4.44/1.94	YES
4.44/1.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.44/1.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.44/1.94	
4.44/1.94	
4.44/1.94	Left Termination of the query pattern
4.44/1.94	
4.44/1.94	flatten(g,a)
4.44/1.94	
4.44/1.94	w.r.t. the given Prolog program could successfully be proven:
4.44/1.94	
4.44/1.94	(0) Prolog
4.44/1.94	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.44/1.94	(2) PiTRS
4.44/1.94	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
4.44/1.94	(4) PiDP
4.44/1.94	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.44/1.94	(6) PiDP
4.44/1.94	(7) UsableRulesProof [EQUIVALENT, 0 ms]
4.44/1.94	(8) PiDP
4.44/1.94	(9) PiDPToQDPProof [SOUND, 0 ms]
4.44/1.94	(10) QDP
4.44/1.94	(11) UsableRulesReductionPairsProof [EQUIVALENT, 8 ms]
4.44/1.94	(12) QDP
4.44/1.94	(13) PisEmptyProof [EQUIVALENT, 0 ms]
4.44/1.94	(14) YES
4.44/1.94	
4.44/1.94	
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(0)
4.44/1.94	Obligation:
4.44/1.94	Clauses:
4.44/1.94	
4.44/1.94	flatten(atom(X), .(X, [])).
4.44/1.94	flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
4.44/1.94	flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).
4.44/1.94	
4.44/1.94	
4.44/1.94	Query: flatten(g,a)
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(1) PrologToPiTRSProof (SOUND)
4.44/1.94	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.44/1.94	
4.44/1.94	flatten_in_2: (b,f)
4.44/1.94	
4.44/1.94	Transforming Prolog into the following Term Rewriting System:
4.44/1.94	
4.44/1.94	Pi-finite rewrite system:
4.44/1.94	The TRS R consists of the following rules:
4.44/1.94	
4.44/1.94	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
4.44/1.94	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
4.44/1.94	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
4.44/1.94	
4.44/1.94	The argument filtering Pi contains the following mapping:
4.44/1.94	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
4.44/1.94	
4.44/1.94	atom(x1)  =  atom(x1)
4.44/1.94	
4.44/1.94	flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
4.44/1.94	
4.44/1.94	cons(x1, x2)  =  cons(x1, x2)
4.44/1.94	
4.44/1.94	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	.(x1, x2)  =  .(x1, x2)
4.44/1.94	
4.44/1.94	
4.44/1.94	
4.44/1.94	
4.44/1.94	
4.44/1.94	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.44/1.94	
4.44/1.94	
4.44/1.94	
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(2)
4.44/1.94	Obligation:
4.44/1.94	Pi-finite rewrite system:
4.44/1.94	The TRS R consists of the following rules:
4.44/1.94	
4.44/1.94	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
4.44/1.94	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
4.44/1.94	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
4.44/1.94	
4.44/1.94	The argument filtering Pi contains the following mapping:
4.44/1.94	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
4.44/1.94	
4.44/1.94	atom(x1)  =  atom(x1)
4.44/1.94	
4.44/1.94	flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
4.44/1.94	
4.44/1.94	cons(x1, x2)  =  cons(x1, x2)
4.44/1.94	
4.44/1.94	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	.(x1, x2)  =  .(x1, x2)
4.44/1.94	
4.44/1.94	
4.44/1.94	
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(3) DependencyPairsProof (EQUIVALENT)
4.44/1.94	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.44/1.94	Pi DP problem:
4.44/1.94	The TRS P consists of the following rules:
4.44/1.94	
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> U1_GA(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
4.44/1.94	
4.44/1.94	The TRS R consists of the following rules:
4.44/1.94	
4.44/1.94	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
4.44/1.94	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
4.44/1.94	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
4.44/1.94	
4.44/1.94	The argument filtering Pi contains the following mapping:
4.44/1.94	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
4.44/1.94	
4.44/1.94	atom(x1)  =  atom(x1)
4.44/1.94	
4.44/1.94	flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
4.44/1.94	
4.44/1.94	cons(x1, x2)  =  cons(x1, x2)
4.44/1.94	
4.44/1.94	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	.(x1, x2)  =  .(x1, x2)
4.44/1.94	
4.44/1.94	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
4.44/1.94	
4.44/1.94	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	
4.44/1.94	We have to consider all (P,R,Pi)-chains
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(4)
4.44/1.94	Obligation:
4.44/1.94	Pi DP problem:
4.44/1.94	The TRS P consists of the following rules:
4.44/1.94	
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> U1_GA(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
4.44/1.94	
4.44/1.94	The TRS R consists of the following rules:
4.44/1.94	
4.44/1.94	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
4.44/1.94	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
4.44/1.94	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
4.44/1.94	
4.44/1.94	The argument filtering Pi contains the following mapping:
4.44/1.94	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
4.44/1.94	
4.44/1.94	atom(x1)  =  atom(x1)
4.44/1.94	
4.44/1.94	flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
4.44/1.94	
4.44/1.94	cons(x1, x2)  =  cons(x1, x2)
4.44/1.94	
4.44/1.94	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	.(x1, x2)  =  .(x1, x2)
4.44/1.94	
4.44/1.94	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
4.44/1.94	
4.44/1.94	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	
4.44/1.94	We have to consider all (P,R,Pi)-chains
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(5) DependencyGraphProof (EQUIVALENT)
4.44/1.94	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(6)
4.44/1.94	Obligation:
4.44/1.94	Pi DP problem:
4.44/1.94	The TRS P consists of the following rules:
4.44/1.94	
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
4.44/1.94	
4.44/1.94	The TRS R consists of the following rules:
4.44/1.94	
4.44/1.94	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
4.44/1.94	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
4.44/1.94	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
4.44/1.94	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
4.44/1.94	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
4.44/1.94	
4.44/1.94	The argument filtering Pi contains the following mapping:
4.44/1.94	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
4.44/1.94	
4.44/1.94	atom(x1)  =  atom(x1)
4.44/1.94	
4.44/1.94	flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
4.44/1.94	
4.44/1.94	cons(x1, x2)  =  cons(x1, x2)
4.44/1.94	
4.44/1.94	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
4.44/1.94	
4.44/1.94	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
4.44/1.94	
4.44/1.94	.(x1, x2)  =  .(x1, x2)
4.44/1.94	
4.44/1.94	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
4.44/1.94	
4.44/1.94	
4.44/1.94	We have to consider all (P,R,Pi)-chains
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(7) UsableRulesProof (EQUIVALENT)
4.44/1.94	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(8)
4.44/1.94	Obligation:
4.44/1.94	Pi DP problem:
4.44/1.94	The TRS P consists of the following rules:
4.44/1.94	
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
4.44/1.94	
4.44/1.94	R is empty.
4.44/1.94	The argument filtering Pi contains the following mapping:
4.44/1.94	atom(x1)  =  atom(x1)
4.44/1.94	
4.44/1.94	cons(x1, x2)  =  cons(x1, x2)
4.44/1.94	
4.44/1.94	.(x1, x2)  =  .(x1, x2)
4.44/1.94	
4.44/1.94	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
4.44/1.94	
4.44/1.94	
4.44/1.94	We have to consider all (P,R,Pi)-chains
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(9) PiDPToQDPProof (SOUND)
4.44/1.94	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(10)
4.44/1.94	Obligation:
4.44/1.94	Q DP problem:
4.44/1.94	The TRS P consists of the following rules:
4.44/1.94	
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W)) -> FLATTEN_IN_GA(cons(U, cons(V, W)))
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U)) -> FLATTEN_IN_GA(U)
4.44/1.94	
4.44/1.94	R is empty.
4.44/1.94	Q is empty.
4.44/1.94	We have to consider all (P,Q,R)-chains.
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(11) UsableRulesReductionPairsProof (EQUIVALENT)
4.44/1.94	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
4.44/1.94	
4.44/1.94	The following dependency pairs can be deleted:
4.44/1.94	
4.44/1.94	   FLATTEN_IN_GA(cons(cons(U, V), W)) -> FLATTEN_IN_GA(cons(U, cons(V, W)))
4.44/1.94	   FLATTEN_IN_GA(cons(atom(X), U)) -> FLATTEN_IN_GA(U)
4.44/1.94	No rules are removed from R.
4.44/1.94	
4.44/1.94	Used ordering: POLO with Polynomial interpretation [POLO]:
4.44/1.94	
4.44/1.94	   POL(FLATTEN_IN_GA(x_1)) = 2*x_1
4.44/1.94	   POL(atom(x_1)) = x_1
4.44/1.94	   POL(cons(x_1, x_2)) = 1 + 2*x_1 + x_2
4.44/1.94	
4.44/1.94	
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(12)
4.44/1.94	Obligation:
4.44/1.94	Q DP problem:
4.44/1.94	P is empty.
4.44/1.94	R is empty.
4.44/1.94	Q is empty.
4.44/1.94	We have to consider all (P,Q,R)-chains.
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(13) PisEmptyProof (EQUIVALENT)
4.44/1.94	The TRS P is empty. Hence, there is no (P,Q,R) chain.
4.44/1.94	----------------------------------------
4.44/1.94	
4.44/1.94	(14)
4.44/1.94	YES
4.55/2.00	EOF