Left Termination of the query pattern
parse(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPiTRSProof (⇒, 52 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 168 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 2 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 3 ms)
        ↳25 QDP
        ↳26 MRRProof (⇔, 40 ms)
        ↳27 QDP
        ↳28 PisEmptyProof (⇔, 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 transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
parse_in: (b,f)
app_in: (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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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 transformation)

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:
U7GA1 > appingga2 > U10gga2 > s2 > appinaag1 > U10aag2 > U8GA1 > a > nil > PARSEINGA1 > U4GA1 > appoutaag2 > U5GA1 > appoutgga1 > b > U1GA1 > U2GA1 > cons2 > s3

and weight map:

nil=2
a=1
b=1
app_out_gga_1=7
app_in_aag_1=3
U1_GA_1=4
U2_GA_1=1
PARSE_IN_GA_1=7
U4_GA_1=4
U5_GA_1=2
U7_GA_1=2
U8_GA_1=1
app_in_gga_2=6
cons_2=5
U10_gga_2=5
app_out_aag_2=0
U10_aag_2=5
s_3=6
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 transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) YES