Left Termination proof of /tmp/tmp9LtHBB/parse.pl

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)

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)

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)

↳ 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)

↳ 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:

APP_3_IN_GGA₂(cons_2₂(X, Xs), Ys) -> APP_3_IN_GGA₂(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ 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)

↳ 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:

APP_3_IN_AAG₁(cons_2₂(X, Zs)) -> APP_3_IN_AAG₁(Zs)
The graph contains the following edges 1 > 1

↳ 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.