7.72/2.81	YES
7.82/2.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
7.82/2.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.82/2.84	
7.82/2.84	
7.82/2.84	Left Termination of the query pattern
7.82/2.84	
7.82/2.84	shanoi(g,g,g,g,a)
7.82/2.84	
7.82/2.84	w.r.t. the given Prolog program could successfully be proven:
7.82/2.84	
7.82/2.84	(0) Prolog
7.82/2.84	(1) PrologToPiTRSProof [SOUND, 0 ms]
7.82/2.84	(2) PiTRS
7.82/2.84	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
7.82/2.84	(4) PiDP
7.82/2.84	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
7.82/2.84	(6) AND
7.82/2.84	    (7) PiDP
7.82/2.84	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
7.82/2.84	        (9) PiDP
7.82/2.84	        (10) PiDPToQDPProof [SOUND, 7 ms]
7.82/2.84	        (11) QDP
7.82/2.84	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.82/2.84	        (13) YES
7.82/2.84	    (14) PiDP
7.82/2.84	        (15) PiDPToQDPProof [SOUND, 0 ms]
7.82/2.84	        (16) QDP
7.82/2.84	        (17) QDPOrderProof [EQUIVALENT, 76 ms]
7.82/2.84	        (18) QDP
7.82/2.84	        (19) DependencyGraphProof [EQUIVALENT, 0 ms]
7.82/2.84	        (20) TRUE
7.82/2.84	
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(0)
7.82/2.84	Obligation:
7.82/2.84	Clauses:
7.82/2.84	
7.82/2.84	shanoi(s(0), A, B, C, .(mv(A, C), [])).
7.82/2.84	shanoi(s(s(X)), A, B, C, M) :- ','(eq(N1, s(X)), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M))))).
7.82/2.84	append([], L, L).
7.82/2.84	append(.(H, L), L1, .(H, R)) :- append(L, L1, R).
7.82/2.84	eq(X, X).
7.82/2.84	
7.82/2.84	
7.82/2.84	Query: shanoi(g,g,g,g,a)
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(1) PrologToPiTRSProof (SOUND)
7.82/2.84	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
7.82/2.84	
7.82/2.84	shanoi_in_5: (b,b,b,b,f)
7.82/2.84	
7.82/2.84	append_in_3: (b,b,f)
7.82/2.84	
7.82/2.84	Transforming Prolog into the following Term Rewriting System:
7.82/2.84	
7.82/2.84	Pi-finite rewrite system:
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   eq_in_ag(X, X) -> eq_out_ag(X, X)
7.82/2.84	   U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   append_in_gga([], L, L) -> append_out_gga([], L, L)
7.82/2.84	   append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R))
7.82/2.84	   U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M)
7.82/2.84	
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	shanoi_in_gggga(x1, x2, x3, x4, x5)  =  shanoi_in_gggga(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	s(x1)  =  s(x1)
7.82/2.84	
7.82/2.84	0  =  0
7.82/2.84	
7.82/2.84	shanoi_out_gggga(x1, x2, x3, x4, x5)  =  shanoi_out_gggga(x5)
7.82/2.84	
7.82/2.84	U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
7.82/2.84	
7.82/2.84	eq_out_ag(x1, x2)  =  eq_out_ag(x1)
7.82/2.84	
7.82/2.84	U2_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggga(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggga(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x6, x7)
7.82/2.84	
7.82/2.84	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
7.82/2.84	
7.82/2.84	[]  =  []
7.82/2.84	
7.82/2.84	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
7.82/2.84	
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
7.82/2.84	
7.82/2.84	mv(x1, x2)  =  mv(x1, x2)
7.82/2.84	
7.82/2.84	U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
7.82/2.84	
7.82/2.84	
7.82/2.84	
7.82/2.84	
7.82/2.84	
7.82/2.84	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
7.82/2.84	
7.82/2.84	
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(2)
7.82/2.84	Obligation:
7.82/2.84	Pi-finite rewrite system:
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   eq_in_ag(X, X) -> eq_out_ag(X, X)
7.82/2.84	   U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   append_in_gga([], L, L) -> append_out_gga([], L, L)
7.82/2.84	   append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R))
7.82/2.84	   U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M)
7.82/2.84	
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	shanoi_in_gggga(x1, x2, x3, x4, x5)  =  shanoi_in_gggga(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	s(x1)  =  s(x1)
7.82/2.84	
7.82/2.84	0  =  0
7.82/2.84	
7.82/2.84	shanoi_out_gggga(x1, x2, x3, x4, x5)  =  shanoi_out_gggga(x5)
7.82/2.84	
7.82/2.84	U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
7.82/2.84	
7.82/2.84	eq_out_ag(x1, x2)  =  eq_out_ag(x1)
7.82/2.84	
7.82/2.84	U2_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggga(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggga(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x6, x7)
7.82/2.84	
7.82/2.84	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
7.82/2.84	
7.82/2.84	[]  =  []
7.82/2.84	
7.82/2.84	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
7.82/2.84	
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
7.82/2.84	
7.82/2.84	mv(x1, x2)  =  mv(x1, x2)
7.82/2.84	
7.82/2.84	U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
7.82/2.84	
7.82/2.84	
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(3) DependencyPairsProof (EQUIVALENT)
7.82/2.84	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
7.82/2.84	Pi DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> U1_GGGGA(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> EQ_IN_AG(N1, s(X))
7.82/2.84	   U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_GGGGA(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> SHANOI_IN_GGGGA(N1, A, C, B, M1)
7.82/2.84	   U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_GGGGA(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> SHANOI_IN_GGGGA(N1, B, A, C, M2)
7.82/2.84	   U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_GGGGA(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> APPEND_IN_GGA(M1, .(mv(A, C), []), T)
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> U6_GGA(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R)
7.82/2.84	   U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_GGGGA(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> APPEND_IN_GGA(T, M2, M)
7.82/2.84	
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   eq_in_ag(X, X) -> eq_out_ag(X, X)
7.82/2.84	   U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   append_in_gga([], L, L) -> append_out_gga([], L, L)
7.82/2.84	   append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R))
7.82/2.84	   U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M)
7.82/2.84	
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	shanoi_in_gggga(x1, x2, x3, x4, x5)  =  shanoi_in_gggga(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	s(x1)  =  s(x1)
7.82/2.84	
7.82/2.84	0  =  0
7.82/2.84	
7.82/2.84	shanoi_out_gggga(x1, x2, x3, x4, x5)  =  shanoi_out_gggga(x5)
7.82/2.84	
7.82/2.84	U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
7.82/2.84	
7.82/2.84	eq_out_ag(x1, x2)  =  eq_out_ag(x1)
7.82/2.84	
7.82/2.84	U2_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggga(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggga(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x6, x7)
7.82/2.84	
7.82/2.84	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
7.82/2.84	
7.82/2.84	[]  =  []
7.82/2.84	
7.82/2.84	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
7.82/2.84	
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
7.82/2.84	
7.82/2.84	mv(x1, x2)  =  mv(x1, x2)
7.82/2.84	
7.82/2.84	U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
7.82/2.84	
7.82/2.84	SHANOI_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI_IN_GGGGA(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
7.82/2.84	
7.82/2.84	U2_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGGGA(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGGGA(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x6, x7)
7.82/2.84	
7.82/2.84	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
7.82/2.84	
7.82/2.84	U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x5)
7.82/2.84	
7.82/2.84	U5_GGGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGGA(x6)
7.82/2.84	
7.82/2.84	
7.82/2.84	We have to consider all (P,R,Pi)-chains
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(4)
7.82/2.84	Obligation:
7.82/2.84	Pi DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> U1_GGGGA(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> EQ_IN_AG(N1, s(X))
7.82/2.84	   U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_GGGGA(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> SHANOI_IN_GGGGA(N1, A, C, B, M1)
7.82/2.84	   U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_GGGGA(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> SHANOI_IN_GGGGA(N1, B, A, C, M2)
7.82/2.84	   U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_GGGGA(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> APPEND_IN_GGA(M1, .(mv(A, C), []), T)
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> U6_GGA(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R)
7.82/2.84	   U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_GGGGA(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> APPEND_IN_GGA(T, M2, M)
7.82/2.84	
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   eq_in_ag(X, X) -> eq_out_ag(X, X)
7.82/2.84	   U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   append_in_gga([], L, L) -> append_out_gga([], L, L)
7.82/2.84	   append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R))
7.82/2.84	   U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M)
7.82/2.84	
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	shanoi_in_gggga(x1, x2, x3, x4, x5)  =  shanoi_in_gggga(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	s(x1)  =  s(x1)
7.82/2.84	
7.82/2.84	0  =  0
7.82/2.84	
7.82/2.84	shanoi_out_gggga(x1, x2, x3, x4, x5)  =  shanoi_out_gggga(x5)
7.82/2.84	
7.82/2.84	U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
7.82/2.84	
7.82/2.84	eq_out_ag(x1, x2)  =  eq_out_ag(x1)
7.82/2.84	
7.82/2.84	U2_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggga(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggga(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x6, x7)
7.82/2.84	
7.82/2.84	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
7.82/2.84	
7.82/2.84	[]  =  []
7.82/2.84	
7.82/2.84	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
7.82/2.84	
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
7.82/2.84	
7.82/2.84	mv(x1, x2)  =  mv(x1, x2)
7.82/2.84	
7.82/2.84	U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
7.82/2.84	
7.82/2.84	SHANOI_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI_IN_GGGGA(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
7.82/2.84	
7.82/2.84	U2_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGGGA(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGGGA(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x6, x7)
7.82/2.84	
7.82/2.84	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
7.82/2.84	
7.82/2.84	U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x5)
7.82/2.84	
7.82/2.84	U5_GGGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGGA(x6)
7.82/2.84	
7.82/2.84	
7.82/2.84	We have to consider all (P,R,Pi)-chains
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(5) DependencyGraphProof (EQUIVALENT)
7.82/2.84	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(6)
7.82/2.84	Complex Obligation (AND)
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(7)
7.82/2.84	Obligation:
7.82/2.84	Pi DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R)
7.82/2.84	
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   eq_in_ag(X, X) -> eq_out_ag(X, X)
7.82/2.84	   U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   append_in_gga([], L, L) -> append_out_gga([], L, L)
7.82/2.84	   append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R))
7.82/2.84	   U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M)
7.82/2.84	
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	shanoi_in_gggga(x1, x2, x3, x4, x5)  =  shanoi_in_gggga(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	s(x1)  =  s(x1)
7.82/2.84	
7.82/2.84	0  =  0
7.82/2.84	
7.82/2.84	shanoi_out_gggga(x1, x2, x3, x4, x5)  =  shanoi_out_gggga(x5)
7.82/2.84	
7.82/2.84	U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
7.82/2.84	
7.82/2.84	eq_out_ag(x1, x2)  =  eq_out_ag(x1)
7.82/2.84	
7.82/2.84	U2_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggga(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggga(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x6, x7)
7.82/2.84	
7.82/2.84	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
7.82/2.84	
7.82/2.84	[]  =  []
7.82/2.84	
7.82/2.84	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
7.82/2.84	
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
7.82/2.84	
7.82/2.84	mv(x1, x2)  =  mv(x1, x2)
7.82/2.84	
7.82/2.84	U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
7.82/2.84	
7.82/2.84	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
7.82/2.84	
7.82/2.84	
7.82/2.84	We have to consider all (P,R,Pi)-chains
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(8) UsableRulesProof (EQUIVALENT)
7.82/2.84	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(9)
7.82/2.84	Obligation:
7.82/2.84	Pi DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R)
7.82/2.84	
7.82/2.84	R is empty.
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
7.82/2.84	
7.82/2.84	
7.82/2.84	We have to consider all (P,R,Pi)-chains
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(10) PiDPToQDPProof (SOUND)
7.82/2.84	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(11)
7.82/2.84	Obligation:
7.82/2.84	Q DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   APPEND_IN_GGA(.(H, L), L1) -> APPEND_IN_GGA(L, L1)
7.82/2.84	
7.82/2.84	R is empty.
7.82/2.84	Q is empty.
7.82/2.84	We have to consider all (P,Q,R)-chains.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(12) QDPSizeChangeProof (EQUIVALENT)
7.82/2.84	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. 
7.82/2.84	
7.82/2.84	From the DPs we obtained the following set of size-change graphs:
7.82/2.84	*APPEND_IN_GGA(.(H, L), L1) -> APPEND_IN_GGA(L, L1)
7.82/2.84	The graph contains the following edges 1 > 1, 2 >= 2
7.82/2.84	
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(13)
7.82/2.84	YES
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(14)
7.82/2.84	Obligation:
7.82/2.84	Pi DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_GGGGA(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> SHANOI_IN_GGGGA(N1, B, A, C, M2)
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> U1_GGGGA(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> SHANOI_IN_GGGGA(N1, A, C, B, M1)
7.82/2.84	
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X)))
7.82/2.84	   eq_in_ag(X, X) -> eq_out_ag(X, X)
7.82/2.84	   U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1))
7.82/2.84	   U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2))
7.82/2.84	   U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T))
7.82/2.84	   append_in_gga([], L, L) -> append_out_gga([], L, L)
7.82/2.84	   append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R))
7.82/2.84	   U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R))
7.82/2.84	   U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M))
7.82/2.84	   U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M)
7.82/2.84	
7.82/2.84	The argument filtering Pi contains the following mapping:
7.82/2.84	shanoi_in_gggga(x1, x2, x3, x4, x5)  =  shanoi_in_gggga(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	s(x1)  =  s(x1)
7.82/2.84	
7.82/2.84	0  =  0
7.82/2.84	
7.82/2.84	shanoi_out_gggga(x1, x2, x3, x4, x5)  =  shanoi_out_gggga(x5)
7.82/2.84	
7.82/2.84	U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
7.82/2.84	
7.82/2.84	eq_out_ag(x1, x2)  =  eq_out_ag(x1)
7.82/2.84	
7.82/2.84	U2_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggga(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	U3_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggga(x2, x4, x6, x7)
7.82/2.84	
7.82/2.84	U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x6, x7)
7.82/2.84	
7.82/2.84	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
7.82/2.84	
7.82/2.84	[]  =  []
7.82/2.84	
7.82/2.84	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
7.82/2.84	
7.82/2.84	.(x1, x2)  =  .(x1, x2)
7.82/2.84	
7.82/2.84	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
7.82/2.84	
7.82/2.84	mv(x1, x2)  =  mv(x1, x2)
7.82/2.84	
7.82/2.84	U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
7.82/2.84	
7.82/2.84	SHANOI_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI_IN_GGGGA(x1, x2, x3, x4)
7.82/2.84	
7.82/2.84	U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x2, x3, x4, x6)
7.82/2.84	
7.82/2.84	U2_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGGGA(x2, x3, x4, x6, x7)
7.82/2.84	
7.82/2.84	
7.82/2.84	We have to consider all (P,R,Pi)-chains
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(15) PiDPToQDPProof (SOUND)
7.82/2.84	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(16)
7.82/2.84	Obligation:
7.82/2.84	Q DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   U1_GGGGA(A, B, C, eq_out_ag(N1)) -> U2_GGGGA(A, B, C, N1, shanoi_in_gggga(N1, A, C, B))
7.82/2.84	   U2_GGGGA(A, B, C, N1, shanoi_out_gggga(M1)) -> SHANOI_IN_GGGGA(N1, B, A, C)
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C) -> U1_GGGGA(A, B, C, eq_in_ag(s(X)))
7.82/2.84	   U1_GGGGA(A, B, C, eq_out_ag(N1)) -> SHANOI_IN_GGGGA(N1, A, C, B)
7.82/2.84	
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C) -> shanoi_out_gggga(.(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C) -> U1_gggga(A, B, C, eq_in_ag(s(X)))
7.82/2.84	   eq_in_ag(X) -> eq_out_ag(X)
7.82/2.84	   U1_gggga(A, B, C, eq_out_ag(N1)) -> U2_gggga(A, B, C, N1, shanoi_in_gggga(N1, A, C, B))
7.82/2.84	   U2_gggga(A, B, C, N1, shanoi_out_gggga(M1)) -> U3_gggga(A, C, M1, shanoi_in_gggga(N1, B, A, C))
7.82/2.84	   U3_gggga(A, C, M1, shanoi_out_gggga(M2)) -> U4_gggga(M2, append_in_gga(M1, .(mv(A, C), [])))
7.82/2.84	   append_in_gga([], L) -> append_out_gga(L)
7.82/2.84	   append_in_gga(.(H, L), L1) -> U6_gga(H, append_in_gga(L, L1))
7.82/2.84	   U6_gga(H, append_out_gga(R)) -> append_out_gga(.(H, R))
7.82/2.84	   U4_gggga(M2, append_out_gga(T)) -> U5_gggga(append_in_gga(T, M2))
7.82/2.84	   U5_gggga(append_out_gga(M)) -> shanoi_out_gggga(M)
7.82/2.84	
7.82/2.84	The set Q consists of the following terms:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(x0, x1, x2, x3)
7.82/2.84	   eq_in_ag(x0)
7.82/2.84	   U1_gggga(x0, x1, x2, x3)
7.82/2.84	   U2_gggga(x0, x1, x2, x3, x4)
7.82/2.84	   U3_gggga(x0, x1, x2, x3)
7.82/2.84	   append_in_gga(x0, x1)
7.82/2.84	   U6_gga(x0, x1)
7.82/2.84	   U4_gggga(x0, x1)
7.82/2.84	   U5_gggga(x0)
7.82/2.84	
7.82/2.84	We have to consider all (P,Q,R)-chains.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(17) QDPOrderProof (EQUIVALENT)
7.82/2.84	We use the reduction pair processor [LPAR04,JAR06].
7.82/2.84	
7.82/2.84	
7.82/2.84	The following pairs can be oriented strictly and are deleted.
7.82/2.84	
7.82/2.84	   SHANOI_IN_GGGGA(s(s(X)), A, B, C) -> U1_GGGGA(A, B, C, eq_in_ag(s(X)))
7.82/2.84	The remaining pairs can at least be oriented weakly.
7.82/2.84	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
7.82/2.84	
7.82/2.84	POL( U2_GGGGA_5(x_1, ..., x_5) ) = x_4
7.82/2.84	POL( shanoi_in_gggga_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4
7.82/2.84	POL( s_1(x_1) ) = x_1 + 2
7.82/2.84	POL( 0 ) = 0
7.82/2.84	POL( shanoi_out_gggga_1(x_1) ) = max{0, 2x_1 - 2}
7.82/2.84	POL( ._2(x_1, x_2) ) = 2x_2 + 2
7.82/2.84	POL( mv_2(x_1, x_2) ) = 2
7.82/2.84	POL( [] ) = 0
7.82/2.84	POL( U1_gggga_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
7.82/2.84	POL( eq_in_ag_1(x_1) ) = x_1 + 1
7.82/2.84	POL( U1_GGGGA_4(x_1, ..., x_4) ) = x_4
7.82/2.84	POL( eq_out_ag_1(x_1) ) = x_1
7.82/2.84	POL( U2_gggga_5(x_1, ..., x_5) ) = max{0, 2x_1 + 2x_2 + 2x_4 + 2x_5 - 2}
7.82/2.84	POL( U3_gggga_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
7.82/2.84	POL( U4_gggga_2(x_1, x_2) ) = max{0, 2x_1 - 2}
7.82/2.84	POL( append_in_gga_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
7.82/2.84	POL( U5_gggga_1(x_1) ) = 2
7.82/2.84	POL( append_out_gga_1(x_1) ) = max{0, -2}
7.82/2.84	POL( U6_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2}
7.82/2.84	POL( SHANOI_IN_GGGGA_4(x_1, ..., x_4) ) = x_1
7.82/2.84	
7.82/2.84	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
7.82/2.84	
7.82/2.84	   eq_in_ag(X) -> eq_out_ag(X)
7.82/2.84	
7.82/2.84	
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(18)
7.82/2.84	Obligation:
7.82/2.84	Q DP problem:
7.82/2.84	The TRS P consists of the following rules:
7.82/2.84	
7.82/2.84	   U1_GGGGA(A, B, C, eq_out_ag(N1)) -> U2_GGGGA(A, B, C, N1, shanoi_in_gggga(N1, A, C, B))
7.82/2.84	   U2_GGGGA(A, B, C, N1, shanoi_out_gggga(M1)) -> SHANOI_IN_GGGGA(N1, B, A, C)
7.82/2.84	   U1_GGGGA(A, B, C, eq_out_ag(N1)) -> SHANOI_IN_GGGGA(N1, A, C, B)
7.82/2.84	
7.82/2.84	The TRS R consists of the following rules:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(s(0), A, B, C) -> shanoi_out_gggga(.(mv(A, C), []))
7.82/2.84	   shanoi_in_gggga(s(s(X)), A, B, C) -> U1_gggga(A, B, C, eq_in_ag(s(X)))
7.82/2.84	   eq_in_ag(X) -> eq_out_ag(X)
7.82/2.84	   U1_gggga(A, B, C, eq_out_ag(N1)) -> U2_gggga(A, B, C, N1, shanoi_in_gggga(N1, A, C, B))
7.82/2.84	   U2_gggga(A, B, C, N1, shanoi_out_gggga(M1)) -> U3_gggga(A, C, M1, shanoi_in_gggga(N1, B, A, C))
7.82/2.84	   U3_gggga(A, C, M1, shanoi_out_gggga(M2)) -> U4_gggga(M2, append_in_gga(M1, .(mv(A, C), [])))
7.82/2.84	   append_in_gga([], L) -> append_out_gga(L)
7.82/2.84	   append_in_gga(.(H, L), L1) -> U6_gga(H, append_in_gga(L, L1))
7.82/2.84	   U6_gga(H, append_out_gga(R)) -> append_out_gga(.(H, R))
7.82/2.84	   U4_gggga(M2, append_out_gga(T)) -> U5_gggga(append_in_gga(T, M2))
7.82/2.84	   U5_gggga(append_out_gga(M)) -> shanoi_out_gggga(M)
7.82/2.84	
7.82/2.84	The set Q consists of the following terms:
7.82/2.84	
7.82/2.84	   shanoi_in_gggga(x0, x1, x2, x3)
7.82/2.84	   eq_in_ag(x0)
7.82/2.84	   U1_gggga(x0, x1, x2, x3)
7.82/2.84	   U2_gggga(x0, x1, x2, x3, x4)
7.82/2.84	   U3_gggga(x0, x1, x2, x3)
7.82/2.84	   append_in_gga(x0, x1)
7.82/2.84	   U6_gga(x0, x1)
7.82/2.84	   U4_gggga(x0, x1)
7.82/2.84	   U5_gggga(x0)
7.82/2.84	
7.82/2.84	We have to consider all (P,Q,R)-chains.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(19) DependencyGraphProof (EQUIVALENT)
7.82/2.84	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
7.82/2.84	----------------------------------------
7.82/2.84	
7.82/2.84	(20)
7.82/2.84	TRUE
8.02/2.96	EOF