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

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1 G-G u l gg gug ll U Ju 5- l k g l l *** gg lgug k u uu/u/ /u/ k R u /u/ u & R u * ** gg u u R u u l l l g u 5 Hll u R u g g l k u R u u k k u & R u R l l ll l gg lgug k u & R u k u & R u

2 l ll-k lk ug ull * l k lg u l H lg l g lg ull l ug uul l lk-rlk * Hll u R u g g l k u R u u k k 5 u & R u 5 5 H u gg u ug 5 5 ug 5 5 ul ul l l g g kg u 5 R K ull llg uk g l ul ll u g 5 -Hll lk lk Gug Rlk l l kg lk Rlk l l k u & R u l ul uug g l k u & R u k u & R u

3 g ll l lgug u g gg l -u l gg lgug g ll l l l l ll l l -u g ll l l g u u ug g u ug ug g uqug l u qug g l g g u ll l u K l l l u l l u ug g u l l- l l g ll- l l g ll- l u u l l l ll -u l u ug g u l l u ug g g ll- l g u u l l qug g l l l l ul / l l l / l l g l l l l l l -l qul ul g ll- l u l l g l ug u g ll- l g uu ul /lk /lk ul l l /lk g g l l l l g -u l l g l /lkl g l l g l ul l g u l l g u l / l ul l ul g g lul l l K /lk ul - u l g l k ug l z u g X /lk l qul g - u ul l - u ul l g -l l g lkg l lul g g X l X X g gu K g ul g g X g u l lul l /lkl l l uu - u ul l g g u u l g u K X g X X g gg g lul g u X l ul lgug lz Gl l u K g ll l l l u ug l z u g u l l l J l 5 5 u l g u ug u l 5lg K ll l ll lg l u l u l l u g ll- l ll lg u l l llg llg l l 5 - llg - - g l 5 ul - - l u 5 g g z g l g g ll lg u l l gg lgug ll lg 5 /lk ll lg l l g g u ul l l lg llg - llg ll u llg l g - g - ll l g lul l - g g llg l l g g ll - g X X g llg g g g g l g l g l g g llg l l l l l g g g ll llg g z g l lg u u qug llg u l l lg g u ug g u l l l l g g g g llg g g lul g llg g llg g u u l u llg g g ll- l l u u l l / l lg u llg g l l u u u llg g l llg 5 u g u llg g u l -u K g ll l g l u k ll llg g ll lg u ll u g g g g g g ll l g l g J l K llg l l - qug - u /lk qug l ll g g g u ul l l l qug l g - l u Q Q K qug l l llg g g u g u u g u llg g l u l lg g g u l u lg l u lg u l u llg g K lz Gl u l /l g l llg g l /l u u l /l l u l -l qul u u X X g l / l l l l / / l l g l l l l l l u llg ul u l lg u ll u llg ul l l uu ul g u l l g Q Q u l l u l l l l u u l Q Q l g g k k u l l l k g g u l k u lg l g g k l g l l ul l g Q Q l ul u l l g l lg l - ul l - l u lg - gu u l l lk gu g g - l /l g l g 5 llg g lul u - - u ul ul l l u l l /l g l u kg/ gu - u ul l g z l ul u l k g l u lg g l l l g l l l u u l u l u l ll llg l ll ul g u g u g g g u g l /l g l ll g g llg l g g l lul u l u u u u l l g lul u l g u g u l lul lz lz Gl Gl k l - g lul lz g Q Q llg l g l Gl lz lz l Gl g l u lu ug l u lz ug lu l llg lz Gl g ug lu l ul gg lgug - l - u g 5 l l u lg lul l gu ll qul Gl l gu ll qul Gl g g 5 l g lul g ll qul Gl ll l /l g l l ll g l ll l l ll lg g g l l g u l l g llg l g - l ll l l l g u g KK u g 5 u l l lz ug lu - - u ug g l l ll g ll l g g g l l g u ug lu K g llg g l - g lul 5 l l l gu ll qul Gl g l ll g u llg l u lz g 5 g lgl l g lu g ug Gl l l l g ll- l qul z u l l lgl l g u l l l l l ll g 5 qul Gl ll l l g llg l ul g lu l u lz ug lz g g z l l l g g /lkl 5 u ug lu l g llg l g g llg l ll g g l l llg g g ll qul Gl u u l l l l gu ll qul Gl g l u g l g g llg l ll g g lul lu l l lgl l 5 u l l g llg l l g lg lgl l u Xu g l X -u u l g l ll l g llg l g k u & R u k u & R u k u R u u - k g/g l l ul u u u-l gu u kg u u l lk u & R u l u l g u g ll g u R u u u l gg Hll u R u g g l k u R u u k J 5 k u R u u l g g l & g u u 5 g U g UK u R u À l l u l g l g lg g U g UK 5 5 u R u l u l u & R u

4 l K g /l gl g k K J K l ul ul u & R u l J uul g l k 5 u & R u l l g l - l g - - k u g g gg Rl gg V 5 - k u & R u k u & R u

5 l g u l U l - u l - g l k u & R u k u & R u l u l g l k u & R u k u & R u

6 g k * «l-k» l uu - uu g Uu u l l - l uu k u & R u * lug l lu k u l l kg R -5 k u & R u / l l k u & R u k u & R u

7 Rl l l ll l l -g * l/l l /k l l ll/l g l -g * gg u * u R u u l l l g u 5 5 k u & R u J u * u é é u éu u u ll l équ g g J g g & * u & R u l lg è É è UG Gl u u k g l g l l u & R u l l k l ll / l g l -g * l ' g J Hll * u R u g g l k u R u u k k u & R u l k u & R u

8 l ' g ul u k lu u ug/l k u u k k l ul l u J u ll R l gg g g l Vl R ug G kg H ï lg u u Hg g u V lulg V g UK k u & R u k u & R u l ' g ul u J ug-l u ll R l gg g g l Vl R ug G kg H ï lg u u Hg g u V lulg V g UK k u & R u k u & R u

9 l l g u V l-u l l - ll-u l u-l u l k l u u l g lg g l/ l l k l l l lg l k l ll-u R ll R ll-u ll-u l l l k k 5 k k k u g lu u u u u H uu llg l uu - l 5 g uu u lg u u l 5 kg l l k Ug u u l kg k k/ l u ll-u l l l k l l 5 llg l u l l u k l l u l ll-u g u ug l l l ug l l 5 u ul u g ll g z g ll-u l k k - k l l/ug l u V u u l u l u u k l u l l u g u V l k ll-u u ll k l ll-u R ll-u - l l 5 u l l G ll-u u k k 5 u k u lg g 5 H lu u u g u gl l/ug g z g ll-u l uu - lu u u g u l k l/ug u ll-u uu - kg u g l ll uu u u lg k l l l k u kg l lu u u u u uu u lg u g k k u l l u kg l - g l uu k ll-u k k ll kg u u g k g l u - uu u lg u k u u k u Ulll l k u k k g 5 u k g u k lg g u H kg u k 5 5 k u Rkg gl kl/ug k g ll-u ll-u l l ll-u k ll-u l ll-u l g u k l l k k k g k ll- 5 k l u k l u l l k u kg k u & R u k u & R u l g R ug u g k k u ll l llg g l l l g l l g g k u g ll-u l g l l g l l g u u Ulll k k l u g l g u u l g k 5 u k glz ll- ug u l g l l ll-u l g u z u u-u l u lu l l u u u ll-u - ll ll l ll u k l l llg R l g ll-u l l u l l -g Y Y l l l -g l ll k u & R u k l l l l u & R u

10 u l G lu u l l l ll l u kl lu u l V kl l u g u l lk l H g l u l g ll l u u Hg u l g ll l u l u- k k u lu u u g u g l - ll Hg llg g u l u lull g l llg u l l ll-u H l l Hg k ll u lk l g l uu - l ll-u l l u H l V u g g l g l l g lk llg g l llg lull l Q k l k H l lk g l k lk uu lg u u u llg g g g l qu g ul l llg - u lull g l lk u l g l g Q kl u l l l - l lull l u u l Hg k l u g u l g l l k gug g kg l u l g u ll l l - qu u g l l l ug u u l l lull g V u lu g g- l u uk g H l g ll-u l uu u g g l k l k - g l u l k l l k g Hg H l u u l l u g ul l lu u u l 5 u lu - H k l g k k u ug/l u l u k ug u l Hg g - g g l l l g lu l g l qu lull l g l g k u l u g llg l lull l l l l g ug u l l ug u k g u lull g g u u lu u l lu l l g ll-u l ug ul l l u lu g g- u uk u l l u l l g g l qu lk g k lull k k l k l lull g l u l H -u l uu l - l lu lk u g - l l g l u g k l u l l u lk l g u l g u l u u g ul lu u l l lk lu u l u l g qul ll gl l lk g lu l u l u - g g k g u ll 5 l lu l g l g qu -l l l g - l lull g l l ll g ll u lu H g ll l ll g g g g u lu ug - g l l Q u lu u l g u g l g ul g g u l k Q lk l - uu u l l lull g l l lk k g l g u u g u g u lk lul l g ll l u l l l g uu u g- lu ul uk gl u lg u ug l u k u l l g l g k u K g ll u u l lk Rll lk l - uu g Q l qu gug kll-u u l l 5 l g -l u ll l kg g ll u ll l u u ll k l qu qul l lul g u u u l l V l u u u- k 5 llg l lull g l l glz ug u g k u u Rll u ug/l u u l l g l u l 5 l l u u l 5 l l u- u u l l ug u l k k 5 l u l u l u k u k l g u Q g u ug u g l 5 5 ug u l ll l glz u gl ugul l lu l u lu l lu ll g u ll-u lu g-g l l l u g g l l g l l l l l - u k l l l lk u g kl u l u u u 5 qu gug g u ug l llg l u 5 55 l l ll u qu g l qul ll gl k - g - u u l l u l V 5 lul u l 5 l ll l u ul l u H k g u ll ll ll u g g uul l -ll l g l uu u u l l l ug g l 5 ll l u V 5 l l g l gl lk u u l k u u u u g l ug g g l l l l g u uu l l Q l g l l u ug/l H u l ll l l g l u ll ul u l u u -lu l V u l u u u g l l l l ug g k u 5 55 l - llg llg l k ug g u g g u llg u g g u g ll ll g ll H u ul u u lu Q -lu l qul ug l l u g l u u l ll H l J l qu lu u u qu u u g g ll k l l u u ll u l l H lu l u ug g g g l u g l l k g l g l u l 5 l qul g u g u l l l l u ulul u lg k l l l u l 5 ll glz ug u g qul ll gl u u g u V ll k u uu l l H l u u l l l u l ullg lz g u g u g V l glz ug k u u u // u g g g 5 u l ν 5 ll l u H g g ll g g l l l u l ug g l l u l u ku Q Q k H l g u g k g l g k g u u l l g u l u g u g l g g u qu l u lu k uk g g u gg ll u l l u l l ν l l uu -ll l H g u l - 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 - uu l k Q k l l u l V l l u H u g l qu gug u u u g lu g l g l u H l ll-u u g g l l l l u ug/l u lu l g ul l l l ug l l lk l lull g l l Hg ug u g l u ug ul l l u lu g l ug lull g g u u l g g- l u uk g l l lu u uu u lu l - l u u l ll l l lk l u gl qul u ll g l lu lu l l ug g lk Q l lull g l l u l u l l g - lu l l lk l l Q k l l g u & R u uu ll u u ll u g H l u g u l l ul l ullg lz l l g */ g H k u l g lg k g H l ug u g l l l lk l u ll-u */ g g ullg lz lk u l l g - l l */ l g 5 l H l k */ 5 u-lk g ul l g ν l - g ug l g u g l u g l */ l l lk */ */ l Hg - 5 ll g g l l l - g g ll Q u lk u - lk u ll g Q g lk u u g */ g l */ l g g υ ν l ug u lull g g u g g l u 5 g g- g k g l u u uk g g l l g u g k l l 5 g u l lk ug u l lk ug u u lk u l lk u ug g - uu 5 uu ll g - g l g l l g l lk l lk lk u u l u l ug uu ll uu ll u u u g u l k g l 5 ll g ll - V k u l lk ug u u-lk g l H l g u-lk uu ll g g u u u u u u uu ll u l u lk H u u l g g ug // u u u u u u l lk - l lk l u g l ll lu g l l g - l l u l l u l ullg lz l R */ - H l ll-u ll lk lk g u g g l l g l l lu l g l g - ul l H k g g k l g g k l l - lk - l lull g l l */ ν g ug u lhg l g g l l ug g l g lull g g u u l l Q g g- l u uk g ν g l l uu u g ν ν lu u l - l - l u l Rll u g lk l - l u lk l ll ug u g l lulk u l */ ll gg l u - V l lk lk l u ullg l u l lz lz ll l 5ullg l lu l u lk ll lk uu lg u ν Rll 5 ll νll l - u lk l lull g l l Q u u u l u g g ll ll g g l l lk lk u u ll lu Q l g l u ll l lk */ ll l lg lk l lk l l Rll Q u u Q u l u Q k l u ll l ll ll g g l k l lk lk l u u ll l lg 5 g Rll l lk l l l l l g l ll g u J l qu lu l l g ν Q k l ll u l ll l g l l qul k Q k l ug u g l glz qu gug l l u l u u l qu gug glz ug g g g g g u g l lk lk u l ug ug u ug/l u l ll l l l R u ug l -g lk g qu-g lk l g ug u g g g ulug/l u l -ll l H u l l u lu l l lk u-lk l lk u-lk k ug ul g lu u l l l g l u lk u l l ug u l u l u u l l u l ll l l l uu qul l lk ug ll gl g u u l u g l u ug u g l k u u g l H l H l l ll l l k -g lk g g g ug ul l l u lu ll g ll g u l lk u-lk u l ug g Q l u H J qu l l lu u u l lu lz lz l g g u l l l l u l u l - 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 - l l l lk l l l */ u */ g l l -ll l H l */ H lk u g */ l lk lk g g u-lk k l g g g u */ l g l l g l l lk l lk u u */ g l ll */ g k g lq u g u υ lk u */ u g lk 5 g lk l */ */ g g k l g - - u k ν l Q Q l ll u ll l l qul g glz ug u g l u l u u glz ug g u u g ll R u qu g l l -ll l H g u l l g l lk u u l l l uu ll k g g l u u g l g g l l ll u g ll g u H J l qu lu l u g u l l l // Q // - ll Q l l l - ll u ll l k l u R u &ullg l qul g u l lz l // // g glz ug u g u l u // u l glz ugq // g u u g R u qu ll g l l u l -ll l H g l - l g l lk u u g g k qu gug g u & R u k Q k l l l - l ug g u l u ll l u l l u u l u g qu gug lu ll u g g l l - u - ll l - g l l u u ll u lk g u ll u lk u ug/l u l ll u lk uul l 5 l lk u ug l l l ug 5 5 lk ug u g lk l uu ll // -lu l lk l lk lk l lk ug ul l l u lu lk u l u u u u l lu lk lk Rll g l u u l u u R -lu l qul ll - Q qul ll Q l g gl qul uu ll g Q u ll u ll l g ll g Q u lk k ll Q ll lg 5 5 u u- l lk Q l - l ug g u g u-lk Q g - Q ll u l - l lk Q l u l g l l - lk u u ug/l u l l l l ug ug u g ug ul l l u lu l lu u u l qul ll gl ll ll u & R u //

11 ug/l' g k l ug ul l l k ν ν l ν ν ν ν ν ν * ν ν uug g è É è UG Gl * u u g l g l l l -Hll u lull k g g u & R u k u & R u * u l l l - Rkg l/ll-u * uu - u uu * R l gg g g l Vl G H ï k u g V V - u & R u l/ug l uug g l u g-l u k u & R u

12 u g 5 l l uul uuul u 5 l l l ul uu u u l g g l g l ul l u k g/g l l ull l ul u ll u λ lg H ull Hgu u λ- u-l u kg - H H - ull -ul ll llu u -ul u R u u u l gg g/g l u R u u Hlll gg u l ul u Hll R u g g l k u u u R u g g l k H u kg u gu u-l u R u u k l lk u u R u u k - u l lk H J 5 H J 5 H u R- u l g H u R u ulh g u H g l & g u u 5 5 H l l l l u ll l lu ll l g l & g u u u 5 U g lu g U g UK UK l u 5 g u Ru À l u l u l u u R u À l l l l g - g l u l l u l u l g lg l g lg uul u u g u g g g u g U g UK 5 g U g UK 5 / u l g 55 u R u ll u l u R u u l ll ll g g qul g g λ g qul g -ul/l u u l k k l g u l l λ- H l g - H kg/ gu H u R u k J 5 kg/ gu u R u k J 5 u g -ul/l l u l 5 l u lk ull H H 5 ull l g Gl g/ u R u l l l - u R u g Gl g/h l g l g 5 H l l l u lk H H H l H g 5 λ l u u R u u - l H u H H u R u u u l g l l l l H H lk H - l l l l λ u R u Àl ull u λ- H - glzg H À u g u g u H Rl ul l ull H l g - glz u glzg ll uul H H 55 H l lk - l g l g λ- l g g l glz ll u R u Hg- u lgug- uul 55 u u g - uul l l R u R glzg u Hg- l lgug- u u u u u ul lgug g 5 g l glzg R l g l g g lu uul u u l u ul Hllgug u g l l g 5 g u u g l l g u u glz ll l u lu Hl l g l u R u ug ll g u u g l g -ul/l g u g u glz ll l V ul k lg ll u l l u R u ug ll g 5 u l l u g V u ul u u gzz g g u lk H H R 5 l ul l ull lg ll u l u ll l 5 g ll k u R u gzz u u R u gzz g g l l ll lg ull l ll V 5 5 H l ull u ll l 5 λ - 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 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 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 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 - 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