5.80/2.38 YES 5.80/2.39 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.80/2.39 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.80/2.39 5.80/2.39 5.80/2.39 Left Termination of the query pattern 5.80/2.39 5.80/2.39 conf(g) 5.80/2.39 5.80/2.39 w.r.t. the given Prolog program could successfully be proven: 5.80/2.39 5.80/2.39 (0) Prolog 5.80/2.39 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.80/2.39 (2) PiTRS 5.80/2.39 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.80/2.39 (4) PiDP 5.80/2.39 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.80/2.39 (6) AND 5.80/2.39 (7) PiDP 5.80/2.39 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.80/2.39 (9) PiDP 5.80/2.39 (10) PiDPToQDPProof [SOUND, 0 ms] 5.80/2.39 (11) QDP 5.80/2.39 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.80/2.39 (13) YES 5.80/2.39 (14) PiDP 5.80/2.39 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.80/2.39 (16) PiDP 5.80/2.39 (17) PiDPToQDPProof [SOUND, 0 ms] 5.80/2.39 (18) QDP 5.80/2.39 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.80/2.39 (20) YES 5.80/2.39 (21) PiDP 5.80/2.39 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.80/2.39 (23) PiDP 5.80/2.39 (24) PiDPToQDPProof [SOUND, 0 ms] 5.80/2.39 (25) QDP 5.80/2.39 (26) MRRProof [EQUIVALENT, 19 ms] 5.80/2.39 (27) QDP 5.80/2.39 (28) DependencyGraphProof [EQUIVALENT, 0 ms] 5.80/2.39 (29) TRUE 5.80/2.39 5.80/2.39 5.80/2.39 ---------------------------------------- 5.80/2.39 5.80/2.39 (0) 5.80/2.39 Obligation: 5.80/2.39 Clauses: 5.80/2.39 5.80/2.39 conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))). 5.80/2.39 del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)). 5.80/2.39 del(X, .(X, T), T). 5.80/2.39 del(X, .(H, T), .(H, T1)) :- del(X, T, T1). 5.80/2.39 s2l(s(X), .(Y, Xs)) :- s2l(X, Xs). 5.80/2.39 s2l(0, []). 5.80/2.39 goal(X) :- ','(s2l(X, XS), conf(XS)). 5.80/2.39 5.80/2.39 5.80/2.39 Query: conf(g) 5.80/2.39 ---------------------------------------- 5.80/2.39 5.80/2.39 (1) PrologToPiTRSProof (SOUND) 5.80/2.39 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.80/2.39 5.80/2.39 conf_in_1: (b) 5.80/2.39 5.80/2.39 del2_in_2: (b,f) 5.80/2.39 5.80/2.39 del_in_3: (f,b,f) (f,f,b) 5.80/2.39 5.80/2.39 Transforming Prolog into the following Term Rewriting System: 5.80/2.39 5.80/2.39 Pi-finite rewrite system: 5.80/2.39 The TRS R consists of the following rules: 5.80/2.39 5.80/2.39 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.39 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.39 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.39 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.39 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.39 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.39 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.39 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.39 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.39 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.39 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.39 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.39 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.39 5.80/2.39 The argument filtering Pi contains the following mapping: 5.80/2.39 conf_in_g(x1) = conf_in_g(x1) 5.80/2.39 5.80/2.39 U1_g(x1, x2) = U1_g(x2) 5.80/2.39 5.80/2.39 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.39 5.80/2.39 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.39 5.80/2.39 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.39 5.80/2.39 .(x1, x2) = .(x2) 5.80/2.39 5.80/2.39 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.39 5.80/2.39 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.39 5.80/2.39 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.39 5.80/2.39 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.39 5.80/2.39 U2_g(x1, x2) = U2_g(x2) 5.80/2.39 5.80/2.39 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.39 5.80/2.39 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.39 5.80/2.39 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.39 5.80/2.39 U3_g(x1, x2) = U3_g(x2) 5.80/2.39 5.80/2.39 conf_out_g(x1) = conf_out_g 5.80/2.39 5.80/2.39 5.80/2.39 5.80/2.39 5.80/2.39 5.80/2.39 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.80/2.39 5.80/2.39 5.80/2.39 5.80/2.39 ---------------------------------------- 5.80/2.39 5.80/2.39 (2) 5.80/2.39 Obligation: 5.80/2.39 Pi-finite rewrite system: 5.80/2.39 The TRS R consists of the following rules: 5.80/2.39 5.80/2.39 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.39 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.39 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.39 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.39 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.39 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.39 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.39 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.39 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.39 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.39 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.39 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.39 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.39 5.80/2.39 The argument filtering Pi contains the following mapping: 5.80/2.39 conf_in_g(x1) = conf_in_g(x1) 5.80/2.39 5.80/2.39 U1_g(x1, x2) = U1_g(x2) 5.80/2.39 5.80/2.39 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.39 5.80/2.39 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.39 5.80/2.39 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.39 5.80/2.39 .(x1, x2) = .(x2) 5.80/2.39 5.80/2.39 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.39 5.80/2.39 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.39 5.80/2.39 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.39 5.80/2.39 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.39 5.80/2.39 U2_g(x1, x2) = U2_g(x2) 5.80/2.39 5.80/2.39 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.39 5.80/2.39 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.39 5.80/2.39 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.39 5.80/2.39 U3_g(x1, x2) = U3_g(x2) 5.80/2.39 5.80/2.39 conf_out_g(x1) = conf_out_g 5.80/2.39 5.80/2.39 5.80/2.39 5.80/2.39 ---------------------------------------- 5.80/2.39 5.80/2.39 (3) DependencyPairsProof (EQUIVALENT) 5.80/2.39 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.80/2.39 Pi DP problem: 5.80/2.39 The TRS P consists of the following rules: 5.80/2.39 5.80/2.39 CONF_IN_G(X) -> U1_G(X, del2_in_ga(X, Z)) 5.80/2.39 CONF_IN_G(X) -> DEL2_IN_GA(X, Z) 5.80/2.39 DEL2_IN_GA(X, Y) -> U4_GA(X, Y, del_in_aga(U, X, Z)) 5.80/2.39 DEL2_IN_GA(X, Y) -> DEL_IN_AGA(U, X, Z) 5.80/2.39 DEL_IN_AGA(X, .(H, T), .(H, T1)) -> U6_AGA(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.39 DEL_IN_AGA(X, .(H, T), .(H, T1)) -> DEL_IN_AGA(X, T, T1) 5.80/2.39 U4_GA(X, Y, del_out_aga(U, X, Z)) -> U5_GA(X, Y, del_in_aga(V, Z, Y)) 5.80/2.39 U4_GA(X, Y, del_out_aga(U, X, Z)) -> DEL_IN_AGA(V, Z, Y) 5.80/2.39 U1_G(X, del2_out_ga(X, Z)) -> U2_G(X, del_in_aag(U, Y, Z)) 5.80/2.39 U1_G(X, del2_out_ga(X, Z)) -> DEL_IN_AAG(U, Y, Z) 5.80/2.39 DEL_IN_AAG(X, .(H, T), .(H, T1)) -> U6_AAG(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.39 DEL_IN_AAG(X, .(H, T), .(H, T1)) -> DEL_IN_AAG(X, T, T1) 5.80/2.39 U2_G(X, del_out_aag(U, Y, Z)) -> U3_G(X, conf_in_g(Y)) 5.80/2.39 U2_G(X, del_out_aag(U, Y, Z)) -> CONF_IN_G(Y) 5.80/2.39 5.80/2.39 The TRS R consists of the following rules: 5.80/2.39 5.80/2.39 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.39 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.39 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.39 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.39 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.39 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.39 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.39 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.39 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.39 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.39 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.39 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.39 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.39 5.80/2.39 The argument filtering Pi contains the following mapping: 5.80/2.39 conf_in_g(x1) = conf_in_g(x1) 5.80/2.39 5.80/2.39 U1_g(x1, x2) = U1_g(x2) 5.80/2.39 5.80/2.39 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.39 5.80/2.39 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.39 5.80/2.39 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.39 5.80/2.39 .(x1, x2) = .(x2) 5.80/2.39 5.80/2.39 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.39 5.80/2.39 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.39 5.80/2.39 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.39 5.80/2.39 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.39 5.80/2.39 U2_g(x1, x2) = U2_g(x2) 5.80/2.39 5.80/2.39 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.39 5.80/2.39 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.39 5.80/2.39 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.39 5.80/2.39 U3_g(x1, x2) = U3_g(x2) 5.80/2.39 5.80/2.39 conf_out_g(x1) = conf_out_g 5.80/2.39 5.80/2.39 CONF_IN_G(x1) = CONF_IN_G(x1) 5.80/2.39 5.80/2.39 U1_G(x1, x2) = U1_G(x2) 5.80/2.39 5.80/2.39 DEL2_IN_GA(x1, x2) = DEL2_IN_GA(x1) 5.80/2.39 5.80/2.39 U4_GA(x1, x2, x3) = U4_GA(x3) 5.80/2.39 5.80/2.39 DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) 5.80/2.39 5.80/2.39 U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5) 5.80/2.39 5.80/2.39 U5_GA(x1, x2, x3) = U5_GA(x3) 5.80/2.39 5.80/2.39 U2_G(x1, x2) = U2_G(x2) 5.80/2.39 5.80/2.39 DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3) 5.80/2.39 5.80/2.39 U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x5) 5.80/2.39 5.80/2.39 U3_G(x1, x2) = U3_G(x2) 5.80/2.39 5.80/2.39 5.80/2.39 We have to consider all (P,R,Pi)-chains 5.80/2.39 ---------------------------------------- 5.80/2.39 5.80/2.39 (4) 5.80/2.39 Obligation: 5.80/2.39 Pi DP problem: 5.80/2.39 The TRS P consists of the following rules: 5.80/2.39 5.80/2.39 CONF_IN_G(X) -> U1_G(X, del2_in_ga(X, Z)) 5.80/2.39 CONF_IN_G(X) -> DEL2_IN_GA(X, Z) 5.80/2.40 DEL2_IN_GA(X, Y) -> U4_GA(X, Y, del_in_aga(U, X, Z)) 5.80/2.40 DEL2_IN_GA(X, Y) -> DEL_IN_AGA(U, X, Z) 5.80/2.40 DEL_IN_AGA(X, .(H, T), .(H, T1)) -> U6_AGA(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.40 DEL_IN_AGA(X, .(H, T), .(H, T1)) -> DEL_IN_AGA(X, T, T1) 5.80/2.40 U4_GA(X, Y, del_out_aga(U, X, Z)) -> U5_GA(X, Y, del_in_aga(V, Z, Y)) 5.80/2.40 U4_GA(X, Y, del_out_aga(U, X, Z)) -> DEL_IN_AGA(V, Z, Y) 5.80/2.40 U1_G(X, del2_out_ga(X, Z)) -> U2_G(X, del_in_aag(U, Y, Z)) 5.80/2.40 U1_G(X, del2_out_ga(X, Z)) -> DEL_IN_AAG(U, Y, Z) 5.80/2.40 DEL_IN_AAG(X, .(H, T), .(H, T1)) -> U6_AAG(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.40 DEL_IN_AAG(X, .(H, T), .(H, T1)) -> DEL_IN_AAG(X, T, T1) 5.80/2.40 U2_G(X, del_out_aag(U, Y, Z)) -> U3_G(X, conf_in_g(Y)) 5.80/2.40 U2_G(X, del_out_aag(U, Y, Z)) -> CONF_IN_G(Y) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.40 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.40 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.40 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.40 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.40 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.40 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.40 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.40 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.40 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.40 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.40 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.40 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.40 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 conf_in_g(x1) = conf_in_g(x1) 5.80/2.40 5.80/2.40 U1_g(x1, x2) = U1_g(x2) 5.80/2.40 5.80/2.40 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.40 5.80/2.40 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.40 5.80/2.40 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.40 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.40 5.80/2.40 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.40 5.80/2.40 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.40 5.80/2.40 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.40 5.80/2.40 U2_g(x1, x2) = U2_g(x2) 5.80/2.40 5.80/2.40 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.40 5.80/2.40 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.40 5.80/2.40 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.40 5.80/2.40 U3_g(x1, x2) = U3_g(x2) 5.80/2.40 5.80/2.40 conf_out_g(x1) = conf_out_g 5.80/2.40 5.80/2.40 CONF_IN_G(x1) = CONF_IN_G(x1) 5.80/2.40 5.80/2.40 U1_G(x1, x2) = U1_G(x2) 5.80/2.40 5.80/2.40 DEL2_IN_GA(x1, x2) = DEL2_IN_GA(x1) 5.80/2.40 5.80/2.40 U4_GA(x1, x2, x3) = U4_GA(x3) 5.80/2.40 5.80/2.40 DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) 5.80/2.40 5.80/2.40 U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5) 5.80/2.40 5.80/2.40 U5_GA(x1, x2, x3) = U5_GA(x3) 5.80/2.40 5.80/2.40 U2_G(x1, x2) = U2_G(x2) 5.80/2.40 5.80/2.40 DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3) 5.80/2.40 5.80/2.40 U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x5) 5.80/2.40 5.80/2.40 U3_G(x1, x2) = U3_G(x2) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (5) DependencyGraphProof (EQUIVALENT) 5.80/2.40 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (6) 5.80/2.40 Complex Obligation (AND) 5.80/2.40 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (7) 5.80/2.40 Obligation: 5.80/2.40 Pi DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 DEL_IN_AAG(X, .(H, T), .(H, T1)) -> DEL_IN_AAG(X, T, T1) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.40 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.40 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.40 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.40 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.40 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.40 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.40 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.40 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.40 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.40 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.40 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.40 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.40 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 conf_in_g(x1) = conf_in_g(x1) 5.80/2.40 5.80/2.40 U1_g(x1, x2) = U1_g(x2) 5.80/2.40 5.80/2.40 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.40 5.80/2.40 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.40 5.80/2.40 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.40 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.40 5.80/2.40 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.40 5.80/2.40 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.40 5.80/2.40 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.40 5.80/2.40 U2_g(x1, x2) = U2_g(x2) 5.80/2.40 5.80/2.40 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.40 5.80/2.40 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.40 5.80/2.40 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.40 5.80/2.40 U3_g(x1, x2) = U3_g(x2) 5.80/2.40 5.80/2.40 conf_out_g(x1) = conf_out_g 5.80/2.40 5.80/2.40 DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (8) UsableRulesProof (EQUIVALENT) 5.80/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (9) 5.80/2.40 Obligation: 5.80/2.40 Pi DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 DEL_IN_AAG(X, .(H, T), .(H, T1)) -> DEL_IN_AAG(X, T, T1) 5.80/2.40 5.80/2.40 R is empty. 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (10) PiDPToQDPProof (SOUND) 5.80/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (11) 5.80/2.40 Obligation: 5.80/2.40 Q DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 DEL_IN_AAG(.(T1)) -> DEL_IN_AAG(T1) 5.80/2.40 5.80/2.40 R is empty. 5.80/2.40 Q is empty. 5.80/2.40 We have to consider all (P,Q,R)-chains. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (12) QDPSizeChangeProof (EQUIVALENT) 5.80/2.40 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. 5.80/2.40 5.80/2.40 From the DPs we obtained the following set of size-change graphs: 5.80/2.40 *DEL_IN_AAG(.(T1)) -> DEL_IN_AAG(T1) 5.80/2.40 The graph contains the following edges 1 > 1 5.80/2.40 5.80/2.40 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (13) 5.80/2.40 YES 5.80/2.40 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (14) 5.80/2.40 Obligation: 5.80/2.40 Pi DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 DEL_IN_AGA(X, .(H, T), .(H, T1)) -> DEL_IN_AGA(X, T, T1) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.40 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.40 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.40 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.40 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.40 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.40 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.40 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.40 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.40 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.40 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.40 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.40 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.40 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 conf_in_g(x1) = conf_in_g(x1) 5.80/2.40 5.80/2.40 U1_g(x1, x2) = U1_g(x2) 5.80/2.40 5.80/2.40 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.40 5.80/2.40 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.40 5.80/2.40 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.40 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.40 5.80/2.40 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.40 5.80/2.40 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.40 5.80/2.40 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.40 5.80/2.40 U2_g(x1, x2) = U2_g(x2) 5.80/2.40 5.80/2.40 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.40 5.80/2.40 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.40 5.80/2.40 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.40 5.80/2.40 U3_g(x1, x2) = U3_g(x2) 5.80/2.40 5.80/2.40 conf_out_g(x1) = conf_out_g 5.80/2.40 5.80/2.40 DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (15) UsableRulesProof (EQUIVALENT) 5.80/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (16) 5.80/2.40 Obligation: 5.80/2.40 Pi DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 DEL_IN_AGA(X, .(H, T), .(H, T1)) -> DEL_IN_AGA(X, T, T1) 5.80/2.40 5.80/2.40 R is empty. 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (17) PiDPToQDPProof (SOUND) 5.80/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (18) 5.80/2.40 Obligation: 5.80/2.40 Q DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 DEL_IN_AGA(.(T)) -> DEL_IN_AGA(T) 5.80/2.40 5.80/2.40 R is empty. 5.80/2.40 Q is empty. 5.80/2.40 We have to consider all (P,Q,R)-chains. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (19) QDPSizeChangeProof (EQUIVALENT) 5.80/2.40 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. 5.80/2.40 5.80/2.40 From the DPs we obtained the following set of size-change graphs: 5.80/2.40 *DEL_IN_AGA(.(T)) -> DEL_IN_AGA(T) 5.80/2.40 The graph contains the following edges 1 > 1 5.80/2.40 5.80/2.40 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (20) 5.80/2.40 YES 5.80/2.40 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (21) 5.80/2.40 Obligation: 5.80/2.40 Pi DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 U1_G(X, del2_out_ga(X, Z)) -> U2_G(X, del_in_aag(U, Y, Z)) 5.80/2.40 U2_G(X, del_out_aag(U, Y, Z)) -> CONF_IN_G(Y) 5.80/2.40 CONF_IN_G(X) -> U1_G(X, del2_in_ga(X, Z)) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 conf_in_g(X) -> U1_g(X, del2_in_ga(X, Z)) 5.80/2.40 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.40 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.40 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.40 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.40 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.40 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.40 U1_g(X, del2_out_ga(X, Z)) -> U2_g(X, del_in_aag(U, Y, Z)) 5.80/2.40 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.40 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.40 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.40 U2_g(X, del_out_aag(U, Y, Z)) -> U3_g(X, conf_in_g(Y)) 5.80/2.40 U3_g(X, conf_out_g(Y)) -> conf_out_g(X) 5.80/2.40 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 conf_in_g(x1) = conf_in_g(x1) 5.80/2.40 5.80/2.40 U1_g(x1, x2) = U1_g(x2) 5.80/2.40 5.80/2.40 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.40 5.80/2.40 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.40 5.80/2.40 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.40 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.40 5.80/2.40 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.40 5.80/2.40 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.40 5.80/2.40 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.40 5.80/2.40 U2_g(x1, x2) = U2_g(x2) 5.80/2.40 5.80/2.40 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.40 5.80/2.40 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.40 5.80/2.40 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.40 5.80/2.40 U3_g(x1, x2) = U3_g(x2) 5.80/2.40 5.80/2.40 conf_out_g(x1) = conf_out_g 5.80/2.40 5.80/2.40 CONF_IN_G(x1) = CONF_IN_G(x1) 5.80/2.40 5.80/2.40 U1_G(x1, x2) = U1_G(x2) 5.80/2.40 5.80/2.40 U2_G(x1, x2) = U2_G(x2) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (22) UsableRulesProof (EQUIVALENT) 5.80/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (23) 5.80/2.40 Obligation: 5.80/2.40 Pi DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 U1_G(X, del2_out_ga(X, Z)) -> U2_G(X, del_in_aag(U, Y, Z)) 5.80/2.40 U2_G(X, del_out_aag(U, Y, Z)) -> CONF_IN_G(Y) 5.80/2.40 CONF_IN_G(X) -> U1_G(X, del2_in_ga(X, Z)) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 del_in_aag(X, .(X, T), T) -> del_out_aag(X, .(X, T), T) 5.80/2.40 del_in_aag(X, .(H, T), .(H, T1)) -> U6_aag(X, H, T, T1, del_in_aag(X, T, T1)) 5.80/2.40 del2_in_ga(X, Y) -> U4_ga(X, Y, del_in_aga(U, X, Z)) 5.80/2.40 U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) -> del_out_aag(X, .(H, T), .(H, T1)) 5.80/2.40 U4_ga(X, Y, del_out_aga(U, X, Z)) -> U5_ga(X, Y, del_in_aga(V, Z, Y)) 5.80/2.40 del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) 5.80/2.40 del_in_aga(X, .(H, T), .(H, T1)) -> U6_aga(X, H, T, T1, del_in_aga(X, T, T1)) 5.80/2.40 U5_ga(X, Y, del_out_aga(V, Z, Y)) -> del2_out_ga(X, Y) 5.80/2.40 U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) -> del_out_aga(X, .(H, T), .(H, T1)) 5.80/2.40 5.80/2.40 The argument filtering Pi contains the following mapping: 5.80/2.40 del2_in_ga(x1, x2) = del2_in_ga(x1) 5.80/2.40 5.80/2.40 U4_ga(x1, x2, x3) = U4_ga(x3) 5.80/2.40 5.80/2.40 del_in_aga(x1, x2, x3) = del_in_aga(x2) 5.80/2.40 5.80/2.40 .(x1, x2) = .(x2) 5.80/2.40 5.80/2.40 del_out_aga(x1, x2, x3) = del_out_aga(x3) 5.80/2.40 5.80/2.40 U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) 5.80/2.40 5.80/2.40 U5_ga(x1, x2, x3) = U5_ga(x3) 5.80/2.40 5.80/2.40 del2_out_ga(x1, x2) = del2_out_ga(x2) 5.80/2.40 5.80/2.40 del_in_aag(x1, x2, x3) = del_in_aag(x3) 5.80/2.40 5.80/2.40 del_out_aag(x1, x2, x3) = del_out_aag(x2) 5.80/2.40 5.80/2.40 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5) 5.80/2.40 5.80/2.40 CONF_IN_G(x1) = CONF_IN_G(x1) 5.80/2.40 5.80/2.40 U1_G(x1, x2) = U1_G(x2) 5.80/2.40 5.80/2.40 U2_G(x1, x2) = U2_G(x2) 5.80/2.40 5.80/2.40 5.80/2.40 We have to consider all (P,R,Pi)-chains 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (24) PiDPToQDPProof (SOUND) 5.80/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (25) 5.80/2.40 Obligation: 5.80/2.40 Q DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 U1_G(del2_out_ga(Z)) -> U2_G(del_in_aag(Z)) 5.80/2.40 U2_G(del_out_aag(Y)) -> CONF_IN_G(Y) 5.80/2.40 CONF_IN_G(X) -> U1_G(del2_in_ga(X)) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 del_in_aag(T) -> del_out_aag(.(T)) 5.80/2.40 del_in_aag(.(T1)) -> U6_aag(del_in_aag(T1)) 5.80/2.40 del2_in_ga(X) -> U4_ga(del_in_aga(X)) 5.80/2.40 U6_aag(del_out_aag(T)) -> del_out_aag(.(T)) 5.80/2.40 U4_ga(del_out_aga(Z)) -> U5_ga(del_in_aga(Z)) 5.80/2.40 del_in_aga(.(T)) -> del_out_aga(T) 5.80/2.40 del_in_aga(.(T)) -> U6_aga(del_in_aga(T)) 5.80/2.40 U5_ga(del_out_aga(Y)) -> del2_out_ga(Y) 5.80/2.40 U6_aga(del_out_aga(T1)) -> del_out_aga(.(T1)) 5.80/2.40 5.80/2.40 The set Q consists of the following terms: 5.80/2.40 5.80/2.40 del_in_aag(x0) 5.80/2.40 del2_in_ga(x0) 5.80/2.40 U6_aag(x0) 5.80/2.40 U4_ga(x0) 5.80/2.40 del_in_aga(x0) 5.80/2.40 U5_ga(x0) 5.80/2.40 U6_aga(x0) 5.80/2.40 5.80/2.40 We have to consider all (P,Q,R)-chains. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (26) MRRProof (EQUIVALENT) 5.80/2.40 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. 5.80/2.40 5.80/2.40 Strictly oriented dependency pairs: 5.80/2.40 5.80/2.40 CONF_IN_G(X) -> U1_G(del2_in_ga(X)) 5.80/2.40 5.80/2.40 5.80/2.40 Used ordering: Polynomial interpretation [POLO]: 5.80/2.40 5.80/2.40 POL(.(x_1)) = 1 + 2*x_1 5.80/2.40 POL(CONF_IN_G(x_1)) = 1 + x_1 5.80/2.40 POL(U1_G(x_1)) = x_1 5.80/2.40 POL(U2_G(x_1)) = 1 + x_1 5.80/2.40 POL(U4_ga(x_1)) = x_1 5.80/2.40 POL(U5_ga(x_1)) = 1 + x_1 5.80/2.40 POL(U6_aag(x_1)) = 1 + 2*x_1 5.80/2.40 POL(U6_aga(x_1)) = 1 + 2*x_1 5.80/2.40 POL(del2_in_ga(x_1)) = x_1 5.80/2.40 POL(del2_out_ga(x_1)) = 2 + 2*x_1 5.80/2.40 POL(del_in_aag(x_1)) = 1 + 2*x_1 5.80/2.40 POL(del_in_aga(x_1)) = x_1 5.80/2.40 POL(del_out_aag(x_1)) = x_1 5.80/2.40 POL(del_out_aga(x_1)) = 1 + 2*x_1 5.80/2.40 5.80/2.40 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (27) 5.80/2.40 Obligation: 5.80/2.40 Q DP problem: 5.80/2.40 The TRS P consists of the following rules: 5.80/2.40 5.80/2.40 U1_G(del2_out_ga(Z)) -> U2_G(del_in_aag(Z)) 5.80/2.40 U2_G(del_out_aag(Y)) -> CONF_IN_G(Y) 5.80/2.40 5.80/2.40 The TRS R consists of the following rules: 5.80/2.40 5.80/2.40 del_in_aag(T) -> del_out_aag(.(T)) 5.80/2.40 del_in_aag(.(T1)) -> U6_aag(del_in_aag(T1)) 5.80/2.40 del2_in_ga(X) -> U4_ga(del_in_aga(X)) 5.80/2.40 U6_aag(del_out_aag(T)) -> del_out_aag(.(T)) 5.80/2.40 U4_ga(del_out_aga(Z)) -> U5_ga(del_in_aga(Z)) 5.80/2.40 del_in_aga(.(T)) -> del_out_aga(T) 5.80/2.40 del_in_aga(.(T)) -> U6_aga(del_in_aga(T)) 5.80/2.40 U5_ga(del_out_aga(Y)) -> del2_out_ga(Y) 5.80/2.40 U6_aga(del_out_aga(T1)) -> del_out_aga(.(T1)) 5.80/2.40 5.80/2.40 The set Q consists of the following terms: 5.80/2.40 5.80/2.40 del_in_aag(x0) 5.80/2.40 del2_in_ga(x0) 5.80/2.40 U6_aag(x0) 5.80/2.40 U4_ga(x0) 5.80/2.40 del_in_aga(x0) 5.80/2.40 U5_ga(x0) 5.80/2.40 U6_aga(x0) 5.80/2.40 5.80/2.40 We have to consider all (P,Q,R)-chains. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (28) DependencyGraphProof (EQUIVALENT) 5.80/2.40 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 5.80/2.40 ---------------------------------------- 5.80/2.40 5.80/2.40 (29) 5.80/2.40 TRUE 5.99/2.47 EOF