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



PROLOG
  ↳ PrologToPiTRSProof

parse2(Xs, T) :- app3(As, cons2(a0, cons2(s3(A, B, C), cons2(b0, Bs))), Xs), app3(As, cons2(s3(a0, s3(A, B, C), b0), Bs), Ys), parse2(Ys, T).
parse2(Xs, T) :- app3(As, cons2(a0, cons2(s2(A, B), cons2(b0, Bs))), Xs), app3(As, cons2(s3(a0, s2(A, B), b0), Bs), Ys), parse2(Ys, T).
parse2(Xs, T) :- app3(As, cons2(a0, cons2(b0, Bs)), Xs), app3(As, cons2(s2(a0, b0), Bs), Ys), parse2(Ys, T).
parse2(cons2(s2(A, B), nil0), s2(A, B)).
parse2(cons2(s3(A, B, C), nil0), s3(A, B, C)).
app3(nil0, X, X).
app3(cons2(X, Xs), Ys, cons2(X, Zs)) :- app3(Xs, Ys, Zs).


With regard to the inferred argument filtering the predicates were used in the following modes:
parse2: (b,f)
app3: (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_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(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_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(x2)


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

PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_1_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
PARSE_2_IN_GA2(Xs, T) -> APP_3_IN_AAG3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)
APP_3_IN_AAG3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> IF_APP_3_IN_1_AAG5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
APP_3_IN_AAG3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_AAG3(Xs, Ys, Zs)
IF_PARSE_2_IN_1_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
IF_PARSE_2_IN_1_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> APP_3_IN_GGA3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)
APP_3_IN_GGA3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)
IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> IF_PARSE_2_IN_3_GA4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_4_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
PARSE_2_IN_GA2(Xs, T) -> APP_3_IN_AAG3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)
IF_PARSE_2_IN_4_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
IF_PARSE_2_IN_4_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> APP_3_IN_GGA3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)
IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> IF_PARSE_2_IN_6_GA4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_7_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
PARSE_2_IN_GA2(Xs, T) -> APP_3_IN_AAG3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)
IF_PARSE_2_IN_7_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
IF_PARSE_2_IN_7_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> APP_3_IN_GGA3(As, cons_22(s_22(a_0, b_0), Bs), Ys)
IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> IF_PARSE_2_IN_9_GA4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)

The TRS R consists of the following rules:

parse_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(x2)
IF_PARSE_2_IN_9_GA4(x1, x2, x3, x4)  =  IF_PARSE_2_IN_9_GA1(x4)
IF_PARSE_2_IN_4_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_4_GA1(x3)
IF_PARSE_2_IN_1_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_1_GA1(x3)
IF_PARSE_2_IN_6_GA4(x1, x2, x3, x4)  =  IF_PARSE_2_IN_6_GA1(x4)
PARSE_2_IN_GA2(x1, x2)  =  PARSE_2_IN_GA1(x1)
IF_APP_3_IN_1_AAG5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_AAG2(x1, x5)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_PARSE_2_IN_2_GA8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_PARSE_2_IN_2_GA1(x8)
IF_PARSE_2_IN_3_GA4(x1, x2, x3, x4)  =  IF_PARSE_2_IN_3_GA1(x4)
IF_PARSE_2_IN_5_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_PARSE_2_IN_5_GA1(x7)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
IF_PARSE_2_IN_7_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_7_GA1(x3)
APP_3_IN_AAG3(x1, x2, x3)  =  APP_3_IN_AAG1(x3)
IF_PARSE_2_IN_8_GA5(x1, x2, x3, x4, x5)  =  IF_PARSE_2_IN_8_GA1(x5)

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_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_1_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
PARSE_2_IN_GA2(Xs, T) -> APP_3_IN_AAG3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)
APP_3_IN_AAG3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> IF_APP_3_IN_1_AAG5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
APP_3_IN_AAG3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_AAG3(Xs, Ys, Zs)
IF_PARSE_2_IN_1_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
IF_PARSE_2_IN_1_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> APP_3_IN_GGA3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)
APP_3_IN_GGA3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)
IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> IF_PARSE_2_IN_3_GA4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_4_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
PARSE_2_IN_GA2(Xs, T) -> APP_3_IN_AAG3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)
IF_PARSE_2_IN_4_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
IF_PARSE_2_IN_4_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> APP_3_IN_GGA3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)
IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> IF_PARSE_2_IN_6_GA4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_7_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
PARSE_2_IN_GA2(Xs, T) -> APP_3_IN_AAG3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)
IF_PARSE_2_IN_7_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
IF_PARSE_2_IN_7_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> APP_3_IN_GGA3(As, cons_22(s_22(a_0, b_0), Bs), Ys)
IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> IF_PARSE_2_IN_9_GA4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)

The TRS R consists of the following rules:

parse_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(x2)
IF_PARSE_2_IN_9_GA4(x1, x2, x3, x4)  =  IF_PARSE_2_IN_9_GA1(x4)
IF_PARSE_2_IN_4_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_4_GA1(x3)
IF_PARSE_2_IN_1_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_1_GA1(x3)
IF_PARSE_2_IN_6_GA4(x1, x2, x3, x4)  =  IF_PARSE_2_IN_6_GA1(x4)
PARSE_2_IN_GA2(x1, x2)  =  PARSE_2_IN_GA1(x1)
IF_APP_3_IN_1_AAG5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_AAG2(x1, x5)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_PARSE_2_IN_2_GA8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_PARSE_2_IN_2_GA1(x8)
IF_PARSE_2_IN_3_GA4(x1, x2, x3, x4)  =  IF_PARSE_2_IN_3_GA1(x4)
IF_PARSE_2_IN_5_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_PARSE_2_IN_5_GA1(x7)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
IF_PARSE_2_IN_7_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_7_GA1(x3)
APP_3_IN_AAG3(x1, x2, x3)  =  APP_3_IN_AAG1(x3)
IF_PARSE_2_IN_8_GA5(x1, x2, x3, x4, x5)  =  IF_PARSE_2_IN_8_GA1(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph 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_3_IN_GGA3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

parse_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(x2)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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_3_IN_GGA3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
cons_22(x1, x2)  =  cons_22(x1, x2)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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_3_IN_GGA2(cons_22(X, Xs), Ys) -> APP_3_IN_GGA2(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APP_3_IN_GGA2}.
By using the subterm criterion together with the size-change analysis 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_3_IN_AAG3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_AAG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

parse_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(x2)
APP_3_IN_AAG3(x1, x2, x3)  =  APP_3_IN_AAG1(x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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_3_IN_AAG3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> APP_3_IN_AAG3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
cons_22(x1, x2)  =  cons_22(x1, x2)
APP_3_IN_AAG3(x1, x2, x3)  =  APP_3_IN_AAG1(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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_3_IN_AAG1(cons_22(X, Zs)) -> APP_3_IN_AAG1(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APP_3_IN_AAG1}.
By using the subterm criterion together with the size-change analysis 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:

IF_PARSE_2_IN_4_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_7_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
IF_PARSE_2_IN_1_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_4_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
IF_PARSE_2_IN_7_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_1_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))

The TRS R consists of the following rules:

parse_2_in_ga2(Xs, T) -> if_parse_2_in_1_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_1_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_parse_2_in_2_ga8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_4_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
if_parse_2_in_4_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
if_parse_2_in_5_ga7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(Xs, T) -> if_parse_2_in_7_ga3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
if_parse_2_in_7_ga3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
if_parse_2_in_8_ga5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_in_ga2(Ys, T))
parse_2_in_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B)) -> parse_2_out_ga2(cons_22(s_22(A, B), nil_0), s_22(A, B))
parse_2_in_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C)) -> parse_2_out_ga2(cons_22(s_33(A, B, C), nil_0), s_33(A, B, C))
if_parse_2_in_9_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_6_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)
if_parse_2_in_3_ga4(Xs, T, Ys, parse_2_out_ga2(Ys, T)) -> parse_2_out_ga2(Xs, T)

The argument filtering Pi contains the following mapping:
parse_2_in_ga2(x1, x2)  =  parse_2_in_ga1(x1)
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
if_parse_2_in_1_ga3(x1, x2, x3)  =  if_parse_2_in_1_ga1(x3)
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
if_parse_2_in_2_ga8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_parse_2_in_2_ga1(x8)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_parse_2_in_3_ga4(x1, x2, x3, x4)  =  if_parse_2_in_3_ga1(x4)
if_parse_2_in_4_ga3(x1, x2, x3)  =  if_parse_2_in_4_ga1(x3)
if_parse_2_in_5_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_parse_2_in_5_ga1(x7)
if_parse_2_in_6_ga4(x1, x2, x3, x4)  =  if_parse_2_in_6_ga1(x4)
if_parse_2_in_7_ga3(x1, x2, x3)  =  if_parse_2_in_7_ga1(x3)
if_parse_2_in_8_ga5(x1, x2, x3, x4, x5)  =  if_parse_2_in_8_ga1(x5)
if_parse_2_in_9_ga4(x1, x2, x3, x4)  =  if_parse_2_in_9_ga1(x4)
parse_2_out_ga2(x1, x2)  =  parse_2_out_ga1(x2)
IF_PARSE_2_IN_4_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_4_GA1(x3)
IF_PARSE_2_IN_1_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_1_GA1(x3)
PARSE_2_IN_GA2(x1, x2)  =  PARSE_2_IN_GA1(x1)
IF_PARSE_2_IN_2_GA8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_PARSE_2_IN_2_GA1(x8)
IF_PARSE_2_IN_5_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_PARSE_2_IN_5_GA1(x7)
IF_PARSE_2_IN_7_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_7_GA1(x3)
IF_PARSE_2_IN_8_GA5(x1, x2, x3, x4, x5)  =  IF_PARSE_2_IN_8_GA1(x5)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

IF_PARSE_2_IN_4_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys))
IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_out_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_7_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs))
IF_PARSE_2_IN_1_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs)) -> IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_in_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys))
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_4_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))), Xs))
IF_PARSE_2_IN_2_GA8(Xs, T, As, A, B, C, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
IF_PARSE_2_IN_7_GA3(Xs, T, app_3_out_aag3(As, cons_22(a_0, cons_22(b_0, Bs)), Xs)) -> IF_PARSE_2_IN_8_GA5(Xs, T, As, Bs, app_3_in_gga3(As, cons_22(s_22(a_0, b_0), Bs), Ys))
IF_PARSE_2_IN_5_GA7(Xs, T, As, A, B, Bs, app_3_out_gga3(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs), Ys)) -> PARSE_2_IN_GA2(Ys, T)
PARSE_2_IN_GA2(Xs, T) -> IF_PARSE_2_IN_1_GA3(Xs, T, app_3_in_aag3(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))), Xs))

The TRS R consists of the following rules:

app_3_in_gga3(nil_0, X, X) -> app_3_out_gga3(nil_0, X, X)
app_3_in_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_aag3(nil_0, X, X) -> app_3_out_aag3(nil_0, X, X)
app_3_in_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs)) -> if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_in_aag3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(cons_22(X, Xs), Ys, cons_22(X, Zs))
if_app_3_in_1_aag5(X, Xs, Ys, Zs, app_3_out_aag3(Xs, Ys, Zs)) -> app_3_out_aag3(cons_22(X, Xs), Ys, cons_22(X, Zs))

The argument filtering Pi contains the following mapping:
cons_22(x1, x2)  =  cons_22(x1, x2)
a_0  =  a_0
s_33(x1, x2, x3)  =  s_33(x1, x2, x3)
b_0  =  b_0
s_22(x1, x2)  =  s_22(x1, x2)
nil_0  =  nil_0
app_3_in_aag3(x1, x2, x3)  =  app_3_in_aag1(x3)
app_3_out_aag3(x1, x2, x3)  =  app_3_out_aag2(x1, x2)
if_app_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aag2(x1, x5)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
IF_PARSE_2_IN_4_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_4_GA1(x3)
IF_PARSE_2_IN_1_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_1_GA1(x3)
PARSE_2_IN_GA2(x1, x2)  =  PARSE_2_IN_GA1(x1)
IF_PARSE_2_IN_2_GA8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_PARSE_2_IN_2_GA1(x8)
IF_PARSE_2_IN_5_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_PARSE_2_IN_5_GA1(x7)
IF_PARSE_2_IN_7_GA3(x1, x2, x3)  =  IF_PARSE_2_IN_7_GA1(x3)
IF_PARSE_2_IN_8_GA5(x1, x2, x3, x4, x5)  =  IF_PARSE_2_IN_8_GA1(x5)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPPoloProof

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

IF_PARSE_2_IN_4_GA1(app_3_out_aag2(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))))) -> IF_PARSE_2_IN_5_GA1(app_3_in_gga2(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs)))
IF_PARSE_2_IN_8_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_7_GA1(app_3_in_aag1(Xs))
IF_PARSE_2_IN_1_GA1(app_3_out_aag2(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))))) -> IF_PARSE_2_IN_2_GA1(app_3_in_gga2(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs)))
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_4_GA1(app_3_in_aag1(Xs))
IF_PARSE_2_IN_2_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
IF_PARSE_2_IN_7_GA1(app_3_out_aag2(As, cons_22(a_0, cons_22(b_0, Bs)))) -> IF_PARSE_2_IN_8_GA1(app_3_in_gga2(As, cons_22(s_22(a_0, b_0), Bs)))
IF_PARSE_2_IN_5_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_1_GA1(app_3_in_aag1(Xs))

The TRS R consists of the following rules:

app_3_in_gga2(nil_0, X) -> app_3_out_gga1(X)
app_3_in_gga2(cons_22(X, Xs), Ys) -> if_app_3_in_1_gga2(X, app_3_in_gga2(Xs, Ys))
app_3_in_aag1(X) -> app_3_out_aag2(nil_0, X)
app_3_in_aag1(cons_22(X, Zs)) -> if_app_3_in_1_aag2(X, app_3_in_aag1(Zs))
if_app_3_in_1_gga2(X, app_3_out_gga1(Zs)) -> app_3_out_gga1(cons_22(X, Zs))
if_app_3_in_1_aag2(X, app_3_out_aag2(Xs, Ys)) -> app_3_out_aag2(cons_22(X, Xs), Ys)

The set Q consists of the following terms:

app_3_in_gga2(x0, x1)
app_3_in_aag1(x0)
if_app_3_in_1_gga2(x0, x1)
if_app_3_in_1_aag2(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_PARSE_2_IN_5_GA1, IF_PARSE_2_IN_4_GA1, PARSE_2_IN_GA1, IF_PARSE_2_IN_8_GA1, IF_PARSE_2_IN_7_GA1, IF_PARSE_2_IN_2_GA1, IF_PARSE_2_IN_1_GA1}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

IF_PARSE_2_IN_4_GA1(app_3_out_aag2(As, cons_22(a_0, cons_22(s_22(A, B), cons_22(b_0, Bs))))) -> IF_PARSE_2_IN_5_GA1(app_3_in_gga2(As, cons_22(s_33(a_0, s_22(A, B), b_0), Bs)))
IF_PARSE_2_IN_1_GA1(app_3_out_aag2(As, cons_22(a_0, cons_22(s_33(A, B, C), cons_22(b_0, Bs))))) -> IF_PARSE_2_IN_2_GA1(app_3_in_gga2(As, cons_22(s_33(a_0, s_33(A, B, C), b_0), Bs)))
IF_PARSE_2_IN_7_GA1(app_3_out_aag2(As, cons_22(a_0, cons_22(b_0, Bs)))) -> IF_PARSE_2_IN_8_GA1(app_3_in_gga2(As, cons_22(s_22(a_0, b_0), Bs)))
The remaining Dependency Pairs were at least non-strictly be oriented.

IF_PARSE_2_IN_8_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_7_GA1(app_3_in_aag1(Xs))
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_4_GA1(app_3_in_aag1(Xs))
IF_PARSE_2_IN_2_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
IF_PARSE_2_IN_5_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_1_GA1(app_3_in_aag1(Xs))
With the implicit AFS we had to orient the following set of usable rules non-strictly.

if_app_3_in_1_aag2(X, app_3_out_aag2(Xs, Ys)) -> app_3_out_aag2(cons_22(X, Xs), Ys)
app_3_in_gga2(cons_22(X, Xs), Ys) -> if_app_3_in_1_gga2(X, app_3_in_gga2(Xs, Ys))
app_3_in_aag1(X) -> app_3_out_aag2(nil_0, X)
app_3_in_aag1(cons_22(X, Zs)) -> if_app_3_in_1_aag2(X, app_3_in_aag1(Zs))
app_3_in_gga2(nil_0, X) -> app_3_out_gga1(X)
if_app_3_in_1_gga2(X, app_3_out_gga1(Zs)) -> app_3_out_gga1(cons_22(X, Zs))
Used ordering: POLO with Polynomial interpretation:

POL(a_0) = 0   
POL(IF_PARSE_2_IN_7_GA1(x1)) = x1   
POL(if_app_3_in_1_gga2(x1, x2)) = x1 + x2   
POL(s_22(x1, x2)) = 0   
POL(IF_PARSE_2_IN_8_GA1(x1)) = x1   
POL(if_app_3_in_1_aag2(x1, x2)) = x1 + x2   
POL(IF_PARSE_2_IN_4_GA1(x1)) = x1   
POL(s_33(x1, x2, x3)) = 0   
POL(nil_0) = 0   
POL(IF_PARSE_2_IN_5_GA1(x1)) = x1   
POL(IF_PARSE_2_IN_2_GA1(x1)) = x1   
POL(PARSE_2_IN_GA1(x1)) = x1   
POL(app_3_out_gga1(x1)) = x1   
POL(app_3_in_aag1(x1)) = x1   
POL(app_3_out_aag2(x1, x2)) = x1 + x2   
POL(b_0) = 1   
POL(IF_PARSE_2_IN_1_GA1(x1)) = x1   
POL(app_3_in_gga2(x1, x2)) = x1 + x2   
POL(cons_22(x1, x2)) = x1 + x2   



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPPoloProof
QDP
                            ↳ DependencyGraphProof

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

IF_PARSE_2_IN_8_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_7_GA1(app_3_in_aag1(Xs))
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_4_GA1(app_3_in_aag1(Xs))
IF_PARSE_2_IN_2_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
IF_PARSE_2_IN_5_GA1(app_3_out_gga1(Ys)) -> PARSE_2_IN_GA1(Ys)
PARSE_2_IN_GA1(Xs) -> IF_PARSE_2_IN_1_GA1(app_3_in_aag1(Xs))

The TRS R consists of the following rules:

app_3_in_gga2(nil_0, X) -> app_3_out_gga1(X)
app_3_in_gga2(cons_22(X, Xs), Ys) -> if_app_3_in_1_gga2(X, app_3_in_gga2(Xs, Ys))
app_3_in_aag1(X) -> app_3_out_aag2(nil_0, X)
app_3_in_aag1(cons_22(X, Zs)) -> if_app_3_in_1_aag2(X, app_3_in_aag1(Zs))
if_app_3_in_1_gga2(X, app_3_out_gga1(Zs)) -> app_3_out_gga1(cons_22(X, Zs))
if_app_3_in_1_aag2(X, app_3_out_aag2(Xs, Ys)) -> app_3_out_aag2(cons_22(X, Xs), Ys)

The set Q consists of the following terms:

app_3_in_gga2(x0, x1)
app_3_in_aag1(x0)
if_app_3_in_1_gga2(x0, x1)
if_app_3_in_1_aag2(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {PARSE_2_IN_GA1, IF_PARSE_2_IN_8_GA1, IF_PARSE_2_IN_7_GA1, IF_PARSE_2_IN_4_GA1, IF_PARSE_2_IN_2_GA1, IF_PARSE_2_IN_5_GA1, IF_PARSE_2_IN_1_GA1}.
The approximation of the Dependency Graph contains 0 SCCs with 6 less nodes.