/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern parse(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) MRRProof [EQUIVALENT, 5 ms] (27) QDP (28) PisEmptyProof [EQUIVALENT, 0 ms] (29) YES ---------------------------------------- (0) Obligation: Clauses: parse(Xs, T) :- ','(app(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs), ','(app(As, cons(s(a, s(A, B, C), b), Bs), Ys), parse(Ys, T))). parse(Xs, T) :- ','(app(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs), ','(app(As, cons(s(a, s(A, B), b), Bs), Ys), parse(Ys, T))). parse(Xs, T) :- ','(app(As, cons(a, cons(b, Bs)), Xs), ','(app(As, cons(s(a, b), Bs), Ys), parse(Ys, T))). parse(cons(s(A, B), nil), s(A, B)). parse(cons(s(A, B, C), nil), s(A, B, C)). app(nil, X, X). app(cons(X, Xs), Ys, cons(X, Zs)) :- app(Xs, Ys, Zs). Query: parse(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: parse_in_2: (b,f) app_in_3: (f,f,b) (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs) APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> APP_IN_GGA(As, cons(s(a, s(A, B, C), b), Bs), Ys) APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_GA(Xs, T, parse_in_ga(Ys, T)) U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs) U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> APP_IN_GGA(As, cons(s(a, s(A, B), b), Bs), Ys) U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_GA(Xs, T, parse_in_ga(Ys, T)) U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(b, Bs)), Xs) U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_GA(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> APP_IN_GGA(As, cons(s(a, b), Bs), Ys) U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_GA(Xs, T, parse_in_ga(Ys, T)) U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U10_AAG(x1, x2, x3, x4, x5) = U10_AAG(x1, x5) U2_GA(x1, x2, x3) = U2_GA(x3) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x1, x5) U3_GA(x1, x2, x3) = U3_GA(x3) U4_GA(x1, x2, x3) = U4_GA(x3) U5_GA(x1, x2, x3) = U5_GA(x3) U6_GA(x1, x2, x3) = U6_GA(x3) U7_GA(x1, x2, x3) = U7_GA(x3) U8_GA(x1, x2, x3) = U8_GA(x3) U9_GA(x1, x2, x3) = U9_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs) APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> APP_IN_GGA(As, cons(s(a, s(A, B, C), b), Bs), Ys) APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_GA(Xs, T, parse_in_ga(Ys, T)) U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs) U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> APP_IN_GGA(As, cons(s(a, s(A, B), b), Bs), Ys) U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_GA(Xs, T, parse_in_ga(Ys, T)) U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(b, Bs)), Xs) U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_GA(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> APP_IN_GGA(As, cons(s(a, b), Bs), Ys) U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_GA(Xs, T, parse_in_ga(Ys, T)) U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U10_AAG(x1, x2, x3, x4, x5) = U10_AAG(x1, x5) U2_GA(x1, x2, x3) = U2_GA(x3) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x1, x5) U3_GA(x1, x2, x3) = U3_GA(x3) U4_GA(x1, x2, x3) = U4_GA(x3) U5_GA(x1, x2, x3) = U5_GA(x3) U6_GA(x1, x2, x3) = U6_GA(x3) U7_GA(x1, x2, x3) = U7_GA(x3) U8_GA(x1, x2, x3) = U8_GA(x3) U9_GA(x1, x2, x3) = U9_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: cons(x1, x2) = cons(x1, x2) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_GGA(cons(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *APP_IN_GGA(cons(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: cons(x1, x2) = cons(x1, x2) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAG(cons(X, Zs)) -> APP_IN_AAG(Zs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *APP_IN_AAG(cons(X, Zs)) -> APP_IN_AAG(Zs) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_GA(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) The TRS R consists of the following rules: parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> U3_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> U6_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) -> parse_out_ga(cons(s(A, B, C), nil), s(A, B, C)) U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) The argument filtering Pi contains the following mapping: parse_in_ga(x1, x2) = parse_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b U2_ga(x1, x2, x3) = U2_ga(x3) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) U3_ga(x1, x2, x3) = U3_ga(x3) U4_ga(x1, x2, x3) = U4_ga(x3) s(x1, x2) = s(x1, x2) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3) = U6_ga(x3) U7_ga(x1, x2, x3) = U7_ga(x3) U8_ga(x1, x2, x3) = U8_ga(x3) U9_ga(x1, x2, x3) = U9_ga(x3) parse_out_ga(x1, x2) = parse_out_ga(x2) PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) U2_GA(x1, x2, x3) = U2_GA(x3) U4_GA(x1, x2, x3) = U4_GA(x3) U5_GA(x1, x2, x3) = U5_GA(x3) U7_GA(x1, x2, x3) = U7_GA(x3) U8_GA(x1, x2, x3) = U8_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) -> U2_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) -> U5_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> U8_GA(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys)) U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) The TRS R consists of the following rules: app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) The argument filtering Pi contains the following mapping: app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) cons(x1, x2) = cons(x1, x2) U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) a = a s(x1, x2, x3) = s(x1, x2, x3) b = b app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) nil = nil app_out_gga(x1, x2, x3) = app_out_gga(x3) U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) s(x1, x2) = s(x1, x2) PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) U2_GA(x1, x2, x3) = U2_GA(x3) U4_GA(x1, x2, x3) = U4_GA(x3) U5_GA(x1, x2, x3) = U5_GA(x3) U7_GA(x1, x2, x3) = U7_GA(x3) U8_GA(x1, x2, x3) = U8_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))))) -> U2_GA(app_in_gga(As, cons(s(a, s(A, B, C), b), Bs))) U2_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) PARSE_IN_GA(Xs) -> U1_GA(app_in_aag(Xs)) PARSE_IN_GA(Xs) -> U4_GA(app_in_aag(Xs)) U4_GA(app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))))) -> U5_GA(app_in_gga(As, cons(s(a, s(A, B), b), Bs))) U5_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) PARSE_IN_GA(Xs) -> U7_GA(app_in_aag(Xs)) U7_GA(app_out_aag(As, cons(a, cons(b, Bs)))) -> U8_GA(app_in_gga(As, cons(s(a, b), Bs))) U8_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) The TRS R consists of the following rules: app_in_gga(nil, X) -> app_out_gga(X) app_in_gga(cons(X, Xs), Ys) -> U10_gga(X, app_in_gga(Xs, Ys)) app_in_aag(X) -> app_out_aag(nil, X) app_in_aag(cons(X, Zs)) -> U10_aag(X, app_in_aag(Zs)) U10_gga(X, app_out_gga(Zs)) -> app_out_gga(cons(X, Zs)) U10_aag(X, app_out_aag(Xs, Ys)) -> app_out_aag(cons(X, Xs), Ys) The set Q consists of the following terms: app_in_gga(x0, x1) app_in_aag(x0) U10_gga(x0, x1) U10_aag(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) MRRProof (EQUIVALENT) 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. Strictly oriented dependency pairs: U1_GA(app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))))) -> U2_GA(app_in_gga(As, cons(s(a, s(A, B, C), b), Bs))) U2_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) PARSE_IN_GA(Xs) -> U1_GA(app_in_aag(Xs)) PARSE_IN_GA(Xs) -> U4_GA(app_in_aag(Xs)) U4_GA(app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))))) -> U5_GA(app_in_gga(As, cons(s(a, s(A, B), b), Bs))) U5_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) PARSE_IN_GA(Xs) -> U7_GA(app_in_aag(Xs)) U7_GA(app_out_aag(As, cons(a, cons(b, Bs)))) -> U8_GA(app_in_gga(As, cons(s(a, b), Bs))) U8_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) Strictly oriented rules of the TRS R: app_in_gga(nil, X) -> app_out_gga(X) app_in_gga(cons(X, Xs), Ys) -> U10_gga(X, app_in_gga(Xs, Ys)) app_in_aag(X) -> app_out_aag(nil, X) app_in_aag(cons(X, Zs)) -> U10_aag(X, app_in_aag(Zs)) U10_gga(X, app_out_gga(Zs)) -> app_out_gga(cons(X, Zs)) U10_aag(X, app_out_aag(Xs, Ys)) -> app_out_aag(cons(X, Xs), Ys) Used ordering: Knuth-Bendix order [KBO] with precedence:U7_GA_1 > app_in_gga_2 > U4_GA_1 > U5_GA_1 > U10_gga_2 > s_2 > app_in_aag_1 > U10_aag_2 > U8_GA_1 > app_out_gga_1 > U2_GA_1 > nil > PARSE_IN_GA_1 > a > cons_2 > app_out_aag_2 > b > s_3 > U1_GA_1 and weight map: nil=2 a=1 b=1 app_out_gga_1=7 app_in_aag_1=3 U1_GA_1=3 U2_GA_1=1 PARSE_IN_GA_1=8 U4_GA_1=3 U5_GA_1=2 U7_GA_1=3 U8_GA_1=2 app_in_gga_2=6 cons_2=5 U10_gga_2=5 app_out_aag_2=0 U10_aag_2=5 s_3=5 s_2=0 The variable weight is 1 ---------------------------------------- (27) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: app_in_gga(x0, x1) app_in_aag(x0) U10_gga(x0, x1) U10_aag(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (29) YES