/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 24 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(10) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy _ x [] = x : [];
insertBy cmp x ys@(y : ys') = case cmp x y of { 
GT-> y : insertBy cmp x ys'; 
_-> x : ys; 
 } ;

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

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

(1) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case cmp x y of {
GT  -> y : insertBy cmp x ys';
_  -> x : ys}
"
is transformed to
"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys _ = x : ys;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy _ x [] = x : [];
insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys _ = x : ys;

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

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

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

Binding Reductions:
The bind variable of the following binding Pattern
"ys@(wu : wv)"
is replaced by the following term
"wu : wv"

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy vz x [] = x : [];
insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys vy = x : ys;

}
module Main where {
import qualified List;
import qualified Maybe;
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 Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy vz x [] = x : [];
insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys vy = x : ys;

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

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.insert",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.insert ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.insert ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.insertBy compare ww3 ww4",fontsize=16,color="burlywood",shape="box"];171[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 171[label="",style="solid", color="burlywood", weight=9];
171 -> 6[label="",style="solid", color="burlywood", weight=3];
172[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="List.insertBy compare ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.insertBy compare ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare ww3 ww40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="ww3 : []",fontsize=16,color="green",shape="box"];10[label="List.insertBy0 ww40 primCmpChar ww3 ww41 (ww40 : ww41) (primCmpChar ww3 ww40)",fontsize=16,color="burlywood",shape="triangle"];173[label="ww3/Char ww30",fontsize=10,color="white",style="solid",shape="box"];10 -> 173[label="",style="solid", color="burlywood", weight=9];
173 -> 11[label="",style="solid", color="burlywood", weight=3];
11[label="List.insertBy0 ww40 primCmpChar (Char ww30) ww41 (ww40 : ww41) (primCmpChar (Char ww30) ww40)",fontsize=16,color="burlywood",shape="box"];174[label="ww40/Char ww400",fontsize=10,color="white",style="solid",shape="box"];11 -> 174[label="",style="solid", color="burlywood", weight=9];
174 -> 12[label="",style="solid", color="burlywood", weight=3];
12[label="List.insertBy0 (Char ww400) primCmpChar (Char ww30) ww41 (Char ww400 : ww41) (primCmpChar (Char ww30) (Char ww400))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="List.insertBy0 (Char ww400) primCmpChar (Char ww30) ww41 (Char ww400 : ww41) (primCmpNat ww30 ww400)",fontsize=16,color="burlywood",shape="box"];175[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];13 -> 175[label="",style="solid", color="burlywood", weight=9];
175 -> 14[label="",style="solid", color="burlywood", weight=3];
176[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 176[label="",style="solid", color="burlywood", weight=9];
176 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="List.insertBy0 (Char ww400) primCmpChar (Char (Succ ww300)) ww41 (Char ww400 : ww41) (primCmpNat (Succ ww300) ww400)",fontsize=16,color="burlywood",shape="box"];177[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];14 -> 177[label="",style="solid", color="burlywood", weight=9];
177 -> 16[label="",style="solid", color="burlywood", weight=3];
178[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 178[label="",style="solid", color="burlywood", weight=9];
178 -> 17[label="",style="solid", color="burlywood", weight=3];
15[label="List.insertBy0 (Char ww400) primCmpChar (Char Zero) ww41 (Char ww400 : ww41) (primCmpNat Zero ww400)",fontsize=16,color="burlywood",shape="box"];179[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 179[label="",style="solid", color="burlywood", weight=9];
179 -> 18[label="",style="solid", color="burlywood", weight=3];
180[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 180[label="",style="solid", color="burlywood", weight=9];
180 -> 19[label="",style="solid", color="burlywood", weight=3];
16[label="List.insertBy0 (Char (Succ ww4000)) primCmpChar (Char (Succ ww300)) ww41 (Char (Succ ww4000) : ww41) (primCmpNat (Succ ww300) (Succ ww4000))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="List.insertBy0 (Char Zero) primCmpChar (Char (Succ ww300)) ww41 (Char Zero : ww41) (primCmpNat (Succ ww300) Zero)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="List.insertBy0 (Char (Succ ww4000)) primCmpChar (Char Zero) ww41 (Char (Succ ww4000) : ww41) (primCmpNat Zero (Succ ww4000))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.insertBy0 (Char Zero) primCmpChar (Char Zero) ww41 (Char Zero : ww41) (primCmpNat Zero Zero)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20 -> 117[label="",style="dashed", color="red", weight=0];
20[label="List.insertBy0 (Char (Succ ww4000)) primCmpChar (Char (Succ ww300)) ww41 (Char (Succ ww4000) : ww41) (primCmpNat ww300 ww4000)",fontsize=16,color="magenta"];20 -> 118[label="",style="dashed", color="magenta", weight=3];
20 -> 119[label="",style="dashed", color="magenta", weight=3];
20 -> 120[label="",style="dashed", color="magenta", weight=3];
20 -> 121[label="",style="dashed", color="magenta", weight=3];
20 -> 122[label="",style="dashed", color="magenta", weight=3];
21[label="List.insertBy0 (Char Zero) primCmpChar (Char (Succ ww300)) ww41 (Char Zero : ww41) GT",fontsize=16,color="black",shape="box"];21 -> 26[label="",style="solid", color="black", weight=3];
22[label="List.insertBy0 (Char (Succ ww4000)) primCmpChar (Char Zero) ww41 (Char (Succ ww4000) : ww41) LT",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
23[label="List.insertBy0 (Char Zero) primCmpChar (Char Zero) ww41 (Char Zero : ww41) EQ",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
118[label="ww4000",fontsize=16,color="green",shape="box"];119[label="ww300",fontsize=16,color="green",shape="box"];120[label="ww300",fontsize=16,color="green",shape="box"];121[label="ww41",fontsize=16,color="green",shape="box"];122[label="ww4000",fontsize=16,color="green",shape="box"];117[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat ww20 ww21)",fontsize=16,color="burlywood",shape="triangle"];181[label="ww20/Succ ww200",fontsize=10,color="white",style="solid",shape="box"];117 -> 181[label="",style="solid", color="burlywood", weight=9];
181 -> 153[label="",style="solid", color="burlywood", weight=3];
182[label="ww20/Zero",fontsize=10,color="white",style="solid",shape="box"];117 -> 182[label="",style="solid", color="burlywood", weight=9];
182 -> 154[label="",style="solid", color="burlywood", weight=3];
26[label="Char Zero : List.insertBy primCmpChar (Char (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];26 -> 33[label="",style="dashed", color="green", weight=3];
27[label="Char Zero : Char (Succ ww4000) : ww41",fontsize=16,color="green",shape="box"];28[label="Char Zero : Char Zero : ww41",fontsize=16,color="green",shape="box"];153[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat (Succ ww200) ww21)",fontsize=16,color="burlywood",shape="box"];183[label="ww21/Succ ww210",fontsize=10,color="white",style="solid",shape="box"];153 -> 183[label="",style="solid", color="burlywood", weight=9];
183 -> 155[label="",style="solid", color="burlywood", weight=3];
184[label="ww21/Zero",fontsize=10,color="white",style="solid",shape="box"];153 -> 184[label="",style="solid", color="burlywood", weight=9];
184 -> 156[label="",style="solid", color="burlywood", weight=3];
154[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat Zero ww21)",fontsize=16,color="burlywood",shape="box"];185[label="ww21/Succ ww210",fontsize=10,color="white",style="solid",shape="box"];154 -> 185[label="",style="solid", color="burlywood", weight=9];
185 -> 157[label="",style="solid", color="burlywood", weight=3];
186[label="ww21/Zero",fontsize=10,color="white",style="solid",shape="box"];154 -> 186[label="",style="solid", color="burlywood", weight=9];
186 -> 158[label="",style="solid", color="burlywood", weight=3];
33[label="List.insertBy primCmpChar (Char (Succ ww300)) ww41",fontsize=16,color="burlywood",shape="triangle"];187[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];33 -> 187[label="",style="solid", color="burlywood", weight=9];
187 -> 38[label="",style="solid", color="burlywood", weight=3];
188[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];33 -> 188[label="",style="solid", color="burlywood", weight=9];
188 -> 39[label="",style="solid", color="burlywood", weight=3];
155[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat (Succ ww200) (Succ ww210))",fontsize=16,color="black",shape="box"];155 -> 159[label="",style="solid", color="black", weight=3];
156[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat (Succ ww200) Zero)",fontsize=16,color="black",shape="box"];156 -> 160[label="",style="solid", color="black", weight=3];
157[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat Zero (Succ ww210))",fontsize=16,color="black",shape="box"];157 -> 161[label="",style="solid", color="black", weight=3];
158[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat Zero Zero)",fontsize=16,color="black",shape="box"];158 -> 162[label="",style="solid", color="black", weight=3];
38[label="List.insertBy primCmpChar (Char (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="List.insertBy primCmpChar (Char (Succ ww300)) []",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
159 -> 117[label="",style="dashed", color="red", weight=0];
159[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) (primCmpNat ww200 ww210)",fontsize=16,color="magenta"];159 -> 163[label="",style="dashed", color="magenta", weight=3];
159 -> 164[label="",style="dashed", color="magenta", weight=3];
160[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) GT",fontsize=16,color="black",shape="box"];160 -> 165[label="",style="solid", color="black", weight=3];
161[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) LT",fontsize=16,color="black",shape="box"];161 -> 166[label="",style="solid", color="black", weight=3];
162[label="List.insertBy0 (Char (Succ ww17)) primCmpChar (Char (Succ ww18)) ww19 (Char (Succ ww17) : ww19) EQ",fontsize=16,color="black",shape="box"];162 -> 167[label="",style="solid", color="black", weight=3];
45 -> 10[label="",style="dashed", color="red", weight=0];
45[label="List.insertBy0 ww410 primCmpChar (Char (Succ ww300)) ww411 (ww410 : ww411) (primCmpChar (Char (Succ ww300)) ww410)",fontsize=16,color="magenta"];45 -> 52[label="",style="dashed", color="magenta", weight=3];
45 -> 53[label="",style="dashed", color="magenta", weight=3];
45 -> 54[label="",style="dashed", color="magenta", weight=3];
46[label="Char (Succ ww300) : []",fontsize=16,color="green",shape="box"];163[label="ww200",fontsize=16,color="green",shape="box"];164[label="ww210",fontsize=16,color="green",shape="box"];165[label="Char (Succ ww17) : List.insertBy primCmpChar (Char (Succ ww18)) ww19",fontsize=16,color="green",shape="box"];165 -> 168[label="",style="dashed", color="green", weight=3];
166[label="Char (Succ ww18) : Char (Succ ww17) : ww19",fontsize=16,color="green",shape="box"];167[label="Char (Succ ww18) : Char (Succ ww17) : ww19",fontsize=16,color="green",shape="box"];52[label="ww411",fontsize=16,color="green",shape="box"];53[label="ww410",fontsize=16,color="green",shape="box"];54[label="Char (Succ ww300)",fontsize=16,color="green",shape="box"];168 -> 33[label="",style="dashed", color="red", weight=0];
168[label="List.insertBy primCmpChar (Char (Succ ww18)) ww19",fontsize=16,color="magenta"];168 -> 169[label="",style="dashed", color="magenta", weight=3];
168 -> 170[label="",style="dashed", color="magenta", weight=3];
169[label="ww18",fontsize=16,color="green",shape="box"];170[label="ww19",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_insertBy00(Char(Succ(ww4000)), Char(Succ(ww300)), ww41) -> new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
   new_insertBy00(Char(Zero), Char(Succ(ww300)), :(ww410, ww411)) -> new_insertBy00(ww410, Char(Succ(ww300)), ww411)
   new_insertBy0(ww17, ww18, ww19, Succ(ww200), Succ(ww210)) -> new_insertBy0(ww17, ww18, ww19, ww200, ww210)
   new_insertBy0(ww17, ww18, ww19, Succ(ww200), Zero) -> new_insertBy(ww18, ww19)
   new_insertBy(ww300, :(ww410, ww411)) -> new_insertBy00(ww410, Char(Succ(ww300)), ww411)

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

(9) 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_insertBy00(Char(Succ(ww4000)), Char(Succ(ww300)), ww41) -> new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 2 > 4, 1 > 5


*new_insertBy00(Char(Zero), Char(Succ(ww300)), :(ww410, ww411)) -> new_insertBy00(ww410, Char(Succ(ww300)), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3


*new_insertBy(ww300, :(ww410, ww411)) -> new_insertBy00(ww410, Char(Succ(ww300)), ww411)
The graph contains the following edges 2 > 1, 2 > 3


*new_insertBy0(ww17, ww18, ww19, Succ(ww200), Succ(ww210)) -> new_insertBy0(ww17, ww18, ww19, ww200, ww210)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5


*new_insertBy0(ww17, ww18, ww19, Succ(ww200), Zero) -> new_insertBy(ww18, ww19)
The graph contains the following edges 2 >= 1, 3 >= 2


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

(10)
YES