4.97/2.16	YES
5.15/2.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.15/2.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.15/2.20	
5.15/2.20	
5.15/2.20	Left Termination of the query pattern
5.15/2.20	
5.15/2.20	goal(g,a,a)
5.15/2.20	
5.15/2.20	w.r.t. the given Prolog program could successfully be proven:
5.15/2.20	
5.15/2.20	(0) Prolog
5.15/2.20	(1) PrologToPiTRSProof [SOUND, 26 ms]
5.15/2.20	(2) PiTRS
5.15/2.20	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
5.15/2.20	(4) PiDP
5.15/2.20	(5) DependencyGraphProof [EQUIVALENT, 4 ms]
5.15/2.20	(6) AND
5.15/2.20	    (7) PiDP
5.15/2.20	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.15/2.20	        (9) PiDP
5.15/2.20	        (10) PiDPToQDPProof [SOUND, 0 ms]
5.15/2.20	        (11) QDP
5.15/2.20	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.15/2.20	        (13) YES
5.15/2.20	    (14) PiDP
5.15/2.20	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
5.15/2.20	        (16) PiDP
5.15/2.20	        (17) PiDPToQDPProof [SOUND, 0 ms]
5.15/2.20	        (18) QDP
5.15/2.20	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.15/2.20	        (20) YES
5.15/2.20	    (21) PiDP
5.15/2.20	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
5.15/2.20	        (23) PiDP
5.15/2.20	        (24) PiDPToQDPProof [SOUND, 0 ms]
5.15/2.20	        (25) QDP
5.15/2.20	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.15/2.20	        (27) YES
5.15/2.20	
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(0)
5.15/2.20	Obligation:
5.15/2.20	Clauses:
5.15/2.20	
5.15/2.20	goal(A, B, C) :- ','(s2t(A, T), tapplast(T, B, C)).
5.15/2.20	tapplast(L, X, Last) :- ','(tappend(L, node(nil, X, nil), LX), tlast(Last, LX)).
5.15/2.20	tlast(X, node(nil, X, nil)).
5.15/2.20	tlast(X, node(L, H, R)) :- tlast(X, L).
5.15/2.20	tlast(X, node(L, H, R)) :- tlast(X, R).
5.15/2.20	tappend(nil, T, T).
5.15/2.20	tappend(node(nil, X, T2), T1, node(T1, X, T2)).
5.15/2.20	tappend(node(T1, X, nil), T2, node(T1, X, T2)).
5.15/2.20	tappend(node(T1, X, T2), T3, node(U, X, T2)) :- tappend(T1, T3, U).
5.15/2.20	tappend(node(T1, X, T2), T3, node(T1, X, U)) :- tappend(T2, T3, U).
5.15/2.20	s2t(s(X), node(T, Y, T)) :- s2t(X, T).
5.15/2.20	s2t(s(X), node(nil, Y, T)) :- s2t(X, T).
5.15/2.20	s2t(s(X), node(T, Y, nil)) :- s2t(X, T).
5.15/2.20	s2t(s(X), node(nil, Y, nil)).
5.15/2.20	s2t(0, nil).
5.15/2.20	
5.15/2.20	
5.15/2.20	Query: goal(g,a,a)
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(1) PrologToPiTRSProof (SOUND)
5.15/2.20	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.15/2.20	
5.15/2.20	goal_in_3: (b,f,f)
5.15/2.20	
5.15/2.20	s2t_in_2: (b,f)
5.15/2.20	
5.15/2.20	tapplast_in_3: (b,f,f)
5.15/2.20	
5.15/2.20	tappend_in_3: (b,b,f)
5.15/2.20	
5.15/2.20	tlast_in_2: (f,b)
5.15/2.20	
5.15/2.20	Transforming Prolog into the following Term Rewriting System:
5.15/2.20	
5.15/2.20	Pi-finite rewrite system:
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	
5.15/2.20	
5.15/2.20	
5.15/2.20	
5.15/2.20	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.15/2.20	
5.15/2.20	
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(2)
5.15/2.20	Obligation:
5.15/2.20	Pi-finite rewrite system:
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(3) DependencyPairsProof (EQUIVALENT)
5.15/2.20	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   GOAL_IN_GAA(A, B, C) -> U1_GAA(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   GOAL_IN_GAA(A, B, C) -> S2T_IN_GA(A, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, T)) -> U9_GA(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(nil, Y, T)) -> U10_GA(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, nil)) -> U11_GA(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T)
5.15/2.20	   U1_GAA(A, B, C, s2t_out_ga(A, T)) -> U2_GAA(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   U1_GAA(A, B, C, s2t_out_ga(A, T)) -> TAPPLAST_IN_GAA(T, B, C)
5.15/2.20	   TAPPLAST_IN_GAA(L, X, Last) -> U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   TAPPLAST_IN_GAA(L, X, Last) -> TAPPEND_IN_GGA(L, node(nil, X, nil), LX)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U)
5.15/2.20	   U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_GAA(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> TLAST_IN_AG(Last, LX)
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> U5_AG(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L)
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> U6_AG(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R)
5.15/2.20	
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
5.15/2.20	
5.15/2.20	U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
5.15/2.20	
5.15/2.20	S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
5.15/2.20	
5.15/2.20	U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
5.15/2.20	
5.15/2.20	U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
5.15/2.20	
5.15/2.20	U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
5.15/2.20	
5.15/2.20	U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
5.15/2.20	
5.15/2.20	TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
5.15/2.20	
5.15/2.20	U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
5.15/2.20	
5.15/2.20	TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
5.15/2.20	
5.15/2.20	U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x3, x6)
5.15/2.20	
5.15/2.20	U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x6)
5.15/2.20	
5.15/2.20	U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
5.15/2.20	
5.15/2.20	TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
5.15/2.20	
5.15/2.20	U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x5)
5.15/2.20	
5.15/2.20	U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(4)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   GOAL_IN_GAA(A, B, C) -> U1_GAA(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   GOAL_IN_GAA(A, B, C) -> S2T_IN_GA(A, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, T)) -> U9_GA(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(nil, Y, T)) -> U10_GA(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, nil)) -> U11_GA(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T)
5.15/2.20	   U1_GAA(A, B, C, s2t_out_ga(A, T)) -> U2_GAA(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   U1_GAA(A, B, C, s2t_out_ga(A, T)) -> TAPPLAST_IN_GAA(T, B, C)
5.15/2.20	   TAPPLAST_IN_GAA(L, X, Last) -> U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   TAPPLAST_IN_GAA(L, X, Last) -> TAPPEND_IN_GGA(L, node(nil, X, nil), LX)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U)
5.15/2.20	   U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_GAA(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> TLAST_IN_AG(Last, LX)
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> U5_AG(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L)
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> U6_AG(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R)
5.15/2.20	
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
5.15/2.20	
5.15/2.20	U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
5.15/2.20	
5.15/2.20	S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
5.15/2.20	
5.15/2.20	U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
5.15/2.20	
5.15/2.20	U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
5.15/2.20	
5.15/2.20	U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
5.15/2.20	
5.15/2.20	U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
5.15/2.20	
5.15/2.20	TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
5.15/2.20	
5.15/2.20	U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
5.15/2.20	
5.15/2.20	TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
5.15/2.20	
5.15/2.20	U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x3, x6)
5.15/2.20	
5.15/2.20	U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x6)
5.15/2.20	
5.15/2.20	U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
5.15/2.20	
5.15/2.20	TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
5.15/2.20	
5.15/2.20	U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x5)
5.15/2.20	
5.15/2.20	U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(5) DependencyGraphProof (EQUIVALENT)
5.15/2.20	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(6)
5.15/2.20	Complex Obligation (AND)
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(7)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R)
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L)
5.15/2.20	
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(8) UsableRulesProof (EQUIVALENT)
5.15/2.20	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(9)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R)
5.15/2.20	   TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L)
5.15/2.20	
5.15/2.20	R is empty.
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(10) PiDPToQDPProof (SOUND)
5.15/2.20	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(11)
5.15/2.20	Obligation:
5.15/2.20	Q DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(R)
5.15/2.20	   TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(L)
5.15/2.20	
5.15/2.20	R is empty.
5.15/2.20	Q is empty.
5.15/2.20	We have to consider all (P,Q,R)-chains.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(12) QDPSizeChangeProof (EQUIVALENT)
5.15/2.20	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. 
5.15/2.20	
5.15/2.20	From the DPs we obtained the following set of size-change graphs:
5.15/2.20	*TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(R)
5.15/2.20	The graph contains the following edges 1 > 1
5.15/2.20	
5.15/2.20	
5.15/2.20	*TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(L)
5.15/2.20	The graph contains the following edges 1 > 1
5.15/2.20	
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(13)
5.15/2.20	YES
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(14)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U)
5.15/2.20	
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(15) UsableRulesProof (EQUIVALENT)
5.15/2.20	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(16)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U)
5.15/2.20	
5.15/2.20	R is empty.
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(17) PiDPToQDPProof (SOUND)
5.15/2.20	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(18)
5.15/2.20	Obligation:
5.15/2.20	Q DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T2, T3)
5.15/2.20	   TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T1, T3)
5.15/2.20	
5.15/2.20	R is empty.
5.15/2.20	Q is empty.
5.15/2.20	We have to consider all (P,Q,R)-chains.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(19) QDPSizeChangeProof (EQUIVALENT)
5.15/2.20	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. 
5.15/2.20	
5.15/2.20	From the DPs we obtained the following set of size-change graphs:
5.15/2.20	*TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T2, T3)
5.15/2.20	The graph contains the following edges 1 > 1, 2 >= 2
5.15/2.20	
5.15/2.20	
5.15/2.20	*TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T1, T3)
5.15/2.20	The graph contains the following edges 1 > 1, 2 >= 2
5.15/2.20	
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(20)
5.15/2.20	YES
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(21)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T)
5.15/2.20	
5.15/2.20	The TRS R consists of the following rules:
5.15/2.20	
5.15/2.20	   goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T))
5.15/2.20	   s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil))
5.15/2.20	   s2t_in_ga(0, nil) -> s2t_out_ga(0, nil)
5.15/2.20	   U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil))
5.15/2.20	   U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T))
5.15/2.20	   U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T))
5.15/2.20	   U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
5.15/2.20	   tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
5.15/2.20	   tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T)
5.15/2.20	   tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
5.15/2.20	   tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
5.15/2.20	   U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
5.15/2.20	   U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
5.15/2.20	   U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
5.15/2.20	   tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L))
5.15/2.20	   tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R))
5.15/2.20	   U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R))
5.15/2.20	   U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last)
5.15/2.20	   U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C)
5.15/2.20	
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
5.15/2.20	
5.15/2.20	U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
5.15/2.20	
5.15/2.20	s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
5.15/2.20	
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
5.15/2.20	
5.15/2.20	U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
5.15/2.20	
5.15/2.20	U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
5.15/2.20	
5.15/2.20	s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	0  =  0
5.15/2.20	
5.15/2.20	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
5.15/2.20	
5.15/2.20	tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
5.15/2.20	
5.15/2.20	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
5.15/2.20	
5.15/2.20	tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
5.15/2.20	
5.15/2.20	U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
5.15/2.20	
5.15/2.20	U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
5.15/2.20	
5.15/2.20	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
5.15/2.20	
5.15/2.20	tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
5.15/2.20	
5.15/2.20	tlast_out_ag(x1, x2)  =  tlast_out_ag
5.15/2.20	
5.15/2.20	U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
5.15/2.20	
5.15/2.20	U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
5.15/2.20	
5.15/2.20	tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
5.15/2.20	
5.15/2.20	goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
5.15/2.20	
5.15/2.20	S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(22) UsableRulesProof (EQUIVALENT)
5.15/2.20	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(23)
5.15/2.20	Obligation:
5.15/2.20	Pi DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T)
5.15/2.20	   S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T)
5.15/2.20	
5.15/2.20	R is empty.
5.15/2.20	The argument filtering Pi contains the following mapping:
5.15/2.20	s(x1)  =  s(x1)
5.15/2.20	
5.15/2.20	node(x1, x2, x3)  =  node(x1, x3)
5.15/2.20	
5.15/2.20	nil  =  nil
5.15/2.20	
5.15/2.20	S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
5.15/2.20	
5.15/2.20	
5.15/2.20	We have to consider all (P,R,Pi)-chains
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(24) PiDPToQDPProof (SOUND)
5.15/2.20	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(25)
5.15/2.20	Obligation:
5.15/2.20	Q DP problem:
5.15/2.20	The TRS P consists of the following rules:
5.15/2.20	
5.15/2.20	   S2T_IN_GA(s(X)) -> S2T_IN_GA(X)
5.15/2.20	
5.15/2.20	R is empty.
5.15/2.20	Q is empty.
5.15/2.20	We have to consider all (P,Q,R)-chains.
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(26) QDPSizeChangeProof (EQUIVALENT)
5.15/2.20	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. 
5.15/2.20	
5.15/2.20	From the DPs we obtained the following set of size-change graphs:
5.15/2.20	*S2T_IN_GA(s(X)) -> S2T_IN_GA(X)
5.15/2.20	The graph contains the following edges 1 > 1
5.15/2.20	
5.15/2.20	
5.15/2.20	----------------------------------------
5.15/2.20	
5.15/2.20	(27)
5.15/2.20	YES
5.15/2.22	EOF