3.89/1.86	YES
3.89/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.89/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.89/1.87	
3.89/1.87	
3.89/1.87	Left Termination of the query pattern
3.89/1.87	
3.89/1.87	rotate(g,a)
3.89/1.87	
3.89/1.87	w.r.t. the given Prolog program could successfully be proven:
3.89/1.87	
3.89/1.87	(0) Prolog
3.89/1.87	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.89/1.87	(2) PiTRS
3.89/1.87	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.89/1.87	(4) PiDP
3.89/1.87	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.89/1.87	(6) AND
3.89/1.87	    (7) PiDP
3.89/1.87	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.89/1.87	        (9) PiDP
3.89/1.87	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.89/1.87	        (11) QDP
3.89/1.87	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.89/1.87	        (13) YES
3.89/1.87	    (14) PiDP
3.89/1.87	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
3.89/1.87	        (16) PiDP
3.89/1.87	        (17) PiDPToQDPProof [SOUND, 0 ms]
3.89/1.87	        (18) QDP
3.89/1.87	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.89/1.87	        (20) YES
3.89/1.87	
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(0)
3.89/1.87	Obligation:
3.89/1.87	Clauses:
3.89/1.87	
3.89/1.87	rotate(X, Y) :- ','(append2(A, B, X), append1(B, A, Y)).
3.89/1.87	append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
3.89/1.87	append1([], Ys, Ys).
3.89/1.87	append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
3.89/1.87	append2([], Ys, Ys).
3.89/1.87	
3.89/1.87	
3.89/1.87	Query: rotate(g,a)
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(1) PrologToPiTRSProof (SOUND)
3.89/1.87	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.89/1.87	
3.89/1.87	rotate_in_2: (b,f)
3.89/1.87	
3.89/1.87	append2_in_3: (f,f,b)
3.89/1.87	
3.89/1.87	append1_in_3: (b,b,f)
3.89/1.87	
3.89/1.87	Transforming Prolog into the following Term Rewriting System:
3.89/1.87	
3.89/1.87	Pi-finite rewrite system:
3.89/1.87	The TRS R consists of the following rules:
3.89/1.87	
3.89/1.87	   rotate_in_ga(X, Y) -> U1_ga(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   append2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   append2_in_aag([], Ys, Ys) -> append2_out_aag([], Ys, Ys)
3.89/1.87	   U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) -> append2_out_aag(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U1_ga(X, Y, append2_out_aag(A, B, X)) -> U2_ga(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   append1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   append1_in_gga([], Ys, Ys) -> append1_out_gga([], Ys, Ys)
3.89/1.87	   U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) -> append1_out_gga(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U2_ga(X, Y, append1_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)
3.89/1.87	
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	rotate_in_ga(x1, x2)  =  rotate_in_ga(x1)
3.89/1.87	
3.89/1.87	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.89/1.87	
3.89/1.87	append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
3.89/1.87	
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
3.89/1.87	
3.89/1.87	append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
3.89/1.87	
3.89/1.87	U2_ga(x1, x2, x3)  =  U2_ga(x3)
3.89/1.87	
3.89/1.87	append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
3.89/1.87	
3.89/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.89/1.87	
3.89/1.87	[]  =  []
3.89/1.87	
3.89/1.87	append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
3.89/1.87	
3.89/1.87	rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)
3.89/1.87	
3.89/1.87	
3.89/1.87	
3.89/1.87	
3.89/1.87	
3.89/1.87	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.89/1.87	
3.89/1.87	
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(2)
3.89/1.87	Obligation:
3.89/1.87	Pi-finite rewrite system:
3.89/1.87	The TRS R consists of the following rules:
3.89/1.87	
3.89/1.87	   rotate_in_ga(X, Y) -> U1_ga(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   append2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   append2_in_aag([], Ys, Ys) -> append2_out_aag([], Ys, Ys)
3.89/1.87	   U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) -> append2_out_aag(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U1_ga(X, Y, append2_out_aag(A, B, X)) -> U2_ga(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   append1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   append1_in_gga([], Ys, Ys) -> append1_out_gga([], Ys, Ys)
3.89/1.87	   U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) -> append1_out_gga(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U2_ga(X, Y, append1_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)
3.89/1.87	
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	rotate_in_ga(x1, x2)  =  rotate_in_ga(x1)
3.89/1.87	
3.89/1.87	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.89/1.87	
3.89/1.87	append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
3.89/1.87	
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
3.89/1.87	
3.89/1.87	append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
3.89/1.87	
3.89/1.87	U2_ga(x1, x2, x3)  =  U2_ga(x3)
3.89/1.87	
3.89/1.87	append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
3.89/1.87	
3.89/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.89/1.87	
3.89/1.87	[]  =  []
3.89/1.87	
3.89/1.87	append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
3.89/1.87	
3.89/1.87	rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)
3.89/1.87	
3.89/1.87	
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(3) DependencyPairsProof (EQUIVALENT)
3.89/1.87	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.89/1.87	Pi DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   ROTATE_IN_GA(X, Y) -> U1_GA(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   ROTATE_IN_GA(X, Y) -> APPEND2_IN_AAG(A, B, X)
3.89/1.87	   APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AAG(Xs, Ys, Zs)
3.89/1.87	   U1_GA(X, Y, append2_out_aag(A, B, X)) -> U2_GA(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   U1_GA(X, Y, append2_out_aag(A, B, X)) -> APPEND1_IN_GGA(B, A, Y)
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_GGA(Xs, Ys, Zs)
3.89/1.87	
3.89/1.87	The TRS R consists of the following rules:
3.89/1.87	
3.89/1.87	   rotate_in_ga(X, Y) -> U1_ga(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   append2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   append2_in_aag([], Ys, Ys) -> append2_out_aag([], Ys, Ys)
3.89/1.87	   U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) -> append2_out_aag(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U1_ga(X, Y, append2_out_aag(A, B, X)) -> U2_ga(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   append1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   append1_in_gga([], Ys, Ys) -> append1_out_gga([], Ys, Ys)
3.89/1.87	   U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) -> append1_out_gga(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U2_ga(X, Y, append1_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)
3.89/1.87	
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	rotate_in_ga(x1, x2)  =  rotate_in_ga(x1)
3.89/1.87	
3.89/1.87	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.89/1.87	
3.89/1.87	append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
3.89/1.87	
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
3.89/1.87	
3.89/1.87	append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
3.89/1.87	
3.89/1.87	U2_ga(x1, x2, x3)  =  U2_ga(x3)
3.89/1.87	
3.89/1.87	append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
3.89/1.87	
3.89/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.89/1.87	
3.89/1.87	[]  =  []
3.89/1.87	
3.89/1.87	append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
3.89/1.87	
3.89/1.87	rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)
3.89/1.87	
3.89/1.87	ROTATE_IN_GA(x1, x2)  =  ROTATE_IN_GA(x1)
3.89/1.87	
3.89/1.87	U1_GA(x1, x2, x3)  =  U1_GA(x3)
3.89/1.87	
3.89/1.87	APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
3.89/1.87	
3.89/1.87	U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x1, x5)
3.89/1.87	
3.89/1.87	U2_GA(x1, x2, x3)  =  U2_GA(x3)
3.89/1.87	
3.89/1.87	APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
3.89/1.87	
3.89/1.87	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
3.89/1.87	
3.89/1.87	
3.89/1.87	We have to consider all (P,R,Pi)-chains
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(4)
3.89/1.87	Obligation:
3.89/1.87	Pi DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   ROTATE_IN_GA(X, Y) -> U1_GA(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   ROTATE_IN_GA(X, Y) -> APPEND2_IN_AAG(A, B, X)
3.89/1.87	   APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AAG(Xs, Ys, Zs)
3.89/1.87	   U1_GA(X, Y, append2_out_aag(A, B, X)) -> U2_GA(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   U1_GA(X, Y, append2_out_aag(A, B, X)) -> APPEND1_IN_GGA(B, A, Y)
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_GGA(Xs, Ys, Zs)
3.89/1.87	
3.89/1.87	The TRS R consists of the following rules:
3.89/1.87	
3.89/1.87	   rotate_in_ga(X, Y) -> U1_ga(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   append2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   append2_in_aag([], Ys, Ys) -> append2_out_aag([], Ys, Ys)
3.89/1.87	   U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) -> append2_out_aag(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U1_ga(X, Y, append2_out_aag(A, B, X)) -> U2_ga(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   append1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   append1_in_gga([], Ys, Ys) -> append1_out_gga([], Ys, Ys)
3.89/1.87	   U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) -> append1_out_gga(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U2_ga(X, Y, append1_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)
3.89/1.87	
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	rotate_in_ga(x1, x2)  =  rotate_in_ga(x1)
3.89/1.87	
3.89/1.87	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.89/1.87	
3.89/1.87	append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
3.89/1.87	
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
3.89/1.87	
3.89/1.87	append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
3.89/1.87	
3.89/1.87	U2_ga(x1, x2, x3)  =  U2_ga(x3)
3.89/1.87	
3.89/1.87	append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
3.89/1.87	
3.89/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.89/1.87	
3.89/1.87	[]  =  []
3.89/1.87	
3.89/1.87	append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
3.89/1.87	
3.89/1.87	rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)
3.89/1.87	
3.89/1.87	ROTATE_IN_GA(x1, x2)  =  ROTATE_IN_GA(x1)
3.89/1.87	
3.89/1.87	U1_GA(x1, x2, x3)  =  U1_GA(x3)
3.89/1.87	
3.89/1.87	APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
3.89/1.87	
3.89/1.87	U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x1, x5)
3.89/1.87	
3.89/1.87	U2_GA(x1, x2, x3)  =  U2_GA(x3)
3.89/1.87	
3.89/1.87	APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
3.89/1.87	
3.89/1.87	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
3.89/1.87	
3.89/1.87	
3.89/1.87	We have to consider all (P,R,Pi)-chains
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(5) DependencyGraphProof (EQUIVALENT)
3.89/1.87	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(6)
3.89/1.87	Complex Obligation (AND)
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(7)
3.89/1.87	Obligation:
3.89/1.87	Pi DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_GGA(Xs, Ys, Zs)
3.89/1.87	
3.89/1.87	The TRS R consists of the following rules:
3.89/1.87	
3.89/1.87	   rotate_in_ga(X, Y) -> U1_ga(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   append2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   append2_in_aag([], Ys, Ys) -> append2_out_aag([], Ys, Ys)
3.89/1.87	   U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) -> append2_out_aag(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U1_ga(X, Y, append2_out_aag(A, B, X)) -> U2_ga(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   append1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   append1_in_gga([], Ys, Ys) -> append1_out_gga([], Ys, Ys)
3.89/1.87	   U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) -> append1_out_gga(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U2_ga(X, Y, append1_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)
3.89/1.87	
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	rotate_in_ga(x1, x2)  =  rotate_in_ga(x1)
3.89/1.87	
3.89/1.87	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.89/1.87	
3.89/1.87	append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
3.89/1.87	
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
3.89/1.87	
3.89/1.87	append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
3.89/1.87	
3.89/1.87	U2_ga(x1, x2, x3)  =  U2_ga(x3)
3.89/1.87	
3.89/1.87	append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
3.89/1.87	
3.89/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.89/1.87	
3.89/1.87	[]  =  []
3.89/1.87	
3.89/1.87	append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
3.89/1.87	
3.89/1.87	rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)
3.89/1.87	
3.89/1.87	APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
3.89/1.87	
3.89/1.87	
3.89/1.87	We have to consider all (P,R,Pi)-chains
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(8) UsableRulesProof (EQUIVALENT)
3.89/1.87	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(9)
3.89/1.87	Obligation:
3.89/1.87	Pi DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_GGA(Xs, Ys, Zs)
3.89/1.87	
3.89/1.87	R is empty.
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
3.89/1.87	
3.89/1.87	
3.89/1.87	We have to consider all (P,R,Pi)-chains
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(10) PiDPToQDPProof (SOUND)
3.89/1.87	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(11)
3.89/1.87	Obligation:
3.89/1.87	Q DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   APPEND1_IN_GGA(.(X, Xs), Ys) -> APPEND1_IN_GGA(Xs, Ys)
3.89/1.87	
3.89/1.87	R is empty.
3.89/1.87	Q is empty.
3.89/1.87	We have to consider all (P,Q,R)-chains.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(12) QDPSizeChangeProof (EQUIVALENT)
3.89/1.87	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. 
3.89/1.87	
3.89/1.87	From the DPs we obtained the following set of size-change graphs:
3.89/1.87	*APPEND1_IN_GGA(.(X, Xs), Ys) -> APPEND1_IN_GGA(Xs, Ys)
3.89/1.87	The graph contains the following edges 1 > 1, 2 >= 2
3.89/1.87	
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(13)
3.89/1.87	YES
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(14)
3.89/1.87	Obligation:
3.89/1.87	Pi DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AAG(Xs, Ys, Zs)
3.89/1.87	
3.89/1.87	The TRS R consists of the following rules:
3.89/1.87	
3.89/1.87	   rotate_in_ga(X, Y) -> U1_ga(X, Y, append2_in_aag(A, B, X))
3.89/1.87	   append2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
3.89/1.87	   append2_in_aag([], Ys, Ys) -> append2_out_aag([], Ys, Ys)
3.89/1.87	   U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) -> append2_out_aag(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U1_ga(X, Y, append2_out_aag(A, B, X)) -> U2_ga(X, Y, append1_in_gga(B, A, Y))
3.89/1.87	   append1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
3.89/1.87	   append1_in_gga([], Ys, Ys) -> append1_out_gga([], Ys, Ys)
3.89/1.87	   U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) -> append1_out_gga(.(X, Xs), Ys, .(X, Zs))
3.89/1.87	   U2_ga(X, Y, append1_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)
3.89/1.87	
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	rotate_in_ga(x1, x2)  =  rotate_in_ga(x1)
3.89/1.87	
3.89/1.87	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.89/1.87	
3.89/1.87	append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
3.89/1.87	
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
3.89/1.87	
3.89/1.87	append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
3.89/1.87	
3.89/1.87	U2_ga(x1, x2, x3)  =  U2_ga(x3)
3.89/1.87	
3.89/1.87	append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
3.89/1.87	
3.89/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.89/1.87	
3.89/1.87	[]  =  []
3.89/1.87	
3.89/1.87	append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
3.89/1.87	
3.89/1.87	rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)
3.89/1.87	
3.89/1.87	APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
3.89/1.87	
3.89/1.87	
3.89/1.87	We have to consider all (P,R,Pi)-chains
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(15) UsableRulesProof (EQUIVALENT)
3.89/1.87	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(16)
3.89/1.87	Obligation:
3.89/1.87	Pi DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AAG(Xs, Ys, Zs)
3.89/1.87	
3.89/1.87	R is empty.
3.89/1.87	The argument filtering Pi contains the following mapping:
3.89/1.87	.(x1, x2)  =  .(x1, x2)
3.89/1.87	
3.89/1.87	APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
3.89/1.87	
3.89/1.87	
3.89/1.87	We have to consider all (P,R,Pi)-chains
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(17) PiDPToQDPProof (SOUND)
3.89/1.87	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(18)
3.89/1.87	Obligation:
3.89/1.87	Q DP problem:
3.89/1.87	The TRS P consists of the following rules:
3.89/1.87	
3.89/1.87	   APPEND2_IN_AAG(.(X, Zs)) -> APPEND2_IN_AAG(Zs)
3.89/1.87	
3.89/1.87	R is empty.
3.89/1.87	Q is empty.
3.89/1.87	We have to consider all (P,Q,R)-chains.
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(19) QDPSizeChangeProof (EQUIVALENT)
3.89/1.87	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. 
3.89/1.87	
3.89/1.87	From the DPs we obtained the following set of size-change graphs:
3.89/1.87	*APPEND2_IN_AAG(.(X, Zs)) -> APPEND2_IN_AAG(Zs)
3.89/1.87	The graph contains the following edges 1 > 1
3.89/1.87	
3.89/1.87	
3.89/1.87	----------------------------------------
3.89/1.87	
3.89/1.87	(20)
3.89/1.87	YES
4.15/2.01	EOF