/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES


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

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

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

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

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

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

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

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

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

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

(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 Main where {
import qualified Prelude;
}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence_ vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr (>>) (return ()) vy3",fontsize=16,color="burlywood",shape="triangle"];28[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 5[label="",style="solid", color="burlywood", weight=3];
29[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr (>>) (return ()) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="return ()",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3];
11[label="() : []",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="triangle"];30[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];13 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 14[label="",style="solid", color="burlywood", weight=3];
31[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="vy300 : vy301 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="[] >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="gtGt0 vy4 vy300 ++ (vy301 >>= gtGt0 vy4)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];19 -> 13[label="",style="dashed", color="red", weight=0];
19[label="vy301 >>= gtGt0 vy4",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
18[label="gtGt0 vy4 vy300 ++ vy5",fontsize=16,color="black",shape="triangle"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="vy301",fontsize=16,color="green",shape="box"];21[label="vy4 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];32[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];21 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 22[label="",style="solid", color="burlywood", weight=3];
33[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 23[label="",style="solid", color="burlywood", weight=3];
22[label="(vy40 : vy41) ++ vy5",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="[] ++ vy5",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="vy40 : vy41 ++ vy5",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
25[label="vy5",fontsize=16,color="green",shape="box"];26 -> 21[label="",style="dashed", color="red", weight=0];
26[label="vy41 ++ vy5",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
27[label="vy41",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(:(vy300, vy301), vy4, h) -> new_gtGtEs(vy301, vy4, 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(:(vy300, vy301), vy4, h) -> new_gtGtEs(vy301, vy4, 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(:(vy40, vy41), vy5) -> new_psPs(vy41, vy5)

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(:(vy40, vy41), vy5) -> new_psPs(vy41, vy5)
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(:(vy30, vy31), h) -> new_foldr(vy31, h)

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

(16) 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(:(vy30, vy31), h) -> new_foldr(vy31, h)
The graph contains the following edges 1 > 1, 2 >= 2


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

(17)
YES