8.53/3.47	YES
8.77/3.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
8.77/3.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.77/3.50	
8.77/3.50	
8.77/3.50	Left Termination of the query pattern
8.77/3.50	
8.77/3.50	qsort(g,a)
8.77/3.50	
8.77/3.50	w.r.t. the given Prolog program could successfully be proven:
8.77/3.50	
8.77/3.50	(0) Prolog
8.77/3.50	(1) PrologToPiTRSProof [SOUND, 0 ms]
8.77/3.50	(2) PiTRS
8.77/3.50	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
8.77/3.50	(4) PiDP
8.77/3.50	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
8.77/3.50	(6) AND
8.77/3.50	    (7) PiDP
8.77/3.50	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
8.77/3.50	        (9) PiDP
8.77/3.50	        (10) PiDPToQDPProof [SOUND, 0 ms]
8.77/3.50	        (11) QDP
8.77/3.50	        (12) QDPSizeChangeProof [EQUIVALENT, 1 ms]
8.77/3.50	        (13) YES
8.77/3.50	    (14) PiDP
8.77/3.50	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
8.77/3.50	        (16) PiDP
8.77/3.50	        (17) PiDPToQDPProof [EQUIVALENT, 0 ms]
8.77/3.50	        (18) QDP
8.77/3.50	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.77/3.50	        (20) YES
8.77/3.50	    (21) PiDP
8.77/3.50	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
8.77/3.50	        (23) PiDP
8.77/3.50	        (24) PiDPToQDPProof [EQUIVALENT, 0 ms]
8.77/3.50	        (25) QDP
8.77/3.50	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.77/3.50	        (27) YES
8.77/3.50	    (28) PiDP
8.77/3.50	        (29) UsableRulesProof [EQUIVALENT, 0 ms]
8.77/3.50	        (30) PiDP
8.77/3.50	        (31) PiDPToQDPProof [SOUND, 0 ms]
8.77/3.50	        (32) QDP
8.77/3.50	        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.77/3.50	        (34) YES
8.77/3.50	    (35) PiDP
8.77/3.50	        (36) PiDPToQDPProof [SOUND, 0 ms]
8.77/3.50	        (37) QDP
8.77/3.50	        (38) QDPOrderProof [EQUIVALENT, 65 ms]
8.77/3.50	        (39) QDP
8.77/3.50	        (40) DependencyGraphProof [EQUIVALENT, 0 ms]
8.77/3.50	        (41) TRUE
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(0)
8.77/3.50	Obligation:
8.77/3.50	Clauses:
8.77/3.50	
8.77/3.50	qsort([], []).
8.77/3.50	qsort(.(H, L), S) :- ','(split(L, H, A, B), ','(qsort(A, A1), ','(qsort(B, B1), append(A1, .(H, B1), S)))).
8.77/3.50	split([], Y, [], []).
8.77/3.50	split(.(X, Xs), Y, .(X, Ls), Bs) :- ','(le(X, Y), split(Xs, Y, Ls, Bs)).
8.77/3.50	split(.(X, Xs), Y, Ls, .(X, Bs)) :- ','(gt(X, Y), split(Xs, Y, Ls, Bs)).
8.77/3.50	append([], L, L).
8.77/3.50	append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
8.77/3.50	gt(s(X), s(Y)) :- gt(X, Y).
8.77/3.50	gt(s(X), 0).
8.77/3.50	le(s(X), s(Y)) :- le(X, Y).
8.77/3.50	le(0, s(Y)).
8.77/3.50	le(0, 0).
8.77/3.50	
8.77/3.50	
8.77/3.50	Query: qsort(g,a)
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(1) PrologToPiTRSProof (SOUND)
8.77/3.50	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
8.77/3.50	
8.77/3.50	qsort_in_2: (b,f)
8.77/3.50	
8.77/3.50	split_in_4: (b,b,f,f)
8.77/3.50	
8.77/3.50	le_in_2: (b,b)
8.77/3.50	
8.77/3.50	gt_in_2: (b,b)
8.77/3.50	
8.77/3.50	append_in_3: (b,b,f)
8.77/3.50	
8.77/3.50	Transforming Prolog into the following Term Rewriting System:
8.77/3.50	
8.77/3.50	Pi-finite rewrite system:
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	
8.77/3.50	
8.77/3.50	
8.77/3.50	
8.77/3.50	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
8.77/3.50	
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(2)
8.77/3.50	Obligation:
8.77/3.50	Pi-finite rewrite system:
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(3) DependencyPairsProof (EQUIVALENT)
8.77/3.50	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   QSORT_IN_GA(.(H, L), S) -> U1_GA(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   QSORT_IN_GA(.(H, L), S) -> SPLIT_IN_GGAA(L, H, A, B)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> LE_IN_GG(X, Y)
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y))
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
8.77/3.50	   U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> GT_IN_GG(X, Y)
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y))
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
8.77/3.50	   U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	   U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_GA(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> QSORT_IN_GA(A, A1)
8.77/3.50	   U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> U3_GA(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> QSORT_IN_GA(B, B1)
8.77/3.50	   U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_GA(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> APPEND_IN_GGA(A1, .(H, B1), S)
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U9_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	QSORT_IN_GA(x1, x2)  =  QSORT_IN_GA(x1)
8.77/3.50	
8.77/3.50	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
8.77/3.50	
8.77/3.50	SPLIT_IN_GGAA(x1, x2, x3, x4)  =  SPLIT_IN_GGAA(x1, x2)
8.77/3.50	
8.77/3.50	U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
8.77/3.50	
8.77/3.50	U11_GG(x1, x2, x3)  =  U11_GG(x3)
8.77/3.50	
8.77/3.50	U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x6)
8.77/3.50	
8.77/3.50	U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
8.77/3.50	
8.77/3.50	U10_GG(x1, x2, x3)  =  U10_GG(x3)
8.77/3.50	
8.77/3.50	U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x6)
8.77/3.50	
8.77/3.50	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
8.77/3.50	
8.77/3.50	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
8.77/3.50	
8.77/3.50	U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x5)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(4)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   QSORT_IN_GA(.(H, L), S) -> U1_GA(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   QSORT_IN_GA(.(H, L), S) -> SPLIT_IN_GGAA(L, H, A, B)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> LE_IN_GG(X, Y)
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y))
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
8.77/3.50	   U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> GT_IN_GG(X, Y)
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y))
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
8.77/3.50	   U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	   U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_GA(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> QSORT_IN_GA(A, A1)
8.77/3.50	   U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> U3_GA(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> QSORT_IN_GA(B, B1)
8.77/3.50	   U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_GA(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> APPEND_IN_GGA(A1, .(H, B1), S)
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U9_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	QSORT_IN_GA(x1, x2)  =  QSORT_IN_GA(x1)
8.77/3.50	
8.77/3.50	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
8.77/3.50	
8.77/3.50	SPLIT_IN_GGAA(x1, x2, x3, x4)  =  SPLIT_IN_GGAA(x1, x2)
8.77/3.50	
8.77/3.50	U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
8.77/3.50	
8.77/3.50	U11_GG(x1, x2, x3)  =  U11_GG(x3)
8.77/3.50	
8.77/3.50	U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x6)
8.77/3.50	
8.77/3.50	U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
8.77/3.50	
8.77/3.50	U10_GG(x1, x2, x3)  =  U10_GG(x3)
8.77/3.50	
8.77/3.50	U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x6)
8.77/3.50	
8.77/3.50	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
8.77/3.50	
8.77/3.50	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
8.77/3.50	
8.77/3.50	U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x5)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(5) DependencyGraphProof (EQUIVALENT)
8.77/3.50	The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 11 less nodes.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(6)
8.77/3.50	Complex Obligation (AND)
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(7)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(8) UsableRulesProof (EQUIVALENT)
8.77/3.50	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(9)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
8.77/3.50	
8.77/3.50	R is empty.
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(10) PiDPToQDPProof (SOUND)
8.77/3.50	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(11)
8.77/3.50	Obligation:
8.77/3.50	Q DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2)
8.77/3.50	
8.77/3.50	R is empty.
8.77/3.50	Q is empty.
8.77/3.50	We have to consider all (P,Q,R)-chains.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(12) QDPSizeChangeProof (EQUIVALENT)
8.77/3.50	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. 
8.77/3.50	
8.77/3.50	From the DPs we obtained the following set of size-change graphs:
8.77/3.50	*APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2)
8.77/3.50	The graph contains the following edges 1 > 1, 2 >= 2
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(13)
8.77/3.50	YES
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(14)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(15) UsableRulesProof (EQUIVALENT)
8.77/3.50	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(16)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
8.77/3.50	
8.77/3.50	R is empty.
8.77/3.50	Pi is empty.
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(17) PiDPToQDPProof (EQUIVALENT)
8.77/3.50	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(18)
8.77/3.50	Obligation:
8.77/3.50	Q DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
8.77/3.50	
8.77/3.50	R is empty.
8.77/3.50	Q is empty.
8.77/3.50	We have to consider all (P,Q,R)-chains.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(19) QDPSizeChangeProof (EQUIVALENT)
8.77/3.50	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. 
8.77/3.50	
8.77/3.50	From the DPs we obtained the following set of size-change graphs:
8.77/3.50	*GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
8.77/3.50	The graph contains the following edges 1 > 1, 2 > 2
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(20)
8.77/3.50	YES
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(21)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(22) UsableRulesProof (EQUIVALENT)
8.77/3.50	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(23)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
8.77/3.50	
8.77/3.50	R is empty.
8.77/3.50	Pi is empty.
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(24) PiDPToQDPProof (EQUIVALENT)
8.77/3.50	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(25)
8.77/3.50	Obligation:
8.77/3.50	Q DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
8.77/3.50	
8.77/3.50	R is empty.
8.77/3.50	Q is empty.
8.77/3.50	We have to consider all (P,Q,R)-chains.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(26) QDPSizeChangeProof (EQUIVALENT)
8.77/3.50	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. 
8.77/3.50	
8.77/3.50	From the DPs we obtained the following set of size-change graphs:
8.77/3.50	*LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
8.77/3.50	The graph contains the following edges 1 > 1, 2 > 2
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(27)
8.77/3.50	YES
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(28)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	SPLIT_IN_GGAA(x1, x2, x3, x4)  =  SPLIT_IN_GGAA(x1, x2)
8.77/3.50	
8.77/3.50	U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(29) UsableRulesProof (EQUIVALENT)
8.77/3.50	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(30)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	SPLIT_IN_GGAA(x1, x2, x3, x4)  =  SPLIT_IN_GGAA(x1, x2)
8.77/3.50	
8.77/3.50	U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(31) PiDPToQDPProof (SOUND)
8.77/3.50	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(32)
8.77/3.50	Obligation:
8.77/3.50	Q DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   U5_GGAA(X, Xs, Y, le_out_gg) -> SPLIT_IN_GGAA(Xs, Y)
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y) -> U5_GGAA(X, Xs, Y, le_in_gg(X, Y))
8.77/3.50	   SPLIT_IN_GGAA(.(X, Xs), Y) -> U7_GGAA(X, Xs, Y, gt_in_gg(X, Y))
8.77/3.50	   U7_GGAA(X, Xs, Y, gt_out_gg) -> SPLIT_IN_GGAA(Xs, Y)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg
8.77/3.50	   U11_gg(le_out_gg) -> le_out_gg
8.77/3.50	   U10_gg(gt_out_gg) -> gt_out_gg
8.77/3.50	
8.77/3.50	The set Q consists of the following terms:
8.77/3.50	
8.77/3.50	   le_in_gg(x0, x1)
8.77/3.50	   gt_in_gg(x0, x1)
8.77/3.50	   U11_gg(x0)
8.77/3.50	   U10_gg(x0)
8.77/3.50	
8.77/3.50	We have to consider all (P,Q,R)-chains.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(33) QDPSizeChangeProof (EQUIVALENT)
8.77/3.50	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. 
8.77/3.50	
8.77/3.50	From the DPs we obtained the following set of size-change graphs:
8.77/3.50	*SPLIT_IN_GGAA(.(X, Xs), Y) -> U5_GGAA(X, Xs, Y, le_in_gg(X, Y))
8.77/3.50	The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
8.77/3.50	
8.77/3.50	
8.77/3.50	*SPLIT_IN_GGAA(.(X, Xs), Y) -> U7_GGAA(X, Xs, Y, gt_in_gg(X, Y))
8.77/3.50	The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
8.77/3.50	
8.77/3.50	
8.77/3.50	*U5_GGAA(X, Xs, Y, le_out_gg) -> SPLIT_IN_GGAA(Xs, Y)
8.77/3.50	The graph contains the following edges 2 >= 1, 3 >= 2
8.77/3.50	
8.77/3.50	
8.77/3.50	*U7_GGAA(X, Xs, Y, gt_out_gg) -> SPLIT_IN_GGAA(Xs, Y)
8.77/3.50	The graph contains the following edges 2 >= 1, 3 >= 2
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(34)
8.77/3.50	YES
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(35)
8.77/3.50	Obligation:
8.77/3.50	Pi DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_GA(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> QSORT_IN_GA(B, B1)
8.77/3.50	   QSORT_IN_GA(.(H, L), S) -> U1_GA(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> QSORT_IN_GA(A, A1)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([], []) -> qsort_out_ga([], [])
8.77/3.50	   qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B))
8.77/3.50	   split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, Ls, Bs, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y))
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg(0, 0)
8.77/3.50	   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
8.77/3.50	   U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0)
8.77/3.50	   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
8.77/3.50	   U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs))
8.77/3.50	   U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs)
8.77/3.50	   U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1))
8.77/3.50	   U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1))
8.77/3.50	   U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S))
8.77/3.50	   append_in_gga([], L, L) -> append_out_gga([], L, L)
8.77/3.50	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
8.77/3.50	   U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
8.77/3.50	   U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S)
8.77/3.50	
8.77/3.50	The argument filtering Pi contains the following mapping:
8.77/3.50	qsort_in_ga(x1, x2)  =  qsort_in_ga(x1)
8.77/3.50	
8.77/3.50	[]  =  []
8.77/3.50	
8.77/3.50	qsort_out_ga(x1, x2)  =  qsort_out_ga(x2)
8.77/3.50	
8.77/3.50	.(x1, x2)  =  .(x1, x2)
8.77/3.50	
8.77/3.50	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
8.77/3.50	
8.77/3.50	split_in_ggaa(x1, x2, x3, x4)  =  split_in_ggaa(x1, x2)
8.77/3.50	
8.77/3.50	split_out_ggaa(x1, x2, x3, x4)  =  split_out_ggaa(x3, x4)
8.77/3.50	
8.77/3.50	U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	s(x1)  =  s(x1)
8.77/3.50	
8.77/3.50	U11_gg(x1, x2, x3)  =  U11_gg(x3)
8.77/3.50	
8.77/3.50	0  =  0
8.77/3.50	
8.77/3.50	le_out_gg(x1, x2)  =  le_out_gg
8.77/3.50	
8.77/3.50	U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
8.77/3.50	
8.77/3.50	gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
8.77/3.50	
8.77/3.50	U10_gg(x1, x2, x3)  =  U10_gg(x3)
8.77/3.50	
8.77/3.50	gt_out_gg(x1, x2)  =  gt_out_gg
8.77/3.50	
8.77/3.50	U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x6)
8.77/3.50	
8.77/3.50	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
8.77/3.50	
8.77/3.50	U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
8.77/3.50	
8.77/3.50	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
8.77/3.50	
8.77/3.50	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
8.77/3.50	
8.77/3.50	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
8.77/3.50	
8.77/3.50	QSORT_IN_GA(x1, x2)  =  QSORT_IN_GA(x1)
8.77/3.50	
8.77/3.50	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
8.77/3.50	
8.77/3.50	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
8.77/3.50	
8.77/3.50	
8.77/3.50	We have to consider all (P,R,Pi)-chains
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(36) PiDPToQDPProof (SOUND)
8.77/3.50	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(37)
8.77/3.50	Obligation:
8.77/3.50	Q DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   U1_GA(H, split_out_ggaa(A, B)) -> U2_GA(H, B, qsort_in_ga(A))
8.77/3.50	   U2_GA(H, B, qsort_out_ga(A1)) -> QSORT_IN_GA(B)
8.77/3.50	   QSORT_IN_GA(.(H, L)) -> U1_GA(H, split_in_ggaa(L, H))
8.77/3.50	   U1_GA(H, split_out_ggaa(A, B)) -> QSORT_IN_GA(A)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([]) -> qsort_out_ga([])
8.77/3.50	   qsort_in_ga(.(H, L)) -> U1_ga(H, split_in_ggaa(L, H))
8.77/3.50	   split_in_ggaa([], Y) -> split_out_ggaa([], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y) -> U5_ggaa(X, Xs, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg
8.77/3.50	   U11_gg(le_out_gg) -> le_out_gg
8.77/3.50	   U5_ggaa(X, Xs, Y, le_out_gg) -> U6_ggaa(X, split_in_ggaa(Xs, Y))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y) -> U7_ggaa(X, Xs, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg
8.77/3.50	   U10_gg(gt_out_gg) -> gt_out_gg
8.77/3.50	   U7_ggaa(X, Xs, Y, gt_out_gg) -> U8_ggaa(X, split_in_ggaa(Xs, Y))
8.77/3.50	   U8_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(.(X, Ls), Bs)
8.77/3.50	   U1_ga(H, split_out_ggaa(A, B)) -> U2_ga(H, B, qsort_in_ga(A))
8.77/3.50	   U2_ga(H, B, qsort_out_ga(A1)) -> U3_ga(H, A1, qsort_in_ga(B))
8.77/3.50	   U3_ga(H, A1, qsort_out_ga(B1)) -> U4_ga(append_in_gga(A1, .(H, B1)))
8.77/3.50	   append_in_gga([], L) -> append_out_gga(L)
8.77/3.50	   append_in_gga(.(H, L1), L2) -> U9_gga(H, append_in_gga(L1, L2))
8.77/3.50	   U9_gga(H, append_out_gga(L3)) -> append_out_gga(.(H, L3))
8.77/3.50	   U4_ga(append_out_gga(S)) -> qsort_out_ga(S)
8.77/3.50	
8.77/3.50	The set Q consists of the following terms:
8.77/3.50	
8.77/3.50	   qsort_in_ga(x0)
8.77/3.50	   split_in_ggaa(x0, x1)
8.77/3.50	   le_in_gg(x0, x1)
8.77/3.50	   U11_gg(x0)
8.77/3.50	   U5_ggaa(x0, x1, x2, x3)
8.77/3.50	   gt_in_gg(x0, x1)
8.77/3.50	   U10_gg(x0)
8.77/3.50	   U7_ggaa(x0, x1, x2, x3)
8.77/3.50	   U8_ggaa(x0, x1)
8.77/3.50	   U6_ggaa(x0, x1)
8.77/3.50	   U1_ga(x0, x1)
8.77/3.50	   U2_ga(x0, x1, x2)
8.77/3.50	   U3_ga(x0, x1, x2)
8.77/3.50	   append_in_gga(x0, x1)
8.77/3.50	   U9_gga(x0, x1)
8.77/3.50	   U4_ga(x0)
8.77/3.50	
8.77/3.50	We have to consider all (P,Q,R)-chains.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(38) QDPOrderProof (EQUIVALENT)
8.77/3.50	We use the reduction pair processor [LPAR04,JAR06].
8.77/3.50	
8.77/3.50	
8.77/3.50	The following pairs can be oriented strictly and are deleted.
8.77/3.50	
8.77/3.50	   U1_GA(H, split_out_ggaa(A, B)) -> U2_GA(H, B, qsort_in_ga(A))
8.77/3.50	   QSORT_IN_GA(.(H, L)) -> U1_GA(H, split_in_ggaa(L, H))
8.77/3.50	   U1_GA(H, split_out_ggaa(A, B)) -> QSORT_IN_GA(A)
8.77/3.50	The remaining pairs can at least be oriented weakly.
8.77/3.50	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
8.77/3.50	
8.77/3.50	POL( U2_GA_3(x_1, ..., x_3) ) = 2x_2
8.77/3.50	POL( qsort_in_ga_1(x_1) ) = 2
8.77/3.50	POL( [] ) = 1
8.77/3.50	POL( qsort_out_ga_1(x_1) ) = max{0, x_1 - 2}
8.77/3.50	POL( ._2(x_1, x_2) ) = x_2 + 2
8.77/3.50	POL( U1_ga_2(x_1, x_2) ) = max{0, x_1 - 2}
8.77/3.50	POL( split_in_ggaa_2(x_1, x_2) ) = x_1 + 2
8.77/3.50	POL( U1_GA_2(x_1, x_2) ) = max{0, 2x_2 - 1}
8.77/3.50	POL( split_out_ggaa_2(x_1, x_2) ) = x_1 + x_2 + 1
8.77/3.50	POL( U5_ggaa_4(x_1, ..., x_4) ) = x_2 + 2x_4
8.77/3.50	POL( le_in_gg_2(x_1, x_2) ) = 2
8.77/3.50	POL( U7_ggaa_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2
8.77/3.50	POL( gt_in_gg_2(x_1, x_2) ) = 1
8.77/3.50	POL( U2_ga_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
8.77/3.50	POL( U3_ga_3(x_1, ..., x_3) ) = 2x_2 + 1
8.77/3.50	POL( U4_ga_1(x_1) ) = 2
8.77/3.50	POL( append_in_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2}
8.77/3.50	POL( s_1(x_1) ) = 0
8.77/3.50	POL( U11_gg_1(x_1) ) = x_1
8.77/3.50	POL( 0 ) = 0
8.77/3.50	POL( le_out_gg ) = 2
8.77/3.50	POL( U6_ggaa_2(x_1, x_2) ) = x_2 + 2
8.77/3.50	POL( U10_gg_1(x_1) ) = x_1
8.77/3.50	POL( gt_out_gg ) = 1
8.77/3.50	POL( U8_ggaa_2(x_1, x_2) ) = x_2 + 2
8.77/3.50	POL( append_out_gga_1(x_1) ) = max{0, 2x_1 - 2}
8.77/3.50	POL( U9_gga_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
8.77/3.50	POL( QSORT_IN_GA_1(x_1) ) = 2x_1
8.77/3.50	
8.77/3.50	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
8.77/3.50	
8.77/3.50	   split_in_ggaa([], Y) -> split_out_ggaa([], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y) -> U5_ggaa(X, Xs, Y, le_in_gg(X, Y))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y) -> U7_ggaa(X, Xs, Y, gt_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg
8.77/3.50	   U5_ggaa(X, Xs, Y, le_out_gg) -> U6_ggaa(X, split_in_ggaa(Xs, Y))
8.77/3.50	   U6_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(.(X, Ls), Bs)
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg
8.77/3.50	   U7_ggaa(X, Xs, Y, gt_out_gg) -> U8_ggaa(X, split_in_ggaa(Xs, Y))
8.77/3.50	   U8_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(Ls, .(X, Bs))
8.77/3.50	   U11_gg(le_out_gg) -> le_out_gg
8.77/3.50	   U10_gg(gt_out_gg) -> gt_out_gg
8.77/3.50	
8.77/3.50	
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(39)
8.77/3.50	Obligation:
8.77/3.50	Q DP problem:
8.77/3.50	The TRS P consists of the following rules:
8.77/3.50	
8.77/3.50	   U2_GA(H, B, qsort_out_ga(A1)) -> QSORT_IN_GA(B)
8.77/3.50	
8.77/3.50	The TRS R consists of the following rules:
8.77/3.50	
8.77/3.50	   qsort_in_ga([]) -> qsort_out_ga([])
8.77/3.50	   qsort_in_ga(.(H, L)) -> U1_ga(H, split_in_ggaa(L, H))
8.77/3.50	   split_in_ggaa([], Y) -> split_out_ggaa([], [])
8.77/3.50	   split_in_ggaa(.(X, Xs), Y) -> U5_ggaa(X, Xs, Y, le_in_gg(X, Y))
8.77/3.50	   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
8.77/3.50	   le_in_gg(0, s(Y)) -> le_out_gg
8.77/3.50	   le_in_gg(0, 0) -> le_out_gg
8.77/3.50	   U11_gg(le_out_gg) -> le_out_gg
8.77/3.50	   U5_ggaa(X, Xs, Y, le_out_gg) -> U6_ggaa(X, split_in_ggaa(Xs, Y))
8.77/3.50	   split_in_ggaa(.(X, Xs), Y) -> U7_ggaa(X, Xs, Y, gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
8.77/3.50	   gt_in_gg(s(X), 0) -> gt_out_gg
8.77/3.50	   U10_gg(gt_out_gg) -> gt_out_gg
8.77/3.50	   U7_ggaa(X, Xs, Y, gt_out_gg) -> U8_ggaa(X, split_in_ggaa(Xs, Y))
8.77/3.50	   U8_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(Ls, .(X, Bs))
8.77/3.50	   U6_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(.(X, Ls), Bs)
8.77/3.50	   U1_ga(H, split_out_ggaa(A, B)) -> U2_ga(H, B, qsort_in_ga(A))
8.77/3.50	   U2_ga(H, B, qsort_out_ga(A1)) -> U3_ga(H, A1, qsort_in_ga(B))
8.77/3.50	   U3_ga(H, A1, qsort_out_ga(B1)) -> U4_ga(append_in_gga(A1, .(H, B1)))
8.77/3.50	   append_in_gga([], L) -> append_out_gga(L)
8.77/3.50	   append_in_gga(.(H, L1), L2) -> U9_gga(H, append_in_gga(L1, L2))
8.77/3.50	   U9_gga(H, append_out_gga(L3)) -> append_out_gga(.(H, L3))
8.77/3.50	   U4_ga(append_out_gga(S)) -> qsort_out_ga(S)
8.77/3.50	
8.77/3.50	The set Q consists of the following terms:
8.77/3.50	
8.77/3.50	   qsort_in_ga(x0)
8.77/3.50	   split_in_ggaa(x0, x1)
8.77/3.50	   le_in_gg(x0, x1)
8.77/3.50	   U11_gg(x0)
8.77/3.50	   U5_ggaa(x0, x1, x2, x3)
8.77/3.50	   gt_in_gg(x0, x1)
8.77/3.50	   U10_gg(x0)
8.77/3.50	   U7_ggaa(x0, x1, x2, x3)
8.77/3.50	   U8_ggaa(x0, x1)
8.77/3.50	   U6_ggaa(x0, x1)
8.77/3.50	   U1_ga(x0, x1)
8.77/3.50	   U2_ga(x0, x1, x2)
8.77/3.50	   U3_ga(x0, x1, x2)
8.77/3.50	   append_in_gga(x0, x1)
8.77/3.50	   U9_gga(x0, x1)
8.77/3.50	   U4_ga(x0)
8.77/3.50	
8.77/3.50	We have to consider all (P,Q,R)-chains.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(40) DependencyGraphProof (EQUIVALENT)
8.77/3.50	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
8.77/3.50	----------------------------------------
8.77/3.50	
8.77/3.50	(41)
8.77/3.50	TRUE
8.89/3.60	EOF