Left Termination of the query pattern parse_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

parse(Xs, T) :- ','(app(As, .(a, .(s(A, B, C), .(b, Bs))), Xs), ','(app(As, .(s(a, s(A, B, C), b), Bs), Ys), parse(Ys, T))).
parse(Xs, T) :- ','(app(As, .(a, .(s(A, B), .(b, Bs))), Xs), ','(app(As, .(s(a, s(A, B), b), Bs), Ys), parse(Ys, T))).
parse(Xs, T) :- ','(app(As, .(a, .(b, Bs)), Xs), ','(app(As, .(s(a, b), Bs), Ys), parse(Ys, T))).
parse(.(s(A, B), []), s(A, B)).
parse(.(s(A, B, C), []), s(A, B, C)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

parse(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B, C), b), Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B), b), Bs), Ys)
U5_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → APP_IN_GGA(As, .(s(a, b), Bs), Ys)
U8_GA(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_GA(Xs, T, parse_in_ga(Ys, T))
U8_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U10_AAG(x1, x2, x3, x4, x5)  =  U10_AAG(x1, x4, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U9_GA(x1, x2, x3)  =  U9_GA(x1, 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B, C), b), Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B), b), Bs), Ys)
U5_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → APP_IN_GGA(As, .(s(a, b), Bs), Ys)
U8_GA(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_GA(Xs, T, parse_in_ga(Ys, T))
U8_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U10_AAG(x1, x2, x3, x4, x5)  =  U10_AAG(x1, x4, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U9_GA(x1, x2, x3)  =  U9_GA(x1, 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(.(X, Xs), Ys, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, 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(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(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(.(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(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) TRUE

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP_IN_AAG(.(X, Xs), Ys, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, 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(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(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(.(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(.(X, Zs)) → APP_IN_AAG(Zs)
  The graph contains the following edges 1 > 1

(20) TRUE

(21) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U2_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → U4_GA(Xs, T, app_in_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_GA(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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(x1, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, 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, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U2_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → U4_GA(Xs, T, app_in_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_GA(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_GA(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → PARSE_IN_GA(Ys, T)

The TRS R consists of the following rules:

app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(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, x3)
.(x1, x2)  =  .(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x4, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
s(x1, x2)  =  s(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, 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(Xs, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, app_in_gga(As, .(s(a, s(A, B, C), b), Bs)))
U2_GA(Xs, app_out_gga(As, .(s(a, s(A, B, C), b), Bs), Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_aag(Xs))
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_aag(Xs))
U4_GA(Xs, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, app_in_gga(As, .(s(a, s(A, B), b), Bs)))
U5_GA(Xs, app_out_gga(As, .(s(a, s(A, B), b), Bs), Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_aag(Xs))
U7_GA(Xs, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, app_in_gga(As, .(s(a, b), Bs)))
U8_GA(Xs, app_out_gga(As, .(s(a, b), Bs), Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys) → U10_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X, X)
app_in_aag(.(X, Zs)) → U10_aag(X, Zs, app_in_aag(Zs))
U10_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U10_aag(X, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U10_gga(x0, x1, x2, x3)
U10_aag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(26) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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

(27) 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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)

(28) 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B, C), b), Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B), b), Bs), Ys)
U5_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → APP_IN_GGA(As, .(s(a, b), Bs), Ys)
U8_GA(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_GA(Xs, T, parse_in_ga(Ys, T))
U8_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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

(29) 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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B, C), b), Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → APP_IN_GGA(As, .(s(a, s(A, B), b), Bs), Ys)
U5_GA(Xs, T, app_out_gga(As, .(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, .(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, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → APP_IN_GGA(As, .(s(a, b), Bs), Ys)
U8_GA(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_GA(Xs, T, parse_in_ga(Ys, T))
U8_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(31) Complex Obligation (AND)

(32) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP_IN_GGA(.(X, Xs), Ys, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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

(33) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(34) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(35) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(37) 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(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
  The graph contains the following edges 1 > 1, 2 >= 2

(38) TRUE

(39) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP_IN_AAG(.(X, Xs), Ys, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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

(40) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(41) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains

(42) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(43) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(44) 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(.(X, Zs)) → APP_IN_AAG(Zs)
  The graph contains the following edges 1 > 1

(45) TRUE

(46) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U2_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → U4_GA(Xs, T, app_in_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_GA(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(.(s(A, B), []), s(A, B)) → parse_out_ga(.(s(A, B), []), s(A, B))
parse_in_ga(.(s(A, B, C), []), s(A, B, C)) → parse_out_ga(.(s(A, B, C), []), 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)
.(x1, x2)  =  .(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)
[]  =  []
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

(47) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(48) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U1_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B, C), b), Bs), Ys))
U2_GA(Xs, T, app_out_gga(As, .(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, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → U4_GA(Xs, T, app_in_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_GA(Xs, T, app_out_aag(As, .(a, .(s(A, B), .(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, .(s(a, s(A, B), b), Bs), Ys))
U5_GA(Xs, T, app_out_gga(As, .(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, .(a, .(b, Bs)), Xs))
U7_GA(Xs, T, app_out_aag(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U8_GA(Xs, T, app_out_gga(As, .(s(a, b), Bs), Ys)) → PARSE_IN_GA(Ys, T)

The TRS R consists of the following rules:

app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(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(.(X, Xs), Ys, .(X, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(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)
.(x1, x2)  =  .(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)
[]  =  []
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

(49) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U1_GA(app_out_aag(As, .(a, .(s(A, B, C), .(b, Bs))))) → U2_GA(app_in_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))))) → U5_GA(app_in_gga(As, .(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, .(a, .(b, Bs)))) → U8_GA(app_in_gga(As, .(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([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U10_gga(X, app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(X, Zs)) → U10_aag(X, app_in_aag(Zs))
U10_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U10_aag(X, app_out_aag(Xs, Ys)) → app_out_aag(.(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.

(51) 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, .(a, .(s(A, B, C), .(b, Bs))))) → U2_GA(app_in_gga(As, .(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, .(a, .(s(A, B), .(b, Bs))))) → U5_GA(app_in_gga(As, .(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, .(a, .(b, Bs)))) → U8_GA(app_in_gga(As, .(s(a, b), Bs)))
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

Strictly oriented rules of the TRS R:

app_in_gga([], X) → app_out_gga(X)
app_in_aag(X) → app_out_aag([], X)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 5 + x₁ + x₂   
POL(PARSE_IN_GA(x₁)) = 2 + x₁   
POL(U10_aag(x₁, x₂)) = 5 + x₁ + x₂   
POL(U10_gga(x₁, x₂)) = 5 + x₁ + x₂   
POL(U1_GA(x₁)) = x₁   
POL(U2_GA(x₁)) = 3 + x₁   
POL(U4_GA(x₁)) = x₁   
POL(U5_GA(x₁)) = 3 + x₁   
POL(U7_GA(x₁)) = x₁   
POL(U8_GA(x₁)) = 3 + x₁   
POL([]) = 0   
POL(a) = 0   
POL(app_in_aag(x₁)) = 1 + x₁   
POL(app_in_gga(x₁, x₂)) = 1 + x₁ + x₂   
POL(app_out_aag(x₁, x₂)) = x₁ + x₂   
POL(app_out_gga(x₁)) = x₁   
POL(b) = 0   
POL(s(x₁, x₂)) = x₁ + x₂   
POL(s(x₁, x₂, x₃)) = x₁ + x₂ + x₃   

(52) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app_in_gga(.(X, Xs), Ys) → U10_gga(X, app_in_gga(Xs, Ys))
app_in_aag(.(X, Zs)) → U10_aag(X, app_in_aag(Zs))
U10_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U10_aag(X, app_out_aag(Xs, Ys)) → app_out_aag(.(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.

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE