Left Termination of the query pattern parse_in_2(g, a) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

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).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
parse_in: (b,f)
app_in: (f,b,b) (f,b,f) (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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
s(x1, x2)  =  s

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
s(x1, x2)  =  s


Using Dependency Pairs [1,30] 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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U10_AGG(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U10_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_agg(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_agg(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_aga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_agg(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_agg(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_agg(As, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_agg(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_agg(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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
s(x1, x2)  =  s
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U10_AGA(x1, x2, x3, x4, x5)  =  U10_AGA(x5)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x5)
APP_IN_AGG(x1, x2, x3)  =  APP_IN_AGG(x2, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U9_GA(x1, x2, x3)  =  U9_GA(x3)
U10_AGG(x1, x2, x3, x4, x5)  =  U10_AGG(x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

PARSE_IN_GA(Xs, T) → U1_GA(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U10_AGG(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U10_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_agg(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_agg(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_aga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_agg(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_agg(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_agg(As, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_agg(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_agg(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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
s(x1, x2)  =  s
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U10_AGA(x1, x2, x3, x4, x5)  =  U10_AGA(x5)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x5)
APP_IN_AGG(x1, x2, x3)  =  APP_IN_AGG(x2, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U9_GA(x1, x2, x3)  =  U9_GA(x3)
U10_AGG(x1, x2, x3, x4, x5)  =  U10_AGG(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 14 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
s(x1, x2)  =  s
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

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

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

The TRS R consists of the following rules:none


s = APP_IN_AGA(Ys) evaluates to t =APP_IN_AGA(Ys)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP_IN_AGA(Ys) to APP_IN_AGA(Ys).





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
s(x1, x2)  =  s
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

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

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_aga(Xs, Ys, Zs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app_in_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x5)
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(x5)
s(x1, x2)  =  s
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U4_GA(app_out_agg(As)) → U5_GA(app_in_gga(As, .))
PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U1_GA(app_out_agg(As)) → U2_GA(app_in_gga(As, .))
U7_GA(app_out_agg(As)) → U8_GA(app_in_gga(As, .))
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
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(., Ys) → U10_gga(app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U4_GA(app_out_agg(As)) → U5_GA(app_in_gga(As, .)) at position [0] we obtained the following new rules:

U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U7_GA(app_out_agg(As)) → U8_GA(app_in_gga(As, .))
U1_GA(app_out_agg(As)) → U2_GA(app_in_gga(As, .))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
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(., Ys) → U10_gga(app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U7_GA(app_out_agg(As)) → U8_GA(app_in_gga(As, .)) at position [0] we obtained the following new rules:

U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U1_GA(app_out_agg(As)) → U2_GA(app_in_gga(As, .))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
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(., Ys) → U10_gga(app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_GA(app_out_agg(As)) → U2_GA(app_in_gga(As, .)) at position [0] we obtained the following new rules:

U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
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(., Ys) → U10_gga(app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))

The TRS R consists of the following rules:

app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

app_in_gga(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))

The TRS R consists of the following rules:

app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PARSE_IN_GA(Xs) → U4_GA(app_in_agg(., Xs)) at position [0] we obtained the following new rules:

PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PARSE_IN_GA(Xs) → U7_GA(app_in_agg(., Xs)) at position [0] we obtained the following new rules:

PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs))
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PARSE_IN_GA(Xs) → U1_GA(app_in_agg(., Xs)) at position [0] we obtained the following new rules:

PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_agg(X, X) → app_out_agg([])
app_in_agg(Ys, .) → U10_agg(app_in_aga(Ys))
app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

app_in_agg(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ ForwardInstantiation
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys) we obtained the following new rules:

U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ ForwardInstantiation
                                                                  ↳ QDP
                                                                    ↳ ForwardInstantiation
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys) we obtained the following new rules:

U2_GA(app_out_gga(.)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ ForwardInstantiation
                                                                  ↳ QDP
                                                                    ↳ ForwardInstantiation
                                                                      ↳ QDP
                                                                        ↳ ForwardInstantiation
  ↳ PrologToPiTRSProof

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

PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(.)) → PARSE_IN_GA(.)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys) we obtained the following new rules:

U8_GA(app_out_gga(.)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ ForwardInstantiation
                                                                  ↳ QDP
                                                                    ↳ ForwardInstantiation
                                                                      ↳ QDP
                                                                        ↳ ForwardInstantiation
                                                                          ↳ QDP
                                                                            ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
U8_GA(app_out_gga(.)) → PARSE_IN_GA(.)
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(.)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)

The set Q consists of the following terms:

U10_gga(x0)
U10_agg(x0)
app_in_aga(x0)
U10_aga(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg([])) → U2_GA(app_out_gga(.))
PARSE_IN_GA(.) → U1_GA(U10_agg(app_in_aga(.)))
U1_GA(app_out_agg(.)) → U2_GA(U10_gga(app_in_aga(.)))
U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
PARSE_IN_GA(.) → U1_GA(app_out_agg([]))
U8_GA(app_out_gga(.)) → PARSE_IN_GA(.)
U4_GA(app_out_agg(.)) → U5_GA(U10_gga(app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(app_out_agg([]))
PARSE_IN_GA(.) → U7_GA(U10_agg(app_in_aga(.)))
U7_GA(app_out_agg([])) → U8_GA(app_out_gga(.))
U7_GA(app_out_agg(.)) → U8_GA(U10_gga(app_in_aga(.)))
U2_GA(app_out_gga(.)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X)
app_in_aga(Ys) → U10_aga(app_in_aga(Ys))
U10_agg(app_out_aga(Xs, Zs)) → app_out_agg(.)
U10_aga(app_out_aga(Xs, Zs)) → app_out_aga(., .)
U10_gga(app_out_aga(Xs, Zs)) → app_out_gga(.)


s = PARSE_IN_GA(.) evaluates to t =PARSE_IN_GA(.)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

PARSE_IN_GA(.) → U4_GA(app_out_agg([]))
with rule PARSE_IN_GA(.) → U4_GA(app_out_agg([])) at position [] and matcher [ ]

U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.))
with rule U4_GA(app_out_agg([])) → U5_GA(app_out_gga(.)) at position [] and matcher [ ]

U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)
with rule U5_GA(app_out_gga(.)) → PARSE_IN_GA(.)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
parse_in: (b,f)
app_in: (f,b,b) (f,b,f) (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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
s(x1, x2)  =  s

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
s(x1, x2)  =  s


Using Dependency Pairs [1,30] 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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U10_AGG(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U10_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_agg(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_agg(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_aga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_agg(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_agg(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_agg(As, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_agg(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_agg(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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
s(x1, x2)  =  s
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U10_AGA(x1, x2, x3, x4, x5)  =  U10_AGA(x3, x5)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x3, x5)
APP_IN_AGG(x1, x2, x3)  =  APP_IN_AGG(x2, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U10_AGG(x1, x2, x3, x4, x5)  =  U10_AGG(x3, x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

PARSE_IN_GA(Xs, T) → U1_GA(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B, C), .(b, Bs))), Xs)
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U10_AGG(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U10_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_agg(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_agg(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_aga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(s(A, B), .(b, Bs))), Xs)
U4_GA(Xs, T, app_out_agg(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_agg(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_agg(As, .(a, .(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AGG(As, .(a, .(b, Bs)), Xs)
U7_GA(Xs, T, app_out_agg(As, .(a, .(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, .(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_agg(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_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
s(x1, x2)  =  s
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U10_AGA(x1, x2, x3, x4, x5)  =  U10_AGA(x3, x5)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x3, x5)
APP_IN_AGG(x1, x2, x3)  =  APP_IN_AGG(x2, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U10_AGG(x1, x2, x3, x4, x5)  =  U10_AGG(x3, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 14 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
s(x1, x2)  =  s
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

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

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

The TRS R consists of the following rules:none


s = APP_IN_AGA(Ys) evaluates to t =APP_IN_AGA(Ys)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP_IN_AGA(Ys) to APP_IN_AGA(Ys).





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_agg(As, .(a, .(s(A, B, C), .(b, Bs))), Xs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, T, app_out_agg(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_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(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_agg(As, .(a, .(s(A, B), .(b, Bs))), Xs))
U4_ga(Xs, T, app_out_agg(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_agg(As, .(a, .(b, Bs)), Xs))
U7_ga(Xs, T, app_out_agg(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_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(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)
parse_out_ga(x1, x2)  =  parse_out_ga(x1, x2)
s(x1, x2)  =  s
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

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

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_aga(Xs, Ys, Zs))
app_in_agg([], X, X) → app_out_agg([], X, X)
app_in_agg(.(X, Xs), Ys, .(X, Zs)) → U10_agg(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U10_agg(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_agg(.(X, Xs), Ys, .(X, Zs))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U10_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U10_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app_in_agg(x1, x2, x3)  =  app_in_agg(x2, x3)
s(x1, x2, x3)  =  s
.(x1, x2)  =  .
app_out_agg(x1, x2, x3)  =  app_out_agg(x1, x2, x3)
U10_agg(x1, x2, x3, x4, x5)  =  U10_agg(x3, x5)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
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(x3, x5)
s(x1, x2)  =  s
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing

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

U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(Xs, app_out_agg(As, ., Xs)) → U2_GA(Xs, app_in_gga(As, .))
U7_GA(Xs, app_out_agg(As, ., Xs)) → U8_GA(Xs, app_in_gga(As, .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(Xs, app_out_agg(As, ., Xs)) → U5_GA(Xs, app_in_gga(As, .))
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(., Ys) → U10_gga(Ys, app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)
app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U4_GA(Xs, app_out_agg(As, ., Xs)) → U5_GA(Xs, app_in_gga(As, .)) at position [1] we obtained the following new rules:

U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing

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

U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U1_GA(Xs, app_out_agg(As, ., Xs)) → U2_GA(Xs, app_in_gga(As, .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(Xs, app_out_agg(As, ., Xs)) → U8_GA(Xs, app_in_gga(As, .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(., Ys) → U10_gga(Ys, app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)
app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U7_GA(Xs, app_out_agg(As, ., Xs)) → U8_GA(Xs, app_in_gga(As, .)) at position [1] we obtained the following new rules:

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(Xs, app_out_agg(As, ., Xs)) → U2_GA(Xs, app_in_gga(As, .))
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(., Ys) → U10_gga(Ys, app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)
app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_GA(Xs, app_out_agg(As, ., Xs)) → U2_GA(Xs, app_in_gga(As, .)) at position [1] we obtained the following new rules:

U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(., Ys) → U10_gga(Ys, app_in_aga(Ys))
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)
app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

app_in_gga(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing

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

U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PARSE_IN_GA(Xs) → U4_GA(Xs, app_in_agg(., Xs)) at position [1] we obtained the following new rules:

PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PARSE_IN_GA(Xs) → U7_GA(Xs, app_in_agg(., Xs)) at position [1] we obtained the following new rules:

PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PARSE_IN_GA(Xs) → U1_GA(Xs, app_in_agg(., Xs)) at position [1] we obtained the following new rules:

PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
app_in_agg(X, X) → app_out_agg([], X, X)
app_in_agg(Ys, .) → U10_agg(Ys, app_in_aga(Ys))
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

app_in_agg(x0, x1)
U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

app_in_agg(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation

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

U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U7_GA(y0, app_out_agg(., ., y0)) → U8_GA(y0, U10_gga(., app_in_aga(.))) we obtained the following new rules:

U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation

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

PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U4_GA(y0, app_out_agg([], ., y0)) → U5_GA(y0, app_out_gga([], ., .)) we obtained the following new rules:

U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation

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

PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U8_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys) we obtained the following new rules:

U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation

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

PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U1_GA(y0, app_out_agg(., ., y0)) → U2_GA(y0, U10_gga(., app_in_aga(.))) we obtained the following new rules:

U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation

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

PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U4_GA(y0, app_out_agg(., ., y0)) → U5_GA(y0, U10_gga(., app_in_aga(.))) we obtained the following new rules:

U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation

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

PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U5_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys) we obtained the following new rules:

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GA(Xs, app_out_gga(As, ., Ys)) → PARSE_IN_GA(Ys) we obtained the following new rules:

U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U2_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U2_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U1_GA(y0, app_out_agg([], ., y0)) → U2_GA(y0, app_out_gga([], ., .)) we obtained the following new rules:

U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U2_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U7_GA(y0, app_out_agg([], ., y0)) → U8_GA(y0, app_out_gga([], ., .)) we obtained the following new rules:

U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation
                                                                                                  ↳ QDP
                                                                                                    ↳ Instantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U2_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U8_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.) we obtained the following new rules:

U8_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation
                                                                                                  ↳ QDP
                                                                                                    ↳ Instantiation
                                                                                                      ↳ QDP
                                                                                                        ↳ Instantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U2_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GA(z0, app_out_gga([], ., .)) → PARSE_IN_GA(.) we obtained the following new rules:

U2_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation
                                                                                                  ↳ QDP
                                                                                                    ↳ Instantiation
                                                                                                      ↳ QDP
                                                                                                        ↳ Instantiation
                                                                                                          ↳ QDP
                                                                                                            ↳ ForwardInstantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U8_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U8_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2) we obtained the following new rules:

U8_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation
                                                                                                  ↳ QDP
                                                                                                    ↳ Instantiation
                                                                                                      ↳ QDP
                                                                                                        ↳ Instantiation
                                                                                                          ↳ QDP
                                                                                                            ↳ ForwardInstantiation
                                                                                                              ↳ QDP
                                                                                                                ↳ ForwardInstantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U8_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U2_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U5_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2) we obtained the following new rules:

U5_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation
                                                                                                  ↳ QDP
                                                                                                    ↳ Instantiation
                                                                                                      ↳ QDP
                                                                                                        ↳ Instantiation
                                                                                                          ↳ QDP
                                                                                                            ↳ ForwardInstantiation
                                                                                                              ↳ QDP
                                                                                                                ↳ ForwardInstantiation
                                                                                                                  ↳ QDP
                                                                                                                    ↳ ForwardInstantiation

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
U8_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U8_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)
U5_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2)
U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
U2_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_GA(., app_out_gga(x1, ., x2)) → PARSE_IN_GA(x2) we obtained the following new rules:

U2_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ UsableRulesProof
                                                          ↳ QDP
                                                            ↳ QReductionProof
                                                              ↳ QDP
                                                                ↳ Instantiation
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ Instantiation
                                                                          ↳ QDP
                                                                            ↳ Instantiation
                                                                              ↳ QDP
                                                                                ↳ Instantiation
                                                                                  ↳ QDP
                                                                                    ↳ Instantiation
                                                                                      ↳ QDP
                                                                                        ↳ Instantiation
                                                                                          ↳ QDP
                                                                                            ↳ Instantiation
                                                                                              ↳ QDP
                                                                                                ↳ Instantiation
                                                                                                  ↳ QDP
                                                                                                    ↳ Instantiation
                                                                                                      ↳ QDP
                                                                                                        ↳ Instantiation
                                                                                                          ↳ QDP
                                                                                                            ↳ ForwardInstantiation
                                                                                                              ↳ QDP
                                                                                                                ↳ ForwardInstantiation
                                                                                                                  ↳ QDP
                                                                                                                    ↳ ForwardInstantiation
                                                                                                                      ↳ QDP

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

U5_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., U10_agg(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., app_out_agg([], ., .))
U8_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)
U7_GA(., app_out_agg(., ., .)) → U8_GA(., U10_gga(., app_in_aga(.)))
U8_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)
PARSE_IN_GA(.) → U1_GA(., app_out_agg([], ., .))
PARSE_IN_GA(.) → U7_GA(., U10_agg(., app_in_aga(.)))
U5_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)
U4_GA(., app_out_agg(., ., .)) → U5_GA(., U10_gga(., app_in_aga(.)))
U7_GA(., app_out_agg([], ., .)) → U8_GA(., app_out_gga([], ., .))
U2_GA(., app_out_gga(x0, ., .)) → PARSE_IN_GA(.)
U1_GA(., app_out_agg([], ., .)) → U2_GA(., app_out_gga([], ., .))
PARSE_IN_GA(.) → U7_GA(., app_out_agg([], ., .))
U4_GA(., app_out_agg([], ., .)) → U5_GA(., app_out_gga([], ., .))
U1_GA(., app_out_agg(., ., .)) → U2_GA(., U10_gga(., app_in_aga(.)))
PARSE_IN_GA(.) → U4_GA(., U10_agg(., app_in_aga(.)))
U2_GA(., app_out_gga([], ., .)) → PARSE_IN_GA(.)

The TRS R consists of the following rules:

app_in_aga(X) → app_out_aga([], X, X)
app_in_aga(Ys) → U10_aga(Ys, app_in_aga(Ys))
U10_gga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_gga(., Ys, .)
U10_aga(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_aga(., Ys, .)
U10_agg(Ys, app_out_aga(Xs, Ys, Zs)) → app_out_agg(., Ys, .)

The set Q consists of the following terms:

U10_gga(x0, x1)
U10_agg(x0, x1)
app_in_aga(x0)
U10_aga(x0, x1)

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