Three principles. POPL 2012, An Abstract Interpretation Framework for Termination. Principle I. Refinement to principle II

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ll 5 R k l llg l Q u u u u ug H u 55 lg g H u g u u u g l g l l lk l g u l l l l uu ll g g Q l u g g u l ll gu g g u qu ug g u l u 5 u H u l u u l g u u ll u ll l l uk g u g l l u 5 H -u ll u g u ll l qu g g g u ll ul g u u g g g gug l qul g g g l qu uu HJ g l l u l lu H l l qu l u ug g l glz u l lg k u u u g u l l 5 ll l u g l u u l l u l u g u ll-u u g u R u k glz ug g H l u u u g l g u H l u l l g l g l u g u u u g ll g H g u g u u l u u g u g V H u u ug/l u 55 l llg l 5 H k ll g l u qu l u g u g qu gug g u g u u g g u l l l ug u l g l -ll H u H g l u u l u l u uu g l u ll u u g uk g u u g lk u l u g l g- uu k ug l l g g u g uu lg u k l u u g k uu ll l l l l u u g l u l u ug ul l l u lu g u u l u k g H l u g u u l u u g u g ll g u l u g k u Q u H kll lu l u g lu qu g k H J l qu u l g u l u u ug/l u u l u u g u l l l H g z 5 g ll-u l l g ll u g g g g- l ul uk g g l H u - uu l u u u l u g u u l lu l g ll-u H l l lu u g g l l l lull g l lk l l g ul l u l u l lu l l lk l - Hg l u k l ug lull g g u g g u uk g l g- l l g - 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u 5 l g l - l 5 l g l - l l H l u k l g u lk l ll g Q l qul - lk gl lk l ll u ll l gll ll ll ll u g u lz l qul g - // ll ug u */ glz */ g u g g u g g l - l l g u5 l u l u g ul g g ul g u & R u k glz ug lk l ug g g u */ */ ll lk lk g lk g Qg g u */R ug g l g l u qu g l g ll l ug g l l l l g ug g l -ll l H gu l k l */u lk */ */ l g l u l lk g g l lk g l g Q u lk l l l ll uu ll u u g lk l l ug u ll g ll g u -g l l l g - l l g - l l lk Q g */ H l J lg qu lu l lk l l l l lk l u g g u-lk */ u l l u-lk l ll ll */ u g R l u g l lu l qul g l lk l // l lk ug g u l u l -g lk g */ k lk lk k g l u-lk g g - ug l u g g l lu glz l H l g u g lk u l g l k uu l l lk ug g g l l g u k l u lk g u -g u lk l H l k lg u lk l l u u lk u l l lk u-lk l g l g l l u u l g- glz g - lz g g l u k lz g u l u ug Hl l l - l l k g u lk l 5 g lk l u- l lk k l l k g lk u l H l ν ll H l g l lz */ 5 g g - ug g l l l l ug l - l g ullg l lz u 5 l g lk l l lk g l g ug ll g-g */ l H l k l u g g-g u lk ll g Q lk g g */ l H l k ll g l lk ug g g l */ ll */ l l lk l R l ug u qu g V */ l l g lz g l ll u-lk Q lk g u g ug gl g l - 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g ul ll u l R u u g/g g l l u R u gzz H Hul l HH u gg u l u 5 R u é u V lé 5 5 l l u u H H k Hll l ll lk - g è Éèg k 5 uul u 5 - u λ- g/g ull l u Rllèl u u u R u g l u g l ll l 5 gll l lg H l l ul l - u-l gu u kg u u l u gg 5 R u é u lé l ul k u u 5 5 55 u g U u É R u u k Hll u u l lk g llèl è è 5 l 5 g u J 5 k Gu l u l l 5 gll l u R u g g l k u H l u u R u u l g g 5 5 5 R u-l gu 5 55 u kg u 5 - uul u u 5 H lg u 5 g l & g u u ul lgug- u u g U -Hll k l gg u R u u k l ul g l u l lk ul lu 5 U g UK l ul uk kg J u k Gu l Jg 5 -g g V J 5 g - u u R u À l 5 5 l u l g u glz 5 5 5 g 5 R u / uul u u l u l l u R u u l g u l Y 5 g l g lg lk u u R l gg g UK g l Vl u g l & g k l gg -Hll g u kg g g U g 5 kg u 5 u ul gl u g qul l ul uk ul J u5 u 5g H lg g lu 5 U g UK u Jg -g g V R u l u l lgug- u u R ull g g ul ll u ll g u 5 G 5 H u ï u g V V uu qul g u R À l l u l g 5 5 g -Hll u g u k k l u l l l 5 l 5 l 5 lk Y u u g l g lg R l gg g g l Vl lk Gug l l kg glz u g Hz J Hk lk R kg/ gu u / uul u u RU u k J 5 kg u 5 g u g g g 5 5 k lg UK lk g 5 lk H Jul u u 5 H uu lgug 5 u & R u k u & R u l Rull ull g g ul ll u - 5 u 5 R u l u l V ll g 5 RG H ï u g V qul u gl u g H u gg u u u g Gl g/ l g g qul g -Hll l l l g k Kk kg g 5 5 ug 5 5 k l u l g l lk l l kg Gug uu u u u u - g R Hz J Hk lk R g ul k l u u 5 g kg/ gu u R u J 5 l 5 l 5 5 ll lg ull k G lk Rlk V g 5 lk H Jul u u l u R uu u l l l l 5 5 k u u u & R u 5 ull g ull g g 5 l u l gg 5 R u l ull 5 K ll À uk l u glzg H u gg u lk g u R u g Gl g/ uu Hll l g l l g k g Gul kg - Rlk g g g k Kk l l l gg l l l lgug H lg ug 5 5 glz ll uul g 5 u 55 l V 5 u R u l g g l k é g H l u ul l g u R u u u R u Hg- l u lgug- V k lk Rlk uz uul u 5 - l l u u ul l ul k G lk Rlk u l glzg R u 5 l l l g u g u uul u R u u k gl 5 g g ull g g 5 ul u l l ul lgug g 5 K ll À uk l 5 u R l ull u glzg k u u l g J 5 l l l lu l u l J u g ugl u u Hl u l g k Gul - Rlk g g l g l l g J u u R u u l g glz glz l g u lg gl u g u ll u l 5 l ll l V 5 55 u g é u u é H u é l u g uul g g l & g u l ul uul u R g ug ll g u 5 k - gg u l l lgug- k lk Rlk uz éu u ll l équ g u Ru Hg- - 5 g U g UK u u l l ul V ul ul g R H ó J ul l gl u u u 5 l glzg R lg ll Gl l l è É è UG u ul lgug- u 5 g uul u g gl uul u u uu u R u À l l u l g l l ul lgug g g 5 l k u u l g - 5 u g l g l l u R u gzz u g l g lg G lk uul l l R J g ugl u g lu u l u u Hl u l g l -Hll ull ll l 5 J u u u l u g g U g UK 5 - u U g ul ll l glz u g Ru u lg gl l l uul u 5 u g u glz ll u gzz u lull g g - u é é u l 5 u R u l u l ul lk g ll- g ll k k l g Rlk l u V ll k - 5 5 5 u Ru R ügg u lull g u g ug éu u ll l équ l u l g qul g g g l H J ul ó V ull gl u gl u g k l ul uul u R 5 l g qul ll g è É è UG Gl lg ll u l l 5 R u é u lé uu 5 u l l k kg R lk uul u 5 5 lk Rlk u R u k J 5 u u g l g l l g llèl è É è 5 l uul u 5 u R u gzz g ul l G lk Rlk uul l R 5 l 5 gll g uul u g 5 uu u g g u lk l l l -Hll 5 H l u l 5 ull u ll 5 u u g - u U g g g l g g ul kg u l 5 5 5 5 55 u R u Gl g/g U V u u lull g gll 5 u R u gzz Gu ul lk g Rlk llg 5 l lk Rlk u u l u 55 k l 5 R ügg lull g k lk g λ l ll V 5 u V V l uu - u qug u 5 l H k k - g lgug uul u g 5 5 5 u R u u 5 &l λ- 5 l qul 5 l X Rl H ug g J g 5 k l gg -Hll R 5 u H g u g g R5 u é u lé u l l ul kg R 5 l lg g l l ul uk kg u k lk Rlk J -g V u R u À l ull u u gg l gg V 5 u 5 ll g 5 g llèl è É è 5 5 5 5 l 5 u λ 5 lk Rlk l l l l l g l l g l 5 gll 5 u ll 5 ug g Y l l lk g uul lk u u 5 5 u u lgug H u RG- U λ- R H l gg g g l Vl uul l - H 55 kg u V 5 - u 5 kg u R u g uu u 5 5 55 H H l g U 5 k l l l J H u R u Hg- l g gg u 55 lk Rlk R ull g ul ll u H g g l G H ï u g V V 5 k Gu l 55 55 u V V l glzg R g -Hll ll l l l 5 - g u R u u l l g 5 5 g g g R u 5 R ug kg lg u u lulg & l ul lgug 5 Hg l l l g l lk Gug l l H u H 5 g kg k X 5 R 5 Hz JRl Hk ug lk g 5 λ λ- H lg uu l u / uul ug / g g l 5 l 5 l H lg uu lgug l u u l gg l l l l l l l qul lg g l ul uul u g uul u g u u l u u g lk uul u g g l - ul uu l uu u qug g l g qulg lg g l H lg 5 - uul u g g u l 5 g ul lgug- u u glz u g qul u gl u g uu u g u lk g 5 g g g g g ll g ull -ul ll k u g u u uu ll u g 5 u g g Jul u H gg k l uu - u qug g UK k l -Hll 5 u g lk H 5 5 5 l l ul uk ul kg u J u k J 5 -g g V u ll 5 gg l gg V g H u gg u - u 5 u g - 5 qul l l k 5 5 g u l u 5 lg l Kk kg l 5 u l 5 ug 5 5 g lk Y u u l lgug RG- U R l gg g g l Vl u ul l g u R u g qu kg u 5 ll u g 5 k l l l J k G lk Rlk V g l l gg u R ull g g ul ll u u g Rlk 5 5 ug qu g ull g 55 G H55 ï u g K ll uk l V V u R u u -Hll u g k Gul - l g l g 5 5 R kg lg u Hg u lulg u l l uul l l u g g u é V lk Gug l l kg u 5 g H g Hz J Hk g lk R UK l k lk H Rlk uz 5 u g 5 5 lk Jul u u u u ul l ul 5 5 u u l -lu u 5 u g 5 gl gg u u g H u u l u l u g g -ul/l k u kg l -ul/lg g u k Kk J g ugl ug 5 5 l J u u k g ul - u lg gl l g u é é u l l u u lk H H 5 u k - ug k G lk ll Rlk V g u éu u l équ g lk H H R H ó J ul l gl ll-u g ull g 5 -lu qu è É è UG Gl 5 K ll uk l l g l uu g k Gul - Rlk g g l u H l - u g g l l l u λ - 5 G lk uul l R l g l V 5 l -Hll - l é g H ll g - u U H l H H H u λ k lk Rlk uz gh u lull g g H uul u l ul - ul ul ll 5 H lk u lk l u H Rlk - u 5 ügg lull R H g H u gl u g J u λ- u - k u l g H H 5 H -lu J g ugl u lk - kg u l l R g l J u 5 lk Rlk λ- gl u lg l 5 é l l u g - 5 u u l g lk Rlk l l - ll-u u g 5 u 5 k 5 u ll u g équ g g k k é kg uv - éu u u l g k R H ó J ul l gl - u qug 5 UG Gl V g 55 lk Rlk è É è l u V uu 5 l g g l l 55 ll-u u u g l & l G lk uul l R g lg X Rl ug g u l -Hll 555 u g g u g l - u U l u ul 5 ull u gg lull g l gg 5 l u g g V lk Rlk g lll l u ll u g 5 R ügg lull g lgug u RG- l U u lu g k u R u g l 5 5 k l l l J ull gg u g u u g u l l lg kg u l 55 5 lk Rlk u R 55 g 5 u R u u l l 55 R l ug kg lg u l Hg u lulg l u & R u lk Rlk l g u u g u ll l 5 u g g UK 5 u u g u & R u k 5 kg u V 5 l u g g l/ug H lug ull k l llg ll l l u g 55 lk Rlk ll lg ull u V V 5 g qul u 5g & 5 u - 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