Left Termination proof of ../tpdb/LP/talp/apt/curry

MAYBE Left Termination proof of ../tpdb/LP/talp/apt/curry_ap.pl
Left Termination of the query pattern type_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

Clauses:

type(E, var(X), T) :- in(E, X, T).
type(E, apply(M, N), T) :- ','(type(E, M, arrow(S, T)), type(E, N, S)).
type(E, lambda(X, M), arrow(S, T)) :- type(.(','(X, S), E), M, T).
in(.(','(X, T), E), X, T).
in(.(','(Y, T), E), X, T) :- ','(\==(X, Y), in(E, X, T)).
\==(X0, X1).

Queries:

type(g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
type_in: (b,b,f) (f,b,f)
in_in: (b,b,f) (b,b,b) (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

      ↳ PrologToPiTRSProof

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

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

TYPE_IN_GGA(E, var(X), T) → U1_GGA(E, X, T, in_in_gga(E, X, T))
TYPE_IN_GGA(E, var(X), T) → IN_IN_GGA(E, X, T)
IN_IN_GGA(.(','(Y, T), E), X, T) → U5_GGA(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGA(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGA(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGG(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGG(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
TYPE_IN_GGA(E, apply(M, N), T) → U2_GGA(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
TYPE_IN_GGA(E, apply(M, N), T) → TYPE_IN_GGA(E, M, arrow(S, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → U4_GGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, var(X), T) → U1_AGA(E, X, T, in_in_aga(E, X, T))
TYPE_IN_AGA(E, var(X), T) → IN_IN_AGA(E, X, T)
IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
IN_IN_AGA(.(','(Y, T), E), X, T) → \==_IN_GA(X, Y)
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → U6_AGA(Y, T, E, X, in_in_aga(E, X, T))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → U4_AGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_AGA(E, M, N, T, type_in_aga(E, N, S))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_GGA(E, M, N, T, type_in_gga(E, N, S))
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → TYPE_IN_GGA(E, N, S)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

The argument filtering Pi contains the following mapping:
type_in_gga(x1, x2, x3) = type_in_gga(x1, x2)
var(x1) = var(x1)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
in_in_gga(x1, x2, x3) = in_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
in_out_gga(x1, x2, x3) = in_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x2, x3, x4, x5)
\==_in_gg(x1, x2) = \==_in_gg(x1, x2)
\==_out_gg(x1, x2) = \==_out_gg
U6_gga(x1, x2, x3, x4, x5) = U6_gga(x2, x5)
in_in_ggg(x1, x2, x3) = in_in_ggg(x1, x2, x3)
in_out_ggg(x1, x2, x3) = in_out_ggg
U5_ggg(x1, x2, x3, x4, x5) = U5_ggg(x2, x3, x4, x5)
U6_ggg(x1, x2, x3, x4, x5) = U6_ggg(x5)
type_out_gga(x1, x2, x3) = type_out_gga
apply(x1, x2) = apply(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x3, x5)
lambda(x1, x2) = lambda(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x6)
type_in_aga(x1, x2, x3) = type_in_aga(x2)
U1_aga(x1, x2, x3, x4) = U1_aga(x4)
in_in_aga(x1, x2, x3) = in_in_aga(x2)
in_out_aga(x1, x2, x3) = in_out_aga
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x4, x5)
\==_in_ga(x1, x2) = \==_in_ga(x1)
\==_out_ga(x1, x2) = \==_out_ga
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
type_out_aga(x1, x2, x3) = type_out_aga
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5)
U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x5)
IN_IN_AGA(x1, x2, x3) = IN_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5, x6) = U4_AGA(x6)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x3, x5)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5)
U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x2, x3, x4, x5)
U1_AGA(x1, x2, x3, x4) = U1_AGA(x4)
\==_IN_GA(x1, x2) = \==_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x2, x5)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x4, x5)
\==_IN_GG(x1, x2) = \==_IN_GG(x1, x2)
U6_GGG(x1, x2, x3, x4, x5) = U6_GGG(x5)
TYPE_IN_GGA(x1, x2, x3) = TYPE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x6)
U5_GGG(x1, x2, x3, x4, x5) = U5_GGG(x2, x3, x4, x5)
U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
IN_IN_GGA(x1, x2, x3) = IN_IN_GGA(x1, x2)
IN_IN_GGG(x1, x2, x3) = IN_IN_GGG(x1, x2, x3)
TYPE_IN_AGA(x1, x2, x3) = TYPE_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x5)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

      ↳ PrologToPiTRSProof

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

TYPE_IN_GGA(E, var(X), T) → U1_GGA(E, X, T, in_in_gga(E, X, T))
TYPE_IN_GGA(E, var(X), T) → IN_IN_GGA(E, X, T)
IN_IN_GGA(.(','(Y, T), E), X, T) → U5_GGA(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGA(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGA(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGG(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGG(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
TYPE_IN_GGA(E, apply(M, N), T) → U2_GGA(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
TYPE_IN_GGA(E, apply(M, N), T) → TYPE_IN_GGA(E, M, arrow(S, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → U4_GGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, var(X), T) → U1_AGA(E, X, T, in_in_aga(E, X, T))
TYPE_IN_AGA(E, var(X), T) → IN_IN_AGA(E, X, T)
IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
IN_IN_AGA(.(','(Y, T), E), X, T) → \==_IN_GA(X, Y)
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → U6_AGA(Y, T, E, X, in_in_aga(E, X, T))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → U4_AGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_AGA(E, M, N, T, type_in_aga(E, N, S))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_GGA(E, M, N, T, type_in_gga(E, N, S))
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → TYPE_IN_GGA(E, N, S)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)

The TRS R consists of the following rules:

\==_in_ga(X0, X1) → \==_out_ga(X0, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
\==_in_ga(x1, x2) = \==_in_ga(x1)
\==_out_ga(x1, x2) = \==_out_ga
IN_IN_AGA(x1, x2, x3) = IN_IN_AGA(x2)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x4, x5)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U5_AGA(X, \==_out_ga) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_in_ga(X))

The TRS R consists of the following rules:

\==_in_ga(X0) → \==_out_ga

The set Q consists of the following terms:

\==_in_ga(x0)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule IN_IN_AGA(X) → U5_AGA(X, \==_in_ga(X)) at position [1] we obtained the following new rules:

IN_IN_AGA(X) → U5_AGA(X, \==_out_ga)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U5_AGA(X, \==_out_ga) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga)

The TRS R consists of the following rules:

\==_in_ga(X0) → \==_out_ga

The set Q consists of the following terms:

\==_in_ga(x0)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U5_AGA(X, \==_out_ga) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga)

R is empty.
The set Q consists of the following terms:

\==_in_ga(x0)

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

\==_in_ga(x0)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

                                        ↳ NonTerminationProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U5_AGA(X, \==_out_ga) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga)

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

The TRS P consists of the following rules:

U5_AGA(X, \==_out_ga) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga)

The TRS R consists of the following rules:none

s = IN_IN_AGA(X') evaluates to t =IN_IN_AGA(X')

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

IN_IN_AGA(X') → U5_AGA(X', \==_out_ga)
with rule IN_IN_AGA(X'') → U5_AGA(X'', \==_out_ga) at position [] and matcher [X'' / X']

U5_AGA(X', \==_out_ga) → IN_IN_AGA(X')
with rule U5_AGA(X, \==_out_ga) → IN_IN_AGA(X)

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

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))

The TRS R consists of the following rules:

type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)

The argument filtering Pi contains the following mapping:
var(x1) = var(x1)
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
apply(x1, x2) = apply(x1, x2)
lambda(x1, x2) = lambda(x1, x2)
type_in_aga(x1, x2, x3) = type_in_aga(x2)
U1_aga(x1, x2, x3, x4) = U1_aga(x4)
in_in_aga(x1, x2, x3) = in_in_aga(x2)
in_out_aga(x1, x2, x3) = in_out_aga
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x4, x5)
\==_in_ga(x1, x2) = \==_in_ga(x1)
\==_out_ga(x1, x2) = \==_out_ga
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
type_out_aga(x1, x2, x3) = type_out_aga
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5)
U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5)
U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5)
TYPE_IN_AGA(x1, x2, x3) = TYPE_IN_AGA(x2)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

TYPE_IN_AGA(apply(M, N)) → U2_AGA(N, type_in_aga(M))
TYPE_IN_AGA(lambda(X, M)) → TYPE_IN_AGA(M)
TYPE_IN_AGA(apply(M, N)) → TYPE_IN_AGA(M)
U2_AGA(N, type_out_aga) → TYPE_IN_AGA(N)

The TRS R consists of the following rules:

type_in_aga(var(X)) → U1_aga(in_in_aga(X))
type_in_aga(apply(M, N)) → U2_aga(N, type_in_aga(M))
type_in_aga(lambda(X, M)) → U4_aga(type_in_aga(M))
U1_aga(in_out_aga) → type_out_aga
U2_aga(N, type_out_aga) → U3_aga(type_in_aga(N))
U4_aga(type_out_aga) → type_out_aga
in_in_aga(X) → in_out_aga
in_in_aga(X) → U5_aga(X, \==_in_ga(X))
U3_aga(type_out_aga) → type_out_aga
U5_aga(X, \==_out_ga) → U6_aga(in_in_aga(X))
\==_in_ga(X0) → \==_out_ga
U6_aga(in_out_aga) → in_out_aga

The set Q consists of the following terms:

type_in_aga(x0)
U1_aga(x0)
U2_aga(x0, x1)
U4_aga(x0)
in_in_aga(x0)
U3_aga(x0)
U5_aga(x0, x1)
\==_in_ga(x0)
U6_aga(x0)

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

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

TYPE_IN_AGA(apply(M, N)) → U2_AGA(N, type_in_aga(M))
The graph contains the following edges 1 > 1
U2_AGA(N, type_out_aga) → TYPE_IN_AGA(N)
The graph contains the following edges 1 >= 1
TYPE_IN_AGA(lambda(X, M)) → TYPE_IN_AGA(M)
The graph contains the following edges 1 > 1
TYPE_IN_AGA(apply(M, N)) → TYPE_IN_AGA(M)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)

The TRS R consists of the following rules:

\==_in_gg(X0, X1) → \==_out_gg(X0, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
\==_in_gg(x1, x2) = \==_in_gg(x1, x2)
\==_out_gg(x1, x2) = \==_out_gg
U5_GGG(x1, x2, x3, x4, x5) = U5_GGG(x2, x3, x4, x5)
IN_IN_GGG(x1, x2, x3) = IN_IN_GGG(x1, x2, x3)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U5_GGG(T, E, X, \==_out_gg) → IN_IN_GGG(E, X, T)
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(T, E, X, \==_in_gg(X, Y))

The TRS R consists of the following rules:

\==_in_gg(X0, X1) → \==_out_gg

The set Q consists of the following terms:

\==_in_gg(x0, x1)

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

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

U5_GGG(T, E, X, \==_out_gg) → IN_IN_GGG(E, X, T)
The graph contains the following edges 2 >= 1, 3 >= 2, 1 >= 3
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(T, E, X, \==_in_gg(X, Y))
The graph contains the following edges 1 > 1, 3 >= 1, 1 > 2, 2 >= 3

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ PiDPToQDPProof

      ↳ PrologToPiTRSProof

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

TYPE_IN_GGA(E, apply(M, N), T) → TYPE_IN_GGA(E, M, arrow(S, T))
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → TYPE_IN_GGA(E, N, S)
TYPE_IN_GGA(E, apply(M, N), T) → U2_GGA(E, M, N, T, type_in_gga(E, M, arrow(S, T)))

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

      ↳ PrologToPiTRSProof

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

U2_GGA(E, N, type_out_gga) → TYPE_IN_GGA(E, N)
TYPE_IN_GGA(E, apply(M, N)) → U2_GGA(E, N, type_in_gga(E, M))
TYPE_IN_GGA(E, apply(M, N)) → TYPE_IN_GGA(E, M)

The TRS R consists of the following rules:

type_in_gga(E, var(X)) → U1_gga(in_in_gga(E, X))
in_in_gga(.(','(X, T), E), X) → in_out_gga(T)
in_in_gga(.(','(Y, T), E), X) → U5_gga(T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg
U5_gga(T, E, X, \==_out_gg) → U6_gga(T, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(T, E, X, \==_in_gg(X, Y))
U5_ggg(T, E, X, \==_out_gg) → U6_ggg(in_in_ggg(E, X, T))
U6_ggg(in_out_ggg) → in_out_ggg
U6_gga(T, in_out_ggg) → in_out_gga(T)
U1_gga(in_out_gga(T)) → type_out_gga
type_in_gga(E, apply(M, N)) → U2_gga(E, N, type_in_gga(E, M))
type_in_gga(E, lambda(X, M)) → U4_gga(type_in_aga(M))
type_in_aga(var(X)) → U1_aga(in_in_aga(X))
in_in_aga(X) → in_out_aga
in_in_aga(X) → U5_aga(X, \==_in_ga(X))
\==_in_ga(X0) → \==_out_ga
U5_aga(X, \==_out_ga) → U6_aga(in_in_aga(X))
U6_aga(in_out_aga) → in_out_aga
U1_aga(in_out_aga) → type_out_aga
type_in_aga(apply(M, N)) → U2_aga(N, type_in_aga(M))
type_in_aga(lambda(X, M)) → U4_aga(type_in_aga(M))
U4_aga(type_out_aga) → type_out_aga
U2_aga(N, type_out_aga) → U3_aga(type_in_aga(N))
U3_aga(type_out_aga) → type_out_aga
U4_gga(type_out_aga) → type_out_gga
U2_gga(E, N, type_out_gga) → U3_gga(type_in_gga(E, N))
U3_gga(type_out_gga) → type_out_gga

The set Q consists of the following terms:

type_in_gga(x0, x1)
in_in_gga(x0, x1)
\==_in_gg(x0, x1)
U5_gga(x0, x1, x2, x3)
in_in_ggg(x0, x1, x2)
U5_ggg(x0, x1, x2, x3)
U6_ggg(x0)
U6_gga(x0, x1)
U1_gga(x0)
type_in_aga(x0)
in_in_aga(x0)
\==_in_ga(x0)
U5_aga(x0, x1)
U6_aga(x0)
U1_aga(x0)
U4_aga(x0)
U2_aga(x0, x1)
U3_aga(x0)
U4_gga(x0)
U2_gga(x0, x1, x2)
U3_gga(x0)

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

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

TYPE_IN_GGA(E, apply(M, N)) → TYPE_IN_GGA(E, M)
The graph contains the following edges 1 >= 1, 2 > 2
TYPE_IN_GGA(E, apply(M, N)) → U2_GGA(E, N, type_in_gga(E, M))
The graph contains the following edges 1 >= 1, 2 > 2
U2_GGA(E, N, type_out_gga) → TYPE_IN_GGA(E, N)
The graph contains the following edges 1 >= 1, 2 >= 2

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
type_in: (b,b,f) (f,b,f)
in_in: (b,b,f) (b,b,b) (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

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

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

TYPE_IN_GGA(E, var(X), T) → U1_GGA(E, X, T, in_in_gga(E, X, T))
TYPE_IN_GGA(E, var(X), T) → IN_IN_GGA(E, X, T)
IN_IN_GGA(.(','(Y, T), E), X, T) → U5_GGA(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGA(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGA(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGG(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGG(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
TYPE_IN_GGA(E, apply(M, N), T) → U2_GGA(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
TYPE_IN_GGA(E, apply(M, N), T) → TYPE_IN_GGA(E, M, arrow(S, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → U4_GGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, var(X), T) → U1_AGA(E, X, T, in_in_aga(E, X, T))
TYPE_IN_AGA(E, var(X), T) → IN_IN_AGA(E, X, T)
IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
IN_IN_AGA(.(','(Y, T), E), X, T) → \==_IN_GA(X, Y)
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → U6_AGA(Y, T, E, X, in_in_aga(E, X, T))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → U4_AGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_AGA(E, M, N, T, type_in_aga(E, N, S))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_GGA(E, M, N, T, type_in_gga(E, N, S))
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → TYPE_IN_GGA(E, N, S)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

The argument filtering Pi contains the following mapping:
type_in_gga(x1, x2, x3) = type_in_gga(x1, x2)
var(x1) = var(x1)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
in_in_gga(x1, x2, x3) = in_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
in_out_gga(x1, x2, x3) = in_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x4, x5)
\==_in_gg(x1, x2) = \==_in_gg(x1, x2)
\==_out_gg(x1, x2) = \==_out_gg(x1, x2)
U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x4, x5)
in_in_ggg(x1, x2, x3) = in_in_ggg(x1, x2, x3)
in_out_ggg(x1, x2, x3) = in_out_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4, x5) = U5_ggg(x1, x2, x3, x4, x5)
U6_ggg(x1, x2, x3, x4, x5) = U6_ggg(x1, x2, x3, x4, x5)
type_out_gga(x1, x2, x3) = type_out_gga(x1, x2)
apply(x1, x2) = apply(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x2, x3, x5)
lambda(x1, x2) = lambda(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x6)
type_in_aga(x1, x2, x3) = type_in_aga(x2)
U1_aga(x1, x2, x3, x4) = U1_aga(x2, x4)
in_in_aga(x1, x2, x3) = in_in_aga(x2)
in_out_aga(x1, x2, x3) = in_out_aga(x2)
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x4, x5)
\==_in_ga(x1, x2) = \==_in_ga(x1)
\==_out_ga(x1, x2) = \==_out_ga(x1)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x4, x5)
type_out_aga(x1, x2, x3) = type_out_aga(x2)
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x2, x3, x5)
U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x2, x3, x6)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x3, x5)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5)
IN_IN_AGA(x1, x2, x3) = IN_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5, x6) = U4_AGA(x2, x3, x6)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x4, x5)
U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x4, x5)
U1_AGA(x1, x2, x3, x4) = U1_AGA(x2, x4)
\==_IN_GA(x1, x2) = \==_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x2, x3, x4, x5)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x4, x5)
\==_IN_GG(x1, x2) = \==_IN_GG(x1, x2)
U6_GGG(x1, x2, x3, x4, x5) = U6_GGG(x1, x2, x3, x4, x5)
TYPE_IN_GGA(x1, x2, x3) = TYPE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x6)
U5_GGG(x1, x2, x3, x4, x5) = U5_GGG(x1, x2, x3, x4, x5)
U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x2, x3, x5)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
IN_IN_GGA(x1, x2, x3) = IN_IN_GGA(x1, x2)
IN_IN_GGG(x1, x2, x3) = IN_IN_GGG(x1, x2, x3)
TYPE_IN_AGA(x1, x2, x3) = TYPE_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x3, x5)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

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

TYPE_IN_GGA(E, var(X), T) → U1_GGA(E, X, T, in_in_gga(E, X, T))
TYPE_IN_GGA(E, var(X), T) → IN_IN_GGA(E, X, T)
IN_IN_GGA(.(','(Y, T), E), X, T) → U5_GGA(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGA(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGA(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGA(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
IN_IN_GGG(.(','(Y, T), E), X, T) → \==_IN_GG(X, Y)
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → U6_GGG(Y, T, E, X, in_in_ggg(E, X, T))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
TYPE_IN_GGA(E, apply(M, N), T) → U2_GGA(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
TYPE_IN_GGA(E, apply(M, N), T) → TYPE_IN_GGA(E, M, arrow(S, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → U4_GGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_GGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, var(X), T) → U1_AGA(E, X, T, in_in_aga(E, X, T))
TYPE_IN_AGA(E, var(X), T) → IN_IN_AGA(E, X, T)
IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
IN_IN_AGA(.(','(Y, T), E), X, T) → \==_IN_GA(X, Y)
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → U6_AGA(Y, T, E, X, in_in_aga(E, X, T))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → U4_AGA(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_AGA(E, M, N, T, type_in_aga(E, N, S))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_GGA(E, M, N, T, type_in_gga(E, N, S))
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → TYPE_IN_GGA(E, N, S)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

IN_IN_AGA(.(','(Y, T), E), X, T) → U5_AGA(Y, T, E, X, \==_in_ga(X, Y))
U5_AGA(Y, T, E, X, \==_out_ga(X, Y)) → IN_IN_AGA(E, X, T)

The TRS R consists of the following rules:

\==_in_ga(X0, X1) → \==_out_ga(X0, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
\==_in_ga(x1, x2) = \==_in_ga(x1)
\==_out_ga(x1, x2) = \==_out_ga(x1)
IN_IN_AGA(x1, x2, x3) = IN_IN_AGA(x2)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x4, x5)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

U5_AGA(X, \==_out_ga(X)) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_in_ga(X))

The TRS R consists of the following rules:

\==_in_ga(X0) → \==_out_ga(X0)

The set Q consists of the following terms:

\==_in_ga(x0)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule IN_IN_AGA(X) → U5_AGA(X, \==_in_ga(X)) at position [1] we obtained the following new rules:

IN_IN_AGA(X) → U5_AGA(X, \==_out_ga(X))

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

U5_AGA(X, \==_out_ga(X)) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga(X))

The TRS R consists of the following rules:

\==_in_ga(X0) → \==_out_ga(X0)

The set Q consists of the following terms:

\==_in_ga(x0)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

U5_AGA(X, \==_out_ga(X)) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga(X))

R is empty.
The set Q consists of the following terms:

\==_in_ga(x0)

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

\==_in_ga(x0)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

                                        ↳ NonTerminationProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

U5_AGA(X, \==_out_ga(X)) → IN_IN_AGA(X)
IN_IN_AGA(X) → U5_AGA(X, \==_out_ga(X))

The TRS R consists of the following rules:none

s = IN_IN_AGA(X') evaluates to t =IN_IN_AGA(X')

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

IN_IN_AGA(X') → U5_AGA(X', \==_out_ga(X'))
with rule IN_IN_AGA(X'') → U5_AGA(X'', \==_out_ga(X'')) at position [] and matcher [X'' / X']

U5_AGA(X', \==_out_ga(X')) → IN_IN_AGA(X')
with rule U5_AGA(X, \==_out_ga(X)) → IN_IN_AGA(X)

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

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

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

TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

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

TYPE_IN_AGA(E, lambda(X, M), arrow(S, T)) → TYPE_IN_AGA(.(','(X, S), E), M, T)
TYPE_IN_AGA(E, apply(M, N), T) → TYPE_IN_AGA(E, M, arrow(S, T))
U2_AGA(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → TYPE_IN_AGA(E, N, S)
TYPE_IN_AGA(E, apply(M, N), T) → U2_AGA(E, M, N, T, type_in_aga(E, M, arrow(S, T)))

The TRS R consists of the following rules:

type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)

The argument filtering Pi contains the following mapping:
var(x1) = var(x1)
.(x1, x2) = .(x1, x2)
','(x1, x2) = ','(x1, x2)
apply(x1, x2) = apply(x1, x2)
lambda(x1, x2) = lambda(x1, x2)
type_in_aga(x1, x2, x3) = type_in_aga(x2)
U1_aga(x1, x2, x3, x4) = U1_aga(x2, x4)
in_in_aga(x1, x2, x3) = in_in_aga(x2)
in_out_aga(x1, x2, x3) = in_out_aga(x2)
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x4, x5)
\==_in_ga(x1, x2) = \==_in_ga(x1)
\==_out_ga(x1, x2) = \==_out_ga(x1)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x4, x5)
type_out_aga(x1, x2, x3) = type_out_aga(x2)
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x2, x3, x5)
U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x2, x3, x6)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x3, x5)
U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x2, x3, x5)
TYPE_IN_AGA(x1, x2, x3) = TYPE_IN_AGA(x2)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

                  ↳ PiDP

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

TYPE_IN_AGA(apply(M, N)) → U2_AGA(M, N, type_in_aga(M))
TYPE_IN_AGA(lambda(X, M)) → TYPE_IN_AGA(M)
U2_AGA(M, N, type_out_aga(M)) → TYPE_IN_AGA(N)
TYPE_IN_AGA(apply(M, N)) → TYPE_IN_AGA(M)

The TRS R consists of the following rules:

type_in_aga(var(X)) → U1_aga(X, in_in_aga(X))
type_in_aga(apply(M, N)) → U2_aga(M, N, type_in_aga(M))
type_in_aga(lambda(X, M)) → U4_aga(X, M, type_in_aga(M))
U1_aga(X, in_out_aga(X)) → type_out_aga(var(X))
U2_aga(M, N, type_out_aga(M)) → U3_aga(M, N, type_in_aga(N))
U4_aga(X, M, type_out_aga(M)) → type_out_aga(lambda(X, M))
in_in_aga(X) → in_out_aga(X)
in_in_aga(X) → U5_aga(X, \==_in_ga(X))
U3_aga(M, N, type_out_aga(N)) → type_out_aga(apply(M, N))
U5_aga(X, \==_out_ga(X)) → U6_aga(X, in_in_aga(X))
\==_in_ga(X0) → \==_out_ga(X0)
U6_aga(X, in_out_aga(X)) → in_out_aga(X)

The set Q consists of the following terms:

type_in_aga(x0)
U1_aga(x0, x1)
U2_aga(x0, x1, x2)
U4_aga(x0, x1, x2)
in_in_aga(x0)
U3_aga(x0, x1, x2)
U5_aga(x0, x1)
\==_in_ga(x0)
U6_aga(x0, x1)

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

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

TYPE_IN_AGA(apply(M, N)) → U2_AGA(M, N, type_in_aga(M))
The graph contains the following edges 1 > 1, 1 > 2
U2_AGA(M, N, type_out_aga(M)) → TYPE_IN_AGA(N)
The graph contains the following edges 2 >= 1
TYPE_IN_AGA(lambda(X, M)) → TYPE_IN_AGA(M)
The graph contains the following edges 1 > 1
TYPE_IN_AGA(apply(M, N)) → TYPE_IN_AGA(M)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

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

IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

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

IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)

The TRS R consists of the following rules:

\==_in_gg(X0, X1) → \==_out_gg(X0, X1)

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

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

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

IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)

The TRS R consists of the following rules:

\==_in_gg(X0, X1) → \==_out_gg(X0, X1)

The set Q consists of the following terms:

\==_in_gg(x0, x1)

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

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

U5_GGG(Y, T, E, X, \==_out_gg(X, Y)) → IN_IN_GGG(E, X, T)
The graph contains the following edges 3 >= 1, 4 >= 2, 5 > 2, 2 >= 3
IN_IN_GGG(.(','(Y, T), E), X, T) → U5_GGG(Y, T, E, X, \==_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 2, 1 > 3, 2 >= 4

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

TYPE_IN_GGA(E, apply(M, N), T) → TYPE_IN_GGA(E, M, arrow(S, T))
U2_GGA(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → TYPE_IN_GGA(E, N, S)
TYPE_IN_GGA(E, apply(M, N), T) → U2_GGA(E, M, N, T, type_in_gga(E, M, arrow(S, T)))

The TRS R consists of the following rules:

type_in_gga(E, var(X), T) → U1_gga(E, X, T, in_in_gga(E, X, T))
in_in_gga(.(','(X, T), E), X, T) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X, T) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, T, in_out_gga(E, X, T)) → type_out_gga(E, var(X), T)
type_in_gga(E, apply(M, N), T) → U2_gga(E, M, N, T, type_in_gga(E, M, arrow(S, T)))
type_in_gga(E, lambda(X, M), arrow(S, T)) → U4_gga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
type_in_aga(E, var(X), T) → U1_aga(E, X, T, in_in_aga(E, X, T))
in_in_aga(.(','(X, T), E), X, T) → in_out_aga(.(','(X, T), E), X, T)
in_in_aga(.(','(Y, T), E), X, T) → U5_aga(Y, T, E, X, \==_in_ga(X, Y))
\==_in_ga(X0, X1) → \==_out_ga(X0, X1)
U5_aga(Y, T, E, X, \==_out_ga(X, Y)) → U6_aga(Y, T, E, X, in_in_aga(E, X, T))
U6_aga(Y, T, E, X, in_out_aga(E, X, T)) → in_out_aga(.(','(Y, T), E), X, T)
U1_aga(E, X, T, in_out_aga(E, X, T)) → type_out_aga(E, var(X), T)
type_in_aga(E, apply(M, N), T) → U2_aga(E, M, N, T, type_in_aga(E, M, arrow(S, T)))
type_in_aga(E, lambda(X, M), arrow(S, T)) → U4_aga(E, X, M, S, T, type_in_aga(.(','(X, S), E), M, T))
U4_aga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_aga(E, lambda(X, M), arrow(S, T))
U2_aga(E, M, N, T, type_out_aga(E, M, arrow(S, T))) → U3_aga(E, M, N, T, type_in_aga(E, N, S))
U3_aga(E, M, N, T, type_out_aga(E, N, S)) → type_out_aga(E, apply(M, N), T)
U4_gga(E, X, M, S, T, type_out_aga(.(','(X, S), E), M, T)) → type_out_gga(E, lambda(X, M), arrow(S, T))
U2_gga(E, M, N, T, type_out_gga(E, M, arrow(S, T))) → U3_gga(E, M, N, T, type_in_gga(E, N, S))
U3_gga(E, M, N, T, type_out_gga(E, N, S)) → type_out_gga(E, apply(M, N), T)

↳ Prolog

  ↳ UndefinedPredicateHandlerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

U2_GGA(E, M, N, type_out_gga(E, M)) → TYPE_IN_GGA(E, N)
TYPE_IN_GGA(E, apply(M, N)) → U2_GGA(E, M, N, type_in_gga(E, M))
TYPE_IN_GGA(E, apply(M, N)) → TYPE_IN_GGA(E, M)

The TRS R consists of the following rules:

type_in_gga(E, var(X)) → U1_gga(E, X, in_in_gga(E, X))
in_in_gga(.(','(X, T), E), X) → in_out_gga(.(','(X, T), E), X, T)
in_in_gga(.(','(Y, T), E), X) → U5_gga(Y, T, E, X, \==_in_gg(X, Y))
\==_in_gg(X0, X1) → \==_out_gg(X0, X1)
U5_gga(Y, T, E, X, \==_out_gg(X, Y)) → U6_gga(Y, T, E, X, in_in_ggg(E, X, T))
in_in_ggg(.(','(X, T), E), X, T) → in_out_ggg(.(','(X, T), E), X, T)
in_in_ggg(.(','(Y, T), E), X, T) → U5_ggg(Y, T, E, X, \==_in_gg(X, Y))
U5_ggg(Y, T, E, X, \==_out_gg(X, Y)) → U6_ggg(Y, T, E, X, in_in_ggg(E, X, T))
U6_ggg(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_ggg(.(','(Y, T), E), X, T)
U6_gga(Y, T, E, X, in_out_ggg(E, X, T)) → in_out_gga(.(','(Y, T), E), X, T)
U1_gga(E, X, in_out_gga(E, X, T)) → type_out_gga(E, var(X))
type_in_gga(E, apply(M, N)) → U2_gga(E, M, N, type_in_gga(E, M))
type_in_gga(E, lambda(X, M)) → U4_gga(E, X, M, type_in_aga(M))
type_in_aga(var(X)) → U1_aga(X, in_in_aga(X))
in_in_aga(X) → in_out_aga(X)
in_in_aga(X) → U5_aga(X, \==_in_ga(X))
\==_in_ga(X0) → \==_out_ga(X0)
U5_aga(X, \==_out_ga(X)) → U6_aga(X, in_in_aga(X))
U6_aga(X, in_out_aga(X)) → in_out_aga(X)
U1_aga(X, in_out_aga(X)) → type_out_aga(var(X))
type_in_aga(apply(M, N)) → U2_aga(M, N, type_in_aga(M))
type_in_aga(lambda(X, M)) → U4_aga(X, M, type_in_aga(M))
U4_aga(X, M, type_out_aga(M)) → type_out_aga(lambda(X, M))
U2_aga(M, N, type_out_aga(M)) → U3_aga(M, N, type_in_aga(N))
U3_aga(M, N, type_out_aga(N)) → type_out_aga(apply(M, N))
U4_gga(E, X, M, type_out_aga(M)) → type_out_gga(E, lambda(X, M))
U2_gga(E, M, N, type_out_gga(E, M)) → U3_gga(E, M, N, type_in_gga(E, N))
U3_gga(E, M, N, type_out_gga(E, N)) → type_out_gga(E, apply(M, N))

The set Q consists of the following terms:

type_in_gga(x0, x1)
in_in_gga(x0, x1)
\==_in_gg(x0, x1)
U5_gga(x0, x1, x2, x3, x4)
in_in_ggg(x0, x1, x2)
U5_ggg(x0, x1, x2, x3, x4)
U6_ggg(x0, x1, x2, x3, x4)
U6_gga(x0, x1, x2, x3, x4)
U1_gga(x0, x1, x2)
type_in_aga(x0)
in_in_aga(x0)
\==_in_ga(x0)
U5_aga(x0, x1)
U6_aga(x0, x1)
U1_aga(x0, x1)
U4_aga(x0, x1, x2)
U2_aga(x0, x1, x2)
U3_aga(x0, x1, x2)
U4_gga(x0, x1, x2, x3)
U2_gga(x0, x1, x2, x3)
U3_gga(x0, x1, x2, x3)

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

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

TYPE_IN_GGA(E, apply(M, N)) → TYPE_IN_GGA(E, M)
The graph contains the following edges 1 >= 1, 2 > 2
TYPE_IN_GGA(E, apply(M, N)) → U2_GGA(E, M, N, type_in_gga(E, M))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
U2_GGA(E, M, N, type_out_gga(E, M)) → TYPE_IN_GGA(E, N)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2