(0) Obligation:

Clauses:

overlap(Xs, Ys) :- ','(member(X, Xs), member(X, Ys)).
member(X1, []) :- ','(!, failure(a)).
member(X, Y) :- head(Y, X).
member(X, Y) :- ','(tail(Y, T), member(X, T)).
head([], X2).
head(.(H, X3), H).
tail([], []).
tail(.(X4, T), T).
failure(b).

Queries:

overlap(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

member4(T7, .(T7, T8)).
member4(X31, .(T10, T11)) :- member4(X31, T11).
member5(T16, .(T16, T17)).
member5(T20, .(T21, T22)) :- member5(T20, T22).
overlap1(T3, T4) :- member4(X9, T3).
overlap1(T3, .(T16, T17)) :- member4(T16, T3).
overlap1(T3, .(T21, T22)) :- ','(member4(T20, T3), member5(T20, T22)).

Queries:

overlap1(g,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
overlap1_in: (b,b)
member4_in: (f,b) (b,b)
member5_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

OVERLAP1_IN_GG(T3, T4) → U3_GG(T3, T4, member4_in_ag(X9, T3))
OVERLAP1_IN_GG(T3, T4) → MEMBER4_IN_AG(X9, T3)
MEMBER4_IN_AG(X31, .(T10, T11)) → U1_AG(X31, T10, T11, member4_in_ag(X31, T11))
MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)
OVERLAP1_IN_GG(T3, .(T16, T17)) → U4_GG(T3, T16, T17, member4_in_gg(T16, T3))
OVERLAP1_IN_GG(T3, .(T16, T17)) → MEMBER4_IN_GG(T16, T3)
MEMBER4_IN_GG(X31, .(T10, T11)) → U1_GG(X31, T10, T11, member4_in_gg(X31, T11))
MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)
OVERLAP1_IN_GG(T3, .(T21, T22)) → U5_GG(T3, T21, T22, member4_in_ag(T20, T3))
OVERLAP1_IN_GG(T3, .(T21, T22)) → MEMBER4_IN_AG(T20, T3)
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → U6_GG(T3, T21, T22, member5_in_gg(T20, T22))
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → MEMBER5_IN_GG(T20, T22)
MEMBER5_IN_GG(T20, .(T21, T22)) → U2_GG(T20, T21, T22, member5_in_gg(T20, T22))
MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)
MEMBER4_IN_GG(x1, x2)  =  MEMBER4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
MEMBER5_IN_GG(x1, x2)  =  MEMBER5_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(6) Obligation:

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

OVERLAP1_IN_GG(T3, T4) → U3_GG(T3, T4, member4_in_ag(X9, T3))
OVERLAP1_IN_GG(T3, T4) → MEMBER4_IN_AG(X9, T3)
MEMBER4_IN_AG(X31, .(T10, T11)) → U1_AG(X31, T10, T11, member4_in_ag(X31, T11))
MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)
OVERLAP1_IN_GG(T3, .(T16, T17)) → U4_GG(T3, T16, T17, member4_in_gg(T16, T3))
OVERLAP1_IN_GG(T3, .(T16, T17)) → MEMBER4_IN_GG(T16, T3)
MEMBER4_IN_GG(X31, .(T10, T11)) → U1_GG(X31, T10, T11, member4_in_gg(X31, T11))
MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)
OVERLAP1_IN_GG(T3, .(T21, T22)) → U5_GG(T3, T21, T22, member4_in_ag(T20, T3))
OVERLAP1_IN_GG(T3, .(T21, T22)) → MEMBER4_IN_AG(T20, T3)
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → U6_GG(T3, T21, T22, member5_in_gg(T20, T22))
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → MEMBER5_IN_GG(T20, T22)
MEMBER5_IN_GG(T20, .(T21, T22)) → U2_GG(T20, T21, T22, member5_in_gg(T20, T22))
MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)
MEMBER4_IN_GG(x1, x2)  =  MEMBER4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
MEMBER5_IN_GG(x1, x2)  =  MEMBER5_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
MEMBER5_IN_GG(x1, x2)  =  MEMBER5_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

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

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

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

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)
    The graph contains the following edges 1 >= 1, 2 > 2

(15) TRUE

(16) Obligation:

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

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
MEMBER4_IN_GG(x1, x2)  =  MEMBER4_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

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

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

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

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)
    The graph contains the following edges 1 >= 1, 2 > 2

(22) TRUE

(23) Obligation:

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

MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

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

MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)

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

(26) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(27) Obligation:

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

MEMBER4_IN_AG(.(T10, T11)) → MEMBER4_IN_AG(T11)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • MEMBER4_IN_AG(.(T10, T11)) → MEMBER4_IN_AG(T11)
    The graph contains the following edges 1 > 1

(29) TRUE