVODTXHOOHOHWHUPH DUELWUDLUHHVWVXEVWLWXpSDUWRX̄SDUXQHYDULDEOH ,OHVWLPSRUWDQWTXHFHWWHVXEVWLWXWLRQVRLW XQLIRUPH
/DUqJOHG¶pOLPLQDWLRQGXTXDQWL¿FDWHXUH[LVWHQWLHOĪa1īFRUUHVSRQGjFHOOHGHO¶pOLPLQDWLRQGH ODGLVMRQFWLRQĪ∨(īHWHVWSDUFRQVpTXHQWSOXVFRPSOLTXpH /¶DSSOLFDWLRQGHODUqJOHGHVSpFLDĥ OLVDWLRQH[LVWHQWLHOOHĪa1īFRPSUHQGTXDWUHpWDSHV
ĪDī ODSUHXYHG¶XQHTXDQWL¿FDWLRQH[LVWHQWLHOOHVRXVFHUWDLQHVVXSSRVLWLRQVHWjSDUWLUGHFHUĥ WDLQHVSUpPLVVHV
ĪEī ODVXSSRVLWLRQGXGLVMRLQWW\SLTXH
ĪFī XQHSUHXYHG¶XQHDXWUHSKUDVHVRXVODVXSSRVLWLRQGXGLVMRLQWW\SLTXH
ĪGī O¶DSSOLFDWLRQGHODUqJOH DYHFODFRQFOXVLRQTXHQRXVSRXYRQVpJDOHPHQWSURXYHUODSKUDVH SURXYpHVRXVODVXSSRVLWLRQGXGLVMRLQWW\SLTXHjSDUWLUGHVVXSSRVLWLRQVHWSUpPLVVHVQpĥ FHVVDLUHVSRXUODSUHXYHGHODTXDQWL¿FDWLRQH[LVWHQWLHOOH
/DUqJOHGHVSpFLDOLVDWLRQH[LVWHQWLHOOHĪa1īHVWVXMHWWHDX[PrPHVGHX[FRQGLWLRQVTXHODUqJOH GHJpQpUDOLVDWLRQXQLYHUVHOOH
ĪDī ODFRQVWDQWHUHPSODoDQWODYDULDEOHGDQVOHGLVMRLQWW\SLTXHQHGRLWDSSDUDvWUHGDQVDXFXQH SUpPLVVHRXVXSSRVLWLRQGHODTXHOOHGpSHQGODTXDQWL¿FDWLRQH[LVWHQWLHOOH
/H1pRĦ)UpJpDQLVPH ĪEī HOOHQHGRLWSDVDYRLUG¶RFFXUUHQFHGDQVODIRUPXOHTXHQRXVYRXORQVSURXYHU
ÿDSSOLFDWLRQGHFHVUqJOHVQRXVSHUPHWGHSURXYHUGHVWKpRUqPHVĪ©# φªīHWGHVVpTXHQWV Ī©φ #ψªī /DGpGXFWLRQQDWXUHOOHHVWXQHPpWKRGHV\QWD[LTXHTXLHVWFRUUHFWHHWFRPSOqWH SDUUDSSRUWjODVpPDQWLTXHGHODORJLTXHGHVSUpGLFDWV WRXWHSKUDVHYDOLGHHVWXQWKpRUqPHHW WRXWWKpRUqPHHVWYDOLGH WRXWVpTXHQWGpGXFWLEOHFRUUHVSRQGjXQHUHODWLRQGHFRQVpTXHQFH VpPDQWLTXHHWWRXWHFRQVpTXHQFHVpPDQWLTXHSHXWrWUHGpGXLWHFRPPHVpTXHQW
'LH(LQIKUXQJQHXHU,QGLYLGXHQNRQVWDQWHQIUµEHOLHELJH¶(OHPHQWHGHV,QGLYLGXHQEHUHLFKV Ī5HIHUHQ]HLQHUµ9DULDEOHĥXQWHUĥHLQHUĥ:HUW]XRUGQXQJ¶īPRWLYLHUWHLQH.ULWLNGHU)UHJHVFKHQ
$XɱDVVXQJGHV6LQQVYRQ(LJHQQDPHQ *HPlVVGHU7KHRULHGHUGLUHNWHQ5HIHUHQ]UHIHULHUHQ (LJHQQDPHQQLFKWYLDGHU(UIOOXQJLKUHVGHVNULSWLYHQ*HKDOWV VRQGHUQµGLUHNW¶ VLHVLQGULJLGH 'HVLJQDWRUHQ GLHLQDOOHQP|JOLFKHQ:HOWHQGDVVHOEH2EMHNWEH]HLFKQHQ