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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

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


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM_ :: Monad c => (d  ->  a  ->  c b)  ->  [d]  ->  [a]  ->  c ();
zipWithM_ f xs ys = sequence_ (zipWith f xs ys);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\_->q"
is transformed to
"gtGt0 q _ = q;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM_ :: Monad c => (b  ->  a  ->  c d)  ->  [b]  ->  [a]  ->  c ();
zipWithM_ f xs ys = sequence_ (zipWith f xs ys);

}

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

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM_ :: Monad c => (d  ->  b  ->  c a)  ->  [d]  ->  [b]  ->  c ();
zipWithM_ f xs ys = sequence_ (zipWith f xs ys);

}

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM_ :: Monad d => (c  ->  b  ->  d a)  ->  [c]  ->  [b]  ->  d ();
zipWithM_ f xs ys = sequence_ (zipWith f xs ys);

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];89[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 8[label="",style="solid", color="burlywood", weight=3];
90[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];91[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 10[label="",style="solid", color="burlywood", weight=3];
92[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) (ww50 : ww51))",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="foldr (>>) (return ()) (ww3 ww40 ww50 : zipWith ww3 ww41 ww51)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14 -> 12[label="",style="dashed", color="red", weight=0];
14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="blue",shape="box"];93[label="return :: () -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];15 -> 93[label="",style="solid", color="blue", weight=9];
93 -> 17[label="",style="solid", color="blue", weight=3];
94[label="return :: () -> IO ()",fontsize=10,color="white",style="solid",shape="box"];15 -> 94[label="",style="solid", color="blue", weight=9];
94 -> 18[label="",style="solid", color="blue", weight=3];
95[label="return :: () -> [] ()",fontsize=10,color="white",style="solid",shape="box"];15 -> 95[label="",style="solid", color="blue", weight=9];
95 -> 19[label="",style="solid", color="blue", weight=3];
16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="blue",shape="box"];96[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];16 -> 96[label="",style="solid", color="blue", weight=9];
96 -> 31[label="",style="solid", color="blue", weight=3];
97[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];16 -> 97[label="",style="solid", color="blue", weight=9];
97 -> 32[label="",style="solid", color="blue", weight=3];
98[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];16 -> 98[label="",style="solid", color="blue", weight=9];
98 -> 33[label="",style="solid", color="blue", weight=3];
17[label="return ()",fontsize=16,color="black",shape="box"];17 -> 22[label="",style="solid", color="black", weight=3];
18[label="return ()",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
19[label="return ()",fontsize=16,color="black",shape="box"];19 -> 24[label="",style="solid", color="black", weight=3];
31 -> 27[label="",style="dashed", color="red", weight=0];
31[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];32 -> 38[label="",style="dashed", color="magenta", weight=3];
33 -> 29[label="",style="dashed", color="red", weight=0];
33[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];33 -> 39[label="",style="dashed", color="magenta", weight=3];
22[label="Just ()",fontsize=16,color="green",shape="box"];23[label="primretIO ()",fontsize=16,color="black",shape="box"];23 -> 30[label="",style="solid", color="black", weight=3];
24[label="() : []",fontsize=16,color="green",shape="box"];37 -> 7[label="",style="dashed", color="red", weight=0];
37[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
37 -> 43[label="",style="dashed", color="magenta", weight=3];
27[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];27 -> 34[label="",style="solid", color="black", weight=3];
38 -> 7[label="",style="dashed", color="red", weight=0];
38[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];38 -> 44[label="",style="dashed", color="magenta", weight=3];
38 -> 45[label="",style="dashed", color="magenta", weight=3];
28[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];28 -> 35[label="",style="solid", color="black", weight=3];
39 -> 7[label="",style="dashed", color="red", weight=0];
39[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];39 -> 46[label="",style="dashed", color="magenta", weight=3];
39 -> 47[label="",style="dashed", color="magenta", weight=3];
29[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];29 -> 36[label="",style="solid", color="black", weight=3];
30[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];42[label="ww51",fontsize=16,color="green",shape="box"];43[label="ww41",fontsize=16,color="green",shape="box"];34 -> 40[label="",style="dashed", color="red", weight=0];
34[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];34 -> 41[label="",style="dashed", color="magenta", weight=3];
44[label="ww51",fontsize=16,color="green",shape="box"];45[label="ww41",fontsize=16,color="green",shape="box"];35[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];35 -> 48[label="",style="solid", color="black", weight=3];
46[label="ww51",fontsize=16,color="green",shape="box"];47[label="ww41",fontsize=16,color="green",shape="box"];36 -> 49[label="",style="dashed", color="red", weight=0];
36[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];36 -> 50[label="",style="dashed", color="magenta", weight=3];
41[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];41 -> 51[label="",style="dashed", color="green", weight=3];
41 -> 52[label="",style="dashed", color="green", weight=3];
40[label="ww7 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];99[label="ww7/Nothing",fontsize=10,color="white",style="solid",shape="box"];40 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 53[label="",style="solid", color="burlywood", weight=3];
100[label="ww7/Just ww70",fontsize=10,color="white",style="solid",shape="box"];40 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 54[label="",style="solid", color="burlywood", weight=3];
48 -> 55[label="",style="dashed", color="red", weight=0];
48[label="primbindIO (ww3 ww40 ww50) (gtGt0 ww6)",fontsize=16,color="magenta"];48 -> 56[label="",style="dashed", color="magenta", weight=3];
50[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];50 -> 57[label="",style="dashed", color="green", weight=3];
50 -> 58[label="",style="dashed", color="green", weight=3];
49[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];101[label="ww8/ww80 : ww81",fontsize=10,color="white",style="solid",shape="box"];49 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 59[label="",style="solid", color="burlywood", weight=3];
102[label="ww8/[]",fontsize=10,color="white",style="solid",shape="box"];49 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 60[label="",style="solid", color="burlywood", weight=3];
51[label="ww40",fontsize=16,color="green",shape="box"];52[label="ww50",fontsize=16,color="green",shape="box"];53[label="Nothing >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
54[label="Just ww70 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];54 -> 62[label="",style="solid", color="black", weight=3];
56[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];56 -> 69[label="",style="dashed", color="green", weight=3];
56 -> 70[label="",style="dashed", color="green", weight=3];
55[label="primbindIO ww9 (gtGt0 ww6)",fontsize=16,color="burlywood",shape="triangle"];103[label="ww9/IO ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 65[label="",style="solid", color="burlywood", weight=3];
104[label="ww9/AProVE_IO ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 66[label="",style="solid", color="burlywood", weight=3];
105[label="ww9/AProVE_Exception ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 67[label="",style="solid", color="burlywood", weight=3];
106[label="ww9/AProVE_Error ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 68[label="",style="solid", color="burlywood", weight=3];
57[label="ww40",fontsize=16,color="green",shape="box"];58[label="ww50",fontsize=16,color="green",shape="box"];59[label="ww80 : ww81 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];59 -> 71[label="",style="solid", color="black", weight=3];
60[label="[] >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];60 -> 72[label="",style="solid", color="black", weight=3];
61[label="Nothing",fontsize=16,color="green",shape="box"];62[label="gtGt0 ww6 ww70",fontsize=16,color="black",shape="box"];62 -> 73[label="",style="solid", color="black", weight=3];
69[label="ww40",fontsize=16,color="green",shape="box"];70[label="ww50",fontsize=16,color="green",shape="box"];65[label="primbindIO (IO ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];65 -> 74[label="",style="solid", color="black", weight=3];
66[label="primbindIO (AProVE_IO ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];66 -> 75[label="",style="solid", color="black", weight=3];
67[label="primbindIO (AProVE_Exception ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];67 -> 76[label="",style="solid", color="black", weight=3];
68[label="primbindIO (AProVE_Error ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];68 -> 77[label="",style="solid", color="black", weight=3];
71 -> 78[label="",style="dashed", color="red", weight=0];
71[label="gtGt0 ww6 ww80 ++ (ww81 >>= gtGt0 ww6)",fontsize=16,color="magenta"];71 -> 79[label="",style="dashed", color="magenta", weight=3];
72[label="[]",fontsize=16,color="green",shape="box"];73[label="ww6",fontsize=16,color="green",shape="box"];74[label="error []",fontsize=16,color="red",shape="box"];75[label="gtGt0 ww6 ww90",fontsize=16,color="black",shape="box"];75 -> 80[label="",style="solid", color="black", weight=3];
76[label="AProVE_Exception ww90",fontsize=16,color="green",shape="box"];77[label="AProVE_Error ww90",fontsize=16,color="green",shape="box"];79 -> 49[label="",style="dashed", color="red", weight=0];
79[label="ww81 >>= gtGt0 ww6",fontsize=16,color="magenta"];79 -> 81[label="",style="dashed", color="magenta", weight=3];
78[label="gtGt0 ww6 ww80 ++ ww10",fontsize=16,color="black",shape="triangle"];78 -> 82[label="",style="solid", color="black", weight=3];
80[label="ww6",fontsize=16,color="green",shape="box"];81[label="ww81",fontsize=16,color="green",shape="box"];82[label="ww6 ++ ww10",fontsize=16,color="burlywood",shape="triangle"];107[label="ww6/ww60 : ww61",fontsize=10,color="white",style="solid",shape="box"];82 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 83[label="",style="solid", color="burlywood", weight=3];
108[label="ww6/[]",fontsize=10,color="white",style="solid",shape="box"];82 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 84[label="",style="solid", color="burlywood", weight=3];
83[label="(ww60 : ww61) ++ ww10",fontsize=16,color="black",shape="box"];83 -> 85[label="",style="solid", color="black", weight=3];
84[label="[] ++ ww10",fontsize=16,color="black",shape="box"];84 -> 86[label="",style="solid", color="black", weight=3];
85[label="ww60 : ww61 ++ ww10",fontsize=16,color="green",shape="box"];85 -> 87[label="",style="dashed", color="green", weight=3];
86[label="ww10",fontsize=16,color="green",shape="box"];87 -> 82[label="",style="dashed", color="red", weight=0];
87[label="ww61 ++ ww10",fontsize=16,color="magenta"];87 -> 88[label="",style="dashed", color="magenta", weight=3];
88[label="ww61",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h)

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

(10) 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_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(11)
YES

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(ww60, ww61), ww10) -> new_psPs(ww61, ww10)

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

(13) 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_psPs(:(ww60, ww61), ww10) -> new_psPs(ww61, ww10)
The graph contains the following edges 1 > 1, 2 >= 2


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)

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

(16) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
----------------------------------------

(17)
Complex Obligation (AND)

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)

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

(19) 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_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7


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

(20)
YES

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

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

   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)

R is empty.
Q is empty.
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_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7


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

(23)
YES

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

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

   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7


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(26)
YES