10.46/4.37	YES
12.22/4.90	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.22/4.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.22/4.90	
12.22/4.90	
12.22/4.90	H-Termination with start terms of the given HASKELL could be proven:
12.22/4.90	
12.22/4.90	(0) HASKELL
12.22/4.90	(1) CR [EQUIVALENT, 0 ms]
12.22/4.90	(2) HASKELL
12.22/4.90	(3) BR [EQUIVALENT, 0 ms]
12.22/4.90	(4) HASKELL
12.22/4.90	(5) COR [EQUIVALENT, 0 ms]
12.22/4.90	(6) HASKELL
12.22/4.90	(7) Narrow [SOUND, 0 ms]
12.22/4.90	(8) QDP
12.22/4.90	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.22/4.90	(10) YES
12.22/4.90	
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(0)
12.22/4.90	Obligation:
12.22/4.90	mainModule Main 
12.22/4.90	module Maybe where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	module List where {
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.22/4.90	insertBy _ x [] = x : [];
12.22/4.90	insertBy cmp x ys@(y : ys') = case cmp x y of { 
12.22/4.90	GT-> y : insertBy cmp x ys'; 
12.22/4.90	_-> x : ys; 
12.22/4.90	 } ;
12.22/4.90	
12.22/4.90	}
12.22/4.90	module Main where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(1) CR (EQUIVALENT)
12.22/4.90	Case Reductions:
12.22/4.90	The following Case expression
12.22/4.90	"case cmp x y of {
12.22/4.90	GT  -> y : insertBy cmp x ys';
12.22/4.90	_  -> x : ys}
12.22/4.90	"
12.22/4.90	is transformed to
12.22/4.90	"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.22/4.90	insertBy0 y cmp x ys' ys _ = x : ys;
12.22/4.90	"
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(2)
12.22/4.90	Obligation:
12.22/4.90	mainModule Main 
12.22/4.90	module Maybe where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	module List where {
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.22/4.90	insertBy _ x [] = x : [];
12.22/4.90	insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);
12.22/4.90	
12.22/4.90	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.22/4.90	insertBy0 y cmp x ys' ys _ = x : ys;
12.22/4.90	
12.22/4.90	}
12.22/4.90	module Main where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(3) BR (EQUIVALENT)
12.22/4.90	Replaced joker patterns by fresh variables and removed binding patterns.
12.22/4.90	
12.22/4.90	Binding Reductions:
12.22/4.90	The bind variable of the following binding Pattern
12.22/4.90	"ys@(wu : wv)"
12.22/4.90	is replaced by the following term
12.22/4.90	"wu : wv"
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(4)
12.22/4.90	Obligation:
12.22/4.90	mainModule Main 
12.22/4.90	module Maybe where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	module List where {
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.22/4.90	insertBy vz x [] = x : [];
12.22/4.90	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
12.22/4.90	
12.22/4.90	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.22/4.90	insertBy0 y cmp x ys' ys vy = x : ys;
12.22/4.90	
12.22/4.90	}
12.22/4.90	module Main where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(5) COR (EQUIVALENT)
12.22/4.90	Cond Reductions:
12.22/4.90	The following Function with conditions
12.22/4.90	"undefined |Falseundefined;
12.22/4.90	"
12.22/4.90	is transformed to
12.22/4.90	"undefined  = undefined1;
12.22/4.90	"
12.22/4.90	"undefined0 True = undefined;
12.22/4.90	"
12.22/4.90	"undefined1  = undefined0 False;
12.22/4.90	"
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(6)
12.22/4.90	Obligation:
12.22/4.90	mainModule Main 
12.22/4.90	module Maybe where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	module List where {
12.22/4.90	import qualified Main;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.22/4.90	insertBy vz x [] = x : [];
12.22/4.90	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
12.22/4.90	
12.22/4.90	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.22/4.90	insertBy0 y cmp x ys' ys vy = x : ys;
12.22/4.90	
12.22/4.90	}
12.22/4.90	module Main where {
12.22/4.90	import qualified List;
12.22/4.90	import qualified Maybe;
12.22/4.90	import qualified Prelude;
12.22/4.90	}
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(7) Narrow (SOUND)
12.22/4.90	Haskell To QDPs
12.22/4.90	
12.22/4.90	digraph dp_graph {
12.22/4.90	node [outthreshold=100, inthreshold=100];1[label="List.insertBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.22/4.90	3[label="List.insertBy ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.22/4.90	4[label="List.insertBy ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
12.22/4.90	5[label="List.insertBy ww3 ww4 ww5",fontsize=16,color="burlywood",shape="triangle"];24[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];5 -> 24[label="",style="solid", color="burlywood", weight=9];
12.22/4.90	24 -> 6[label="",style="solid", color="burlywood", weight=3];
12.22/4.90	25[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 25[label="",style="solid", color="burlywood", weight=9];
12.22/4.90	25 -> 7[label="",style="solid", color="burlywood", weight=3];
12.22/4.90	6[label="List.insertBy ww3 ww4 (ww50 : ww51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.22/4.90	7[label="List.insertBy ww3 ww4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.22/4.90	8 -> 10[label="",style="dashed", color="red", weight=0];
12.22/4.90	8[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) (ww3 ww4 ww50)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
12.22/4.90	9[label="ww4 : []",fontsize=16,color="green",shape="box"];11[label="ww3 ww4 ww50",fontsize=16,color="green",shape="box"];11 -> 17[label="",style="dashed", color="green", weight=3];
12.22/4.90	11 -> 18[label="",style="dashed", color="green", weight=3];
12.22/4.90	10[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) ww6",fontsize=16,color="burlywood",shape="triangle"];26[label="ww6/LT",fontsize=10,color="white",style="solid",shape="box"];10 -> 26[label="",style="solid", color="burlywood", weight=9];
12.22/4.90	26 -> 14[label="",style="solid", color="burlywood", weight=3];
12.22/4.90	27[label="ww6/EQ",fontsize=10,color="white",style="solid",shape="box"];10 -> 27[label="",style="solid", color="burlywood", weight=9];
12.22/4.90	27 -> 15[label="",style="solid", color="burlywood", weight=3];
12.22/4.90	28[label="ww6/GT",fontsize=10,color="white",style="solid",shape="box"];10 -> 28[label="",style="solid", color="burlywood", weight=9];
12.22/4.90	28 -> 16[label="",style="solid", color="burlywood", weight=3];
12.22/4.90	17[label="ww4",fontsize=16,color="green",shape="box"];18[label="ww50",fontsize=16,color="green",shape="box"];14[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) LT",fontsize=16,color="black",shape="box"];14 -> 19[label="",style="solid", color="black", weight=3];
12.22/4.90	15[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) EQ",fontsize=16,color="black",shape="box"];15 -> 20[label="",style="solid", color="black", weight=3];
12.22/4.90	16[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) GT",fontsize=16,color="black",shape="box"];16 -> 21[label="",style="solid", color="black", weight=3];
12.22/4.90	19[label="ww4 : ww50 : ww51",fontsize=16,color="green",shape="box"];20[label="ww4 : ww50 : ww51",fontsize=16,color="green",shape="box"];21[label="ww50 : List.insertBy ww3 ww4 ww51",fontsize=16,color="green",shape="box"];21 -> 22[label="",style="dashed", color="green", weight=3];
12.22/4.90	22 -> 5[label="",style="dashed", color="red", weight=0];
12.22/4.90	22[label="List.insertBy ww3 ww4 ww51",fontsize=16,color="magenta"];22 -> 23[label="",style="dashed", color="magenta", weight=3];
12.22/4.90	23[label="ww51",fontsize=16,color="green",shape="box"];}
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(8)
12.22/4.90	Obligation:
12.22/4.90	Q DP problem:
12.22/4.90	The TRS P consists of the following rules:
12.22/4.90	
12.22/4.90	   new_insertBy0(ww50, ww3, ww4, ww51, ba) -> new_insertBy(ww3, ww4, ww51, ba)
12.22/4.90	   new_insertBy(ww3, ww4, :(ww50, ww51), ba) -> new_insertBy0(ww50, ww3, ww4, ww51, ba)
12.22/4.90	
12.22/4.90	R is empty.
12.22/4.90	Q is empty.
12.22/4.90	We have to consider all minimal (P,Q,R)-chains.
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(9) QDPSizeChangeProof (EQUIVALENT)
12.22/4.90	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. 
12.22/4.90	
12.22/4.90	From the DPs we obtained the following set of size-change graphs:
12.22/4.90	*new_insertBy(ww3, ww4, :(ww50, ww51), ba) -> new_insertBy0(ww50, ww3, ww4, ww51, ba)
12.22/4.90	The graph contains the following edges 3 > 1, 1 >= 2, 2 >= 3, 3 > 4, 4 >= 5
12.22/4.90	
12.22/4.90	
12.22/4.90	*new_insertBy0(ww50, ww3, ww4, ww51, ba) -> new_insertBy(ww3, ww4, ww51, ba)
12.22/4.90	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4
12.22/4.90	
12.22/4.90	
12.22/4.90	----------------------------------------
12.22/4.90	
12.22/4.90	(10)
12.22/4.90	YES
12.44/4.95	EOF