/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) TransformationProof [EQUIVALENT, 0 ms]
(8) QDP
(9) UsableRulesProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QReductionProof [EQUIVALENT, 0 ms]
(12) QDP
(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(14) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
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 Main where {
import qualified Prelude;
}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="reverse",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="reverse vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl (flip (:)) [] vx3",fontsize=16,color="burlywood",shape="box"];35[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 5[label="",style="solid", color="burlywood", weight=3];
36[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl (flip (:)) [] (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl (flip (:)) [] []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl (flip (:)) (flip (:) [] vx30) vx31",fontsize=16,color="burlywood",shape="box"];37[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 9[label="",style="solid", color="burlywood", weight=3];
38[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9[label="foldl (flip (:)) (flip (:) [] vx30) (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl (flip (:)) (flip (:) [] vx30) []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 18[label="",style="dashed", color="red", weight=0];
11[label="foldl (flip (:)) (flip (:) (flip (:) [] vx30) vx310) vx311",fontsize=16,color="magenta"];11 -> 19[label="",style="dashed", color="magenta", weight=3];
11 -> 20[label="",style="dashed", color="magenta", weight=3];
11 -> 21[label="",style="dashed", color="magenta", weight=3];
11 -> 22[label="",style="dashed", color="magenta", weight=3];
12[label="flip (:) [] vx30",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
19[label="[]",fontsize=16,color="green",shape="box"];20[label="vx311",fontsize=16,color="green",shape="box"];21[label="vx30",fontsize=16,color="green",shape="box"];22[label="vx310",fontsize=16,color="green",shape="box"];18[label="foldl (flip (:)) (flip (:) (flip (:) vx4 vx310) vx3110) vx3111",fontsize=16,color="burlywood",shape="triangle"];39[label="vx3111/vx31110 : vx31111",fontsize=10,color="white",style="solid",shape="box"];18 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 24[label="",style="solid", color="burlywood", weight=3];
40[label="vx3111/[]",fontsize=10,color="white",style="solid",shape="box"];18 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 25[label="",style="solid", color="burlywood", weight=3];
15[label="(:) vx30 []",fontsize=16,color="green",shape="box"];24[label="foldl (flip (:)) (flip (:) (flip (:) vx4 vx310) vx3110) (vx31110 : vx31111)",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="foldl (flip (:)) (flip (:) (flip (:) vx4 vx310) vx3110) []",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26 -> 18[label="",style="dashed", color="red", weight=0];
26[label="foldl (flip (:)) (flip (:) (flip (:) (flip (:) vx4 vx310) vx3110) vx31110) vx31111",fontsize=16,color="magenta"];26 -> 28[label="",style="dashed", color="magenta", weight=3];
26 -> 29[label="",style="dashed", color="magenta", weight=3];
26 -> 30[label="",style="dashed", color="magenta", weight=3];
26 -> 31[label="",style="dashed", color="magenta", weight=3];
27[label="flip (:) (flip (:) vx4 vx310) vx3110",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3];
28[label="flip (:) vx4 vx310",fontsize=16,color="black",shape="triangle"];28 -> 33[label="",style="solid", color="black", weight=3];
29[label="vx31111",fontsize=16,color="green",shape="box"];30[label="vx3110",fontsize=16,color="green",shape="box"];31[label="vx31110",fontsize=16,color="green",shape="box"];32[label="(:) vx3110 flip (:) vx4 vx310",fontsize=16,color="green",shape="box"];32 -> 34[label="",style="dashed", color="green", weight=3];
33[label="(:) vx310 vx4",fontsize=16,color="green",shape="box"];34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="flip (:) vx4 vx310",fontsize=16,color="magenta"];}

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

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

   new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(new_flip(vx4, vx310, h), vx3110, vx31110, vx31111, h)

The TRS R consists of the following rules:

   new_flip(vx4, vx310, h) -> :(vx310, vx4)

The set Q consists of the following terms:

   new_flip(x0, x1, x2)

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

(7) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(new_flip(vx4, vx310, h), vx3110, vx31110, vx31111, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(:(vx310, vx4), vx3110, vx31110, vx31111, h),new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(:(vx310, vx4), vx3110, vx31110, vx31111, h))


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

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

   new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(:(vx310, vx4), vx3110, vx31110, vx31111, h)

The TRS R consists of the following rules:

   new_flip(vx4, vx310, h) -> :(vx310, vx4)

The set Q consists of the following terms:

   new_flip(x0, x1, x2)

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

(9) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(:(vx310, vx4), vx3110, vx31110, vx31111, h)

R is empty.
The set Q consists of the following terms:

   new_flip(x0, x1, x2)

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

(11) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_flip(x0, x1, x2)


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

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

   new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(:(vx310, vx4), vx3110, vx31110, vx31111, h)

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_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), h) -> new_foldl(:(vx310, vx4), vx3110, vx31110, vx31111, h)
The graph contains the following edges 3 >= 2, 4 > 3, 4 > 4, 5 >= 5


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

(14)
YES