YES Left Termination proof of ../tpdb/LP/SGST06/parse.pl
Left Termination of the query pattern parse_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

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

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,f,b) (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ 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_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)


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_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)
APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) → U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → APP_IN_GGA(As, cons(s(a, s(A, B, C), b), Bs), Ys)
APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) → U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_GA(Xs, T, parse_in_ga(Ys, T))
U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → PARSE_IN_GA(Ys, T)
PARSE_IN_GA(Xs, T) → U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)
U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → APP_IN_GGA(As, cons(s(a, s(A, B), b), Bs), Ys)
U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_GA(Xs, T, parse_in_ga(Ys, T))
U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → PARSE_IN_GA(Ys, T)
PARSE_IN_GA(Xs, T) → U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, cons(a, cons(b, Bs)), Xs)
U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → APP_IN_GGA(As, cons(s(a, b), Bs), Ys)
U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_GA(Xs, T, parse_in_ga(Ys, T))
U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → PARSE_IN_GA(Ys, T)

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U10_AAG(x1, x2, x3, x4, x5)  =  U10_AAG(x1, x5)
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)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x5)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(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)

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

↳ Prolog
  ↳ 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_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)
APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) → U10_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(cons(X, Xs), Ys, cons(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
U1_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → APP_IN_GGA(As, cons(s(a, s(A, B, C), b), Bs), Ys)
APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) → U10_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(cons(X, Xs), Ys, cons(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_GA(Xs, T, parse_in_ga(Ys, T))
U2_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → PARSE_IN_GA(Ys, T)
PARSE_IN_GA(Xs, T) → U4_GA(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)
U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_GA(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U4_GA(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → APP_IN_GGA(As, cons(s(a, s(A, B), b), Bs), Ys)
U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_GA(Xs, T, parse_in_ga(Ys, T))
U5_GA(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → PARSE_IN_GA(Ys, T)
PARSE_IN_GA(Xs, T) → U7_GA(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
PARSE_IN_GA(Xs, T) → APP_IN_AAG(As, cons(a, cons(b, Bs)), Xs)
U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_GA(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U7_GA(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → APP_IN_GGA(As, cons(s(a, b), Bs), Ys)
U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_GA(Xs, T, parse_in_ga(Ys, T))
U8_GA(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → PARSE_IN_GA(Ys, T)

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
U10_AAG(x1, x2, x3, x4, x5)  =  U10_AAG(x1, x5)
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)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x5)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(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)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 11 less nodes.

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

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

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

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

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

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

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

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

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

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

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

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

APP_IN_AAG(cons(X, Zs)) → APP_IN_AAG(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

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

The TRS R consists of the following rules:

parse_in_ga(Xs, T) → U1_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs))
app_in_aag(nil, X, X) → app_out_aag(nil, X, X)
app_in_aag(cons(X, Xs), Ys, cons(X, Zs)) → U10_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U10_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(cons(X, Xs), Ys, cons(X, Zs))
U1_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))), Xs)) → U2_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys))
app_in_gga(nil, X, X) → app_out_gga(nil, X, X)
app_in_gga(cons(X, Xs), Ys, cons(X, Zs)) → U10_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U10_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(cons(X, Xs), Ys, cons(X, Zs))
U2_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B, C), b), Bs), Ys)) → U3_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U4_ga(Xs, T, app_in_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs))
U4_ga(Xs, T, app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))), Xs)) → U5_ga(Xs, T, app_in_gga(As, cons(s(a, s(A, B), b), Bs), Ys))
U5_ga(Xs, T, app_out_gga(As, cons(s(a, s(A, B), b), Bs), Ys)) → U6_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(Xs, T) → U7_ga(Xs, T, app_in_aag(As, cons(a, cons(b, Bs)), Xs))
U7_ga(Xs, T, app_out_aag(As, cons(a, cons(b, Bs)), Xs)) → U8_ga(Xs, T, app_in_gga(As, cons(s(a, b), Bs), Ys))
U8_ga(Xs, T, app_out_gga(As, cons(s(a, b), Bs), Ys)) → U9_ga(Xs, T, parse_in_ga(Ys, T))
parse_in_ga(cons(s(A, B), nil), s(A, B)) → parse_out_ga(cons(s(A, B), nil), s(A, B))
parse_in_ga(cons(s(A, B, C), nil), s(A, B, C)) → parse_out_ga(cons(s(A, B, C), nil), s(A, B, C))
U9_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U6_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)
U3_ga(Xs, T, parse_out_ga(Ys, T)) → parse_out_ga(Xs, T)

The argument filtering Pi contains the following mapping:
parse_in_ga(x1, x2)  =  parse_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
s(x1, x2)  =  s(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
parse_out_ga(x1, x2)  =  parse_out_ga(x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

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

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U10_aag(x1, x2, x3, x4, x5)  =  U10_aag(x1, x5)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
nil  =  nil
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
s(x1, x2)  =  s(x1, x2)
PARSE_IN_GA(x1, x2)  =  PARSE_IN_GA(x1)
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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U5_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U7_GA(app_out_aag(As, cons(a, cons(b, Bs)))) → U8_GA(app_in_gga(As, cons(s(a, b), Bs)))
U1_GA(app_out_aag(As, cons(a, cons(s(A, B, C), cons(b, Bs))))) → U2_GA(app_in_gga(As, cons(s(a, s(A, B, C), b), Bs)))
The remaining pairs can at least be oriented weakly.

U4_GA(app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))))) → U5_GA(app_in_gga(As, cons(s(a, s(A, B), b), Bs)))
PARSE_IN_GA(Xs) → U7_GA(app_in_aag(Xs))
PARSE_IN_GA(Xs) → U1_GA(app_in_aag(Xs))
PARSE_IN_GA(Xs) → U4_GA(app_in_aag(Xs))
U2_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
U8_GA(app_out_gga(Ys)) → PARSE_IN_GA(Ys)
Used ordering: Polynomial interpretation [25]:

POL(PARSE_IN_GA(x1)) = x1   
POL(U10_aag(x1, x2)) = 1 + x1 + x2   
POL(U10_gga(x1, x2)) = 1 + x1 + x2   
POL(U1_GA(x1)) = x1   
POL(U2_GA(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL(U5_GA(x1)) = 1 + x1   
POL(U7_GA(x1)) = x1   
POL(U8_GA(x1)) = x1   
POL(a) = 1   
POL(app_in_aag(x1)) = x1   
POL(app_in_gga(x1, x2)) = x1 + x2   
POL(app_out_aag(x1, x2)) = x1 + x2   
POL(app_out_gga(x1)) = x1   
POL(b) = 0   
POL(cons(x1, x2)) = 1 + x1 + x2   
POL(nil) = 0   
POL(s(x1, x2)) = 0   
POL(s(x1, x2, x3)) = 1 + x1   

The following usable rules [17] were oriented:

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



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

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

PARSE_IN_GA(Xs) → U7_GA(app_in_aag(Xs))
U4_GA(app_out_aag(As, cons(a, cons(s(A, B), cons(b, Bs))))) → U5_GA(app_in_gga(As, cons(s(a, s(A, B), b), Bs)))
PARSE_IN_GA(Xs) → U1_GA(app_in_aag(Xs))
PARSE_IN_GA(Xs) → U4_GA(app_in_aag(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_aag(X) → app_out_aag(nil, X)
app_in_aag(cons(X, Zs)) → U10_aag(X, app_in_aag(Zs))
app_in_gga(nil, X) → app_out_gga(X)
app_in_gga(cons(X, Xs), Ys) → U10_gga(X, app_in_gga(Xs, Ys))
U10_aag(X, app_out_aag(Xs, Ys)) → app_out_aag(cons(X, Xs), Ys)
U10_gga(X, app_out_gga(Zs)) → app_out_gga(cons(X, Zs))

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 6 less nodes.