5.75/2.36	YES
5.75/2.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.75/2.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.75/2.37	
5.75/2.37	
5.75/2.37	Left Termination of the query pattern
5.75/2.37	
5.75/2.37	count(g,a)
5.75/2.37	
5.75/2.37	w.r.t. the given Prolog program could successfully be proven:
5.75/2.37	
5.75/2.37	(0) Prolog
5.75/2.37	(1) PrologToPiTRSProof [SOUND, 0 ms]
5.75/2.37	(2) PiTRS
5.75/2.37	(3) DependencyPairsProof [EQUIVALENT, 3 ms]
5.75/2.37	(4) PiDP
5.75/2.37	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.75/2.37	(6) AND
5.75/2.37	    (7) PiDP
5.75/2.37	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.75/2.37	        (9) PiDP
5.75/2.37	        (10) PiDPToQDPProof [SOUND, 0 ms]
5.75/2.37	        (11) QDP
5.75/2.37	        (12) UsableRulesReductionPairsProof [EQUIVALENT, 23 ms]
5.75/2.37	        (13) QDP
5.75/2.37	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
5.75/2.37	        (15) YES
5.75/2.37	    (16) PiDP
5.75/2.37	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
5.75/2.37	        (18) PiDP
5.75/2.37	        (19) PiDPToQDPProof [SOUND, 0 ms]
5.75/2.37	        (20) QDP
5.75/2.37	        (21) UsableRulesReductionPairsProof [EQUIVALENT, 19 ms]
5.75/2.37	        (22) QDP
5.75/2.37	        (23) DependencyGraphProof [EQUIVALENT, 0 ms]
5.75/2.37	        (24) TRUE
5.75/2.37	
5.75/2.37	
5.75/2.37	----------------------------------------
5.75/2.37	
5.75/2.37	(0)
5.75/2.37	Obligation:
5.75/2.37	Clauses:
5.75/2.37	
5.75/2.37	flatten(atom(X), .(X, [])).
5.75/2.37	flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
5.75/2.37	flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).
5.75/2.37	count(atom(X), s(0)).
5.75/2.37	count(cons(atom(X), Y), s(Z)) :- count(Y, Z).
5.75/2.37	count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z)).
5.75/2.37	
5.75/2.37	
5.75/2.37	Query: count(g,a)
5.75/2.37	----------------------------------------
5.75/2.37	
5.75/2.37	(1) PrologToPiTRSProof (SOUND)
5.75/2.37	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.75/2.37	
5.75/2.37	count_in_2: (b,f)
5.75/2.37	
5.75/2.37	flatten_in_2: (b,f)
5.75/2.37	
5.75/2.37	Transforming Prolog into the following Term Rewriting System:
5.75/2.37	
5.75/2.37	Pi-finite rewrite system:
5.75/2.37	The TRS R consists of the following rules:
5.75/2.37	
5.75/2.37	   count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0))
5.75/2.37	   count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z))
5.75/2.37	   count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.37	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.37	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.37	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.37	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.37	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.37	   U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z))
5.75/2.37	   U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z)
5.75/2.37	   U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z))
5.75/2.37	
5.75/2.37	The argument filtering Pi contains the following mapping:
5.75/2.37	count_in_ga(x1, x2)  =  count_in_ga(x1)
5.75/2.37	
5.75/2.37	atom(x1)  =  atom(x1)
5.75/2.37	
5.75/2.37	count_out_ga(x1, x2)  =  count_out_ga(x2)
5.75/2.37	
5.75/2.37	cons(x1, x2)  =  cons(x1, x2)
5.75/2.37	
5.75/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
5.75/2.37	
5.75/2.37	U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
5.75/2.37	
5.75/2.37	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.37	
5.75/2.37	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.37	
5.75/2.37	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.37	
5.75/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.37	
5.75/2.37	U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
5.75/2.37	
5.75/2.37	.(x1, x2)  =  .(x1, x2)
5.75/2.37	
5.75/2.37	s(x1)  =  s(x1)
5.75/2.37	
5.75/2.37	
5.75/2.37	
5.75/2.37	
5.75/2.37	
5.75/2.37	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.75/2.37	
5.75/2.37	
5.75/2.37	
5.75/2.37	----------------------------------------
5.75/2.37	
5.75/2.37	(2)
5.75/2.37	Obligation:
5.75/2.37	Pi-finite rewrite system:
5.75/2.37	The TRS R consists of the following rules:
5.75/2.37	
5.75/2.37	   count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0))
5.75/2.37	   count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z))
5.75/2.37	   count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.37	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.37	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.37	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.37	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.37	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.37	   U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z))
5.75/2.37	   U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z)
5.75/2.37	   U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z))
5.75/2.37	
5.75/2.37	The argument filtering Pi contains the following mapping:
5.75/2.37	count_in_ga(x1, x2)  =  count_in_ga(x1)
5.75/2.37	
5.75/2.37	atom(x1)  =  atom(x1)
5.75/2.37	
5.75/2.37	count_out_ga(x1, x2)  =  count_out_ga(x2)
5.75/2.37	
5.75/2.37	cons(x1, x2)  =  cons(x1, x2)
5.75/2.37	
5.75/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
5.75/2.37	
5.75/2.37	U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
5.75/2.37	
5.75/2.37	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.37	
5.75/2.37	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.37	
5.75/2.37	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.37	
5.75/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.37	
5.75/2.37	U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
5.75/2.37	
5.75/2.37	.(x1, x2)  =  .(x1, x2)
5.75/2.37	
5.75/2.37	s(x1)  =  s(x1)
5.75/2.37	
5.75/2.37	
5.75/2.37	
5.75/2.37	----------------------------------------
5.75/2.37	
5.75/2.37	(3) DependencyPairsProof (EQUIVALENT)
5.75/2.37	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.75/2.37	Pi DP problem:
5.75/2.37	The TRS P consists of the following rules:
5.75/2.37	
5.75/2.37	   COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> U3_GA(X, Y, Z, count_in_ga(Y, Z))
5.75/2.37	   COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z)
5.75/2.37	   COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.37	   COUNT_IN_GA(cons(cons(U, V), W), Z) -> FLATTEN_IN_GA(cons(cons(U, V), W), X)
5.75/2.37	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> U1_GA(X, U, Y, flatten_in_ga(U, Y))
5.75/2.37	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
5.75/2.37	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.37	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
5.75/2.37	   U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_GA(U, V, W, Z, count_in_ga(X, Z))
5.75/2.37	   U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z)
5.75/2.37	
5.75/2.37	The TRS R consists of the following rules:
5.75/2.37	
5.75/2.37	   count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0))
5.75/2.37	   count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z))
5.75/2.37	   count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.37	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.37	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.37	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.37	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.37	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.37	   U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z))
5.75/2.37	   U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z)
5.75/2.37	   U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z))
5.75/2.37	
5.75/2.37	The argument filtering Pi contains the following mapping:
5.75/2.37	count_in_ga(x1, x2)  =  count_in_ga(x1)
5.75/2.37	
5.75/2.37	atom(x1)  =  atom(x1)
5.75/2.37	
5.75/2.37	count_out_ga(x1, x2)  =  count_out_ga(x2)
5.75/2.37	
5.75/2.37	cons(x1, x2)  =  cons(x1, x2)
5.75/2.37	
5.75/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
5.75/2.37	
5.75/2.37	U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
5.75/2.37	
5.75/2.37	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.37	
5.75/2.37	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.37	
5.75/2.37	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.37	
5.75/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.37	
5.75/2.37	U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
5.75/2.37	
5.75/2.37	.(x1, x2)  =  .(x1, x2)
5.75/2.37	
5.75/2.37	s(x1)  =  s(x1)
5.75/2.37	
5.75/2.37	COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
5.75/2.37	
5.75/2.37	U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
5.75/2.37	
5.75/2.37	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
5.75/2.37	
5.75/2.37	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
5.75/2.37	
5.75/2.37	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
5.75/2.37	
5.75/2.37	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
5.75/2.37	
5.75/2.37	U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
5.75/2.37	
5.75/2.37	
5.75/2.37	We have to consider all (P,R,Pi)-chains
5.75/2.37	----------------------------------------
5.75/2.37	
5.75/2.37	(4)
5.75/2.37	Obligation:
5.75/2.37	Pi DP problem:
5.75/2.37	The TRS P consists of the following rules:
5.75/2.37	
5.75/2.37	   COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> U3_GA(X, Y, Z, count_in_ga(Y, Z))
5.75/2.37	   COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z)
5.75/2.37	   COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.37	   COUNT_IN_GA(cons(cons(U, V), W), Z) -> FLATTEN_IN_GA(cons(cons(U, V), W), X)
5.75/2.37	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> U1_GA(X, U, Y, flatten_in_ga(U, Y))
5.75/2.37	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
5.75/2.37	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.37	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
5.75/2.37	   U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_GA(U, V, W, Z, count_in_ga(X, Z))
5.75/2.37	   U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z)
5.75/2.37	
5.75/2.37	The TRS R consists of the following rules:
5.75/2.37	
5.75/2.37	   count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0))
5.75/2.37	   count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z))
5.75/2.37	   count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.37	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.37	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.37	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.37	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.37	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.37	   U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z))
5.75/2.37	   U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z)
5.75/2.37	   U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z))
5.75/2.37	
5.75/2.37	The argument filtering Pi contains the following mapping:
5.75/2.37	count_in_ga(x1, x2)  =  count_in_ga(x1)
5.75/2.37	
5.75/2.37	atom(x1)  =  atom(x1)
5.75/2.37	
5.75/2.37	count_out_ga(x1, x2)  =  count_out_ga(x2)
5.75/2.37	
5.75/2.37	cons(x1, x2)  =  cons(x1, x2)
5.75/2.37	
5.75/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
5.75/2.37	
5.75/2.37	U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
5.75/2.37	
5.75/2.37	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.37	
5.75/2.37	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.37	
5.75/2.37	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.37	
5.75/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.37	
5.75/2.37	U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
5.75/2.38	
5.75/2.38	.(x1, x2)  =  .(x1, x2)
5.75/2.38	
5.75/2.38	s(x1)  =  s(x1)
5.75/2.38	
5.75/2.38	COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
5.75/2.38	
5.75/2.38	U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
5.75/2.38	
5.75/2.38	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
5.75/2.38	
5.75/2.38	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
5.75/2.38	
5.75/2.38	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
5.75/2.38	
5.75/2.38	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
5.75/2.38	
5.75/2.38	U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
5.75/2.38	
5.75/2.38	
5.75/2.38	We have to consider all (P,R,Pi)-chains
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(5) DependencyGraphProof (EQUIVALENT)
5.75/2.38	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(6)
5.75/2.38	Complex Obligation (AND)
5.75/2.38	
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(7)
5.75/2.38	Obligation:
5.75/2.38	Pi DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
5.75/2.38	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
5.75/2.38	
5.75/2.38	The TRS R consists of the following rules:
5.75/2.38	
5.75/2.38	   count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0))
5.75/2.38	   count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z))
5.75/2.38	   count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.38	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.38	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.38	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.38	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.38	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.38	   U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z))
5.75/2.38	   U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z)
5.75/2.38	   U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z))
5.75/2.38	
5.75/2.38	The argument filtering Pi contains the following mapping:
5.75/2.38	count_in_ga(x1, x2)  =  count_in_ga(x1)
5.75/2.38	
5.75/2.38	atom(x1)  =  atom(x1)
5.75/2.38	
5.75/2.38	count_out_ga(x1, x2)  =  count_out_ga(x2)
5.75/2.38	
5.75/2.38	cons(x1, x2)  =  cons(x1, x2)
5.75/2.38	
5.75/2.38	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
5.75/2.38	
5.75/2.38	U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
5.75/2.38	
5.75/2.38	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.38	
5.75/2.38	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.38	
5.75/2.38	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.38	
5.75/2.38	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.38	
5.75/2.38	U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
5.75/2.38	
5.75/2.38	.(x1, x2)  =  .(x1, x2)
5.75/2.38	
5.75/2.38	s(x1)  =  s(x1)
5.75/2.38	
5.75/2.38	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
5.75/2.38	
5.75/2.38	
5.75/2.38	We have to consider all (P,R,Pi)-chains
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(8) UsableRulesProof (EQUIVALENT)
5.75/2.38	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(9)
5.75/2.38	Obligation:
5.75/2.38	Pi DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X)
5.75/2.38	   FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y)
5.75/2.38	
5.75/2.38	R is empty.
5.75/2.38	The argument filtering Pi contains the following mapping:
5.75/2.38	atom(x1)  =  atom(x1)
5.75/2.38	
5.75/2.38	cons(x1, x2)  =  cons(x1, x2)
5.75/2.38	
5.75/2.38	.(x1, x2)  =  .(x1, x2)
5.75/2.38	
5.75/2.38	FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
5.75/2.38	
5.75/2.38	
5.75/2.38	We have to consider all (P,R,Pi)-chains
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(10) PiDPToQDPProof (SOUND)
5.75/2.38	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(11)
5.75/2.38	Obligation:
5.75/2.38	Q DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   FLATTEN_IN_GA(cons(cons(U, V), W)) -> FLATTEN_IN_GA(cons(U, cons(V, W)))
5.75/2.38	   FLATTEN_IN_GA(cons(atom(X), U)) -> FLATTEN_IN_GA(U)
5.75/2.38	
5.75/2.38	R is empty.
5.75/2.38	Q is empty.
5.75/2.38	We have to consider all (P,Q,R)-chains.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(12) UsableRulesReductionPairsProof (EQUIVALENT)
5.75/2.38	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.
5.75/2.38	
5.75/2.38	The following dependency pairs can be deleted:
5.75/2.38	
5.75/2.38	   FLATTEN_IN_GA(cons(cons(U, V), W)) -> FLATTEN_IN_GA(cons(U, cons(V, W)))
5.75/2.38	   FLATTEN_IN_GA(cons(atom(X), U)) -> FLATTEN_IN_GA(U)
5.75/2.38	No rules are removed from R.
5.75/2.38	
5.75/2.38	Used ordering: POLO with Polynomial interpretation [POLO]:
5.75/2.38	
5.75/2.38	   POL(FLATTEN_IN_GA(x_1)) = 2*x_1
5.75/2.38	   POL(atom(x_1)) = x_1
5.75/2.38	   POL(cons(x_1, x_2)) = 1 + 2*x_1 + x_2
5.75/2.38	
5.75/2.38	
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(13)
5.75/2.38	Obligation:
5.75/2.38	Q DP problem:
5.75/2.38	P is empty.
5.75/2.38	R is empty.
5.75/2.38	Q is empty.
5.75/2.38	We have to consider all (P,Q,R)-chains.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(14) PisEmptyProof (EQUIVALENT)
5.75/2.38	The TRS P is empty. Hence, there is no (P,Q,R) chain.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(15)
5.75/2.38	YES
5.75/2.38	
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(16)
5.75/2.38	Obligation:
5.75/2.38	Pi DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.38	   U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z)
5.75/2.38	   COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z)
5.75/2.38	
5.75/2.38	The TRS R consists of the following rules:
5.75/2.38	
5.75/2.38	   count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0))
5.75/2.38	   count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z))
5.75/2.38	   count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.38	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.38	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.38	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.38	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.38	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.38	   U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z))
5.75/2.38	   U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z)
5.75/2.38	   U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z))
5.75/2.38	
5.75/2.38	The argument filtering Pi contains the following mapping:
5.75/2.38	count_in_ga(x1, x2)  =  count_in_ga(x1)
5.75/2.38	
5.75/2.38	atom(x1)  =  atom(x1)
5.75/2.38	
5.75/2.38	count_out_ga(x1, x2)  =  count_out_ga(x2)
5.75/2.38	
5.75/2.38	cons(x1, x2)  =  cons(x1, x2)
5.75/2.38	
5.75/2.38	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
5.75/2.38	
5.75/2.38	U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
5.75/2.38	
5.75/2.38	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.38	
5.75/2.38	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.38	
5.75/2.38	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.38	
5.75/2.38	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.38	
5.75/2.38	U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
5.75/2.38	
5.75/2.38	.(x1, x2)  =  .(x1, x2)
5.75/2.38	
5.75/2.38	s(x1)  =  s(x1)
5.75/2.38	
5.75/2.38	COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
5.75/2.38	
5.75/2.38	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
5.75/2.38	
5.75/2.38	
5.75/2.38	We have to consider all (P,R,Pi)-chains
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(17) UsableRulesProof (EQUIVALENT)
5.75/2.38	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(18)
5.75/2.38	Obligation:
5.75/2.38	Pi DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
5.75/2.38	   U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z)
5.75/2.38	   COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z)
5.75/2.38	
5.75/2.38	The TRS R consists of the following rules:
5.75/2.38	
5.75/2.38	   flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
5.75/2.38	   U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X)
5.75/2.38	   flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y))
5.75/2.38	   U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y))
5.75/2.38	   flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, []))
5.75/2.38	
5.75/2.38	The argument filtering Pi contains the following mapping:
5.75/2.38	atom(x1)  =  atom(x1)
5.75/2.38	
5.75/2.38	cons(x1, x2)  =  cons(x1, x2)
5.75/2.38	
5.75/2.38	flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
5.75/2.38	
5.75/2.38	flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
5.75/2.38	
5.75/2.38	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
5.75/2.38	
5.75/2.38	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
5.75/2.38	
5.75/2.38	.(x1, x2)  =  .(x1, x2)
5.75/2.38	
5.75/2.38	s(x1)  =  s(x1)
5.75/2.38	
5.75/2.38	COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
5.75/2.38	
5.75/2.38	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
5.75/2.38	
5.75/2.38	
5.75/2.38	We have to consider all (P,R,Pi)-chains
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(19) PiDPToQDPProof (SOUND)
5.75/2.38	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(20)
5.75/2.38	Obligation:
5.75/2.38	Q DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   COUNT_IN_GA(cons(cons(U, V), W)) -> U4_GA(flatten_in_ga(cons(cons(U, V), W)))
5.75/2.38	   U4_GA(flatten_out_ga(X)) -> COUNT_IN_GA(X)
5.75/2.38	   COUNT_IN_GA(cons(atom(X), Y)) -> COUNT_IN_GA(Y)
5.75/2.38	
5.75/2.38	The TRS R consists of the following rules:
5.75/2.38	
5.75/2.38	   flatten_in_ga(cons(cons(U, V), W)) -> U2_ga(flatten_in_ga(cons(U, cons(V, W))))
5.75/2.38	   U2_ga(flatten_out_ga(X)) -> flatten_out_ga(X)
5.75/2.38	   flatten_in_ga(cons(atom(X), U)) -> U1_ga(X, flatten_in_ga(U))
5.75/2.38	   U1_ga(X, flatten_out_ga(Y)) -> flatten_out_ga(.(X, Y))
5.75/2.38	   flatten_in_ga(atom(X)) -> flatten_out_ga(.(X, []))
5.75/2.38	
5.75/2.38	The set Q consists of the following terms:
5.75/2.38	
5.75/2.38	   flatten_in_ga(x0)
5.75/2.38	   U2_ga(x0)
5.75/2.38	   U1_ga(x0, x1)
5.75/2.38	
5.75/2.38	We have to consider all (P,Q,R)-chains.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(21) UsableRulesReductionPairsProof (EQUIVALENT)
5.75/2.38	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.
5.75/2.38	
5.75/2.38	The following dependency pairs can be deleted:
5.75/2.38	
5.75/2.38	   COUNT_IN_GA(cons(cons(U, V), W)) -> U4_GA(flatten_in_ga(cons(cons(U, V), W)))
5.75/2.38	   COUNT_IN_GA(cons(atom(X), Y)) -> COUNT_IN_GA(Y)
5.75/2.38	The following rules are removed from R:
5.75/2.38	
5.75/2.38	   flatten_in_ga(cons(atom(X), U)) -> U1_ga(X, flatten_in_ga(U))
5.75/2.38	   flatten_in_ga(atom(X)) -> flatten_out_ga(.(X, []))
5.75/2.38	Used ordering: POLO with Polynomial interpretation [POLO]:
5.75/2.38	
5.75/2.38	   POL(.(x_1, x_2)) = 1 + x_1 + x_2
5.75/2.38	   POL(COUNT_IN_GA(x_1)) = 1 + x_1
5.75/2.38	   POL(U1_ga(x_1, x_2)) = 1 + x_1 + x_2
5.75/2.38	   POL(U2_ga(x_1)) = x_1
5.75/2.38	   POL(U4_GA(x_1)) = x_1
5.75/2.38	   POL([]) = 0
5.75/2.38	   POL(atom(x_1)) = 2 + x_1
5.75/2.38	   POL(cons(x_1, x_2)) = x_1 + x_2
5.75/2.38	   POL(flatten_in_ga(x_1)) = x_1
5.75/2.38	   POL(flatten_out_ga(x_1)) = 1 + x_1
5.75/2.38	
5.75/2.38	
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(22)
5.75/2.38	Obligation:
5.75/2.38	Q DP problem:
5.75/2.38	The TRS P consists of the following rules:
5.75/2.38	
5.75/2.38	   U4_GA(flatten_out_ga(X)) -> COUNT_IN_GA(X)
5.75/2.38	
5.75/2.38	The TRS R consists of the following rules:
5.75/2.38	
5.75/2.38	   flatten_in_ga(cons(cons(U, V), W)) -> U2_ga(flatten_in_ga(cons(U, cons(V, W))))
5.75/2.38	   U2_ga(flatten_out_ga(X)) -> flatten_out_ga(X)
5.75/2.38	   U1_ga(X, flatten_out_ga(Y)) -> flatten_out_ga(.(X, Y))
5.75/2.38	
5.75/2.38	The set Q consists of the following terms:
5.75/2.38	
5.75/2.38	   flatten_in_ga(x0)
5.75/2.38	   U2_ga(x0)
5.75/2.38	   U1_ga(x0, x1)
5.75/2.38	
5.75/2.38	We have to consider all (P,Q,R)-chains.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(23) DependencyGraphProof (EQUIVALENT)
5.75/2.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
5.75/2.38	----------------------------------------
5.75/2.38	
5.75/2.38	(24)
5.75/2.38	TRUE
6.08/2.47	EOF