5.23/2.09 YES 5.23/2.10 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.23/2.10 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.23/2.10 5.23/2.10 5.23/2.10 Left Termination of the query pattern 5.23/2.10 5.23/2.10 t(g) 5.23/2.10 5.23/2.10 w.r.t. the given Prolog program could successfully be proven: 5.23/2.10 5.23/2.10 (0) Prolog 5.23/2.10 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.23/2.10 (2) PiTRS 5.23/2.10 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.23/2.10 (4) PiDP 5.23/2.10 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.23/2.10 (6) AND 5.23/2.10 (7) PiDP 5.23/2.10 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.10 (9) PiDP 5.23/2.10 (10) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.10 (11) QDP 5.23/2.10 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.23/2.10 (13) YES 5.23/2.10 (14) PiDP 5.23/2.10 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.10 (16) PiDP 5.23/2.10 (17) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.10 (18) QDP 5.23/2.10 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.23/2.10 (20) YES 5.23/2.10 (21) PiDP 5.23/2.10 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.10 (23) PiDP 5.23/2.10 (24) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.10 (25) QDP 5.23/2.10 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.23/2.10 (27) YES 5.23/2.10 (28) PiDP 5.23/2.10 (29) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.10 (30) PiDP 5.23/2.10 (31) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.10 (32) QDP 5.23/2.10 (33) MRRProof [EQUIVALENT, 22 ms] 5.23/2.10 (34) QDP 5.23/2.10 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 5.23/2.10 (36) TRUE 5.23/2.10 5.23/2.10 5.23/2.10 ---------------------------------------- 5.23/2.10 5.23/2.10 (0) 5.23/2.10 Obligation: 5.23/2.10 Clauses: 5.23/2.10 5.23/2.10 t(N) :- ','(ll(N, Xs), ','(select(X1, Xs, Xs1), ','(ll(M, Xs1), t(M)))). 5.23/2.10 t(0). 5.23/2.10 ll(s(N), .(X, Xs)) :- ll(N, Xs). 5.23/2.10 ll(0, []). 5.23/2.10 select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys). 5.23/2.10 select(X, .(X, Xs), Xs). 5.23/2.10 5.23/2.10 5.23/2.10 Query: t(g) 5.23/2.10 ---------------------------------------- 5.23/2.10 5.23/2.10 (1) PrologToPiTRSProof (SOUND) 5.23/2.10 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.23/2.10 5.23/2.10 t_in_1: (b) 5.23/2.10 5.23/2.10 ll_in_2: (b,f) (f,b) 5.23/2.10 5.23/2.10 select_in_3: (f,b,f) 5.23/2.10 5.23/2.10 Transforming Prolog into the following Term Rewriting System: 5.23/2.10 5.23/2.10 Pi-finite rewrite system: 5.23/2.10 The TRS R consists of the following rules: 5.23/2.10 5.23/2.10 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.10 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.10 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.10 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.10 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.10 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.10 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.10 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.10 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.10 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.10 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.10 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.10 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.10 t_in_g(0) -> t_out_g(0) 5.23/2.10 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.10 5.23/2.10 The argument filtering Pi contains the following mapping: 5.23/2.10 t_in_g(x1) = t_in_g(x1) 5.23/2.10 5.23/2.10 U1_g(x1, x2) = U1_g(x2) 5.23/2.10 5.23/2.10 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.10 5.23/2.10 s(x1) = s(x1) 5.23/2.10 5.23/2.10 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.10 5.23/2.10 0 = 0 5.23/2.10 5.23/2.10 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.10 5.23/2.10 .(x1, x2) = .(x2) 5.23/2.10 5.23/2.10 U2_g(x1, x2) = U2_g(x2) 5.23/2.10 5.23/2.10 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.10 5.23/2.10 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.10 5.23/2.10 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.10 5.23/2.10 U3_g(x1, x2) = U3_g(x2) 5.23/2.10 5.23/2.10 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.10 5.23/2.10 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.10 5.23/2.10 [] = [] 5.23/2.10 5.23/2.10 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.10 5.23/2.10 U4_g(x1, x2) = U4_g(x2) 5.23/2.10 5.23/2.10 t_out_g(x1) = t_out_g 5.23/2.10 5.23/2.10 5.23/2.10 5.23/2.10 5.23/2.10 5.23/2.10 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.23/2.10 5.23/2.10 5.23/2.10 5.23/2.10 ---------------------------------------- 5.23/2.10 5.23/2.10 (2) 5.23/2.10 Obligation: 5.23/2.10 Pi-finite rewrite system: 5.23/2.10 The TRS R consists of the following rules: 5.23/2.10 5.23/2.10 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.10 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.10 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.10 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.10 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.10 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.10 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.10 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.10 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.10 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.10 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.10 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.10 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.10 t_in_g(0) -> t_out_g(0) 5.23/2.10 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.10 5.23/2.10 The argument filtering Pi contains the following mapping: 5.23/2.10 t_in_g(x1) = t_in_g(x1) 5.23/2.10 5.23/2.10 U1_g(x1, x2) = U1_g(x2) 5.23/2.10 5.23/2.10 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.10 5.23/2.10 s(x1) = s(x1) 5.23/2.10 5.23/2.10 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.10 5.23/2.10 0 = 0 5.23/2.10 5.23/2.10 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.10 5.23/2.10 .(x1, x2) = .(x2) 5.23/2.10 5.23/2.10 U2_g(x1, x2) = U2_g(x2) 5.23/2.10 5.23/2.10 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.10 5.23/2.10 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.10 5.23/2.10 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.10 5.23/2.10 U3_g(x1, x2) = U3_g(x2) 5.23/2.10 5.23/2.10 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.10 5.23/2.10 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.10 5.23/2.10 [] = [] 5.23/2.10 5.23/2.10 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.10 5.23/2.10 U4_g(x1, x2) = U4_g(x2) 5.23/2.10 5.23/2.10 t_out_g(x1) = t_out_g 5.23/2.10 5.23/2.10 5.23/2.10 5.23/2.10 ---------------------------------------- 5.23/2.10 5.23/2.10 (3) DependencyPairsProof (EQUIVALENT) 5.23/2.10 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.23/2.10 Pi DP problem: 5.23/2.10 The TRS P consists of the following rules: 5.23/2.10 5.23/2.10 T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) 5.23/2.10 T_IN_G(N) -> LL_IN_GA(N, Xs) 5.23/2.10 LL_IN_GA(s(N), .(X, Xs)) -> U5_GA(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.10 LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) 5.23/2.10 U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.10 U1_G(N, ll_out_ga(N, Xs)) -> SELECT_IN_AGA(X1, Xs, Xs1) 5.23/2.10 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> U6_AGA(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.10 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) 5.23/2.10 U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) 5.23/2.10 U2_G(N, select_out_aga(X1, Xs, Xs1)) -> LL_IN_AG(M, Xs1) 5.23/2.10 LL_IN_AG(s(N), .(X, Xs)) -> U5_AG(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.10 LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) 5.23/2.10 U3_G(N, ll_out_ag(M, Xs1)) -> U4_G(N, t_in_g(M)) 5.23/2.10 U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) 5.23/2.10 5.23/2.10 The TRS R consists of the following rules: 5.23/2.10 5.23/2.10 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.10 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.10 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.10 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.10 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.10 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.10 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.10 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.10 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.10 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.10 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.10 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.10 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.10 t_in_g(0) -> t_out_g(0) 5.23/2.10 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.10 5.23/2.10 The argument filtering Pi contains the following mapping: 5.23/2.10 t_in_g(x1) = t_in_g(x1) 5.23/2.10 5.23/2.10 U1_g(x1, x2) = U1_g(x2) 5.23/2.10 5.23/2.10 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.10 5.23/2.10 s(x1) = s(x1) 5.23/2.10 5.23/2.10 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.10 5.23/2.10 0 = 0 5.23/2.10 5.23/2.10 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.10 5.23/2.10 .(x1, x2) = .(x2) 5.23/2.10 5.23/2.10 U2_g(x1, x2) = U2_g(x2) 5.23/2.10 5.23/2.10 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.10 5.23/2.10 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.10 5.23/2.10 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.10 5.23/2.10 U3_g(x1, x2) = U3_g(x2) 5.23/2.10 5.23/2.10 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.10 5.23/2.10 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.10 5.23/2.10 [] = [] 5.23/2.10 5.23/2.10 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.10 5.23/2.10 U4_g(x1, x2) = U4_g(x2) 5.23/2.10 5.23/2.10 t_out_g(x1) = t_out_g 5.23/2.10 5.23/2.10 T_IN_G(x1) = T_IN_G(x1) 5.23/2.10 5.23/2.10 U1_G(x1, x2) = U1_G(x2) 5.23/2.10 5.23/2.10 LL_IN_GA(x1, x2) = LL_IN_GA(x1) 5.23/2.10 5.23/2.10 U5_GA(x1, x2, x3, x4) = U5_GA(x4) 5.23/2.10 5.23/2.10 U2_G(x1, x2) = U2_G(x2) 5.23/2.10 5.23/2.10 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 5.23/2.10 5.23/2.10 U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5) 5.23/2.10 5.23/2.10 U3_G(x1, x2) = U3_G(x2) 5.23/2.10 5.23/2.10 LL_IN_AG(x1, x2) = LL_IN_AG(x2) 5.23/2.10 5.23/2.10 U5_AG(x1, x2, x3, x4) = U5_AG(x4) 5.23/2.11 5.23/2.11 U4_G(x1, x2) = U4_G(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (4) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) 5.23/2.11 T_IN_G(N) -> LL_IN_GA(N, Xs) 5.23/2.11 LL_IN_GA(s(N), .(X, Xs)) -> U5_GA(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) 5.23/2.11 U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 U1_G(N, ll_out_ga(N, Xs)) -> SELECT_IN_AGA(X1, Xs, Xs1) 5.23/2.11 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> U6_AGA(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) 5.23/2.11 U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) 5.23/2.11 U2_G(N, select_out_aga(X1, Xs, Xs1)) -> LL_IN_AG(M, Xs1) 5.23/2.11 LL_IN_AG(s(N), .(X, Xs)) -> U5_AG(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) 5.23/2.11 U3_G(N, ll_out_ag(M, Xs1)) -> U4_G(N, t_in_g(M)) 5.23/2.11 U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.11 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.11 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.11 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.11 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.11 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.11 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.11 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.11 t_in_g(0) -> t_out_g(0) 5.23/2.11 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.11 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 t_in_g(x1) = t_in_g(x1) 5.23/2.11 5.23/2.11 U1_g(x1, x2) = U1_g(x2) 5.23/2.11 5.23/2.11 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.11 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.11 5.23/2.11 0 = 0 5.23/2.11 5.23/2.11 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 U2_g(x1, x2) = U2_g(x2) 5.23/2.11 5.23/2.11 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.11 5.23/2.11 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.11 5.23/2.11 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.11 5.23/2.11 U3_g(x1, x2) = U3_g(x2) 5.23/2.11 5.23/2.11 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.11 5.23/2.11 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.11 5.23/2.11 [] = [] 5.23/2.11 5.23/2.11 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.11 5.23/2.11 U4_g(x1, x2) = U4_g(x2) 5.23/2.11 5.23/2.11 t_out_g(x1) = t_out_g 5.23/2.11 5.23/2.11 T_IN_G(x1) = T_IN_G(x1) 5.23/2.11 5.23/2.11 U1_G(x1, x2) = U1_G(x2) 5.23/2.11 5.23/2.11 LL_IN_GA(x1, x2) = LL_IN_GA(x1) 5.23/2.11 5.23/2.11 U5_GA(x1, x2, x3, x4) = U5_GA(x4) 5.23/2.11 5.23/2.11 U2_G(x1, x2) = U2_G(x2) 5.23/2.11 5.23/2.11 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 5.23/2.11 5.23/2.11 U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5) 5.23/2.11 5.23/2.11 U3_G(x1, x2) = U3_G(x2) 5.23/2.11 5.23/2.11 LL_IN_AG(x1, x2) = LL_IN_AG(x2) 5.23/2.11 5.23/2.11 U5_AG(x1, x2, x3, x4) = U5_AG(x4) 5.23/2.11 5.23/2.11 U4_G(x1, x2) = U4_G(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (5) DependencyGraphProof (EQUIVALENT) 5.23/2.11 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 7 less nodes. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (6) 5.23/2.11 Complex Obligation (AND) 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (7) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.11 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.11 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.11 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.11 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.11 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.11 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.11 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.11 t_in_g(0) -> t_out_g(0) 5.23/2.11 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.11 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 t_in_g(x1) = t_in_g(x1) 5.23/2.11 5.23/2.11 U1_g(x1, x2) = U1_g(x2) 5.23/2.11 5.23/2.11 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.11 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.11 5.23/2.11 0 = 0 5.23/2.11 5.23/2.11 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 U2_g(x1, x2) = U2_g(x2) 5.23/2.11 5.23/2.11 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.11 5.23/2.11 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.11 5.23/2.11 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.11 5.23/2.11 U3_g(x1, x2) = U3_g(x2) 5.23/2.11 5.23/2.11 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.11 5.23/2.11 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.11 5.23/2.11 [] = [] 5.23/2.11 5.23/2.11 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.11 5.23/2.11 U4_g(x1, x2) = U4_g(x2) 5.23/2.11 5.23/2.11 t_out_g(x1) = t_out_g 5.23/2.11 5.23/2.11 LL_IN_AG(x1, x2) = LL_IN_AG(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (8) UsableRulesProof (EQUIVALENT) 5.23/2.11 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (9) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) 5.23/2.11 5.23/2.11 R is empty. 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 LL_IN_AG(x1, x2) = LL_IN_AG(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (10) PiDPToQDPProof (SOUND) 5.23/2.11 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (11) 5.23/2.11 Obligation: 5.23/2.11 Q DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 LL_IN_AG(.(Xs)) -> LL_IN_AG(Xs) 5.23/2.11 5.23/2.11 R is empty. 5.23/2.11 Q is empty. 5.23/2.11 We have to consider all (P,Q,R)-chains. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (12) QDPSizeChangeProof (EQUIVALENT) 5.23/2.11 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 5.23/2.11 5.23/2.11 From the DPs we obtained the following set of size-change graphs: 5.23/2.11 *LL_IN_AG(.(Xs)) -> LL_IN_AG(Xs) 5.23/2.11 The graph contains the following edges 1 > 1 5.23/2.11 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (13) 5.23/2.11 YES 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (14) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.11 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.11 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.11 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.11 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.11 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.11 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.11 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.11 t_in_g(0) -> t_out_g(0) 5.23/2.11 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.11 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 t_in_g(x1) = t_in_g(x1) 5.23/2.11 5.23/2.11 U1_g(x1, x2) = U1_g(x2) 5.23/2.11 5.23/2.11 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.11 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.11 5.23/2.11 0 = 0 5.23/2.11 5.23/2.11 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 U2_g(x1, x2) = U2_g(x2) 5.23/2.11 5.23/2.11 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.11 5.23/2.11 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.11 5.23/2.11 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.11 5.23/2.11 U3_g(x1, x2) = U3_g(x2) 5.23/2.11 5.23/2.11 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.11 5.23/2.11 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.11 5.23/2.11 [] = [] 5.23/2.11 5.23/2.11 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.11 5.23/2.11 U4_g(x1, x2) = U4_g(x2) 5.23/2.11 5.23/2.11 t_out_g(x1) = t_out_g 5.23/2.11 5.23/2.11 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (15) UsableRulesProof (EQUIVALENT) 5.23/2.11 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (16) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) 5.23/2.11 5.23/2.11 R is empty. 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (17) PiDPToQDPProof (SOUND) 5.23/2.11 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (18) 5.23/2.11 Obligation: 5.23/2.11 Q DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 SELECT_IN_AGA(.(Xs)) -> SELECT_IN_AGA(Xs) 5.23/2.11 5.23/2.11 R is empty. 5.23/2.11 Q is empty. 5.23/2.11 We have to consider all (P,Q,R)-chains. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (19) QDPSizeChangeProof (EQUIVALENT) 5.23/2.11 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 5.23/2.11 5.23/2.11 From the DPs we obtained the following set of size-change graphs: 5.23/2.11 *SELECT_IN_AGA(.(Xs)) -> SELECT_IN_AGA(Xs) 5.23/2.11 The graph contains the following edges 1 > 1 5.23/2.11 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (20) 5.23/2.11 YES 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (21) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.11 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.11 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.11 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.11 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.11 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.11 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.11 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.11 t_in_g(0) -> t_out_g(0) 5.23/2.11 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.11 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 t_in_g(x1) = t_in_g(x1) 5.23/2.11 5.23/2.11 U1_g(x1, x2) = U1_g(x2) 5.23/2.11 5.23/2.11 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.11 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.11 5.23/2.11 0 = 0 5.23/2.11 5.23/2.11 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 U2_g(x1, x2) = U2_g(x2) 5.23/2.11 5.23/2.11 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.11 5.23/2.11 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.11 5.23/2.11 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.11 5.23/2.11 U3_g(x1, x2) = U3_g(x2) 5.23/2.11 5.23/2.11 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.11 5.23/2.11 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.11 5.23/2.11 [] = [] 5.23/2.11 5.23/2.11 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.11 5.23/2.11 U4_g(x1, x2) = U4_g(x2) 5.23/2.11 5.23/2.11 t_out_g(x1) = t_out_g 5.23/2.11 5.23/2.11 LL_IN_GA(x1, x2) = LL_IN_GA(x1) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (22) UsableRulesProof (EQUIVALENT) 5.23/2.11 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (23) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) 5.23/2.11 5.23/2.11 R is empty. 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 LL_IN_GA(x1, x2) = LL_IN_GA(x1) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (24) PiDPToQDPProof (SOUND) 5.23/2.11 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (25) 5.23/2.11 Obligation: 5.23/2.11 Q DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 LL_IN_GA(s(N)) -> LL_IN_GA(N) 5.23/2.11 5.23/2.11 R is empty. 5.23/2.11 Q is empty. 5.23/2.11 We have to consider all (P,Q,R)-chains. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (26) QDPSizeChangeProof (EQUIVALENT) 5.23/2.11 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 5.23/2.11 5.23/2.11 From the DPs we obtained the following set of size-change graphs: 5.23/2.11 *LL_IN_GA(s(N)) -> LL_IN_GA(N) 5.23/2.11 The graph contains the following edges 1 > 1 5.23/2.11 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (27) 5.23/2.11 YES 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (28) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) 5.23/2.11 U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) 5.23/2.11 T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.11 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.11 U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.11 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.11 U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) 5.23/2.11 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.11 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.11 U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) 5.23/2.11 t_in_g(0) -> t_out_g(0) 5.23/2.11 U4_g(N, t_out_g(M)) -> t_out_g(N) 5.23/2.11 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 t_in_g(x1) = t_in_g(x1) 5.23/2.11 5.23/2.11 U1_g(x1, x2) = U1_g(x2) 5.23/2.11 5.23/2.11 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.11 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.11 5.23/2.11 0 = 0 5.23/2.11 5.23/2.11 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 U2_g(x1, x2) = U2_g(x2) 5.23/2.11 5.23/2.11 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.11 5.23/2.11 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.11 5.23/2.11 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.11 5.23/2.11 U3_g(x1, x2) = U3_g(x2) 5.23/2.11 5.23/2.11 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.11 5.23/2.11 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.11 5.23/2.11 [] = [] 5.23/2.11 5.23/2.11 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.11 5.23/2.11 U4_g(x1, x2) = U4_g(x2) 5.23/2.11 5.23/2.11 t_out_g(x1) = t_out_g 5.23/2.11 5.23/2.11 T_IN_G(x1) = T_IN_G(x1) 5.23/2.11 5.23/2.11 U1_G(x1, x2) = U1_G(x2) 5.23/2.11 5.23/2.11 U2_G(x1, x2) = U2_G(x2) 5.23/2.11 5.23/2.11 U3_G(x1, x2) = U3_G(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (29) UsableRulesProof (EQUIVALENT) 5.23/2.11 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (30) 5.23/2.11 Obligation: 5.23/2.11 Pi DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) 5.23/2.11 U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) 5.23/2.11 U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) 5.23/2.11 T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) 5.23/2.11 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 5.23/2.11 ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) 5.23/2.11 ll_in_ag(0, []) -> ll_out_ag(0, []) 5.23/2.11 ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) 5.23/2.11 ll_in_ga(0, []) -> ll_out_ga(0, []) 5.23/2.11 U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) 5.23/2.11 U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) 5.23/2.11 U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) 5.23/2.11 5.23/2.11 The argument filtering Pi contains the following mapping: 5.23/2.11 ll_in_ga(x1, x2) = ll_in_ga(x1) 5.23/2.11 5.23/2.11 s(x1) = s(x1) 5.23/2.11 5.23/2.11 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.23/2.11 5.23/2.11 0 = 0 5.23/2.11 5.23/2.11 ll_out_ga(x1, x2) = ll_out_ga(x2) 5.23/2.11 5.23/2.11 .(x1, x2) = .(x2) 5.23/2.11 5.23/2.11 select_in_aga(x1, x2, x3) = select_in_aga(x2) 5.23/2.11 5.23/2.11 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.23/2.11 5.23/2.11 select_out_aga(x1, x2, x3) = select_out_aga(x3) 5.23/2.11 5.23/2.11 ll_in_ag(x1, x2) = ll_in_ag(x2) 5.23/2.11 5.23/2.11 U5_ag(x1, x2, x3, x4) = U5_ag(x4) 5.23/2.11 5.23/2.11 [] = [] 5.23/2.11 5.23/2.11 ll_out_ag(x1, x2) = ll_out_ag(x1) 5.23/2.11 5.23/2.11 T_IN_G(x1) = T_IN_G(x1) 5.23/2.11 5.23/2.11 U1_G(x1, x2) = U1_G(x2) 5.23/2.11 5.23/2.11 U2_G(x1, x2) = U2_G(x2) 5.23/2.11 5.23/2.11 U3_G(x1, x2) = U3_G(x2) 5.23/2.11 5.23/2.11 5.23/2.11 We have to consider all (P,R,Pi)-chains 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (31) PiDPToQDPProof (SOUND) 5.23/2.11 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (32) 5.23/2.11 Obligation: 5.23/2.11 Q DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 U1_G(ll_out_ga(Xs)) -> U2_G(select_in_aga(Xs)) 5.23/2.11 U2_G(select_out_aga(Xs1)) -> U3_G(ll_in_ag(Xs1)) 5.23/2.11 U3_G(ll_out_ag(M)) -> T_IN_G(M) 5.23/2.11 T_IN_G(N) -> U1_G(ll_in_ga(N)) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 select_in_aga(.(Xs)) -> U6_aga(select_in_aga(Xs)) 5.23/2.11 select_in_aga(.(Xs)) -> select_out_aga(Xs) 5.23/2.11 ll_in_ag(.(Xs)) -> U5_ag(ll_in_ag(Xs)) 5.23/2.11 ll_in_ag([]) -> ll_out_ag(0) 5.23/2.11 ll_in_ga(s(N)) -> U5_ga(ll_in_ga(N)) 5.23/2.11 ll_in_ga(0) -> ll_out_ga([]) 5.23/2.11 U6_aga(select_out_aga(Ys)) -> select_out_aga(.(Ys)) 5.23/2.11 U5_ag(ll_out_ag(N)) -> ll_out_ag(s(N)) 5.23/2.11 U5_ga(ll_out_ga(Xs)) -> ll_out_ga(.(Xs)) 5.23/2.11 5.23/2.11 The set Q consists of the following terms: 5.23/2.11 5.23/2.11 select_in_aga(x0) 5.23/2.11 ll_in_ag(x0) 5.23/2.11 ll_in_ga(x0) 5.23/2.11 U6_aga(x0) 5.23/2.11 U5_ag(x0) 5.23/2.11 U5_ga(x0) 5.23/2.11 5.23/2.11 We have to consider all (P,Q,R)-chains. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (33) MRRProof (EQUIVALENT) 5.23/2.11 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. 5.23/2.11 5.23/2.11 Strictly oriented dependency pairs: 5.23/2.11 5.23/2.11 T_IN_G(N) -> U1_G(ll_in_ga(N)) 5.23/2.11 5.23/2.11 Strictly oriented rules of the TRS R: 5.23/2.11 5.23/2.11 select_in_aga(.(Xs)) -> select_out_aga(Xs) 5.23/2.11 U6_aga(select_out_aga(Ys)) -> select_out_aga(.(Ys)) 5.23/2.11 5.23/2.11 Used ordering: Polynomial interpretation [POLO]: 5.23/2.11 5.23/2.11 POL(.(x_1)) = 1 + 2*x_1 5.23/2.11 POL(0) = 0 5.23/2.11 POL(T_IN_G(x_1)) = 1 + 2*x_1 5.23/2.11 POL(U1_G(x_1)) = 2*x_1 5.23/2.11 POL(U2_G(x_1)) = x_1 5.23/2.11 POL(U3_G(x_1)) = 1 + 2*x_1 5.23/2.11 POL(U5_ag(x_1)) = 1 + 2*x_1 5.23/2.11 POL(U5_ga(x_1)) = 1 + 2*x_1 5.23/2.11 POL(U6_aga(x_1)) = 2 + 2*x_1 5.23/2.11 POL([]) = 0 5.23/2.11 POL(ll_in_ag(x_1)) = x_1 5.23/2.11 POL(ll_in_ga(x_1)) = x_1 5.23/2.11 POL(ll_out_ag(x_1)) = x_1 5.23/2.11 POL(ll_out_ga(x_1)) = x_1 5.23/2.11 POL(s(x_1)) = 1 + 2*x_1 5.23/2.11 POL(select_in_aga(x_1)) = 2*x_1 5.23/2.11 POL(select_out_aga(x_1)) = 1 + 2*x_1 5.23/2.11 5.23/2.11 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (34) 5.23/2.11 Obligation: 5.23/2.11 Q DP problem: 5.23/2.11 The TRS P consists of the following rules: 5.23/2.11 5.23/2.11 U1_G(ll_out_ga(Xs)) -> U2_G(select_in_aga(Xs)) 5.23/2.11 U2_G(select_out_aga(Xs1)) -> U3_G(ll_in_ag(Xs1)) 5.23/2.11 U3_G(ll_out_ag(M)) -> T_IN_G(M) 5.23/2.11 5.23/2.11 The TRS R consists of the following rules: 5.23/2.11 5.23/2.11 select_in_aga(.(Xs)) -> U6_aga(select_in_aga(Xs)) 5.23/2.11 ll_in_ag(.(Xs)) -> U5_ag(ll_in_ag(Xs)) 5.23/2.11 ll_in_ag([]) -> ll_out_ag(0) 5.23/2.11 ll_in_ga(s(N)) -> U5_ga(ll_in_ga(N)) 5.23/2.11 ll_in_ga(0) -> ll_out_ga([]) 5.23/2.11 U5_ag(ll_out_ag(N)) -> ll_out_ag(s(N)) 5.23/2.11 U5_ga(ll_out_ga(Xs)) -> ll_out_ga(.(Xs)) 5.23/2.11 5.23/2.11 The set Q consists of the following terms: 5.23/2.11 5.23/2.11 select_in_aga(x0) 5.23/2.11 ll_in_ag(x0) 5.23/2.11 ll_in_ga(x0) 5.23/2.11 U6_aga(x0) 5.23/2.11 U5_ag(x0) 5.23/2.11 U5_ga(x0) 5.23/2.11 5.23/2.11 We have to consider all (P,Q,R)-chains. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (35) DependencyGraphProof (EQUIVALENT) 5.23/2.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.23/2.11 ---------------------------------------- 5.23/2.11 5.23/2.11 (36) 5.23/2.11 TRUE 5.23/2.13 EOF