5.79/2.35	YES
6.06/2.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.06/2.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.06/2.37	
6.06/2.37	
6.06/2.37	Left Termination of the query pattern
6.06/2.37	
6.06/2.37	perm(g,a)
6.06/2.37	
6.06/2.37	w.r.t. the given Prolog program could successfully be proven:
6.06/2.37	
6.06/2.37	(0) Prolog
6.06/2.37	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.06/2.37	(2) PiTRS
6.06/2.37	(3) DependencyPairsProof [EQUIVALENT, 20 ms]
6.06/2.37	(4) PiDP
6.06/2.37	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.37	(6) AND
6.06/2.37	    (7) PiDP
6.06/2.37	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.06/2.37	        (9) PiDP
6.06/2.37	        (10) PiDPToQDPProof [SOUND, 0 ms]
6.06/2.37	        (11) QDP
6.06/2.37	        (12) UsableRulesReductionPairsProof [EQUIVALENT, 2 ms]
6.06/2.37	        (13) QDP
6.06/2.37	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
6.06/2.37	        (15) YES
6.06/2.37	    (16) PiDP
6.06/2.37	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
6.06/2.37	        (18) PiDP
6.06/2.37	        (19) PiDPToQDPProof [SOUND, 0 ms]
6.06/2.37	        (20) QDP
6.06/2.37	        (21) UsableRulesReductionPairsProof [EQUIVALENT, 4 ms]
6.06/2.37	        (22) QDP
6.06/2.37	        (23) PisEmptyProof [EQUIVALENT, 0 ms]
6.06/2.37	        (24) YES
6.06/2.37	    (25) PiDP
6.06/2.37	        (26) UsableRulesProof [EQUIVALENT, 0 ms]
6.06/2.37	        (27) PiDP
6.06/2.37	        (28) PiDPToQDPProof [SOUND, 0 ms]
6.06/2.37	        (29) QDP
6.06/2.37	        (30) MRRProof [EQUIVALENT, 0 ms]
6.06/2.37	        (31) QDP
6.06/2.37	        (32) PisEmptyProof [EQUIVALENT, 0 ms]
6.06/2.37	        (33) YES
6.06/2.37	
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(0)
6.06/2.37	Obligation:
6.06/2.37	Clauses:
6.06/2.37	
6.06/2.37	append2(parts([], Y), is(sum(Y))).
6.06/2.37	append2(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append2(parts(X, Y), is(sum(Z))).
6.06/2.37	append1(parts([], Y), is(sum(Y))).
6.06/2.37	append1(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append1(parts(X, Y), is(sum(Z))).
6.06/2.37	perm([], []).
6.06/2.37	perm(L, .(H, T)) :- ','(append2(parts(V, .(H, U)), is(sum(L))), ','(append1(parts(V, U), is(sum(W))), perm(W, T))).
6.06/2.37	
6.06/2.37	
6.06/2.37	Query: perm(g,a)
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(1) PrologToPiTRSProof (SOUND)
6.06/2.37	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.06/2.37	
6.06/2.37	perm_in_2: (b,f)
6.06/2.37	
6.06/2.37	append2_in_2: (f,b)
6.06/2.37	
6.06/2.37	append1_in_2: (b,f)
6.06/2.37	
6.06/2.37	Transforming Prolog into the following Term Rewriting System:
6.06/2.37	
6.06/2.37	Pi-finite rewrite system:
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	
6.06/2.37	
6.06/2.37	
6.06/2.37	
6.06/2.37	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.06/2.37	
6.06/2.37	
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(2)
6.06/2.37	Obligation:
6.06/2.37	Pi-finite rewrite system:
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(3) DependencyPairsProof (EQUIVALENT)
6.06/2.37	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   PERM_IN_GA(L, .(H, T)) -> U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   PERM_IN_GA(L, .(H, T)) -> APPEND2_IN_AG(parts(V, .(H, U)), is(sum(L)))
6.06/2.37	   APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_AG(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND2_IN_AG(parts(X, Y), is(sum(Z)))
6.06/2.37	   U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> APPEND1_IN_GA(parts(V, U), is(sum(W)))
6.06/2.37	   APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_GA(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND1_IN_GA(parts(X, Y), is(sum(Z)))
6.06/2.37	   U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_GA(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> PERM_IN_GA(W, T)
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
6.06/2.37	
6.06/2.37	U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
6.06/2.37	
6.06/2.37	APPEND2_IN_AG(x1, x2)  =  APPEND2_IN_AG(x2)
6.06/2.37	
6.06/2.37	U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x5)
6.06/2.37	
6.06/2.37	U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
6.06/2.37	
6.06/2.37	APPEND1_IN_GA(x1, x2)  =  APPEND1_IN_GA(x1)
6.06/2.37	
6.06/2.37	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
6.06/2.37	
6.06/2.37	U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(4)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   PERM_IN_GA(L, .(H, T)) -> U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   PERM_IN_GA(L, .(H, T)) -> APPEND2_IN_AG(parts(V, .(H, U)), is(sum(L)))
6.06/2.37	   APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_AG(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND2_IN_AG(parts(X, Y), is(sum(Z)))
6.06/2.37	   U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> APPEND1_IN_GA(parts(V, U), is(sum(W)))
6.06/2.37	   APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_GA(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND1_IN_GA(parts(X, Y), is(sum(Z)))
6.06/2.37	   U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_GA(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> PERM_IN_GA(W, T)
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
6.06/2.37	
6.06/2.37	U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
6.06/2.37	
6.06/2.37	APPEND2_IN_AG(x1, x2)  =  APPEND2_IN_AG(x2)
6.06/2.37	
6.06/2.37	U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x5)
6.06/2.37	
6.06/2.37	U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
6.06/2.37	
6.06/2.37	APPEND1_IN_GA(x1, x2)  =  APPEND1_IN_GA(x1)
6.06/2.37	
6.06/2.37	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
6.06/2.37	
6.06/2.37	U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(5) DependencyGraphProof (EQUIVALENT)
6.06/2.37	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(6)
6.06/2.37	Complex Obligation (AND)
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(7)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND1_IN_GA(parts(X, Y), is(sum(Z)))
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	APPEND1_IN_GA(x1, x2)  =  APPEND1_IN_GA(x1)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(8) UsableRulesProof (EQUIVALENT)
6.06/2.37	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(9)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND1_IN_GA(parts(X, Y), is(sum(Z)))
6.06/2.37	
6.06/2.37	R is empty.
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	APPEND1_IN_GA(x1, x2)  =  APPEND1_IN_GA(x1)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(10) PiDPToQDPProof (SOUND)
6.06/2.37	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(11)
6.06/2.37	Obligation:
6.06/2.37	Q DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   APPEND1_IN_GA(parts(.(X), Y)) -> APPEND1_IN_GA(parts(X, Y))
6.06/2.37	
6.06/2.37	R is empty.
6.06/2.37	Q is empty.
6.06/2.37	We have to consider all (P,Q,R)-chains.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(12) UsableRulesReductionPairsProof (EQUIVALENT)
6.06/2.37	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
6.06/2.37	
6.06/2.37	The following dependency pairs can be deleted:
6.06/2.37	
6.06/2.37	   APPEND1_IN_GA(parts(.(X), Y)) -> APPEND1_IN_GA(parts(X, Y))
6.06/2.37	No rules are removed from R.
6.06/2.37	
6.06/2.37	Used ordering: POLO with Polynomial interpretation [POLO]:
6.06/2.37	
6.06/2.37	   POL(.(x_1)) = x_1
6.06/2.37	   POL(APPEND1_IN_GA(x_1)) = 2*x_1
6.06/2.37	   POL(parts(x_1, x_2)) = 2*x_1 + x_2
6.06/2.37	
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(13)
6.06/2.37	Obligation:
6.06/2.37	Q DP problem:
6.06/2.37	P is empty.
6.06/2.37	R is empty.
6.06/2.37	Q is empty.
6.06/2.37	We have to consider all (P,Q,R)-chains.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(14) PisEmptyProof (EQUIVALENT)
6.06/2.37	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(15)
6.06/2.37	YES
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(16)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND2_IN_AG(parts(X, Y), is(sum(Z)))
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	APPEND2_IN_AG(x1, x2)  =  APPEND2_IN_AG(x2)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(17) UsableRulesProof (EQUIVALENT)
6.06/2.37	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(18)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) -> APPEND2_IN_AG(parts(X, Y), is(sum(Z)))
6.06/2.37	
6.06/2.37	R is empty.
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	APPEND2_IN_AG(x1, x2)  =  APPEND2_IN_AG(x2)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(19) PiDPToQDPProof (SOUND)
6.06/2.37	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(20)
6.06/2.37	Obligation:
6.06/2.37	Q DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   APPEND2_IN_AG(is(sum(.(Z)))) -> APPEND2_IN_AG(is(sum(Z)))
6.06/2.37	
6.06/2.37	R is empty.
6.06/2.37	Q is empty.
6.06/2.37	We have to consider all (P,Q,R)-chains.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(21) UsableRulesReductionPairsProof (EQUIVALENT)
6.06/2.37	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
6.06/2.37	
6.06/2.37	The following dependency pairs can be deleted:
6.06/2.37	
6.06/2.37	   APPEND2_IN_AG(is(sum(.(Z)))) -> APPEND2_IN_AG(is(sum(Z)))
6.06/2.37	No rules are removed from R.
6.06/2.37	
6.06/2.37	Used ordering: POLO with Polynomial interpretation [POLO]:
6.06/2.37	
6.06/2.37	   POL(.(x_1)) = 2*x_1
6.06/2.37	   POL(APPEND2_IN_AG(x_1)) = 2*x_1
6.06/2.37	   POL(is(x_1)) = x_1
6.06/2.37	   POL(sum(x_1)) = x_1
6.06/2.37	
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(22)
6.06/2.37	Obligation:
6.06/2.37	Q DP problem:
6.06/2.37	P is empty.
6.06/2.37	R is empty.
6.06/2.37	Q is empty.
6.06/2.37	We have to consider all (P,Q,R)-chains.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(23) PisEmptyProof (EQUIVALENT)
6.06/2.37	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(24)
6.06/2.37	YES
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(25)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> PERM_IN_GA(W, T)
6.06/2.37	   PERM_IN_GA(L, .(H, T)) -> U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   perm_in_ga([], []) -> perm_out_ga([], [])
6.06/2.37	   perm_in_ga(L, .(H, T)) -> U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> U5_ga(L, H, T, perm_in_ga(W, T))
6.06/2.37	   U5_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
6.06/2.37	
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
6.06/2.37	
6.06/2.37	U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
6.06/2.37	
6.06/2.37	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
6.06/2.37	
6.06/2.37	U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
6.06/2.37	
6.06/2.37	U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(26) UsableRulesProof (EQUIVALENT)
6.06/2.37	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(27)
6.06/2.37	Obligation:
6.06/2.37	Pi DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) -> U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
6.06/2.37	   U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) -> PERM_IN_GA(W, T)
6.06/2.37	   PERM_IN_GA(L, .(H, T)) -> U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   append1_in_ga(parts([], Y), is(sum(Y))) -> append1_out_ga(parts([], Y), is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
6.06/2.37	   append2_in_ag(parts([], Y), is(sum(Y))) -> append2_out_ag(parts([], Y), is(sum(Y)))
6.06/2.37	   append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) -> U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
6.06/2.37	   U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) -> append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	   U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) -> append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
6.06/2.37	
6.06/2.37	The argument filtering Pi contains the following mapping:
6.06/2.37	[]  =  []
6.06/2.37	
6.06/2.37	append2_in_ag(x1, x2)  =  append2_in_ag(x2)
6.06/2.37	
6.06/2.37	.(x1, x2)  =  .(x2)
6.06/2.37	
6.06/2.37	is(x1)  =  is(x1)
6.06/2.37	
6.06/2.37	sum(x1)  =  sum(x1)
6.06/2.37	
6.06/2.37	append2_out_ag(x1, x2)  =  append2_out_ag(x1)
6.06/2.37	
6.06/2.37	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
6.06/2.37	
6.06/2.37	parts(x1, x2)  =  parts(x1, x2)
6.06/2.37	
6.06/2.37	append1_in_ga(x1, x2)  =  append1_in_ga(x1)
6.06/2.37	
6.06/2.37	append1_out_ga(x1, x2)  =  append1_out_ga(x2)
6.06/2.37	
6.06/2.37	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.06/2.37	
6.06/2.37	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
6.06/2.37	
6.06/2.37	U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
6.06/2.37	
6.06/2.37	U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
6.06/2.37	
6.06/2.37	
6.06/2.37	We have to consider all (P,R,Pi)-chains
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(28) PiDPToQDPProof (SOUND)
6.06/2.37	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(29)
6.06/2.37	Obligation:
6.06/2.37	Q DP problem:
6.06/2.37	The TRS P consists of the following rules:
6.06/2.37	
6.06/2.37	   U3_GA(append2_out_ag(parts(V, .(U)))) -> U4_GA(append1_in_ga(parts(V, U)))
6.06/2.37	   U4_GA(append1_out_ga(is(sum(W)))) -> PERM_IN_GA(W)
6.06/2.37	   PERM_IN_GA(L) -> U3_GA(append2_in_ag(is(sum(L))))
6.06/2.37	
6.06/2.37	The TRS R consists of the following rules:
6.06/2.37	
6.06/2.37	   append1_in_ga(parts([], Y)) -> append1_out_ga(is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(X), Y)) -> U2_ga(append1_in_ga(parts(X, Y)))
6.06/2.37	   append2_in_ag(is(sum(Y))) -> append2_out_ag(parts([], Y))
6.06/2.37	   append2_in_ag(is(sum(.(Z)))) -> U1_ag(append2_in_ag(is(sum(Z))))
6.06/2.37	   U2_ga(append1_out_ga(is(sum(Z)))) -> append1_out_ga(is(sum(.(Z))))
6.06/2.37	   U1_ag(append2_out_ag(parts(X, Y))) -> append2_out_ag(parts(.(X), Y))
6.06/2.37	
6.06/2.37	The set Q consists of the following terms:
6.06/2.37	
6.06/2.37	   append1_in_ga(x0)
6.06/2.37	   append2_in_ag(x0)
6.06/2.37	   U2_ga(x0)
6.06/2.37	   U1_ag(x0)
6.06/2.37	
6.06/2.37	We have to consider all (P,Q,R)-chains.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(30) MRRProof (EQUIVALENT)
6.06/2.37	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
6.06/2.37	
6.06/2.37	Strictly oriented dependency pairs:
6.06/2.37	
6.06/2.37	   U3_GA(append2_out_ag(parts(V, .(U)))) -> U4_GA(append1_in_ga(parts(V, U)))
6.06/2.37	   U4_GA(append1_out_ga(is(sum(W)))) -> PERM_IN_GA(W)
6.06/2.37	   PERM_IN_GA(L) -> U3_GA(append2_in_ag(is(sum(L))))
6.06/2.37	
6.06/2.37	Strictly oriented rules of the TRS R:
6.06/2.37	
6.06/2.37	   append1_in_ga(parts([], Y)) -> append1_out_ga(is(sum(Y)))
6.06/2.37	   append1_in_ga(parts(.(X), Y)) -> U2_ga(append1_in_ga(parts(X, Y)))
6.06/2.37	   append2_in_ag(is(sum(Y))) -> append2_out_ag(parts([], Y))
6.06/2.37	   append2_in_ag(is(sum(.(Z)))) -> U1_ag(append2_in_ag(is(sum(Z))))
6.06/2.37	   U2_ga(append1_out_ga(is(sum(Z)))) -> append1_out_ga(is(sum(.(Z))))
6.06/2.37	   U1_ag(append2_out_ag(parts(X, Y))) -> append2_out_ag(parts(.(X), Y))
6.06/2.37	
6.06/2.37	Used ordering: Knuth-Bendix order [KBO] with precedence:parts_2 > append1_in_ga_1 > U2_ga_1 > PERM_IN_GA_1 > ._1 > is_1 > append1_out_ga_1 > append2_in_ag_1 > U1_ag_1 > append2_out_ag_1 > U3_GA_1 > U4_GA_1 > sum_1 > []
6.06/2.37	
6.06/2.37	and weight map:
6.06/2.37	
6.06/2.37	   []=15
6.06/2.37	   append1_in_ga_1=8
6.06/2.37	   append1_out_ga_1=5
6.06/2.37	   is_1=7
6.06/2.37	   sum_1=7
6.06/2.37	   ._1=6
6.06/2.37	   U2_ga_1=6
6.06/2.37	   append2_in_ag_1=2
6.06/2.37	   append2_out_ag_1=1
6.06/2.37	   U1_ag_1=6
6.06/2.37	   U3_GA_1=2
6.06/2.37	   U4_GA_1=1
6.06/2.37	   PERM_IN_GA_1=19
6.06/2.37	   parts_2=0
6.06/2.37	
6.06/2.37	The variable weight is 5
6.06/2.37	
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(31)
6.06/2.37	Obligation:
6.06/2.37	Q DP problem:
6.06/2.37	P is empty.
6.06/2.37	R is empty.
6.06/2.37	The set Q consists of the following terms:
6.06/2.37	
6.06/2.37	   append1_in_ga(x0)
6.06/2.37	   append2_in_ag(x0)
6.06/2.37	   U2_ga(x0)
6.06/2.37	   U1_ag(x0)
6.06/2.37	
6.06/2.37	We have to consider all (P,Q,R)-chains.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(32) PisEmptyProof (EQUIVALENT)
6.06/2.37	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.06/2.37	----------------------------------------
6.06/2.37	
6.06/2.37	(33)
6.06/2.37	YES
6.06/2.40	EOF