/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

scanr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  List b;
scanr f q0 Nil = Cons q0 Nil;
scanr f q0 (Cons x xs) = Cons (f x (scanrQ f q0 xs)) (scanrQs f q0 xs);

scanrQ vw vx vy = scanrQ1 vw vx vy (scanrVu40 vw vx vy);

scanrQ1 vw vx vy (Cons q vv) = q;

scanrQs vw vx vy = scanrQs0 vw vx vy (scanrVu40 vw vx vy);

scanrQs0 vw vx vy qs = qs;

scanrVu40 vw vx vy = scanr vw vx vy;

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

scanr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  List b;
scanr f q0 Nil = Cons q0 Nil;
scanr f q0 (Cons x xs) = Cons (f x (scanrQ f q0 xs)) (scanrQs f q0 xs);

scanrQ vw vx vy = scanrQ1 vw vx vy (scanrVu40 vw vx vy);

scanrQ1 vw vx vy (Cons q vv) = q;

scanrQs vw vx vy = scanrQs0 vw vx vy (scanrVu40 vw vx vy);

scanrQs0 vw vx vy qs = qs;

scanrVu40 vw vx vy = scanr vw vx vy;

}

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

(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 Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

scanr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  List b;
scanr f q0 Nil = Cons q0 Nil;
scanr f q0 (Cons x xs) = Cons (f x (scanrQ f q0 xs)) (scanrQs f q0 xs);

scanrQ vw vx vy = scanrQ1 vw vx vy (scanrVu40 vw vx vy);

scanrQ1 vw vx vy (Cons q vv) = q;

scanrQs vw vx vy = scanrQs0 vw vx vy (scanrVu40 vw vx vy);

scanrQs0 vw vx vy qs = qs;

scanrVu40 vw vx vy = scanr vw vx vy;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="scanr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="scanr wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="scanr wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="scanr wv3 wv4 wv5",fontsize=16,color="burlywood",shape="triangle"];27[label="wv5/Cons wv50 wv51",fontsize=10,color="white",style="solid",shape="box"];5 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 6[label="",style="solid", color="burlywood", weight=3];
28[label="wv5/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="scanr wv3 wv4 (Cons wv50 wv51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="scanr wv3 wv4 Nil",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="Cons (wv3 wv50 (scanrQ wv3 wv4 wv51)) (scanrQs wv3 wv4 wv51)",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
8 -> 11[label="",style="dashed", color="green", weight=3];
9[label="Cons wv4 Nil",fontsize=16,color="green",shape="box"];10[label="wv3 wv50 (scanrQ wv3 wv4 wv51)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 13[label="",style="dashed", color="green", weight=3];
11[label="scanrQs wv3 wv4 wv51",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="wv50",fontsize=16,color="green",shape="box"];13[label="scanrQ wv3 wv4 wv51",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="scanrQs0 wv3 wv4 wv51 (scanrVu40 wv3 wv4 wv51)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15 -> 19[label="",style="dashed", color="red", weight=0];
15[label="scanrQ1 wv3 wv4 wv51 (scanrVu40 wv3 wv4 wv51)",fontsize=16,color="magenta"];15 -> 20[label="",style="dashed", color="magenta", weight=3];
16[label="scanrVu40 wv3 wv4 wv51",fontsize=16,color="black",shape="triangle"];16 -> 18[label="",style="solid", color="black", weight=3];
20 -> 16[label="",style="dashed", color="red", weight=0];
20[label="scanrVu40 wv3 wv4 wv51",fontsize=16,color="magenta"];19[label="scanrQ1 wv3 wv4 wv51 wv6",fontsize=16,color="burlywood",shape="triangle"];29[label="wv6/Cons wv60 wv61",fontsize=10,color="white",style="solid",shape="box"];19 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 22[label="",style="solid", color="burlywood", weight=3];
30[label="wv6/Nil",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 23[label="",style="solid", color="burlywood", weight=3];
18 -> 5[label="",style="dashed", color="red", weight=0];
18[label="scanr wv3 wv4 wv51",fontsize=16,color="magenta"];18 -> 24[label="",style="dashed", color="magenta", weight=3];
22[label="scanrQ1 wv3 wv4 wv51 (Cons wv60 wv61)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="scanrQ1 wv3 wv4 wv51 Nil",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="wv51",fontsize=16,color="green",shape="box"];25[label="wv60",fontsize=16,color="green",shape="box"];26[label="error []",fontsize=16,color="red",shape="box"];}

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

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

   new_scanr(wv3, wv4, Cons(wv50, wv51), h, ba) -> new_scanrVu40(wv3, wv4, wv51, h, ba)
   new_scanrVu40(wv3, wv4, wv51, h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
   new_scanr(wv3, wv4, Cons(wv50, wv51), h, ba) -> new_scanr(wv3, wv4, wv51, 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_scanrVu40(wv3, wv4, wv51, h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5


*new_scanr(wv3, wv4, Cons(wv50, wv51), h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_scanr(wv3, wv4, Cons(wv50, wv51), h, ba) -> new_scanrVu40(wv3, wv4, wv51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


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(8)
YES