YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/fWXdi.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

perm1(g,g)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) PiDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) PiDP
        (10) PiDPToQDPProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) PiDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) PiDP
        (17) PiDPToQDPProof [SOUND, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) PiDP
        (22) UsableRulesProof [EQUIVALENT, 0 ms]
        (23) PiDP
        (24) PiDPToQDPProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (27) YES


----------------------------------------

(0)
Obligation:
Clauses:

perm1(L, M) :- ','(eq_len1(L, M), same_sets(L, M)).
eq_len1([], []).
eq_len1(.(X1, Xs), .(X2, Ys)) :- eq_len1(Xs, Ys).
member(X, .(X, X3)).
member(X, .(X4, T)) :- member(X, T).
same_sets([], X5).
same_sets(.(X, Xs), L) :- ','(member(X, L), same_sets(Xs, L)).


Query: perm1(g,g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

perm1_in_2: (b,b)

eq_len1_in_2: (b,b)

same_sets_in_2: (b,b)

member_in_2: (b,b)

Transforming Prolog into the following Term Rewriting System:

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

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg



----------------------------------------

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

   PERM1_IN_GG(L, M) -> U1_GG(L, M, eq_len1_in_gg(L, M))
   PERM1_IN_GG(L, M) -> EQ_LEN1_IN_GG(L, M)
   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> U3_GG(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> EQ_LEN1_IN_GG(Xs, Ys)
   U1_GG(L, M, eq_len1_out_gg(L, M)) -> U2_GG(L, M, same_sets_in_gg(L, M))
   U1_GG(L, M, eq_len1_out_gg(L, M)) -> SAME_SETS_IN_GG(L, M)
   SAME_SETS_IN_GG(.(X, Xs), L) -> U5_GG(X, Xs, L, member_in_gg(X, L))
   SAME_SETS_IN_GG(.(X, Xs), L) -> MEMBER_IN_GG(X, L)
   MEMBER_IN_GG(X, .(X4, T)) -> U4_GG(X, X4, T, member_in_gg(X, T))
   MEMBER_IN_GG(X, .(X4, T)) -> MEMBER_IN_GG(X, T)
   U5_GG(X, Xs, L, member_out_gg(X, L)) -> U6_GG(X, Xs, L, same_sets_in_gg(Xs, L))
   U5_GG(X, Xs, L, member_out_gg(X, L)) -> SAME_SETS_IN_GG(Xs, L)

The TRS R consists of the following rules:

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg

PERM1_IN_GG(x1, x2)  =  PERM1_IN_GG(x1, x2)

U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)

EQ_LEN1_IN_GG(x1, x2)  =  EQ_LEN1_IN_GG(x1, x2)

U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)

U2_GG(x1, x2, x3)  =  U2_GG(x3)

SAME_SETS_IN_GG(x1, x2)  =  SAME_SETS_IN_GG(x1, x2)

U5_GG(x1, x2, x3, x4)  =  U5_GG(x2, x3, x4)

MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)

U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)

U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)


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

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PERM1_IN_GG(L, M) -> U1_GG(L, M, eq_len1_in_gg(L, M))
   PERM1_IN_GG(L, M) -> EQ_LEN1_IN_GG(L, M)
   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> U3_GG(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> EQ_LEN1_IN_GG(Xs, Ys)
   U1_GG(L, M, eq_len1_out_gg(L, M)) -> U2_GG(L, M, same_sets_in_gg(L, M))
   U1_GG(L, M, eq_len1_out_gg(L, M)) -> SAME_SETS_IN_GG(L, M)
   SAME_SETS_IN_GG(.(X, Xs), L) -> U5_GG(X, Xs, L, member_in_gg(X, L))
   SAME_SETS_IN_GG(.(X, Xs), L) -> MEMBER_IN_GG(X, L)
   MEMBER_IN_GG(X, .(X4, T)) -> U4_GG(X, X4, T, member_in_gg(X, T))
   MEMBER_IN_GG(X, .(X4, T)) -> MEMBER_IN_GG(X, T)
   U5_GG(X, Xs, L, member_out_gg(X, L)) -> U6_GG(X, Xs, L, same_sets_in_gg(Xs, L))
   U5_GG(X, Xs, L, member_out_gg(X, L)) -> SAME_SETS_IN_GG(Xs, L)

The TRS R consists of the following rules:

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg

PERM1_IN_GG(x1, x2)  =  PERM1_IN_GG(x1, x2)

U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)

EQ_LEN1_IN_GG(x1, x2)  =  EQ_LEN1_IN_GG(x1, x2)

U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)

U2_GG(x1, x2, x3)  =  U2_GG(x3)

SAME_SETS_IN_GG(x1, x2)  =  SAME_SETS_IN_GG(x1, x2)

U5_GG(x1, x2, x3, x4)  =  U5_GG(x2, x3, x4)

MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)

U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)

U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MEMBER_IN_GG(X, .(X4, T)) -> MEMBER_IN_GG(X, T)

The TRS R consists of the following rules:

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg

MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)


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

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

(9)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MEMBER_IN_GG(X, .(X4, T)) -> MEMBER_IN_GG(X, T)

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

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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MEMBER_IN_GG(X, .(X4, T)) -> MEMBER_IN_GG(X, T)

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

(12) QDPSizeChangeProof (EQUIVALENT)
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:
*MEMBER_IN_GG(X, .(X4, T)) -> MEMBER_IN_GG(X, T)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(13)
YES

----------------------------------------

(14)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GG(X, Xs, L, member_out_gg(X, L)) -> SAME_SETS_IN_GG(Xs, L)
   SAME_SETS_IN_GG(.(X, Xs), L) -> U5_GG(X, Xs, L, member_in_gg(X, L))

The TRS R consists of the following rules:

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg

SAME_SETS_IN_GG(x1, x2)  =  SAME_SETS_IN_GG(x1, x2)

U5_GG(x1, x2, x3, x4)  =  U5_GG(x2, x3, x4)


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

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

(16)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GG(X, Xs, L, member_out_gg(X, L)) -> SAME_SETS_IN_GG(Xs, L)
   SAME_SETS_IN_GG(.(X, Xs), L) -> U5_GG(X, Xs, L, member_in_gg(X, L))

The TRS R consists of the following rules:

   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))

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

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

SAME_SETS_IN_GG(x1, x2)  =  SAME_SETS_IN_GG(x1, x2)

U5_GG(x1, x2, x3, x4)  =  U5_GG(x2, x3, x4)


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

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GG(Xs, L, member_out_gg) -> SAME_SETS_IN_GG(Xs, L)
   SAME_SETS_IN_GG(.(X, Xs), L) -> U5_GG(Xs, L, member_in_gg(X, L))

The TRS R consists of the following rules:

   member_in_gg(X, .(X, X3)) -> member_out_gg
   member_in_gg(X, .(X4, T)) -> U4_gg(member_in_gg(X, T))
   U4_gg(member_out_gg) -> member_out_gg

The set Q consists of the following terms:

   member_in_gg(x0, x1)
   U4_gg(x0)

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

(19) QDPSizeChangeProof (EQUIVALENT)
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:
*SAME_SETS_IN_GG(.(X, Xs), L) -> U5_GG(Xs, L, member_in_gg(X, L))
The graph contains the following edges 1 > 1, 2 >= 2


*U5_GG(Xs, L, member_out_gg) -> SAME_SETS_IN_GG(Xs, L)
The graph contains the following edges 1 >= 1, 2 >= 2


----------------------------------------

(20)
YES

----------------------------------------

(21)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> EQ_LEN1_IN_GG(Xs, Ys)

The TRS R consists of the following rules:

   perm1_in_gg(L, M) -> U1_gg(L, M, eq_len1_in_gg(L, M))
   eq_len1_in_gg([], []) -> eq_len1_out_gg([], [])
   eq_len1_in_gg(.(X1, Xs), .(X2, Ys)) -> U3_gg(X1, Xs, X2, Ys, eq_len1_in_gg(Xs, Ys))
   U3_gg(X1, Xs, X2, Ys, eq_len1_out_gg(Xs, Ys)) -> eq_len1_out_gg(.(X1, Xs), .(X2, Ys))
   U1_gg(L, M, eq_len1_out_gg(L, M)) -> U2_gg(L, M, same_sets_in_gg(L, M))
   same_sets_in_gg([], X5) -> same_sets_out_gg([], X5)
   same_sets_in_gg(.(X, Xs), L) -> U5_gg(X, Xs, L, member_in_gg(X, L))
   member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3))
   member_in_gg(X, .(X4, T)) -> U4_gg(X, X4, T, member_in_gg(X, T))
   U4_gg(X, X4, T, member_out_gg(X, T)) -> member_out_gg(X, .(X4, T))
   U5_gg(X, Xs, L, member_out_gg(X, L)) -> U6_gg(X, Xs, L, same_sets_in_gg(Xs, L))
   U6_gg(X, Xs, L, same_sets_out_gg(Xs, L)) -> same_sets_out_gg(.(X, Xs), L)
   U2_gg(L, M, same_sets_out_gg(L, M)) -> perm1_out_gg(L, M)

The argument filtering Pi contains the following mapping:
perm1_in_gg(x1, x2)  =  perm1_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)

eq_len1_in_gg(x1, x2)  =  eq_len1_in_gg(x1, x2)

[]  =  []

eq_len1_out_gg(x1, x2)  =  eq_len1_out_gg

.(x1, x2)  =  .(x1, x2)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

same_sets_in_gg(x1, x2)  =  same_sets_in_gg(x1, x2)

same_sets_out_gg(x1, x2)  =  same_sets_out_gg

U5_gg(x1, x2, x3, x4)  =  U5_gg(x2, x3, x4)

member_in_gg(x1, x2)  =  member_in_gg(x1, x2)

member_out_gg(x1, x2)  =  member_out_gg

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)

perm1_out_gg(x1, x2)  =  perm1_out_gg

EQ_LEN1_IN_GG(x1, x2)  =  EQ_LEN1_IN_GG(x1, x2)


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

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

(23)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> EQ_LEN1_IN_GG(Xs, Ys)

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

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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> EQ_LEN1_IN_GG(Xs, Ys)

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

(26) QDPSizeChangeProof (EQUIVALENT)
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:
*EQ_LEN1_IN_GG(.(X1, Xs), .(X2, Ys)) -> EQ_LEN1_IN_GG(Xs, Ys)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(27)
YES
