Toward a geometric view on computations
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Toward a geometric view on computations Philippe Audebaud. N° 5492 February 2005. N 0249-6399. ISRN INRIA/RR--5492--FR+ENG. Thème SYM. apport de recherche.
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(13) U@ e?SVUVAOJ a ∈ π RT[ b ∈ π (π prime) ∀a, b.a ∗ b ∈ π `iwfKAO[ LP=?UJdTJNJXw@be?LPUVRTiaKA¨}sAL K=?A iwg ALP=?A+gRsiwfKULPUVRTi e 6∈ V (I ∗ J) =?UVgh=$A i|JX?[NAOJ M (e) ∗ M (e) 6∈ π b ? u s R ¨ [ N L ? = A A @ K e M L t H P J A ' L P L $ R ; Q b A [NA ew[PAYJAOisLNAOf4LN=?UJ'd³HsU¥L+UJJPX gUVA i9LLPR·dTJNJPX?@ A V (I) ∪ V (J) = V (I ∗ J) J V U w i g A _ s d M J Y L T d ` J J I R T R · i dsJLP=wA [NA+AªKUVJLNJ¨JRs@ A e ∈ π |LNdTTA i4UVi9LPR%dsg gRsX?i9L 1 6∈ π dsJJRs@ A'g RTiw∅JLNdT=i9L V@ ({1}) Rsi?RT@ Ud^S. E = V ({e}) <&R6JPX?@ X?e;ew[PRpIUfKAOf4LN=?AbAªILPAOiwfKAOfg dTSVg X?SX|J dTiwf·LN=?Ab³dT[PUVRTX|J`gRsiwfKULPUVRTiwJ¨RTi lLMHIe|AYJ¨gOd^i Q;AUi9LPAO[Pew[PA LPAOf6dTJgSVRsJPAOf6JA LNJR^&JPRT@ A¨ewd^[PLPUgX?Sd^[LPERse|RsSRs}THb§RT[=?Ugh=6dTi9Hg SR9JAYf%JPAL πUVJ⊆LP=wEAJPAL V (I) R^ g RT@ @ RTit¯OA [NRIAOJ`§RT['d²d^@ UVSH·RT2@ RTi?Rs@ UVdTSVJO< =wA[NAOdTf?A [RsX?SVft=wd³TA[NAOg RT}si?UVJPAOf$LN=wdpL'Abg RT@ Ag SR9JA LNR·LP=wA6fKA Mwi?ULPUVRTi RTdTS}sA Q?[hd^UgJPALNJdTJ¯ A [NRIAOJJPALhJRTe|RsSHIi?Rs@bUd^S2UVfKAYd^SJUVi®LP=wA6gRsi9LPAªILRTµSV}TA Qw[NdTUVg ¶AORT@ ALN[PHs ¤A §RT[NA`}TRsUiw}d^iIHb§X?[PLP=?AO[UViLN=?UVJfKUV[PAYgLNURsiaISA LXwJ dTJPbRTX?[hJAOSsAOJ2=?ALN=?A [U¥L UJ2RT[PLP=I=?UVSKA J µ¨JPJPX?@ A π fKAOi?R^LNALN=?AJA L'RTgRsiITA [N}TAOi9LLPAO[P@J¨§RT[+JPRT@ Abg RTiITAO[P}sA iwg A+e?[NAOfKUg d^LPA • ⇓ < =?A (π prime) g RTiwf?U¥LNURsi$UJ LN[NdTiwJSdpLNAOf6Ui9LNRww=wA i?AOTA [ a ∗ b ⇓ KLN=?A i a ⇓ Rs[ b ⇓ \i4LN=?UVJewd^[PLPUgX?Sd^[[NAOdsfKUVi?}wILN=?A a ∗ b g RTUViwgUfKAYJULP=·LN=?AewdT[NdTSSVA S&gRTi|JMLN[PXwgL UVisLN[PRKfKX|gAOftQ9H4¤RTXwf?RTS D o I L`RpAOTA [Y?UL¨UJ`RT[PLP=4i?RTLPUgUVi?} LN=wdpLRTX?[ ∗Rse|AO[Nd^LPUVRTif?R9AYJ+i?RTL[PAOSHRTi akb dTi9HUi9LNA iwfKAYf@ AOd^iwUi?}|&dsJ'²d^[dsJJPH9i9LhdpªUJg RTiwg A [Ni?AOfa6D·Rs[PA ew[PAYgUJAOSHsnRTX?[X?iwfKAO[NJLNdTiwfKUVi?}UJLN=wdpLbLP=wA%ewdT[NdTSSVA SQ|AO=wd³IURsX?[UJ[Nd^LP=?AO[d4lP£Nw¢ ¦LP=|d^i ewdT[LbR^LN=?A f? A Mwi?ULPUVRTi RT;LN=?UJnRTe;A [hdpLNRT[YiwJLPAYdTfTJPUViwgA LN=?A gOd^SgX?SVXwJng dT[P[NUVAOJe?[NAOg UVJPA SVH+LP=wAi?R^LNURsiRTlg RT@ e?XKLhdpLNURsia LN=?Abi?A Rse|AO[Nd^LPUVRTi a ∗ b JP=?RTX?Sf·[NA e?[NAOJPA i9L'd%i?λAO gRs@ e?XKLNd^LPUVRTiJNgh=?AO@bAbRpTAO[LNA [N@ JOg dTSSVAOftLP=?AO[PAYdp¡LPAO[ K ; h ²£ & C\A LO LP=wA%JA L UJ+i?R^Lbd·[PUVi?}|LN=?A ²£ Rse|AO[Nd^LPUVRTi¦Q|AOUi?}@bUJNJUVi?}wtBIRSA LXwJUVi9LP[NRKfKXwgA LNRT}sALP=wA [ULP=$LP=?EAi?A gRsiwJMLhd^i9L 0 JXwgh=$LN=wdpLLN=?A+[PAYJXwS¥LNUi?}¢ F ¢ s¢ hpN K H E LNX?[NiwJUVisLNRdgRT@ a @+X b a. −1 a. −1 u ¯. k k. k. k. k k. 1. I. I. J. I. I. J. I. 2. J. J. J. ɧÁ&ÊÉ¡Ë.
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(19) . . . . . . . . . . "!$#&%('*)"+. ¤AYJUfKAOJ&LN=?A XwJX|d^S [PAYfKXwgLPUVRTiapA Ui9LP[NRKfKXwg A2LMRJPALhJR^;[PAYfKXwgLNURsi[NX?SVAOJO ¨ 9¢ P`N¢ ²£ g RTiwJPUJML VU i%SVA¡LLPRb[NU}s=sLRT[NUAOβis LhdpLNURsiR^LN=?A¨X|JXwdTS|AYW9XwdpLNURsiwJ 9AªKe?[NAOJNJPUi?} (E, +, ∗) Q;A UVi?} dg RT@ @XKLhdpLNUsA`X?i?ULNdT[PH [NUVi?}
(20) . Ê&Ê ÆPQSR TU.
(21) T¢h {. (0+). e+0→e. (C+) (A+). e 1 + e2 → e2 + e1 (e1 + e2 ) + e3 → e1 + (e2 + e3 ). (I+) (−0). e1 + (−e1 ) → 0 −0 → 0. (−−) (−+). −(−e) → e −(e1 + e2 ) → (−e1 ) + (−e2 ). (0∗) (1∗). e∗0→0 e∗1→e. (C∗) (A∗). e 1 ∗ e2 → e2 ∗ e1 e1 ∗ (e2 ∗ e3 ) → e1 ∗ (e2 ∗ e3 ). (D). e ∗ (e1 + e2 ) → e ∗ e1 + e ∗ e2. (∗−). e1 ∗ (−e2 ) → −(e1 ∗ e2 ). < =wAJA LR^d^SV}TA Qw[NdTUVg[PAYfKXwgLPUVRTiwJ'UJfKAOi?R^LNAOf 6< =?Ai?R^LhdpLNURsi e → e USVS\Q;AXwJPAOf§RT[+LN=?A [NA´|AªKUsAdTiwf6LP[hd^iwJPULPUVTA'gSVRsJPX?[PA'RTLP=wUVJJPALRT&[NAOfKALG X|gLPUVRTi|J gOd^eKLNX?[PAYJLP=?A=?RT@ Rs@bRs[Pew=?UVg`iwdpLNX?[NA`R^ < =wAJPAOg RTiwfJPAL R^&[PAYfKXwgLNURsi%[NX?SVAOJO9fKAOi?R^LNAOf dTiwf%LP=?A'²dsgLLP=|dpL 0 d^iwf 1 d^[NA¨RpsA [NSR9dTfKAYfdTJg RTHOM iwJLNdTi9L§XwiwgLNURsiwJUVi%LN=?AgRsisLNAªILR^ λ g dTSVg X?SVXwJ •
(22) 7→ (•c) alg. (∗ app) (+ app). (a ∗ bc) → (ac) ∗ (bc) (a + bc) → (ac) + (bc). (− app). (−ac) → −(ac). (1 app) (0 app). (1c) → 1 (0c) → 0. 0. BKR+²d^[2§RT[\LN=?A[NAOfKX|gLPUVRTi[NX?SVAOJO ASVAL JLNdTiwf?J\§RT[2QwUiwdT[PHb[NA SdpLPUVRTi =?Ugh=UJnLN=?ALN[NdTiwJULPUVTAg SR9JXw[PA RT&LN=?A'e?[NA gRTiw}T[NX?A iwg A¨QwX?USLRTi4d^SVSLN=?Ad^Q;RpTA[PAYfKXwgLPUVRTi$[PAOSVd^LPUVRTiwJO ¾ 2¾ ² ¾
(23) ¾ º º?¢$N¢ ²£ ¢ £ K h£ w¢7h¢£p·p ¡ e, e , e ∈ E
(24) ?¢;¢ p¢ ¢ e e e ?¢ P¢b¢ F e ? e e e e ¾a¾ "!\d^[NUVdTisLRTLP=wA+XwJX|d^S< d^UL Dtd^[PLPUVi6$_ #^@ ALP=wRIfawd^SVRTi?}LP=wA+AªKe?Sd^iwd^LPUVRTiwJ}TUVTAOi6UVi D yYKx I < =wUVJ e?[NRTe;A [PLMHUVSSaQ;AfKUJPg XwJNJAYf6dTSVJPRdpL LP=?A+AOiwf$R^JAYgLPUVRTi4o? 1. 1. &%. . 2. ')(. + #. 3. +* -# ,/. '. 10 ). +. . 1. 2.43. % #&+. 53. '. 3. 63. ". 2. 2. 3. . ¶UVTAOi·dfKA iIX?@ A [hd^QwSA+JPAL879;:-< RTLMH9e;A+pd^[NUVdTQ?SVAOJOK[NdTi?}TAYf%RpsA [QIH wdTiwf6LP=?ALMH9e;Ag RTiwJLNdTisL ω ?A f?A Mwi?A'LN=?AJPAL=79>:-<RTLMHIe;AOJ σ }TAOi?A [hdpLNAOfQIH%LP=?A'§RsSSVRpUiw}}s[NdT@b@dT[ t
(25) . σ. ::= ω|t|σ ∧ σ|σ → σ. ɧÁ&ÊÉ¡Ë.
(26) ¢h£¢/ ²¢
(27) £h£` ²£ G. < =wAJA L/79;:-<UJ e?[PRpIUfKAOf6ULP=$LP=?A+ew[PAORT[hfKA [ 6 ew[PAYJAOisLNAOf$dTJLP=wAJ@d^SVSVAOJL Q?UViwdT[PH[NA SdpLNURsi$JPXwgh=6LP=wd^L U. 6 UJ Q;R^LP=4[NA´wA ªKUsA'dTiwf%LP[hd^iwJPU¥LNUsAT¸ UU. σ ∧ τ 6 ρ U σ 6 ρ dTiwf τ 6 ρ ¸ UVUU. §RT[dTSSlLMHIe;A σ σ 6 ω ¸ Ul σ 6 σ dTiwf τ 6 τ UV@ e?SH σ → τ 6 σ → τ ¸ l σ → ω 6 ω → ω ¸ 9U. (σ → τ ) ∧ (σ → ρ) 6 σ → (τ ∧ ρ) < =wA[NX?SVA4°²?±`UJRs[LN=ti?RTLPUgUVi?}w LUJ'i?R^L+d6ewd^[PLPUgX?Sd^[UViwJMLhd^iwg ART[NX?SVAUVUU¡± L+=wdTJQ;A A iUi9LN[PRKfKXwg AOf UVi D o I\§RT[LP=?AJLP[NUVgLg dTSVg X?SX|J ALhd^TAbLN=?UVJ[PAOSVd^LPUVRTi¦Ui9LPRtdsg g RTX?i9L+UiLN=?UJd^[PLPUgSVA §RT[+A ªIew[PAYJPJPUiw}6LN=?A UfKAYd+LP=|dpL ω → ω [NALNdTUi|JnLP=wA ¢hK l@ ²S¢ N¢£ £p ²£ 4LN=wdpL A`gRsX?SVf}TAL§RT[ d'LPAO[P@4¸9LMHIe|AYfULP=?UVi ω RTLP=?AO[PUJPAT iR^LN=?A [Rs[NfwJ sSA L = JLNd^i|f?Jn§Rs[2LN=?A¨JPHI@b@ A LP[NUVg`g SR9JXw[PART 6 < =wA i σ 6= ω U σ 6 ω → ω 0. . ' ,/.. 0. 4* 6 3& ). %). 0. ! . 0. < =wAtfKAOJPU}si¬RT`LN=?A·Ui?§A [NA iwg A·[PX?SVAOJ UJd^i UV@ e|Rs[Lhd^i9L@ d^LLNA [Y i¬LN=?UJd^[PLPUgSVATA$LNdTTA$LP=wA·e|RsUi9LR^ IUVA LN=wdpL4d^iIH LMHIe;AJP=?RTXwSVf Q;AUVi9LPAO[Pe?[NALNAOf dsJ%d JPAL$R^+LNA [N@ J6=?Ugh= UJUVi9pdT[PUd^i9L%X?i|fKA [4d^SV}TAOQ?[NdTUVg g RT@ e?XKLhdpLNURsiwJ n< =IXwJO?dLMHIe;A'UVJ @ AOdTi9LLNR Q|A+d^i4dTfwfKU¥LNUsAJPX?Q }T[NRTXwea µ¨J§RT[ LN=?A'@X?SLPUVe?SVUVgOdpLPUVRTiK}TUVTA i R^ LMHIe|A wdTiwf d^iIHLNA [N@4?A'A ªIe;AOgL LPR Q|AR^LMHIe;A LPRIRw\< =?AO[PA §RT[NAT?dbLMHIe;A+UVJ @ AOdTi9LLPRQ;A a dTi4UVfKAYd^S.< σ =wUVJ`UVJ eLhd^sA i4Ui9LNR%dTgOgRTXwisLQIH$e[PX?∗SVAOaJ (R . . . .) BIR LNσ=?Ag RTiwJLNdTisL 0 Q;A SVRTiw}sJ LNR%d^iIHLMHIe;AT ¤XKL Q;A SVRTi?}9JLPRbi?RTi?AsKAª?gAOeKL ω ¸IdsJPJPX?@ UVi?}RTLP=?AO[PUJPA¨RTX?SfSVAOdsfLNR e ∗ 1 ∈ σ §Rs[ dTSS e ?dTJ JPR9Rsi6dsJ 1 ∈ σ 1 L`A iwg AT wJPUViwgALMHIe|AYJd^[NA'd^SJR AªKe;AOgLPAOf6LNRbQ;AJLNd^QwSA'QIH%[NAOfKX|gLPUVRTi R^LNUVeg A∈d^σSJR LN=wdpL`A=wd³sA'[PAYJMLN[PUgLNAOf6LP=?A[PX?SVA (ω) UitJPXwgh=td d³H§[NA A+pd^[NUVdTQ?SVAOJRT&LNA [N@ e @XwJL`Q;A f?AOgSd^[NAOfUi6LP=?A+g RTi9LPA ª9L ∆ §RT[LN=?A2]MX|fK}TAO@bAOi9L ∆ ` e : ω LNRQ;A+d^SVSRpAOf< =?AR^LN=?A [ [NX?SAYJdT[PA¨LP=wA¨XwJPXwdTS Rsi?AOJO (Hyp) (→ I) (→ E) (T ω) (T 6) (R0) (R−) (R+). Ê&Ê ÆPQS(R∗) R TU (T ∧). ∆ ` (x : σ) ∈ ∆ ∆`x:σ ∆, x : σ ` e : τ ∆ ` λx.e : σ → τ ∆ ` a : σ → τ∆ ` b : σ ∆ ` (ab) : τ ∆` FV(e) ⊆ FV(∆) ∆`e:ω ∆ ` e : σσ 6 σ 0 ∆ ` e : σ0 ∆` ∆`0:σ ∆`a:σ ∆ ` −a : σ ∆ ` a : σ∆ ` b : σ ∆`a+b:σ ∆ ` a : σ∆ ` b : τ ∆ `a∗b: σ ∆ ` e : σ∆ ` e : τ ∆`e:σ∧τ.
(28) T¢h E. µ¨J2X|JXwdTS/9A[PULPA ULP=?AO[PA¨AªKUVJLNJde?[NRIR^A i|fKUiw}ULP=LN=?UVJ]MXwfK}sA @ A i9L d^SVRTi?}+LN=?A`SVUVi?AOJR^ LN=?A`dTQ|RpsAUiK§AO[PAOiwgA¨JHKJ∆LPA @4` e µ`: ρJNJXw@bUVi?}+LN=?ASdTJL2[NX?SAdTe?e?SVUAYf UVJ2i?A ULP=wA [¨°/< ∧±°²< 6±iwRT[`°/< ω ±9Aª?dTgLPSVH Rsi?AUViK§AO[PAOiwgA[PX?SVAdTe?e?SVUAYJ lfKAOe|AOiwfKUVi?}6Rsi·LP=wAbJLP[NXwgLNX?[NAR^\LNA [N@ e _aA LXwJfKAMwi?A ∆ ` e : ρ §Rs[dTiIH f?A [NUpdpLNURsi6=wUVgh=$A iwfwJU¥LN=·dbJLP[NXwgLPX?[hd^S[NX?SVAT ¾ 2¾ ² ¾ " O &T
(29) &³ *~ ! * » ² ?¢ ;¢ p¢ ?¢ ? ρ ∧ . . . ∧ ρ 6 ρ ∀i.∆ ` e : ρ ∆ ` e : ρ ∃n, ρ . . . ρ ^
(30) 2 K ∆` 0:ρ s. 1. s. . . . . . 1. n. s. n. i. £p £ ¢ £p £ ¢. ∆ `s x : ρ (x : ρ) ∈ ∆ O ∆, x : σ ` e : τ σ, τ ∆ `s λx.e : ρ ρ = σ → τ. ( s ∆`b:σ σ ∆ ` (ab) : ρ ∆ ` a : σ → ρ s ∆ ` a + b : ρ ∆ ` a : ρ ∆ ` b : ρ s ∆ ` −a : ρ ∆ ` a : ρ ∆ `s a ∗ b : ρ ∆ ` a : ρ ∆ ` b : ρ . . . . . . . p£ ¾a¾ < =?AJPA iwJPUQwSA+ewdT[LUJe;RTUVisLU¡± =?Ugh=4dTgLNXwd^SVSVH%[NA SVUAYJ RTi·JRsSIUVi?}LP=?A+LMHIe;AOJUVi?AYWsX|d^SVU¥LNUAYJ γ ∧ δ 6 dTiwf®° V A=?UVf?A+LP=?AfKALhd^UVSVJ`=?UVgh=·fKAOTAOSRse4LP=?AO@ JPA SVTAYJdTJXwJX|d^SUVi4LP=wA ρ→τ g RTi9LPA ªIL RT&LY dw JOV γ → δ ) 6 ρ → τ º 6º ¿ ^ ¾ ∆ ` e : σ e e ?¢ ∆ ` e : σ n i=1. i. i. 0. +(
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(37) £h?£^ £ 4£ E E[X] ·¾ 2º ¤HtgRsiwJLP[NXwgLNURsiaad^iIH4e|RsSHIi?Rs@ UVdTS UJ'gRsisLNUiIX?RsXwJ[L'=?UJ¨LPRTe;RTSVRT}sHTBIR$UJ (•e) §RT[+d^SVS e ∈ E dsJ`de|d^[PLPUgX?Sd^[g dTJPAT< =wUVJ¨[PAYJXwS¥L'gRT@ ewdT[PAYJULP=¶=?UVSVA ¯OdTi D yY r I |=wA [NAdLNRTe;RTSVRT}TH6RTi Λ UVJ¨Q?X?UVSLb°¡§Rs[ AYdTgh=·gRsi9LPAªIL ∆± ULP=·Q|dTJPUVJ (V ) d^iwf i·Q;R^LN=tg dsJAYJ d^QwJLP[hdTgLNURsi f?R9AYJi?R^LJPA A @ LNRQ;A+gRsisLNUiIX?RsXwJOµ`iwf$dTJ2²d^[dTJVLP=wA'dT≡e?e?{eSVUVgOdp| LN∆URsi%` UeJ:g RTσ}iwg A [Ni?AOfasLN=?A+JA LNJ Vλ g RTX?Sf%Q;A [NAOdsf6dsJ Q|AOUiw} g SR9JAYf6dsJ AOSS.KULP=$LP=?AJNd^@ A+gRsi9LPUVi9XwU¥LMH[NAOJPX?SLO . . . ∆,σ σ∈Type. ∆,σ. ∆,σ. . $ &!$ '& . _AL JPX?Q [NUi?}6gSVRsJPAOf4QIH4[PAYfKXwgLNURsiwJOldTiwf e?[NUV@bAbd^iwfe?[NRTe;A [`UfKAYd^S R^ R ;ULP=[PAYJe;AOgL`LNRLP=wA UViwfKX|RgAOf$⊆[NEUiw} JLP[N XwgLPX?[NAT ¨A Mwi?A'LN=?A'§RTSVSVRpUi?}g RTiwπfKULPUVRTi|J YK * º uwRT[d^SVS JPXwgh=6LP=wd^L FV(e) ⊆ {x , . . . , x } e(R ) ⊆ R e∈E ¿ ²OK » π t=wA i?AOTA [ (e¯u) ∈ π LP=wA i e ∈ π &º º O? uwRT[dTSS ¯ (0¯ π0 u ¯∈R u) ∈ π O? =?AOi?A sA [ LP=?AOi (−a¯u) ∈ π π− (a¯ u) ∈ π O? =?AOi?A sA [ d^iwf (b¯u) ∈ π LP=wA i (a + b¯u) ∈ π π+ (a¯ u) ∈ π O? =wA i?AOTA [ dTiwf (b¯u) ∈ R LN=?A i (a ∗ b¯u) ∈ π `°²dTiwfLN=?AJPH9@ @ ALN[PUg'g dsJAs ± π∗ (a¯ u) ∈ π O? =wA i?AOTA [ U¥LN= a ∈ R LP=?AOi (λx.ea¯u) ∈ π. πβ (e[a/x]¯ u) ∈ π < =wA2JPd^LPX?[hdpLNURsi'e?[PRse|AO[LNUAYJ (sat π . . .) gRs@ A\U¥LN=i?R`JPX?[Ne?[NUVJPA§RT[LP=?A[PAYdTfKAO[O< =?AJMLhd^Q?UVSULMH'gRTi|fKU¥LNURsi Rsi LP=wA¦[NUVi?} `UJ$[PAYW9X?U[NAOf §RT[$LP=wALMHIe?UVi?}©[PX?SVA (ω) °§=?AO[PAYfKULNd^[NH π ±6A iwJPX?[NAOJ6LP=?A¦i?Rsi (stable R) AO@ eKLMHIi?AOJNJ eK LP=?A'§RsSSVRpURiw}bUVi9LPAO[Pe?[NALhdpLNURsia A Mw[hJMLe?[NRpIUVf?AdTi6UVi9LPA [Ne?[NALhdpLPUVRTi$R^&LMHIe|AYJd^SVRTi?}LP=wA+SUVi?AOJR^
(38) U. [ω] = R dTiwf [t] = π §Rs[AOdsgh=LMHIe|A+pdT[PUd^Q?SVA t ¸ 1. Ê&Ê ÆPQSR TU. n. n.
(39) T¢h. yYx. U U. [σ ∧ τ ] = [σ] ∩ [τ ] ¸ UVUU. [σ → τ ] = {e ∈ π|∀a ∈ [σ].(ea) ∈ [τ ]} = [σ] → [τ ] < =wAUi9LNA [Ne?[PA LNd^LPUVRTi4UVJ`AªILPAOiwfKAYf4LPR%g RTi9LPA ªILNJQIH = {id} d^iwf [∆, x : σ] = {s[a/x]|s ∈ [∆], a ∈ [σ]} 2SAO@bAOi9LNJ s ∈ [∆] dT[PAbJX?Q|JMLNU¥LNXKLPUVRTiwJOQ?XKL+AdTSSV[ε] RpjRTX?[hJPA SVTAOJLPR$XwJPALN=?AbdTe?e?SVUVgOdpLNUsAi?RTLNdpLNURsi e(s) = T d J A V S / S e(a , . . . , a ) ¾ 2¾ ² ¾ &£p p ¡ σ [σ] R s¢hp
(40) w [σ] ⊆ π σ 6= ω &£pbp ¡ σ, τ σ 6 τ w¢ 1. n. [σ] ⊆ [τ ] . ¶UVTAOi dTiwf JX|gh=LN=wdpL KfKAMwi?A U¥ n< =?AUVisLNA [Ne?[NALNd^LPUVRTi RT|]MX|fK}TAO@b∆AOi9L e∆ ` eσ: σ UVJ LN=IXwJ FV(e) AOSS&fKA Mw⊆iwFV(∆) AOf$QIH [∆ ` e : σ]∆ ≡ e∆: σ e :e([∆]) σ ⊆ [σ] ¾ 2¾ ² ¾ " $º&º ² º º OK ¾ ∆ ` e : σ ?¢ ∆ e : σ ¾a¾ iwfKAO[LP=wAd^Q;RpTA'dTJNJX?@ eKLNURsi6Rsi%LN=?A+ewdTU[ (R, π) KLN=?UVJe?[PRIRT&e?[NRKgA AYf?JdTJXwJPXwdTS/ ·¾ º µ¨JPJPX?@ AtA$ d^i9LLNR®XwJA4LN=?UJ§[hd^@ A RT[NLPR®e?[NRpTA·g RTiK´wXwA iwg AT < =?Atg RTiwf?U¥LNURsi (stable R) gOd^i?iwR^L2Q;A§X?NS MwSVSAYfX?i?SVAOJNJ\ALNdTTA n< =?A iTLN=?A`[NAOW9X?UV[PAO@ A i9LnLP=wd^L UJ2d^iUfKAYd^Swew[PAOTA i9LhJnXwJn§[NRT@ g RTiwJPUfKA [NUiw}'LMH9e;AUVi9LPA [Ne?[NALhdpLPUVRTi|J\dTRJJ=X?Q|EJA LNJ\RTLPAO[P@J\=wUVgh==wd³sA LP=?AgRsπiK´wX?AOiwgAew[PRse|AO[LMHs&< =?AO[PA §RT[NAT f?X?ALNRLN=?Ae?[NAOJPA i|gARTnLP=?Ab@XwS¥LNUe?SVUg dpLNURsi lAgOd^i?i?RTLdTe?e?SVH6LN=?A@ A LP=?RKfJPX?}s}TAOJLPAYf4Ui D ysy I ;=wUVgh= ]MX|JMLNU MwAYJ LP=wAfKU[NAOgLe?[PRIRT&UVi4JPAOgLPUVRTi¦yT ∗ i D yOo I 2Znd^RsSUVi?U d^iwf¬cRTiwgh=w(U ¨A SVSdcRKgOg de?[PRse|R9JA$dt§Rs[P@dTSe?[NAOJPA i9LhdpLPUVRTi R^¨fKU lA [NA i9L g d^SgXwSU. Q|dTJPAOf6RTi·dbi?R^LNURsi$RT+ ^ ³^ w¢ sBIXwgh=·d JA LUVJ`gSVRsJPAOf6Q9H6JX?Q|JMLNU¥LNXKLPUVRTiwJUiITRsSIUVi?} pd^SVX?AOJO?gλSV RsJPAOf6QIH [NAOf?XwgLNURsi dTiwfgRsi9LNd^UViwJnpdT[PUd^Q?SVAOJO&< =wASVd^LLNA [2g RTiwfKULPUVRTi UJ\i?RTL2ewdT[L\RT;RsX?[2gRsiwfKULPUVRTiwJ2RTi R UVi}TAOi?A [hd^S. QwXKLUVJ`JPd^LPUJ MwAYf%UVi$LP=?A+iwAªILJAYgLPUVRTi. & $
(41) $+(+& . UVi?}%LPR6LP=?Abe?[NA IURsXwJ`dTiwd^SVHKJUJO|LN=?A JMLN[PRsi?}i?Rs[P@d^SVUJPd^LPUVRTi·UJPJPX?A§Rs[ g dTitQ|A fKAYd^SL'dTSRsi?}LN=?AbJNd^@ A SVUVi?AOJdsJ§RT[e?X?[NA λ gOd^SgX?SVXwJO_ALXwJg RTiwJPUfKA [ LN=?AfKU ;AO[PAOiwgAYJ Mw[NJLO E ¨X?A'LNR LP=?A+e?[NAOJPA i|gA'R^ndTJNJRKg UVd^LPUVTA gRs@ @XKLhdpLPUVTA¨[PXwSAYJ Ui ?Ag dTi?i?RTL A ªKe|AYgL¨d JLP[hd^UV}T=9L§Rs[P d^[hf iwR^LPUVRTi$R^nJLP[NRTi?}bi?RT[N@d^SVUVJNdpLNURsia QwJPA [N TA¨LP=wd^L}TUVTA i4d^iIHLNA [N@ Ee KLP=wAJA L {e |e → e } UVJ MwiwU¥LNAT2_AL X|J`fKA Mwi?A e B e U e → e ld^i|f e ⇒ e U e → e d^e?ewSHIUVi?}%d^iIH6[PAYfKXwgLNURsi·[PXwSAswQ?XKLi?R%dsJPJPRKgUdpLPUVTA Rs[`gRs@ @XKLhdpLPUVTARTi?As SVAOdT[PSVHT;d^iIH$[PAYfKXwgLNURsi4ewdpLN= e → e gOd^i·Q;AfKAYJPg [PUVQ|AYf·dTJ¨d^itd^SLPAO[Pi|dpLPUVRTi4RT B dTiwf JMLNA ewJO|UVitdX?iwUVW9X?Ad³Hs _AL `(e → e ) JLNdTiwf?J§Rs[LN=?AiIX?@Q|AO[`R^ ⇒ JMLNA ewJ`Ui·LN=?A[NAOfKX|gLPUVRTi dTiwf·⇒ g dTSSaLP=?UJ`UVi9LPAO}TA [`U¥LhJ ¢ a< =?AOi e ∈ E UOJ .P£ ^ ;£p p S 4U LN=?A [NAA ªIUJLNJ`dTi·X?e?e;A [¨Q|RsX?iwf LNRLN=?A'SAOi?}^LN=wJ R^ [PAYfKXwgLNURsi%ewd^LP=wJ Rs[PUV}TUViwd^LPAOf%dpL \< =wUVJ fKA Mwi?ULPUVRTi$AªILPAOiwf?JLP=wA'g SVdsJPJPUg d^S;RTiwA kek ∈ JPUViwgA e B e UJ LP=?A'Rsi?SH6e|R9JPJPUVQ?USVULMH§RT[e?Xw[PA λ LNA [N@ JO e A [PAYJMLN[PUgL'LN=?A LMHIe?Uiw}$[NX?SVAOJ'QIHfK[NRTewe?Uiw} µ`Jd$@dpLPLPAO[+R^g RTiwJPAOW9X?AOiwgAsLMHIe|Ag RTiwJLNdTi9L iwA TAO[`RKg g X?[hJUVid^iIH]MXwfK}TAO@ A i9L ∆ ` e : σ UVi·LP=?(TAω)[PAYJXwS¥LNUi?}6JHKJLPA @4 LA i|gATlgRsiwfKULPUVRTi (stable R) g dTωi Q;A6f?[PRse?e|AYf®dTJAOSS. RTLPUgA6@bRs[PAORpTAO['LP=wd^L ∆ ` 1 : σ U σ = ω < =?A [NA§Rs[PA%RTX?[+LMHIe;A6dsJPJPU}si?@ A i9L JPHKJMLNA @ fKRIAOJ¨i?RTLdTgOgAOeKLLP=wAbg RTiwJLNdTisL d^iIH$@ RT[NATld^SLP=?RsX?}T=·ULUJJMLN[PRsi?}TSVH%i?Rs[P@dTSUJUVi?}%dsg g RT[hfKUiw} LPR LN=?A'e?[NA IURsXwJ fK A M|i?U¥LNURsia X?[ e;RsJPU¥LNURsi%U1JLP=wd^L 1 gOd^i?i?RTLQ;A+dTgOgA e?LPAOfLPR =wd³sAdTi9He|d^[PLPUgX?Sd^[e?[NRTe;A [PLMHT < =wUVJ'UJgRsiwJPUVJLPAOisL+ULP=¦LP=wAXwJX|d^S\d^ewe?[PR9dTgh=?AYJ`=wA [NA d4JPe|AYgUd^S\g RTiwJLNdTisL ⊥ RT[ Ω UJ+UVi9LP[NRIf?XwgAYf< =?A f?U lA [NA iwg A%UJ=?RpA sA [+LN=wdpLbA6fKRi?RTLXwJPA%dTi9H¦RT[hfKAO[PAYf¦JLP[NXwgLPX?[NAT\dsJ+LP=wA%[NAOdTf?A [@d³Hd^SV[PAYdTfKH=wd³TA [NA @dT[PsAOf . . 0. ∗ AC. 0. 0. 0. 0. ∗. ∗. . ∗ AC. 0. 0. 0. . . . ɧÁ&ÊÉ¡Ë.
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(43) £h£` ²£. º . yTy. ?¢ ¢ R £ /N£ T ;£^ ^ ¢ ; H 1 %£ E ¾a¾©< =?AJX?Q|JA LRT;[NX?SVAOJ A gh=wRsJPA§Rs[\d^SV}TAOQ?[NdTUVg2[PAYfKXwgLPUVRTiwJe?[NA sA i9LNJ XwJ §[NRT@XwJUVi?}+d °§@ RT[NAY± JPU@ e?SVA e;RTSVHIi?RT@ Ud^S;Ui9LNA [Ne?[PA LNd^LPUVRTia X?[gh=wRTUgA`JPA AO@bAYf@ RT[NAiwd^LPX?[N[hd^Sw[PL@d^LP=?AO@ d^LPUg dTS|XwJNd^}sAT9d^iwf[NAOW9X?UV[NAOJ UViwJLPAYdTf%LP=?A+XwJPXwdTS[PAYgX?[hJUVTAewd^LP=4RT[hfKA [NUVi?}Rsi6=wUVgh=4ULUVJLN=?A i4e;RsJNJUVQ?SVA¨LNR[PAYdTJPRTi6Uiwf?XwgLNUsA SVHT _AL XwJgRsiwJPUVfKAO[LN=?A UfKAOdTS }TAOi?A [hdpLNAOf QIHLN=?A$LPA [N@J u) U¥LN= u¯ ∈ R¯ ©< =?UVJbUfKAOdTSUVJbA SVS f?A Mwi?AYf·d^iwf4ew[PRse|AO[Owg RTX?SRf4 Q;Ae?[NU@ π AdsJAOSS.wQwXKL¨UJi?RTL`}TX|d^[hd^i9(x¯ LPAOAOf%LPR%JNdpLNUVJ§HLN=?A+§RT[N@ A [`JNdpLNX?[hdpLPUVRTi ew[PRse|AO[LNUAYJ ALhd^TA§RT[ π JPRT@ Ab@ d^ªKU@d^S UfKAOdTS&=?Ugh=UViwg SXwf?AOJ π d^i|fd³TRsUVf$LN=?Ab@X?SLPUVe?SVUVgOdpLPUVTA JPAL < =?UJUfKAOdTSAªKUVJLNJdsJd }TAOi?A [hd^Sl[NAOJPX?S¥LRT µQ|JMLN[NdsgL`µSV}TAOQ?[hd?D·RT[NA RpsA [Y {1} º ?4¢ T¢hp π l ¢b 'lP£Nw¢ R T¢h^
(44) H ¢O ?(¢ KN ²£ h£ ²£ ¾a¾ π UVJ@d^ªIUV@d^SI=?AOiwgA e?[NU@ As 1 6∈ π QIHgRsiwJMLN[PX|gLPUVRTi
(45) π UJe?[NRTe;A [Y&_ALnXwJgRsiwJUfKAO[&LN=?AgRsiwfKULPUVRTi JPXwgh=bLP=wd^L & (λx.ea¯u) 6∈ π 9JPUi|gA π UVJ\@ d^ªKU@d^S.pLN=?A [NAAªKUJMLhJ p ∈ π (sat πβ). Take a ∈ R u) ∈ π dTiwf b ∈ R JPXwgh= LP=wd^L 1 = (λx.ea¯(e[a/x]¯. ¤ K X LLP=?AOi 1 = (e[a/x]¯u) ∗ b + p ∈ π =?Ugh=UVJd^Q|JX?[hf LP=?AO[ u) ∗ b + p gOdTJPAOJ d^[NA+JUV@ USd^[Y ¾ 2¾ ² ¾ ∆ ` e : σ
(46) H ?¢ P ¢ / ¢ N?¢%K S ¢ & K ¢ w ¢ e ∈ R w ¢ @h¢ e .P£ ^ ;£p p S ¾a¾ u&UV[hJML [σ] ⊆ π UJd^SVd³HKJ LN[PX?AUitLP=wd^L'gOdTJPAT`¤H$e?[PRse|R9JULPUVRTimK;A}TAL e([∆]) ⊆ [σ] ⊆ π ⊆ R Ag dTi$gh=wAOgh%dTSVJPRLN=wdpL π ⊆ [σ] =?Ugh=$@bAYd^iwJLN=wdpLAOTAO[PH [σ] has variables \< =IXwJ e ∈ R dTJ A SVS. ( . . . . . 0. 0. . . . . . . 0. $ 6# $(a \& i LP=?UJJPAOgLPUVRTiaA%d^[NA%g RTiwg A [Ni?AOf¦ULP= ew[PRs}T[hd^@J &=?Ugh=®@bAYd^iwJLP=|dpLA%UVi9LPA [Ne?[NALRsX?[gOd^SgX?SVXwJUi LN=?AJPAL E R^\gSVRsJPAOf6LNA [N@ JO SAYd^[NSHs E UJd[NUi?}|¸wJPRA'Lhd^TA R ≡ E µgOgRT[hfKUVi?}bLPR%JPRT@ A+AO³dTSX|dpLPUVRTi @ AYgh=wd^i?UJP@$ AAªKe|AYgLew[PRs}T[hd^@J¨LPR·Q;AA pd^SVXwd^LPAOf¦LNR·pd^SVX?AOJO$uwRTSVSRpUVi?}$LP=wA%gSdTJNJUg dTS\d^e?ew[PR9dTgh=aaA @d³H%f? A Mwi?A'LN=?AJPAL π ⊆ R R^ ³^ w ¢ +dsg g RT[hfKUiw}LNRLN=?A+}T[hd^@ @d^[
(47) . . . .
(48) . . 0. 0. . 0. 0. QwXKLUL+J=?RsX?Sf·Q|A gSVAOdT[LN=wdpL d^QwJLP[hdTgLPUVRTiwJO λ. Ê&Ê ÆPQSR TU. v. π0. ::= λx.e|e ∗ v|v ∗ e| − v|v + v|0. UVJ'fKAMwi?AYf·AOW9X?UVpd^SVA i9LPSVH·dTJ`LP=?AbUVf?AOd^S }TAOi?A [hdpLNAOf4QIH4LP=wAJPAL'R^`°²g SR9JAYfw±.
(49) T¢h. y³r A i?RpfKA Mwi?ALP=?AAOpd^SVXwdpLNURsitR^e?[PRs}T[hd^@JX|JUVi?}4i?R^LhdpLNURsi g RTiwf?U¥LNURsiwJd^[NA+dTgLNXwd^SVSVHewd^[PLR^&LP=?AfKAMwi?ULPUVRTi4R^ π
(50) (Value) (App −λ) (App −0) (App −+) (App −−). e⇓v. UVJ'JPXwgh=¦d6 d³H6LN=wdpLJNdpLNX?[Nd^LPUVRTi. v ∈ π0 v⇓v (e[a/x]˘ u) ⇓ v (λx.ea¯ u) ⇓ v (0¯ u) ∈ R (0¯ u) ⇓ 0 (a¯ u) + (b¯ u) ⇓ v (a + b¯ u) ⇓ v −(a¯ u) ⇓ v (−a¯ u) ⇓ v (a¯ u) ⇓ v(b¯ u) ∈ R (a ∗ b¯ u) ⇓ v. . Td iwf6LP=?AJPHI@ @bA LP[NUVg'gOdTJPA AJNd³HLN=wdpLLN=?Aew[PRs}T[hd^@ e ?K pp w¢Rs[ h£ p¢ s¢ TlfKA iwR^LPAYf e ⇓ wUnLP=?AO[PAA ªKUVJLNJ v ∈ π JPXwgh=4LP=|dpL dTg g RT[hfKUVi?}%LNR·LP=wAe?[NA IURsXwJ'[NX?SAYJ LP=?AO[PUJPATaLN=?Ae?[NRT}s[NdT@ ³¢ 9¢ T §Rs[J=wRT[PLO6_ALXwJ e⇓v iwR^LPUgALN=wdpL 1 UJQ|RTLP=·gSVRsJPAOf$d^i|f6fKUVTAO[P}sA i9LdTg g RT[hfKUVi?}LNRbLP=?AYJAfKA Mwi?eULPUVRTiwJO e ⇑ º &¢ π ≡ {e ∈ R|∃v ∈ π .e ⇓ v} ?¢ π l ¢ lP£Nw¢4 T¢hp £ R ¾2¾ ² ¾"
(51) ¾;º ;º º °yY± ∀e ∈ R `e:ω→ω e ∈ π iff e ⇓ ¾a¾ Z\[PRse|R9JULPUVRTi mte?[NRp9UfKAYJ'LP=?A6i?AOg AOJNJPdT[PH¦gRsiwfKULPUVRTia¦<Re?[PRpsA LP=?A$gRsiwfKULPUVRTi UJJPX gUVA i9LY A ew[PRKgAOAOf QIH©Uiwf?XwgLNURsi RTi LN=?AAO³dTSX|dpLPUVRTi [NX?SVAOJO¸dsJ§Rs[LP=wAQwdsJAgOdTJPATQIH©Uiwf?XwgLNURsi JMLN[PX|gLPXw[NdTSRTi pdTSX?AYJ (App −∗). . 0. . 0. . . )0. . 0. )"')"% !. (. )*#&+. < =wA`}TAOi?A [NUg@ RIf?A SlJ=?RpJ\LP=?A = V (I) §RT[ JPRT@ AUfKAOdTS|n=?Ugh=UJ2}sA i?AO[Nd^LPAYf Q9H AOSAO@bAOi9LNd^[NHLPAOJLNJOnµ ew[PUVRT[NU/ I UJ`iwR^LMwiwU¥LNA SVH$}sA i?AO[Nd^[σ] LPAYf|fKXwALPR%LP=?AUi9LNA [Ne?[PA LNd^LPUVRTi4R^nLN=?AdT[P[NRp LMHIe;AT`uwRTSVSRpUVi?} D y^l{I&A SVRIRT§Rs[d²d^@ USVH (e , ∇ )
(52) K=wA [NA e ∈ R d^iwf ∇ ∈ R[X] ?JPXwgh=%LN=wdpL ` e : τ U σ 6 τ ¸ ∀e ∈ R.∇ (e) ⇓ U ` e : σ < =wAOJPALMR4gRTi|fKU¥LNURsiwJ'AªKe?[NAOJNJ¨LP=?A ²dsgL'LP=|dpL+Q|RTLP=®JPALNJ'RT[NX?SVAOJ'=?Ugh=RKgOgX?[+RsitLP=wAJHIi9LNdsgLPUg dTSnJUfKA ° d^iwf [NAOJPe|AYgLNUsA SVHK±d^[NA·AªKe?[NAOJNJUVQ?SVA·Ui LN=?Ag dTSVg X?SVXwJULNJPA SM µ`J²d^[%dsJ%gRsiITA [N}TAOiwgA$UJ g σRTiw6g A [Nτi?AOfaKA e ;:¢h¢ σ6d^SJRbLPR [NA e?[NAOJPA i9LLP=?A+ew[PAYfKUVgOdpLNA e ⇓ _ALaXwJ=wd³TAndg SR9JAO[SRIRs¨dpL&e?[PRse|AO[LMH` L dTJNJAO[LhJ;LP=?A\A ªKUVJLPAOiwgA\RT JXwgh='LN=wdpL U ` e : ω → ω A+=|d³TA¨dTS[NAOdsfKH π = V (hXi) wd^i|fLN=?ALPAOJL X JP=?RTXwSV∇f6Q|A+JP(X) X gUVA i9L §RT[LP=w∇ UVJewX?[Pe;(e)RsJPAT⇓ µgLPXwdTSSVHTKA'i?A AYf4dQwU¥L@ RT[NAT?JPUi|gA'AOdsgh= gOd^[N[PUVAOJd^SJPRLN=?A'e?[NRTe;A [PLMH
(53) U ∇ (e) ⇓ LP=?AOi ∇ (e) ⇒ I =wA [NA I UVJLN=?A+UfKA i9LPULMH6g RT@Q?Ui|dpLPRs[O ∇ σ. σ σ∈. σ. σ. σ. σ. ω→ω. σ. ω→ω. σ. σ. ɧÁ&ÊÉ¡Ë.
(54) ¢h£¢/ ²¢
(55) £h£` ²£ µ¨JPJPX?@ ALP=?A+A ªKUVJLPAOiwgA'R^ ∀e ∈ R. y/. PJ Xwgh=6LP=wd^L UJdb[PUVi?} =?Rs@bRs@ RT[Ne?=?UJ@ ∇(•). ∇(X) ∈ R[X]. °.rT± ° s±. : E → R with kernel π. ∇(e) ⇓ iff ` e : π. < =wA i4A+fKAMwi?A'LN=?A+A SVA @ A i9LhJ (e , ∇ ) QIH%UViwfKXwgLPUVRTi4RTi σ dsJ d^i|f ∇ (X) ≡ I ¸ • e ≡1 d^iwf ∇ (X) ≡ ∇(X) + ∇ (Xe ) ¸ • e ≡ λx.(∇ (x)e ) d^i|f ∇ (X) ≡ ∇ (X) + ∇ (X) − I • e ≡ e ∗e ¾ 2¾ ² ¾ w¢ ¡ (e , ∇ ) ¢4h£ ²£ ¾a¾ u?RsSSVRpJ D yOKx I Ie?[NRp9UVi?}+LN=?A@bRs[PA`e?[PAYgUJAgRsiwfKULPUVRTi°² Ò±2JLNd^LPUVi?} ` ∇ (e) : τ → γ → δ U σ 6 τ dTiwf γ 6 δ ¾ 2¾ ² ¾
(56) ¾ * º ;º º O &£pp ¡ ∆ σ e ∈ E 2 w FV(e) ⊆ FV(∆) ∆ ` e : σ e([∆]) ⊆ [σ] .
(57) P ¢ K 5£^( w ¢ Yl ¢ 2 ?%¢ h£ ²£ £ O ?¢ 9 ¢ l¢ £ s¢ ?¢ h£ p¢ S¢ wp ;⇒P£ ³¢ £ @N¢b 9 K ( (R,π) <&Rp d^[hf?J§X?SVSd^QwJLP[hdTgLPUVRTia9A¨[PAYJMLN[PUgLRTX?[hJAOSsAOJ\LPRde?[NAOgUJPA`i?RTLPUVRTiRTRTQwJPA [NpdpLNURsigRTi9LNAªILOn¶UsA i ULP= ^d`pd^SVUVf'LNAOJL&§Rs[ UJ&dTiIH'g RTi9LPA ªIL ≡ P ([](s)) =?AO[PA P (X) ∈ R[X] e∈E dTiwf s JPX?QwJLPFV(e) ULPXKLNURsi¦≡JP{x Xwgh=·, LN. =w. .dp,L xs(x} ) ∈ R §RT[AOdsgh= ei = 1, . . . , n < C[] =?A gh=?RsJPA ifK A MwiwU¥LNURsit§RT[¨LPAYJMLhJ@ d³H JPA AO@@ RT[NA [PAYJMLN[PUgLNUsATaQwXKLULUVJdTS[NAOdsfKHtIi?RpiLP=wd^LLP=wAOJPAgRTi9LNAªILNJ+dT[PAdTgLPXwdTSSVHJPX gUVA i9L%°/JAOA §RT[ A ª?d^@ e?SVA¨LN=?AfKA LNd^UVSVAOf$e?[NR9RT²JR^ D o I§±_aALXwJ`fK A MwiwA+RTi E LN=?A+e?[NA RT[hfKA [hJ » K * ² » a v b U t§RT[dTSS ∆, σ ∆ ` a : σ UV@ e?SUVAOJ ∆ ` b : σ ¸ º K ¾ a v b U C[a] ⇓ U@ e?SVUVAOJ C[b] ⇓ §RT[d^iIHpdTSUf6LPAOJL C[] _AL Γ ` s : ∆ §RT[ Γ ` s(x) : σ §Rs[d^SVS (x : σ) ∈ ∆ \< =?A i º ?O º Γ ` e(s) : σ w ¢ ∃∆.Γ ` s : ∆ ∆ ` e : σ ¾a¾ ¤H%Uiwf?XwgLNURsi$Rsi$LP=?AfKAO[PUVpdpLNURsiLP[NA A Γ ` e(s) : σ &ºI¾ º * *K O³w^ ¾ v =v ¾a¾ ∆ ` a : σ d^iwf ∆ 0 b : σ A$J=wRp LP=?A$LPAYJML C[] = ∇ ([]s ) =?AO[PA s UVJ fKA Mwi?AOf QIH nJAOewd^[hdpLNAOJ a dTiwf b
(58) C[a] ⇓ dTiwf C[b] ⇑ RTiITAO[NJPA SVHT&LP=?A6e?[NR9RTUVJQ9H¦UViwfKXwgLPUVRTi Rsi s ≡ s ,e XwJPUiw}6gRs@ e?SA LPAOi?AOJNJ¨dTiwf4LN=?A²dsgL [ω → ω] = π UJ¨d%e?[NU@ AUfKAYd^Sd^iwf4LN=?A P ::= e|(X u ew[PAO9UVRTX|J°²Zn¯dT)|P JLPAY±∗SVA P@ |P@d?+ P σ. ω. σ. ω. σ→τ. σ. σ∧τ. σ. τ. τ. σ→τ. σ∧τ. τ. σ. σ. σ. τ. σ σ∈. σ. . . . 1. n. i. T. O. T. O. . Γ,x:τ. Ê&Ê ÆPQSR TU. Γ. τ. σ. ∆. ∆.
(59) T¢h. yOo. ·¾ º i D o? yOKx I lg d^SVS QIH pdTSX?A+UJUVi9LP[NRKfKXwgAYf4UitLP=?AbJHIi9LNd^ªlµ`S¥LN=?RTX?}s=4§RT[fKU lA [NA i9Le?X?[Ne|R9JAYJ wLP=?UJ U J[NAOW9X?UV[PAYf$Ui·Q;R^LN=tg dTJPAO J UVi· RT[hfKAO[ LPRAOJLNdTQ?SVUVJP=4§X?SVS&d^Q|JMLN[NdsgLPUVRTi$[NAOJPX?S¥LhJ < =?AfKAOSVd³HsAOf6Ui9LN[PRKfKXwgLPUVRTi4R^ LN=?AA SVA @ AOisL ∇ ∈ R[X] dTiwfLP=IXwJ+UVi E dsJ+d4@dpLLNA [RTg RTiwJPAOW9X?AOiwgAsUJ'i?R^LUVi9LPA i|fKAOf¦LNR4dT[P}sX?AbLP=wd^L JPXwgh= d$§AYdpLPXw[PAUJ+XwJPA SVAOJNJO A[hdpLN=?A [A ªKe|AYgLLNRt}TA LdtgSVRsJPA [+SVRIRTdpL+LP=wA%e?[NAOg UVJPA@ AOgh=wdTi?UVJP@=wUVgh= UJ[NAOW9X?UV[NAOf6LPRp d^[hf6LP=?UJ[NAOJPX?SLO i4²dTgLO ∇(a) + b A ªKe?[PAYJPJPAOJLP=?A[NAOW9X?UV[NA @ A i9LLP=|dpLg RT@ e?XKLhdpLNURsiwJR^ a g RTiITAO[P}sAOJOT=?UVSVAULfKUsA [N}TAYJ2RTLP=?AO[PUJAsI=?UVgh=wA TAO[2LN=?A+gRs@bewXKLNd^LPUVRTi%RT i?A'gRsX?SVf%=wd³TA¨Ui9LN[PRKfKXwg AOf dTi?R^LN=?A [RTe;A [hdpLNRT[dTJ¨AOSS
(60) `JNd³H ∇ (e) [NAOf?XwgUVi?}LNR I UViJPe?ULPAbR^ 0 +< =?AObi (∇ (a)b) RsX?SVftAªKe?[NAOJNJLN=?A ²dsgLLP=?A+g RT@ e?XKLhdpLPUVRTi|JdT[PA¨@dTfKAUi ¢ w¢ 7h¢sn< =?A¨SdpLPLPAO[=|dTJLP=?A+JNd^@ A`Rse|AO[Nd^LPUVRTiwdTS|Q;A =|d³9UVRTXw[dsJ2LN=?A g RTiwJLP[NXwgL e?[NAOJPA i9LPAYf·UVi D m I µ¨J¨²d^[+dTJi?R4JUfKAbA lAOgL'UJ'dTf?f?AOf·LNR%LN=?Ag dTSVg X?SX|J lLN=?A A ;AYgL+R^ LN=?A'§RT[N@ A [ seq(a; LNA [N@ b)UVJLP=?A+JNd^@ AT
(61) ?£ g RTiwJLP[hd^UVi?UVi?} d^iIHRT[hfKAO[Rsi%LN=?A+ d³H%gRs@bewXKLNd^LPUVRTiwJdT[PA'fKRTiwAT BKUiwg A`LN=?ALP=?A§X?SVSad^Q|JMLN[NdsgLPUVRTi%[PAYJX?SLNJ[NAOW9X?UV[PAYJLP=?A'UVisLN[PRKfKX|gLPUVRTi6R^aLN=?A RTe;A [hdpLNRT[YT§Rs[LP=?A'e?[NRIR^²J LNRQ;A³dTSUfIRTiwA`i?AOAOf?+J Mw[hJMLAªILPAOiwfKUVi?}LP=?A'JHIi9LNd^ªU¥LN=LP=wA¨i?AO LPAO[P@ g RTiwJLP∇[NXwgLNRT[ ∇(e) I=?A [NA e ∈ E < =wUVJ%Ui9LP[NRKfKXwgLPUVRTi [NAOW9X?UV[PAYJd^SJR dsf?fKULPUVRTiwdTS`[NAOfKX|gLPUVRTi [NX?SVAOJOj< =?A=?RsSAtJPHKJMLNA @"UVJ6e?[NAOJPA i9LPAYf d^iwf JLPX|fKUAYf$Ui$LP=wA'§X?SSTA [hJPURsi%R^ LP=wUVJdTQwJMLN[NdsgLY . 0. 0. 6 #$( < =wA}sA i?AO[PUg+@ RKfKA Se?[NAOJPA i9LPAYftUViJPAOgLPUVRTi 6e?[NRp9UfKAYJ`d6X?i?U§RT[N@ §[NdT@bAORs[P6§RT[¨LP=wAbJLPXwf?H·R^\LNA [N@ ew[PRse|AO[LNUAYJ < =?Ad^LLNA i9LPUVTA[PAYdTfKAO[@d³Hd^SV[NAOdTf?H¨=wd³sA2RsQwJAO[PsAOf'LN=wdpL LP=wA2§X?SVS?d^ewewd^[hdpLNXwJUJ i?R^Lnd^Q|JRsSXKLNA SVH [NAOW9X?UV[NAOf6§RT[`LP=?Ag X?[N[PAOisLewX?[Pe;RsJPAT;JUViwg A+LMHIe|AUVi9LPAO[Pe?[NALhdpLNURsi4f?R9AYJi?RTL`UVi9sRTSVTA+@ RT[NA'LP=|d^i·A SVA @ AOisLhd^[NH @ Rsi?RT@ Ud^SJ (X u¯) \=wUVgh=¬@ dTTALN=?A @ dTe?e;AOd^[ dTJbJPU@ e?SVA6dsJ=IHIe|AO[ e?Sd^i®UVi9LPA [hJPAOgLNURsiwJO¦µgLPXwdTSSVHTnRsX?[ @ RTLPUV³d^LPUVRTiwJ§Rs[Ui9LN[PRKfKXwg Uiw}b[NUV}T=9Li?Rp LN=?A'§X?SVSJMLN[PX|gLPXw[PA+d^[NA@ dTi?U§RTSf µ¨J`JP=?Rpi4Ui4LN=?AUVi9LP[NRIf?XwgLNURsia?LN=?AewdT[NdTSSVA SaRse|AO[Nd^LPRs[d^e?e;AOdT[NJW9X?ULPAiwdpLNX?[NdTSSVHT|Ui9LN[PRKfKXwg AOf$QIH6RsX?[ LNRTe;RTSVRT}sUVgOd^Snd^iwdTSHKJPUVJO&dTiwf¦JX?}s}TAYJMLhJ+d·fKU lA [NA i9LdTe?e?[NRsdTgh=LPRtg RTiwg X?[P[NA i|gHT=?A [NA LPAO[P@J+d^[NA e?UVAOgAYJ+R^ g RT@ e?XKLhdpLNURsi =?Ugh= d^[NAg RT@QwUi?AYf d^SVRTiw}®LP=?AX?i?UVTAO[NJNd^S¨[PX?SVAOJ6R^dTS}sA Q?[hd
(62) ¦LNA [N@ J$g dTi Q;A¦dsf?fKAOf RT[ JPX?QKLN[NdsgLNAOf °²[PAOTAO[NJPAOfw±¨dTJUVi¦dTiIH·}T[NRTX?eJLP[NXwgLPX?[NATZ\[PRKfKX|gL+R^LPA [N@JSAYdTf?J'XwJOad^SVRTiw}%LN=?AJPdT@bAbSVUi?AYJ dsJ+§Rs[µSV}TAOQ?[hd^Ug%¶A Rs@bA LP[NHT LPRSRKg dTSUJNdpLPUVRTi®RTgRT@ e?X?LNdpLNURsiwJQIH@d^IUVi?}·LPAO[P@JR^SVAOJNJUVisLNA [NAOJL6°¡§Rs[ UViwJLNdTiwgAf?UsA [N}TA i9Le?[NRT}s[NdT@ Jh±UiIsA [PLPUVQ?SAs < =wAgSdTJNJUg dTS LPAYgh=?i?UWsXwAtR^+SVRIgOd^SVUVJNdpLNURsi R^+d®[PUVi?}®=wdsJd UVi9LPX?ULPUVTA¨@ AOd^iwUi?}Ui%LP=?Ae?[NAOJPA i9Lg RTi9LPA ª9L
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