7.43/2.72 YES 7.43/2.74 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 7.43/2.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.43/2.74 7.43/2.74 7.43/2.74 Left Termination of the query pattern 7.43/2.74 7.43/2.74 parse(g,a) 7.43/2.74 7.43/2.74 w.r.t. the given Prolog program could successfully be proven: 7.43/2.74 7.43/2.74 (0) Prolog 7.43/2.74 (1) PrologToPiTRSProof [SOUND, 0 ms] 7.43/2.74 (2) PiTRS 7.43/2.74 (3) DependencyPairsProof [EQUIVALENT, 6 ms] 7.43/2.74 (4) PiDP 7.43/2.74 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 7.43/2.74 (6) AND 7.43/2.74 (7) PiDP 7.43/2.74 (8) UsableRulesProof [EQUIVALENT, 0 ms] 7.43/2.74 (9) PiDP 7.43/2.74 (10) PiDPToQDPProof [SOUND, 6 ms] 7.43/2.74 (11) QDP 7.43/2.74 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.43/2.74 (13) YES 7.43/2.74 (14) PiDP 7.43/2.74 (15) UsableRulesProof [EQUIVALENT, 0 ms] 7.43/2.74 (16) PiDP 7.43/2.74 (17) PiDPToQDPProof [SOUND, 0 ms] 7.43/2.74 (18) QDP 7.43/2.74 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.43/2.74 (20) YES 7.43/2.74 (21) PiDP 7.43/2.74 (22) UsableRulesProof [EQUIVALENT, 0 ms] 7.43/2.74 (23) PiDP 7.43/2.74 (24) PiDPToQDPProof [SOUND, 0 ms] 7.43/2.74 (25) QDP 7.43/2.74 (26) MRRProof [EQUIVALENT, 117 ms] 7.43/2.74 (27) QDP 7.43/2.74 (28) DependencyGraphProof [EQUIVALENT, 0 ms] 7.43/2.74 (29) TRUE 7.43/2.74 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (0) 7.43/2.74 Obligation: 7.43/2.74 Clauses: 7.43/2.74 7.43/2.74 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))). 7.43/2.74 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))). 7.43/2.74 parse(Xs, T) :- ','(app(As, cons(a, cons(b, Bs)), Xs), ','(app(As, cons(s(a, b), Bs), Ys), parse(Ys, T))). 7.43/2.74 parse(cons(s(A, B), nil), s(A, B)). 7.43/2.74 parse(cons(s(A, B, C), nil), s(A, B, C)). 7.43/2.74 app(nil, X, X). 7.43/2.74 app(cons(X, Xs), Ys, cons(X, Zs)) :- app(Xs, Ys, Zs). 7.43/2.74 7.43/2.74 7.43/2.74 Query: parse(g,a) 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (1) PrologToPiTRSProof (SOUND) 7.43/2.74 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 7.43/2.74 7.43/2.74 parse_in_2: (b,f) 7.43/2.74 7.43/2.74 app_in_3: (f,f,b) (b,b,f) 7.43/2.74 7.43/2.74 Transforming Prolog into the following Term Rewriting System: 7.43/2.74 7.43/2.74 Pi-finite rewrite system: 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 7.43/2.74 7.43/2.74 7.43/2.74 7.43/2.74 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 7.43/2.74 7.43/2.74 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (2) 7.43/2.74 Obligation: 7.43/2.74 Pi-finite rewrite system: 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (3) DependencyPairsProof (EQUIVALENT) 7.43/2.74 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs) 7.43/2.74 APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 7.43/2.74 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)) 7.43/2.74 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) 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.43/2.74 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)) 7.43/2.74 U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs) 7.43/2.74 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)) 7.43/2.74 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) 7.43/2.74 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)) 7.43/2.74 U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(b, Bs)), Xs) 7.43/2.74 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)) 7.43/2.74 U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> APP_IN_GGA(As, cons(s(a, b), Bs), Ys) 7.43/2.74 U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_GA(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) 7.43/2.74 7.43/2.74 U1_GA(x1, x2, x3) = U1_GA(x3) 7.43/2.74 7.43/2.74 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 7.43/2.74 7.43/2.74 U10_AAG(x1, x2, x3, x4, x5) = U10_AAG(x1, x5) 7.43/2.74 7.43/2.74 U2_GA(x1, x2, x3) = U2_GA(x3) 7.43/2.74 7.43/2.74 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.43/2.74 7.43/2.74 U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x1, x5) 7.43/2.74 7.43/2.74 U3_GA(x1, x2, x3) = U3_GA(x3) 7.43/2.74 7.43/2.74 U4_GA(x1, x2, x3) = U4_GA(x3) 7.43/2.74 7.43/2.74 U5_GA(x1, x2, x3) = U5_GA(x3) 7.43/2.74 7.43/2.74 U6_GA(x1, x2, x3) = U6_GA(x3) 7.43/2.74 7.43/2.74 U7_GA(x1, x2, x3) = U7_GA(x3) 7.43/2.74 7.43/2.74 U8_GA(x1, x2, x3) = U8_GA(x3) 7.43/2.74 7.43/2.74 U9_GA(x1, x2, x3) = U9_GA(x3) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (4) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs) 7.43/2.74 APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 7.43/2.74 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)) 7.43/2.74 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) 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.43/2.74 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)) 7.43/2.74 U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs) 7.43/2.74 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)) 7.43/2.74 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) 7.43/2.74 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)) 7.43/2.74 U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> APP_IN_AAG(As, cons(a, cons(b, Bs)), Xs) 7.43/2.74 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)) 7.43/2.74 U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) -> APP_IN_GGA(As, cons(s(a, b), Bs), Ys) 7.43/2.74 U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_GA(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) 7.43/2.74 7.43/2.74 U1_GA(x1, x2, x3) = U1_GA(x3) 7.43/2.74 7.43/2.74 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 7.43/2.74 7.43/2.74 U10_AAG(x1, x2, x3, x4, x5) = U10_AAG(x1, x5) 7.43/2.74 7.43/2.74 U2_GA(x1, x2, x3) = U2_GA(x3) 7.43/2.74 7.43/2.74 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.43/2.74 7.43/2.74 U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x1, x5) 7.43/2.74 7.43/2.74 U3_GA(x1, x2, x3) = U3_GA(x3) 7.43/2.74 7.43/2.74 U4_GA(x1, x2, x3) = U4_GA(x3) 7.43/2.74 7.43/2.74 U5_GA(x1, x2, x3) = U5_GA(x3) 7.43/2.74 7.43/2.74 U6_GA(x1, x2, x3) = U6_GA(x3) 7.43/2.74 7.43/2.74 U7_GA(x1, x2, x3) = U7_GA(x3) 7.43/2.74 7.43/2.74 U8_GA(x1, x2, x3) = U8_GA(x3) 7.43/2.74 7.43/2.74 U9_GA(x1, x2, x3) = U9_GA(x3) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (5) DependencyGraphProof (EQUIVALENT) 7.43/2.74 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (6) 7.43/2.74 Complex Obligation (AND) 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (7) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (8) UsableRulesProof (EQUIVALENT) 7.43/2.74 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (9) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.43/2.74 7.43/2.74 R is empty. 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (10) PiDPToQDPProof (SOUND) 7.43/2.74 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (11) 7.43/2.74 Obligation: 7.43/2.74 Q DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 APP_IN_GGA(cons(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.43/2.74 7.43/2.74 R is empty. 7.43/2.74 Q is empty. 7.43/2.74 We have to consider all (P,Q,R)-chains. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (12) QDPSizeChangeProof (EQUIVALENT) 7.43/2.74 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. 7.43/2.74 7.43/2.74 From the DPs we obtained the following set of size-change graphs: 7.43/2.74 *APP_IN_GGA(cons(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.43/2.74 The graph contains the following edges 1 > 1, 2 >= 2 7.43/2.74 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (13) 7.43/2.74 YES 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (14) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (15) UsableRulesProof (EQUIVALENT) 7.43/2.74 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (16) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 7.43/2.74 7.43/2.74 R is empty. 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (17) PiDPToQDPProof (SOUND) 7.43/2.74 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (18) 7.43/2.74 Obligation: 7.43/2.74 Q DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 APP_IN_AAG(cons(X, Zs)) -> APP_IN_AAG(Zs) 7.43/2.74 7.43/2.74 R is empty. 7.43/2.74 Q is empty. 7.43/2.74 We have to consider all (P,Q,R)-chains. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (19) QDPSizeChangeProof (EQUIVALENT) 7.43/2.74 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. 7.43/2.74 7.43/2.74 From the DPs we obtained the following set of size-change graphs: 7.43/2.74 *APP_IN_AAG(cons(X, Zs)) -> APP_IN_AAG(Zs) 7.43/2.74 The graph contains the following edges 1 > 1 7.43/2.74 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (20) 7.43/2.74 YES 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (21) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 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)) 7.43/2.74 U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 parse_in_ga(Xs, T) -> U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 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)) 7.43/2.74 parse_in_ga(Xs, T) -> U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> U9_ga(Xs, T, parse_in_ga(Ys, T)) 7.43/2.74 parse_in_ga(cons(s(A, B), nil), s(A, B)) -> parse_out_ga(cons(s(A, B), nil), s(A, B)) 7.43/2.74 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)) 7.43/2.74 U9_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U6_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 U3_ga(Xs, T, parse_out_ga(Ys, T)) -> parse_out_ga(Xs, T) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 parse_in_ga(x1, x2) = parse_in_ga(x1) 7.43/2.74 7.43/2.74 U1_ga(x1, x2, x3) = U1_ga(x3) 7.43/2.74 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 U2_ga(x1, x2, x3) = U2_ga(x3) 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 U3_ga(x1, x2, x3) = U3_ga(x3) 7.43/2.74 7.43/2.74 U4_ga(x1, x2, x3) = U4_ga(x3) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 U5_ga(x1, x2, x3) = U5_ga(x3) 7.43/2.74 7.43/2.74 U6_ga(x1, x2, x3) = U6_ga(x3) 7.43/2.74 7.43/2.74 U7_ga(x1, x2, x3) = U7_ga(x3) 7.43/2.74 7.43/2.74 U8_ga(x1, x2, x3) = U8_ga(x3) 7.43/2.74 7.43/2.74 U9_ga(x1, x2, x3) = U9_ga(x3) 7.43/2.74 7.43/2.74 parse_out_ga(x1, x2) = parse_out_ga(x2) 7.43/2.74 7.43/2.74 PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) 7.43/2.74 7.43/2.74 U1_GA(x1, x2, x3) = U1_GA(x3) 7.43/2.74 7.43/2.74 U2_GA(x1, x2, x3) = U2_GA(x3) 7.43/2.74 7.43/2.74 U4_GA(x1, x2, x3) = U4_GA(x3) 7.43/2.74 7.43/2.74 U5_GA(x1, x2, x3) = U5_GA(x3) 7.43/2.74 7.43/2.74 U7_GA(x1, x2, x3) = U7_GA(x3) 7.43/2.74 7.43/2.74 U8_GA(x1, x2, x3) = U8_GA(x3) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (22) UsableRulesProof (EQUIVALENT) 7.43/2.74 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (23) 7.43/2.74 Obligation: 7.43/2.74 Pi DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 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)) 7.43/2.74 U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U1_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) 7.43/2.74 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)) 7.43/2.74 U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 PARSE_IN_GA(Xs, T) -> U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs)) 7.43/2.74 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)) 7.43/2.74 U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) -> PARSE_IN_GA(Ys, T) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 app_in_gga(nil, X, X) -> app_out_gga(nil, X, X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) -> U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.43/2.74 app_in_aag(nil, X, X) -> app_out_aag(nil, X, X) 7.43/2.74 app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) -> U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 7.43/2.74 U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(cons(X, Xs), Ys, cons(X, Zs)) 7.43/2.74 7.43/2.74 The argument filtering Pi contains the following mapping: 7.43/2.74 app_in_aag(x1, x2, x3) = app_in_aag(x3) 7.43/2.74 7.43/2.74 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 7.43/2.74 7.43/2.74 cons(x1, x2) = cons(x1, x2) 7.43/2.74 7.43/2.74 U10_aag(x1, x2, x3, x4, x5) = U10_aag(x1, x5) 7.43/2.74 7.43/2.74 a = a 7.43/2.74 7.43/2.74 s(x1, x2, x3) = s(x1, x2, x3) 7.43/2.74 7.43/2.74 b = b 7.43/2.74 7.43/2.74 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.43/2.74 7.43/2.74 nil = nil 7.43/2.74 7.43/2.74 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.43/2.74 7.43/2.74 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5) 7.43/2.74 7.43/2.74 s(x1, x2) = s(x1, x2) 7.43/2.74 7.43/2.74 PARSE_IN_GA(x1, x2) = PARSE_IN_GA(x1) 7.43/2.74 7.43/2.74 U1_GA(x1, x2, x3) = U1_GA(x3) 7.43/2.74 7.43/2.74 U2_GA(x1, x2, x3) = U2_GA(x3) 7.43/2.74 7.43/2.74 U4_GA(x1, x2, x3) = U4_GA(x3) 7.43/2.74 7.43/2.74 U5_GA(x1, x2, x3) = U5_GA(x3) 7.43/2.74 7.43/2.74 U7_GA(x1, x2, x3) = U7_GA(x3) 7.43/2.74 7.43/2.74 U8_GA(x1, x2, x3) = U8_GA(x3) 7.43/2.74 7.43/2.74 7.43/2.74 We have to consider all (P,R,Pi)-chains 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (24) PiDPToQDPProof (SOUND) 7.43/2.74 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (25) 7.43/2.74 Obligation: 7.43/2.74 Q DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 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))) 7.43/2.74 U2_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) 7.43/2.74 PARSE_IN_GA(Xs) -> U1_GA(app_in_aag(Xs)) 7.43/2.74 PARSE_IN_GA(Xs) -> U4_GA(app_in_aag(Xs)) 7.43/2.74 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))) 7.43/2.74 U5_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) 7.43/2.74 PARSE_IN_GA(Xs) -> U7_GA(app_in_aag(Xs)) 7.43/2.74 U7_GA(app_out_aag(As, cons(a, cons(b, Bs)))) -> U8_GA(app_in_gga(As, cons(s(a, b), Bs))) 7.43/2.74 U8_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 app_in_gga(nil, X) -> app_out_gga(X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys) -> U10_gga(X, app_in_gga(Xs, Ys)) 7.43/2.74 app_in_aag(X) -> app_out_aag(nil, X) 7.43/2.74 app_in_aag(cons(X, Zs)) -> U10_aag(X, app_in_aag(Zs)) 7.43/2.74 U10_gga(X, app_out_gga(Zs)) -> app_out_gga(cons(X, Zs)) 7.43/2.74 U10_aag(X, app_out_aag(Xs, Ys)) -> app_out_aag(cons(X, Xs), Ys) 7.43/2.74 7.43/2.74 The set Q consists of the following terms: 7.43/2.74 7.43/2.74 app_in_gga(x0, x1) 7.43/2.74 app_in_aag(x0) 7.43/2.74 U10_gga(x0, x1) 7.43/2.74 U10_aag(x0, x1) 7.43/2.74 7.43/2.74 We have to consider all (P,Q,R)-chains. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (26) MRRProof (EQUIVALENT) 7.43/2.74 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. 7.43/2.74 7.43/2.74 Strictly oriented dependency pairs: 7.43/2.74 7.43/2.74 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))) 7.43/2.74 U2_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) 7.43/2.74 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))) 7.43/2.74 U5_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) 7.43/2.74 U7_GA(app_out_aag(As, cons(a, cons(b, Bs)))) -> U8_GA(app_in_gga(As, cons(s(a, b), Bs))) 7.43/2.74 U8_GA(app_out_gga(Ys)) -> PARSE_IN_GA(Ys) 7.43/2.74 7.43/2.74 7.43/2.74 Used ordering: Polynomial interpretation [POLO]: 7.43/2.74 7.43/2.74 POL(PARSE_IN_GA(x_1)) = 2*x_1 7.43/2.74 POL(U10_aag(x_1, x_2)) = 1 + 2*x_1 + x_2 7.43/2.74 POL(U10_gga(x_1, x_2)) = 1 + 2*x_1 + x_2 7.43/2.74 POL(U1_GA(x_1)) = 2*x_1 7.43/2.74 POL(U2_GA(x_1)) = 1 + 2*x_1 7.43/2.74 POL(U4_GA(x_1)) = 2*x_1 7.43/2.74 POL(U5_GA(x_1)) = 1 + 2*x_1 7.43/2.74 POL(U7_GA(x_1)) = 2*x_1 7.43/2.74 POL(U8_GA(x_1)) = 1 + 2*x_1 7.43/2.74 POL(a) = 0 7.43/2.74 POL(app_in_aag(x_1)) = x_1 7.43/2.74 POL(app_in_gga(x_1, x_2)) = x_1 + x_2 7.43/2.74 POL(app_out_aag(x_1, x_2)) = x_1 + x_2 7.43/2.74 POL(app_out_gga(x_1)) = x_1 7.43/2.74 POL(b) = 0 7.43/2.74 POL(cons(x_1, x_2)) = 1 + 2*x_1 + x_2 7.43/2.74 POL(nil) = 0 7.43/2.74 POL(s(x_1, x_2)) = x_1 + 2*x_2 7.43/2.74 POL(s(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 7.43/2.74 7.43/2.74 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (27) 7.43/2.74 Obligation: 7.43/2.74 Q DP problem: 7.43/2.74 The TRS P consists of the following rules: 7.43/2.74 7.43/2.74 PARSE_IN_GA(Xs) -> U1_GA(app_in_aag(Xs)) 7.43/2.74 PARSE_IN_GA(Xs) -> U4_GA(app_in_aag(Xs)) 7.43/2.74 PARSE_IN_GA(Xs) -> U7_GA(app_in_aag(Xs)) 7.43/2.74 7.43/2.74 The TRS R consists of the following rules: 7.43/2.74 7.43/2.74 app_in_gga(nil, X) -> app_out_gga(X) 7.43/2.74 app_in_gga(cons(X, Xs), Ys) -> U10_gga(X, app_in_gga(Xs, Ys)) 7.43/2.74 app_in_aag(X) -> app_out_aag(nil, X) 7.43/2.74 app_in_aag(cons(X, Zs)) -> U10_aag(X, app_in_aag(Zs)) 7.43/2.74 U10_gga(X, app_out_gga(Zs)) -> app_out_gga(cons(X, Zs)) 7.43/2.74 U10_aag(X, app_out_aag(Xs, Ys)) -> app_out_aag(cons(X, Xs), Ys) 7.43/2.74 7.43/2.74 The set Q consists of the following terms: 7.43/2.74 7.43/2.74 app_in_gga(x0, x1) 7.43/2.74 app_in_aag(x0) 7.43/2.74 U10_gga(x0, x1) 7.43/2.74 U10_aag(x0, x1) 7.43/2.74 7.43/2.74 We have to consider all (P,Q,R)-chains. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (28) DependencyGraphProof (EQUIVALENT) 7.43/2.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 7.43/2.74 ---------------------------------------- 7.43/2.74 7.43/2.74 (29) 7.43/2.74 TRUE 7.43/2.78 EOF