/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) CR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) LetRed [EQUIVALENT, 10 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) AND
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) TransformationProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) YES
    (24) QDP
        (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (26) YES
    (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\vu68->case vu68 of {
(cs@(_ : _),t)  -> (cs,t) : [];
_  -> []}
"
is transformed to
"nonnull0 vu68 = case vu68 of {
(cs@(_ : _),t)  -> (cs,t) : [];
_  -> []}
;
"
The following Lambda expression
"\nd->n * radix + d"
is transformed to
"readInt0 radix n d = n * radix + d;
"
The following Lambda expression
"\vu77->case vu77 of {
(ds,r)  -> (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
_  -> []}
"
is transformed to
"readInt1 radix digToInt vu77 = case vu77 of {
(ds,r)  -> (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
_  -> []}
;
"
The following Lambda expression
"\(_,zs)->zs"
is transformed to
"zs0 (_,zs) = zs;
"
The following Lambda expression
"\(ys,_)->ys"
is transformed to
"ys0 (ys,_) = ys;
"

----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(3) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case vu68 of {
(cs@(_ : _),t)  -> (cs,t) : [];
_  -> []}
"
is transformed to
"nonnull00 (cs@(_ : _),t) = (cs,t) : [];
nonnull00 _ = [];
"
The following Case expression
"case vu77 of {
(ds,r)  -> (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
_  -> []}
"
is transformed to
"readInt10 radix digToInt (ds,r) = (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
readInt10 radix digToInt _ = [];
"

----------------------------------------

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(5) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.

Binding Reductions:
The bind variable of the following binding Pattern
"cs@(vx : vy)"
is replaced by the following term
"vx : vy"
The bind variable of the following binding Pattern
"xs@(ww : wx)"
is replaced by the following term
"ww : wx"

----------------------------------------

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(7) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"span p [] = ([],[]);
span p (ww : wx)|p ww(ww : ys,zs)|otherwise([],ww : wx) where {
vu43  = span p wx;
;
ys  = ys0 vu43;
;
ys0 (ys,wz) = ys;
;
zs  = zs0 vu43;
;
zs0 (wy,zs) = zs;
} 
;
"
is transformed to
"span p [] = span3 p [];
span p (ww : wx) = span2 p (ww : wx);
"
"span2 p (ww : wx) = span1 p ww wx (p ww) where {
span0 p ww wx True = ([],ww : wx);
;
span1 p ww wx True = (ww : ys,zs);
span1 p ww wx False = span0 p ww wx otherwise;
;
vu43  = span p wx;
;
ys  = ys0 vu43;
;
ys0 (ys,wz) = ys;
;
zs  = zs0 vu43;
;
zs0 (wy,zs) = zs;
} 
;
"
"span3 p [] = ([],[]);
span3 xx xy = span2 xx xy;
"

----------------------------------------

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(9) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"span1 p ww wx (p ww) where {
span0 p ww wx True = ([],ww : wx);
;
span1 p ww wx True = (ww : ys,zs);
span1 p ww wx False = span0 p ww wx otherwise;
;
vu43  = span p wx;
;
ys  = ys0 vu43;
;
ys0 (ys,wz) = ys;
;
zs  = zs0 vu43;
;
zs0 (wy,zs) = zs;
} 
"
are unpacked to the following functions on top level
"span2Vu43 xz yu = span xz yu;
"
"span2Ys0 xz yu (ys,wz) = ys;
"
"span2Span1 xz yu p ww wx True = (ww : span2Ys xz yu,span2Zs xz yu);
span2Span1 xz yu p ww wx False = span2Span0 xz yu p ww wx otherwise;
"
"span2Ys xz yu = span2Ys0 xz yu (span2Vu43 xz yu);
"
"span2Zs xz yu = span2Zs0 xz yu (span2Vu43 xz yu);
"
"span2Zs0 xz yu (wy,zs) = zs;
"
"span2Span0 xz yu p ww wx True = ([],ww : wx);
"

----------------------------------------

(10)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="readInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="readInt yv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="readInt yv3 yv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="readInt yv3 yv4 yv5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="readInt yv3 yv4 yv5 yv6",fontsize=16,color="black",shape="triangle"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concatMap (readInt1 yv3 yv5) (nonnull yv4 yv6)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="concat . map (readInt1 yv3 yv5)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="concat (map (readInt1 yv3 yv5) (nonnull yv4 yv6))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="foldr (++) [] (map (readInt1 yv3 yv5) (nonnull yv4 yv6))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] (map (readInt1 yv3 yv5) (concatMap nonnull0 (span yv4 yv6 : [])))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] (map (readInt1 yv3 yv5) (concat . map nonnull0))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="foldr (++) [] (map (readInt1 yv3 yv5) (concat (map nonnull0 (span yv4 yv6 : []))))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (map nonnull0 (span yv4 yv6 : []))))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (nonnull0 (span yv4 yv6) : map nonnull0 [])))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull0 (span yv4 yv6) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 yv6) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="burlywood",shape="box"];278[label="yv6/yv60 : yv61",fontsize=10,color="white",style="solid",shape="box"];17 -> 278[label="",style="solid", color="burlywood", weight=9];
278 -> 18[label="",style="solid", color="burlywood", weight=3];
279[label="yv6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 279[label="",style="solid", color="burlywood", weight=9];
279 -> 19[label="",style="solid", color="burlywood", weight=3];
18[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 (yv60 : yv61)) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2 yv4 (yv60 : yv61)) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span3 yv4 []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22 -> 24[label="",style="dashed", color="red", weight=0];
22[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 (yv4 yv60)) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="magenta"];22 -> 25[label="",style="dashed", color="magenta", weight=3];
23[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 ([],[]) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
25[label="yv4 yv60",fontsize=16,color="green",shape="box"];25 -> 30[label="",style="dashed", color="green", weight=3];
24[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 yv7) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="burlywood",shape="triangle"];280[label="yv7/False",fontsize=10,color="white",style="solid",shape="box"];24 -> 280[label="",style="solid", color="burlywood", weight=9];
280 -> 28[label="",style="solid", color="burlywood", weight=3];
281[label="yv7/True",fontsize=10,color="white",style="solid",shape="box"];24 -> 281[label="",style="solid", color="burlywood", weight=9];
281 -> 29[label="",style="solid", color="burlywood", weight=3];
26[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) [] foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="triangle"];26 -> 31[label="",style="solid", color="black", weight=3];
30[label="yv60",fontsize=16,color="green",shape="box"];28[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 False) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 True) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
31[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
32[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span0 yv4 yv61 yv4 yv60 yv61 otherwise) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
33[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3];
34[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] []))",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
35[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span0 yv4 yv61 yv4 yv60 yv61 True) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
36[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) ((yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
37[label="foldr (++) [] (map (readInt1 yv3 yv5) [])",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
38[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 ([],yv60 : yv61) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];38 -> 41[label="",style="solid", color="black", weight=3];
39[label="foldr (++) [] (map (readInt1 yv3 yv5) ((yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : [] ++ foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3];
40[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3];
41 -> 26[label="",style="dashed", color="red", weight=0];
41[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) [] foldr (++) [] (map nonnull0 [])))",fontsize=16,color="magenta"];42[label="foldr (++) [] (readInt1 yv3 yv5 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : map (readInt1 yv3 yv5) ([] ++ foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="[]",fontsize=16,color="green",shape="box"];44 -> 45[label="",style="dashed", color="red", weight=0];
44[label="(++) readInt1 yv3 yv5 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) foldr (++) [] (map (readInt1 yv3 yv5) ([] ++ foldr (++) [] (map nonnull0 [])))",fontsize=16,color="magenta"];44 -> 46[label="",style="dashed", color="magenta", weight=3];
46 -> 26[label="",style="dashed", color="red", weight=0];
46[label="foldr (++) [] (map (readInt1 yv3 yv5) ([] ++ foldr (++) [] (map nonnull0 [])))",fontsize=16,color="magenta"];45[label="(++) readInt1 yv3 yv5 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) yv8",fontsize=16,color="black",shape="triangle"];45 -> 47[label="",style="solid", color="black", weight=3];
47[label="(++) readInt10 yv3 yv5 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) yv8",fontsize=16,color="black",shape="box"];47 -> 48[label="",style="solid", color="black", weight=3];
48[label="(++) ((foldl1 (readInt0 yv3) (map (fromIntegral . yv5) (yv60 : span2Ys yv4 yv61)),span2Zs yv4 yv61) : []) yv8",fontsize=16,color="black",shape="box"];48 -> 49[label="",style="solid", color="black", weight=3];
49[label="(foldl1 (readInt0 yv3) (map (fromIntegral . yv5) (yv60 : span2Ys yv4 yv61)),span2Zs yv4 yv61) : [] ++ yv8",fontsize=16,color="green",shape="box"];49 -> 50[label="",style="dashed", color="green", weight=3];
49 -> 51[label="",style="dashed", color="green", weight=3];
49 -> 52[label="",style="dashed", color="green", weight=3];
50[label="foldl1 (readInt0 yv3) (map (fromIntegral . yv5) (yv60 : span2Ys yv4 yv61))",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
51[label="span2Zs yv4 yv61",fontsize=16,color="black",shape="triangle"];51 -> 54[label="",style="solid", color="black", weight=3];
52[label="[] ++ yv8",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
53[label="foldl1 (readInt0 yv3) (fromIntegral . yv5 : map (fromIntegral . yv5) (span2Ys yv4 yv61))",fontsize=16,color="black",shape="box"];53 -> 56[label="",style="solid", color="black", weight=3];
54[label="span2Zs0 yv4 yv61 (span2Vu43 yv4 yv61)",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="yv8",fontsize=16,color="green",shape="box"];56[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys yv4 yv61))",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
57[label="span2Zs0 yv4 yv61 (span yv4 yv61)",fontsize=16,color="burlywood",shape="box"];282[label="yv61/yv610 : yv611",fontsize=10,color="white",style="solid",shape="box"];57 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 59[label="",style="solid", color="burlywood", weight=3];
283[label="yv61/[]",fontsize=10,color="white",style="solid",shape="box"];57 -> 283[label="",style="solid", color="burlywood", weight=9];
283 -> 60[label="",style="solid", color="burlywood", weight=3];
58[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 yv61 (span2Vu43 yv4 yv61)))",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="span2Zs0 yv4 (yv610 : yv611) (span yv4 (yv610 : yv611))",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
60[label="span2Zs0 yv4 [] (span yv4 [])",fontsize=16,color="black",shape="box"];60 -> 63[label="",style="solid", color="black", weight=3];
61[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 yv61 (span yv4 yv61)))",fontsize=16,color="burlywood",shape="box"];284[label="yv61/yv610 : yv611",fontsize=10,color="white",style="solid",shape="box"];61 -> 284[label="",style="solid", color="burlywood", weight=9];
284 -> 64[label="",style="solid", color="burlywood", weight=3];
285[label="yv61/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 285[label="",style="solid", color="burlywood", weight=9];
285 -> 65[label="",style="solid", color="burlywood", weight=3];
62[label="span2Zs0 yv4 (yv610 : yv611) (span2 yv4 (yv610 : yv611))",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63[label="span2Zs0 yv4 [] (span3 yv4 [])",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
64[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 (yv610 : yv611) (span yv4 (yv610 : yv611))))",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
65[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 [] (span yv4 [])))",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
66 -> 70[label="",style="dashed", color="red", weight=0];
66[label="span2Zs0 yv4 (yv610 : yv611) (span2Span1 yv4 yv611 yv4 yv610 yv611 (yv4 yv610))",fontsize=16,color="magenta"];66 -> 71[label="",style="dashed", color="magenta", weight=3];
67[label="span2Zs0 yv4 [] ([],[])",fontsize=16,color="black",shape="box"];67 -> 72[label="",style="solid", color="black", weight=3];
68[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 (yv610 : yv611) (span2 yv4 (yv610 : yv611))))",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3];
69[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 [] (span3 yv4 [])))",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3];
71[label="yv4 yv610",fontsize=16,color="green",shape="box"];71 -> 78[label="",style="dashed", color="green", weight=3];
70[label="span2Zs0 yv4 (yv610 : yv611) (span2Span1 yv4 yv611 yv4 yv610 yv611 yv9)",fontsize=16,color="burlywood",shape="triangle"];286[label="yv9/False",fontsize=10,color="white",style="solid",shape="box"];70 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 76[label="",style="solid", color="burlywood", weight=3];
287[label="yv9/True",fontsize=10,color="white",style="solid",shape="box"];70 -> 287[label="",style="solid", color="burlywood", weight=9];
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183 -> 234[label="",style="dashed", color="red", weight=0];
183[label="primPlusNat (primMulNat yv140 yv30) yv150",fontsize=16,color="magenta"];183 -> 235[label="",style="dashed", color="magenta", weight=3];
222[label="primMulNat yv140 yv30",fontsize=16,color="burlywood",shape="triangle"];304[label="yv140/Succ yv1400",fontsize=10,color="white",style="solid",shape="box"];222 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 229[label="",style="solid", color="burlywood", weight=3];
305[label="yv140/Zero",fontsize=10,color="white",style="solid",shape="box"];222 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 230[label="",style="solid", color="burlywood", weight=3];
221[label="primMinusNat yv18 yv150",fontsize=16,color="burlywood",shape="triangle"];306[label="yv18/Succ yv180",fontsize=10,color="white",style="solid",shape="box"];221 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 231[label="",style="solid", color="burlywood", weight=3];
307[label="yv18/Zero",fontsize=10,color="white",style="solid",shape="box"];221 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 232[label="",style="solid", color="burlywood", weight=3];
223 -> 222[label="",style="dashed", color="red", weight=0];
223[label="primMulNat yv140 yv30",fontsize=16,color="magenta"];223 -> 233[label="",style="dashed", color="magenta", weight=3];
224[label="yv150",fontsize=16,color="green",shape="box"];188 -> 234[label="",style="dashed", color="red", weight=0];
188[label="primPlusNat (primMulNat yv140 yv30) yv150",fontsize=16,color="magenta"];188 -> 236[label="",style="dashed", color="magenta", weight=3];
188 -> 237[label="",style="dashed", color="magenta", weight=3];
235 -> 222[label="",style="dashed", color="red", weight=0];
235[label="primMulNat yv140 yv30",fontsize=16,color="magenta"];234[label="primPlusNat yv19 yv150",fontsize=16,color="burlywood",shape="triangle"];308[label="yv19/Succ yv190",fontsize=10,color="white",style="solid",shape="box"];234 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 242[label="",style="solid", color="burlywood", weight=3];
309[label="yv19/Zero",fontsize=10,color="white",style="solid",shape="box"];234 -> 309[label="",style="solid", color="burlywood", weight=9];
309 -> 243[label="",style="solid", color="burlywood", weight=3];
229[label="primMulNat (Succ yv1400) yv30",fontsize=16,color="burlywood",shape="box"];310[label="yv30/Succ yv300",fontsize=10,color="white",style="solid",shape="box"];229 -> 310[label="",style="solid", color="burlywood", weight=9];
310 -> 244[label="",style="solid", color="burlywood", weight=3];
311[label="yv30/Zero",fontsize=10,color="white",style="solid",shape="box"];229 -> 311[label="",style="solid", color="burlywood", weight=9];
311 -> 245[label="",style="solid", color="burlywood", weight=3];
230[label="primMulNat Zero yv30",fontsize=16,color="burlywood",shape="box"];312[label="yv30/Succ yv300",fontsize=10,color="white",style="solid",shape="box"];230 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 246[label="",style="solid", color="burlywood", weight=3];
313[label="yv30/Zero",fontsize=10,color="white",style="solid",shape="box"];230 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 247[label="",style="solid", color="burlywood", weight=3];
231[label="primMinusNat (Succ yv180) yv150",fontsize=16,color="burlywood",shape="box"];314[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];231 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 248[label="",style="solid", color="burlywood", weight=3];
315[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];231 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 249[label="",style="solid", color="burlywood", weight=3];
232[label="primMinusNat Zero yv150",fontsize=16,color="burlywood",shape="box"];316[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];232 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 250[label="",style="solid", color="burlywood", weight=3];
317[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];232 -> 317[label="",style="solid", color="burlywood", weight=9];
317 -> 251[label="",style="solid", color="burlywood", weight=3];
233[label="yv30",fontsize=16,color="green",shape="box"];236[label="yv150",fontsize=16,color="green",shape="box"];237 -> 222[label="",style="dashed", color="red", weight=0];
237[label="primMulNat yv140 yv30",fontsize=16,color="magenta"];237 -> 252[label="",style="dashed", color="magenta", weight=3];
242[label="primPlusNat (Succ yv190) yv150",fontsize=16,color="burlywood",shape="box"];318[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];242 -> 318[label="",style="solid", color="burlywood", weight=9];
318 -> 253[label="",style="solid", color="burlywood", weight=3];
319[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];242 -> 319[label="",style="solid", color="burlywood", weight=9];
319 -> 254[label="",style="solid", color="burlywood", weight=3];
243[label="primPlusNat Zero yv150",fontsize=16,color="burlywood",shape="box"];320[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];243 -> 320[label="",style="solid", color="burlywood", weight=9];
320 -> 255[label="",style="solid", color="burlywood", weight=3];
321[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];243 -> 321[label="",style="solid", color="burlywood", weight=9];
321 -> 256[label="",style="solid", color="burlywood", weight=3];
244[label="primMulNat (Succ yv1400) (Succ yv300)",fontsize=16,color="black",shape="box"];244 -> 257[label="",style="solid", color="black", weight=3];
245[label="primMulNat (Succ yv1400) Zero",fontsize=16,color="black",shape="box"];245 -> 258[label="",style="solid", color="black", weight=3];
246[label="primMulNat Zero (Succ yv300)",fontsize=16,color="black",shape="box"];246 -> 259[label="",style="solid", color="black", weight=3];
247[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];247 -> 260[label="",style="solid", color="black", weight=3];
248[label="primMinusNat (Succ yv180) (Succ yv1500)",fontsize=16,color="black",shape="box"];248 -> 261[label="",style="solid", color="black", weight=3];
249[label="primMinusNat (Succ yv180) Zero",fontsize=16,color="black",shape="box"];249 -> 262[label="",style="solid", color="black", weight=3];
250[label="primMinusNat Zero (Succ yv1500)",fontsize=16,color="black",shape="box"];250 -> 263[label="",style="solid", color="black", weight=3];
251[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];251 -> 264[label="",style="solid", color="black", weight=3];
252[label="yv30",fontsize=16,color="green",shape="box"];253[label="primPlusNat (Succ yv190) (Succ yv1500)",fontsize=16,color="black",shape="box"];253 -> 265[label="",style="solid", color="black", weight=3];
254[label="primPlusNat (Succ yv190) Zero",fontsize=16,color="black",shape="box"];254 -> 266[label="",style="solid", color="black", weight=3];
255[label="primPlusNat Zero (Succ yv1500)",fontsize=16,color="black",shape="box"];255 -> 267[label="",style="solid", color="black", weight=3];
256[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];256 -> 268[label="",style="solid", color="black", weight=3];
257 -> 234[label="",style="dashed", color="red", weight=0];
257[label="primPlusNat (primMulNat yv1400 (Succ yv300)) (Succ yv300)",fontsize=16,color="magenta"];257 -> 269[label="",style="dashed", color="magenta", weight=3];
257 -> 270[label="",style="dashed", color="magenta", weight=3];
258[label="Zero",fontsize=16,color="green",shape="box"];259[label="Zero",fontsize=16,color="green",shape="box"];260[label="Zero",fontsize=16,color="green",shape="box"];261 -> 221[label="",style="dashed", color="red", weight=0];
261[label="primMinusNat yv180 yv1500",fontsize=16,color="magenta"];261 -> 271[label="",style="dashed", color="magenta", weight=3];
261 -> 272[label="",style="dashed", color="magenta", weight=3];
262[label="Pos (Succ yv180)",fontsize=16,color="green",shape="box"];263[label="Neg (Succ yv1500)",fontsize=16,color="green",shape="box"];264[label="Pos Zero",fontsize=16,color="green",shape="box"];265[label="Succ (Succ (primPlusNat yv190 yv1500))",fontsize=16,color="green",shape="box"];265 -> 273[label="",style="dashed", color="green", weight=3];
266[label="Succ yv190",fontsize=16,color="green",shape="box"];267[label="Succ yv1500",fontsize=16,color="green",shape="box"];268[label="Zero",fontsize=16,color="green",shape="box"];269[label="Succ yv300",fontsize=16,color="green",shape="box"];270 -> 222[label="",style="dashed", color="red", weight=0];
270[label="primMulNat yv1400 (Succ yv300)",fontsize=16,color="magenta"];270 -> 274[label="",style="dashed", color="magenta", weight=3];
270 -> 275[label="",style="dashed", color="magenta", weight=3];
271[label="yv1500",fontsize=16,color="green",shape="box"];272[label="yv180",fontsize=16,color="green",shape="box"];273 -> 234[label="",style="dashed", color="red", weight=0];
273[label="primPlusNat yv190 yv1500",fontsize=16,color="magenta"];273 -> 276[label="",style="dashed", color="magenta", weight=3];
273 -> 277[label="",style="dashed", color="magenta", weight=3];
274[label="Succ yv300",fontsize=16,color="green",shape="box"];275[label="yv1400",fontsize=16,color="green",shape="box"];276[label="yv1500",fontsize=16,color="green",shape="box"];277[label="yv190",fontsize=16,color="green",shape="box"];}

----------------------------------------

(12)
Complex Obligation (AND)

----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_span2Zs0(yv4, yv610, yv611) -> new_span2Zs(yv4, yv611)
   new_span2Zs(yv4, :(yv610, yv611)) -> new_span2Zs0(yv4, yv610, yv611)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_span2Zs(yv4, :(yv610, yv611)) -> new_span2Zs0(yv4, yv610, yv611)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*new_span2Zs0(yv4, yv610, yv611) -> new_span2Zs(yv4, yv611)
The graph contains the following edges 1 >= 1, 3 >= 2


----------------------------------------

(15)
YES

----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMulNat(Succ(yv1400), Succ(yv300)) -> new_primMulNat(yv1400, Succ(yv300))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primMulNat(Succ(yv1400), Succ(yv300)) -> new_primMulNat(yv1400, Succ(yv300))
The graph contains the following edges 1 > 1, 2 >= 2


----------------------------------------

(18)
YES

----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldl(yv3, yv14, yv5, yv4, yv610, yv611, yv12) -> new_foldl0(yv3, yv14, new_pt(yv5, yv610), yv5, yv4, yv611)
   new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) -> new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
   new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) -> new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)

The TRS R consists of the following rules:

   new_primPlusInt0(yv140, yv30, Neg(yv150)) -> new_primMinusNat0(new_primMulNat0(yv140, yv30), yv150)
   new_primPlusNat0(Succ(yv190), Zero) -> Succ(yv190)
   new_primPlusNat0(Zero, Succ(yv1500)) -> Succ(yv1500)
   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
   new_primMulNat0(Zero, Zero) -> Zero
   new_primPlusNat0(Zero, Zero) -> Zero
   new_primMinusNat0(Zero, Succ(yv1500)) -> Neg(Succ(yv1500))
   new_primPlusInt(yv140, yv30, Neg(yv150)) -> Neg(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
   new_primMulNat0(Succ(yv1400), Zero) -> Zero
   new_primMulNat0(Zero, Succ(yv300)) -> Zero
   new_readInt0(Neg(yv30), Neg(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
   new_primPlusInt(yv140, yv30, Pos(yv150)) -> new_primMinusNat0(yv150, new_primMulNat0(yv140, yv30))
   new_readInt0(Pos(yv30), Pos(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
   new_primMulNat0(Succ(yv1400), Succ(yv300)) -> new_primPlusNat0(new_primMulNat0(yv1400, Succ(yv300)), Succ(yv300))
   new_readInt0(Neg(yv30), Pos(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
   new_readInt0(Pos(yv30), Neg(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
   new_primPlusNat0(Succ(yv190), Succ(yv1500)) -> Succ(Succ(new_primPlusNat0(yv190, yv1500)))
   new_primMinusNat0(Succ(yv180), Succ(yv1500)) -> new_primMinusNat0(yv180, yv1500)
   new_primPlusInt0(yv140, yv30, Pos(yv150)) -> Pos(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
   new_primMinusNat0(Succ(yv180), Zero) -> Pos(Succ(yv180))

The set Q consists of the following terms:

   new_primPlusNat0(Zero, Succ(x0))
   new_primMulNat0(Zero, Zero)
   new_readInt0(Pos(x0), Pos(x1), x2)
   new_primMulNat0(Succ(x0), Succ(x1))
   new_primPlusInt0(x0, x1, Pos(x2))
   new_primPlusInt(x0, x1, Pos(x2))
   new_primPlusInt0(x0, x1, Neg(x2))
   new_primMinusNat0(Zero, Zero)
   new_primPlusNat0(Succ(x0), Zero)
   new_readInt0(Neg(x0), Neg(x1), x2)
   new_primMinusNat0(Succ(x0), Succ(x1))
   new_primMinusNat0(Zero, Succ(x0))
   new_readInt0(Neg(x0), Pos(x1), x2)
   new_readInt0(Pos(x0), Neg(x1), x2)
   new_primPlusInt(x0, x1, Neg(x2))
   new_primPlusNat0(Succ(x0), Succ(x1))
   new_primMulNat0(Zero, Succ(x0))
   new_primMinusNat0(Succ(x0), Zero)
   new_primPlusNat0(Zero, Zero)
   new_primMulNat0(Succ(x0), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(20) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_foldl(yv3, yv14, yv5, yv4, yv610, yv611, yv12) -> new_foldl0(yv3, yv14, new_pt(yv5, yv610), yv5, yv4, yv611) we obtained the following new rules [LPAR04]:

   (new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6),new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6))


----------------------------------------

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) -> new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
   new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) -> new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)
   new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6)

The TRS R consists of the following rules:

   new_primPlusInt0(yv140, yv30, Neg(yv150)) -> new_primMinusNat0(new_primMulNat0(yv140, yv30), yv150)
   new_primPlusNat0(Succ(yv190), Zero) -> Succ(yv190)
   new_primPlusNat0(Zero, Succ(yv1500)) -> Succ(yv1500)
   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
   new_primMulNat0(Zero, Zero) -> Zero
   new_primPlusNat0(Zero, Zero) -> Zero
   new_primMinusNat0(Zero, Succ(yv1500)) -> Neg(Succ(yv1500))
   new_primPlusInt(yv140, yv30, Neg(yv150)) -> Neg(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
   new_primMulNat0(Succ(yv1400), Zero) -> Zero
   new_primMulNat0(Zero, Succ(yv300)) -> Zero
   new_readInt0(Neg(yv30), Neg(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
   new_primPlusInt(yv140, yv30, Pos(yv150)) -> new_primMinusNat0(yv150, new_primMulNat0(yv140, yv30))
   new_readInt0(Pos(yv30), Pos(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
   new_primMulNat0(Succ(yv1400), Succ(yv300)) -> new_primPlusNat0(new_primMulNat0(yv1400, Succ(yv300)), Succ(yv300))
   new_readInt0(Neg(yv30), Pos(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
   new_readInt0(Pos(yv30), Neg(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
   new_primPlusNat0(Succ(yv190), Succ(yv1500)) -> Succ(Succ(new_primPlusNat0(yv190, yv1500)))
   new_primMinusNat0(Succ(yv180), Succ(yv1500)) -> new_primMinusNat0(yv180, yv1500)
   new_primPlusInt0(yv140, yv30, Pos(yv150)) -> Pos(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
   new_primMinusNat0(Succ(yv180), Zero) -> Pos(Succ(yv180))

The set Q consists of the following terms:

   new_primPlusNat0(Zero, Succ(x0))
   new_primMulNat0(Zero, Zero)
   new_readInt0(Pos(x0), Pos(x1), x2)
   new_primMulNat0(Succ(x0), Succ(x1))
   new_primPlusInt0(x0, x1, Pos(x2))
   new_primPlusInt(x0, x1, Pos(x2))
   new_primPlusInt0(x0, x1, Neg(x2))
   new_primMinusNat0(Zero, Zero)
   new_primPlusNat0(Succ(x0), Zero)
   new_readInt0(Neg(x0), Neg(x1), x2)
   new_primMinusNat0(Succ(x0), Succ(x1))
   new_primMinusNat0(Zero, Succ(x0))
   new_readInt0(Neg(x0), Pos(x1), x2)
   new_readInt0(Pos(x0), Neg(x1), x2)
   new_primPlusInt(x0, x1, Neg(x2))
   new_primPlusNat0(Succ(x0), Succ(x1))
   new_primMulNat0(Zero, Succ(x0))
   new_primMinusNat0(Succ(x0), Zero)
   new_primPlusNat0(Zero, Zero)
   new_primMulNat0(Succ(x0), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(22) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5, 7 > 5, 6 >= 6, 7 > 6


*new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) -> new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 6 > 7


*new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) -> new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
The graph contains the following edges 1 >= 1, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6


----------------------------------------

(23)
YES

----------------------------------------

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNat(Succ(yv180), Succ(yv1500)) -> new_primMinusNat(yv180, yv1500)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(25) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primMinusNat(Succ(yv180), Succ(yv1500)) -> new_primMinusNat(yv180, yv1500)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(26)
YES

----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primPlusNat(Succ(yv190), Succ(yv1500)) -> new_primPlusNat(yv190, yv1500)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(28) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primPlusNat(Succ(yv190), Succ(yv1500)) -> new_primPlusNat(yv190, yv1500)
The graph contains the following edges 1 > 1, 2 > 2


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(29)
YES