/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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

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

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
foldFM_GE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  c;
foldFM_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt _ fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = foldFM_GE k z fr fm_r;

}
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 a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
foldFM_GE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
foldFM_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = foldFM_GE k z fr fm_r;

}
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_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r)|key >= frfoldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l|otherwisefoldFM_GE k z fr fm_r;
"
is transformed to
"foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r);
"
"foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;
"
"foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l;
foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise;
"
"foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr);
"
"foldFM_GE3 k z fr EmptyFM = z;
foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;
"

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

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

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
foldFM_GE :: Ord c => (c  ->  b  ->  a  ->  a)  ->  a  ->  c  ->  FiniteMap c b  ->  a;
foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r);

foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;

foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l;
foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise;

foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr);

foldFM_GE3 k z fr EmptyFM = z;
foldFM_GE3 wv ww wx wy = foldFM_GE2 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_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.foldFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.foldFM_GE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="FiniteMap.foldFM_GE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="FiniteMap.foldFM_GE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];29[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 7[label="",style="solid", color="burlywood", weight=3];
30[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="FiniteMap.foldFM_GE wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.foldFM_GE 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_GE3 wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.foldFM_GE2 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_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 >= wz5)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= LT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == LT))",fontsize=16,color="burlywood",shape="box"];31[label="wz60/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 15[label="",style="solid", color="burlywood", weight=3];
15[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 () wz61 wz62 wz63 wz64 (not (compare () wz5 == LT))",fontsize=16,color="burlywood",shape="box"];32[label="wz5/()",fontsize=10,color="white",style="solid",shape="box"];15 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 16[label="",style="solid", color="burlywood", weight=3];
16[label="FiniteMap.foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="FiniteMap.foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not (EQ == LT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="FiniteMap.foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="FiniteMap.foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20 -> 6[label="",style="dashed", color="red", weight=0];
20[label="FiniteMap.foldFM_GE wz3 (wz3 () wz61 (FiniteMap.foldFM_GE wz3 wz4 () wz64)) () wz63",fontsize=16,color="magenta"];20 -> 21[label="",style="dashed", color="magenta", weight=3];
20 -> 22[label="",style="dashed", color="magenta", weight=3];
20 -> 23[label="",style="dashed", color="magenta", weight=3];
21[label="wz3 () wz61 (FiniteMap.foldFM_GE wz3 wz4 () wz64)",fontsize=16,color="green",shape="box"];21 -> 24[label="",style="dashed", color="green", weight=3];
21 -> 25[label="",style="dashed", color="green", weight=3];
21 -> 26[label="",style="dashed", color="green", weight=3];
22[label="wz63",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="()",fontsize=16,color="green",shape="box"];25[label="wz61",fontsize=16,color="green",shape="box"];26 -> 6[label="",style="dashed", color="red", weight=0];
26[label="FiniteMap.foldFM_GE wz3 wz4 () wz64",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
26 -> 28[label="",style="dashed", color="magenta", weight=3];
27[label="wz64",fontsize=16,color="green",shape="box"];28[label="()",fontsize=16,color="green",shape="box"];}

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

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

   new_foldFM_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, @0, wz64, h, ba)
   new_foldFM_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, @0, wz63, h, ba)

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

(7) 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_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, @0, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, @0, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


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

(8)
YES