/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES


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

(0)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

foldFM_LE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
foldFM_LE k z fr EmptyFM = z;
foldFM_LE k z fr (Branch key elt _ fm_l fm_r)  | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = foldFM_LE k z fr fm_l;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

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

(2)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

foldFM_LE :: Ord c => (c  ->  b  ->  a  ->  a)  ->  a  ->  c  ->  FiniteMap c b  ->  a;
foldFM_LE k z fr EmptyFM = z;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r)  | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = foldFM_LE k z fr fm_l;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(3) 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
"foldFM_LE k z fr EmptyFM = z;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r)|key <= frfoldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r|otherwisefoldFM_LE k z fr fm_l;
"
is transformed to
"foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);
"
"foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;
"
"foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
"
"foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);
"
"foldFM_LE3 k z fr EmptyFM = z;
foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
"

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

(4)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

foldFM_LE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  c;
foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);

foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;

foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;

foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);

foldFM_LE3 k z fr EmptyFM = z;
foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.foldFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.foldFM_LE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="FiniteMap.foldFM_LE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="FiniteMap.foldFM_LE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];55[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 7[label="",style="solid", color="burlywood", weight=3];
56[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="FiniteMap.foldFM_LE wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.foldFM_LE wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.foldFM_LE3 wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.foldFM_LE2 wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="wz4",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 <= wz5)",fontsize=16,color="burlywood",shape="box"];57[label="wz60/False",fontsize=10,color="white",style="solid",shape="box"];12 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 13[label="",style="solid", color="burlywood", weight=3];
58[label="wz60/True",fontsize=10,color="white",style="solid",shape="box"];12 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 False wz61 wz62 wz63 wz64 (False <= wz5)",fontsize=16,color="burlywood",shape="box"];59[label="wz5/False",fontsize=10,color="white",style="solid",shape="box"];13 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 15[label="",style="solid", color="burlywood", weight=3];
60[label="wz5/True",fontsize=10,color="white",style="solid",shape="box"];13 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 True wz61 wz62 wz63 wz64 (True <= wz5)",fontsize=16,color="burlywood",shape="box"];61[label="wz5/False",fontsize=10,color="white",style="solid",shape="box"];14 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 17[label="",style="solid", color="burlywood", weight=3];
62[label="wz5/True",fontsize=10,color="white",style="solid",shape="box"];14 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="FiniteMap.foldFM_LE1 wz3 wz4 False False wz61 wz62 wz63 wz64 (False <= False)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="FiniteMap.foldFM_LE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (False <= True)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="FiniteMap.foldFM_LE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (True <= False)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="FiniteMap.foldFM_LE1 wz3 wz4 True True wz61 wz62 wz63 wz64 (True <= True)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="FiniteMap.foldFM_LE1 wz3 wz4 False False wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="FiniteMap.foldFM_LE1 wz3 wz4 True False wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="FiniteMap.foldFM_LE1 wz3 wz4 False True wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="FiniteMap.foldFM_LE1 wz3 wz4 True True wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23 -> 6[label="",style="dashed", color="red", weight=0];
23[label="FiniteMap.foldFM_LE wz3 (wz3 False wz61 (FiniteMap.foldFM_LE wz3 wz4 False wz63)) False wz64",fontsize=16,color="magenta"];23 -> 27[label="",style="dashed", color="magenta", weight=3];
23 -> 28[label="",style="dashed", color="magenta", weight=3];
23 -> 29[label="",style="dashed", color="magenta", weight=3];
24 -> 6[label="",style="dashed", color="red", weight=0];
24[label="FiniteMap.foldFM_LE wz3 (wz3 False wz61 (FiniteMap.foldFM_LE wz3 wz4 True wz63)) True wz64",fontsize=16,color="magenta"];24 -> 30[label="",style="dashed", color="magenta", weight=3];
24 -> 31[label="",style="dashed", color="magenta", weight=3];
24 -> 32[label="",style="dashed", color="magenta", weight=3];
25[label="FiniteMap.foldFM_LE0 wz3 wz4 False True wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
26 -> 6[label="",style="dashed", color="red", weight=0];
26[label="FiniteMap.foldFM_LE wz3 (wz3 True wz61 (FiniteMap.foldFM_LE wz3 wz4 True wz63)) True wz64",fontsize=16,color="magenta"];26 -> 34[label="",style="dashed", color="magenta", weight=3];
26 -> 35[label="",style="dashed", color="magenta", weight=3];
26 -> 36[label="",style="dashed", color="magenta", weight=3];
27[label="wz3 False wz61 (FiniteMap.foldFM_LE wz3 wz4 False wz63)",fontsize=16,color="green",shape="box"];27 -> 37[label="",style="dashed", color="green", weight=3];
27 -> 38[label="",style="dashed", color="green", weight=3];
27 -> 39[label="",style="dashed", color="green", weight=3];
28[label="False",fontsize=16,color="green",shape="box"];29[label="wz64",fontsize=16,color="green",shape="box"];30[label="wz3 False wz61 (FiniteMap.foldFM_LE wz3 wz4 True wz63)",fontsize=16,color="green",shape="box"];30 -> 40[label="",style="dashed", color="green", weight=3];
30 -> 41[label="",style="dashed", color="green", weight=3];
30 -> 42[label="",style="dashed", color="green", weight=3];
31[label="True",fontsize=16,color="green",shape="box"];32[label="wz64",fontsize=16,color="green",shape="box"];33[label="FiniteMap.foldFM_LE0 wz3 wz4 False True wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];33 -> 43[label="",style="solid", color="black", weight=3];
34[label="wz3 True wz61 (FiniteMap.foldFM_LE wz3 wz4 True wz63)",fontsize=16,color="green",shape="box"];34 -> 44[label="",style="dashed", color="green", weight=3];
34 -> 45[label="",style="dashed", color="green", weight=3];
34 -> 46[label="",style="dashed", color="green", weight=3];
35[label="True",fontsize=16,color="green",shape="box"];36[label="wz64",fontsize=16,color="green",shape="box"];37[label="False",fontsize=16,color="green",shape="box"];38[label="wz61",fontsize=16,color="green",shape="box"];39 -> 6[label="",style="dashed", color="red", weight=0];
39[label="FiniteMap.foldFM_LE wz3 wz4 False wz63",fontsize=16,color="magenta"];39 -> 47[label="",style="dashed", color="magenta", weight=3];
39 -> 48[label="",style="dashed", color="magenta", weight=3];
40[label="False",fontsize=16,color="green",shape="box"];41[label="wz61",fontsize=16,color="green",shape="box"];42 -> 6[label="",style="dashed", color="red", weight=0];
42[label="FiniteMap.foldFM_LE wz3 wz4 True wz63",fontsize=16,color="magenta"];42 -> 49[label="",style="dashed", color="magenta", weight=3];
42 -> 50[label="",style="dashed", color="magenta", weight=3];
43 -> 6[label="",style="dashed", color="red", weight=0];
43[label="FiniteMap.foldFM_LE wz3 wz4 False wz63",fontsize=16,color="magenta"];43 -> 51[label="",style="dashed", color="magenta", weight=3];
43 -> 52[label="",style="dashed", color="magenta", weight=3];
44[label="True",fontsize=16,color="green",shape="box"];45[label="wz61",fontsize=16,color="green",shape="box"];46 -> 6[label="",style="dashed", color="red", weight=0];
46[label="FiniteMap.foldFM_LE wz3 wz4 True wz63",fontsize=16,color="magenta"];46 -> 53[label="",style="dashed", color="magenta", weight=3];
46 -> 54[label="",style="dashed", color="magenta", weight=3];
47[label="False",fontsize=16,color="green",shape="box"];48[label="wz63",fontsize=16,color="green",shape="box"];49[label="True",fontsize=16,color="green",shape="box"];50[label="wz63",fontsize=16,color="green",shape="box"];51[label="False",fontsize=16,color="green",shape="box"];52[label="wz63",fontsize=16,color="green",shape="box"];53[label="True",fontsize=16,color="green",shape="box"];54[label="wz63",fontsize=16,color="green",shape="box"];}

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz64, h, ba)
   new_foldFM_LE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz64, h, ba)
   new_foldFM_LE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz63, h, ba)
   new_foldFM_LE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz63, h, ba)
   new_foldFM_LE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz63, h, ba)
   new_foldFM_LE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz63, h, ba)
   new_foldFM_LE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz64, h, ba)

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

   new_foldFM_LE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz63, h, ba)
   new_foldFM_LE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz64, h, ba)
   new_foldFM_LE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz63, h, ba)
   new_foldFM_LE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz64, h, ba)

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_foldFM_LE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, True, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


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

(11)
YES

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

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

   new_foldFM_LE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz63, h, ba)
   new_foldFM_LE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz64, h, ba)
   new_foldFM_LE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz63, h, ba)

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_foldFM_LE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, False, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


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

(14)
YES