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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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(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 :: (c  ->  a  ->  b  ->  b)  ->  b  ->  FiniteMap c a  ->  b;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt _ fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) 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 a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

foldFM :: (a  ->  b  ->  c  ->  c)  ->  c  ->  FiniteMap a b  ->  c;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) 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;
"

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

(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) ;

foldFM :: (a  ->  c  ->  b  ->  b)  ->  b  ->  FiniteMap a c  ->  b;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) 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;
}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.foldFM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.foldFM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="FiniteMap.foldFM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];16[label="vz5/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 6[label="",style="solid", color="burlywood", weight=3];
17[label="vz5/FiniteMap.Branch vz50 vz51 vz52 vz53 vz54",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="FiniteMap.foldFM vz3 vz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="FiniteMap.foldFM vz3 vz4 (FiniteMap.Branch vz50 vz51 vz52 vz53 vz54)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="vz4",fontsize=16,color="green",shape="box"];9 -> 5[label="",style="dashed", color="red", weight=0];
9[label="FiniteMap.foldFM vz3 (vz3 vz50 vz51 (FiniteMap.foldFM vz3 vz4 vz54)) vz53",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
9 -> 11[label="",style="dashed", color="magenta", weight=3];
10[label="vz53",fontsize=16,color="green",shape="box"];11[label="vz3 vz50 vz51 (FiniteMap.foldFM vz3 vz4 vz54)",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
11 -> 13[label="",style="dashed", color="green", weight=3];
11 -> 14[label="",style="dashed", color="green", weight=3];
12[label="vz50",fontsize=16,color="green",shape="box"];13[label="vz51",fontsize=16,color="green",shape="box"];14 -> 5[label="",style="dashed", color="red", weight=0];
14[label="FiniteMap.foldFM vz3 vz4 vz54",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
15[label="vz54",fontsize=16,color="green",shape="box"];}

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

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

   new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz54, h, ba, bb)
   new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz53, h, ba, bb)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz54, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5


*new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz53, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5


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(8)
YES