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# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

q(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) PiDP
(7) PiDPToQDPProof [SOUND, 5 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 68 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) QDP
(13) UsableRulesProof [EQUIVALENT, 0 ms]
(14) QDP
(15) QReductionProof [EQUIVALENT, 0 ms]
(16) QDP
(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(18) YES


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

(0)
Obligation:
Clauses:

e(a, b).
q(X, Y) :- e(X, Y).
q(X, f(f(X))) :- ','(p(X, f(f(X))), q(X, f(X))).
q(X, f(f(Y))) :- p(X, f(Y)).
p(X, Y) :- e(X, Y).
p(X, f(Y)) :- ','(r(X, f(Y)), p(X, Y)).
r(X, Y) :- e(X, Y).
r(X, f(Y)) :- ','(q(X, Y), r(X, Y)).
r(f(X), f(X)) :- t(f(X), f(X)).
t(X, Y) :- e(X, Y).
t(f(X), f(Y)) :- ','(q(f(X), f(Y)), t(X, Y)).


Query: q(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:

q_in_2: (b,b)

p_in_2: (b,b)

r_in_2: (b,b)

t_in_2: (b,b)

Transforming Prolog into the following Term Rewriting System:

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

   q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg(a, b)
   U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y)
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y))
   U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y)
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y))
   U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y)
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y)))
   U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y)))
   U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y))
   U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y)
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y))
   U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y))
   U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X))
   U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y))
   U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y))
   U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y))
   U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X)))
   U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X)))

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

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

e_in_gg(x1, x2)  =  e_in_gg(x1, x2)

a  =  a

b  =  b

e_out_gg(x1, x2)  =  e_out_gg

q_out_gg(x1, x2)  =  q_out_gg

f(x1)  =  f(x1)

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

p_in_gg(x1, x2)  =  p_in_gg(x1, x2)

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

p_out_gg(x1, x2)  =  p_out_gg

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

r_in_gg(x1, x2)  =  r_in_gg(x1, x2)

U8_gg(x1, x2, x3)  =  U8_gg(x3)

r_out_gg(x1, x2)  =  r_out_gg

U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

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

U10_gg(x1, x2, x3)  =  U10_gg(x3)

U11_gg(x1, x2)  =  U11_gg(x2)

t_in_gg(x1, x2)  =  t_in_gg(x1, x2)

U12_gg(x1, x2, x3)  =  U12_gg(x3)

t_out_gg(x1, x2)  =  t_out_gg

U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)

U14_gg(x1, x2, x3)  =  U14_gg(x3)

U7_gg(x1, x2, x3)  =  U7_gg(x3)

U3_gg(x1, x2)  =  U3_gg(x2)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg(a, b)
   U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y)
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y))
   U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y)
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y))
   U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y)
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y)))
   U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y)))
   U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y))
   U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y)
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y))
   U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y))
   U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X))
   U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y))
   U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y))
   U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y))
   U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X)))
   U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X)))

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

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

e_in_gg(x1, x2)  =  e_in_gg(x1, x2)

a  =  a

b  =  b

e_out_gg(x1, x2)  =  e_out_gg

q_out_gg(x1, x2)  =  q_out_gg

f(x1)  =  f(x1)

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

p_in_gg(x1, x2)  =  p_in_gg(x1, x2)

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

p_out_gg(x1, x2)  =  p_out_gg

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

r_in_gg(x1, x2)  =  r_in_gg(x1, x2)

U8_gg(x1, x2, x3)  =  U8_gg(x3)

r_out_gg(x1, x2)  =  r_out_gg

U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

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

U10_gg(x1, x2, x3)  =  U10_gg(x3)

U11_gg(x1, x2)  =  U11_gg(x2)

t_in_gg(x1, x2)  =  t_in_gg(x1, x2)

U12_gg(x1, x2, x3)  =  U12_gg(x3)

t_out_gg(x1, x2)  =  t_out_gg

U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)

U14_gg(x1, x2, x3)  =  U14_gg(x3)

U7_gg(x1, x2, x3)  =  U7_gg(x3)

U3_gg(x1, x2)  =  U3_gg(x2)



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

(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:

   Q_IN_GG(X, Y) -> U1_GG(X, Y, e_in_gg(X, Y))
   Q_IN_GG(X, Y) -> E_IN_GG(X, Y)
   Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X))))
   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, Y) -> U5_GG(X, Y, e_in_gg(X, Y))
   P_IN_GG(X, Y) -> E_IN_GG(X, Y)
   P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y)))
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(X, Y) -> U8_GG(X, Y, e_in_gg(X, Y))
   R_IN_GG(X, Y) -> E_IN_GG(X, Y)
   R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y))
   R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y)
   Q_IN_GG(X, f(f(Y))) -> U4_GG(X, Y, p_in_gg(X, f(Y)))
   Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y))
   U9_GG(X, Y, q_out_gg(X, Y)) -> U10_GG(X, Y, r_in_gg(X, Y))
   U9_GG(X, Y, q_out_gg(X, Y)) -> R_IN_GG(X, Y)
   R_IN_GG(f(X), f(X)) -> U11_GG(X, t_in_gg(f(X), f(X)))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(X, Y) -> U12_GG(X, Y, e_in_gg(X, Y))
   T_IN_GG(X, Y) -> E_IN_GG(X, Y)
   T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y)))
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))
   U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> U14_GG(X, Y, t_in_gg(X, Y))
   U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> T_IN_GG(X, Y)
   U6_GG(X, Y, r_out_gg(X, f(Y))) -> U7_GG(X, Y, p_in_gg(X, Y))
   U6_GG(X, Y, r_out_gg(X, f(Y))) -> P_IN_GG(X, Y)
   U2_GG(X, p_out_gg(X, f(f(X)))) -> U3_GG(X, q_in_gg(X, f(X)))
   U2_GG(X, p_out_gg(X, f(f(X)))) -> Q_IN_GG(X, f(X))

The TRS R consists of the following rules:

   q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg(a, b)
   U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y)
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y))
   U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y)
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y))
   U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y)
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y)))
   U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y)))
   U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y))
   U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y)
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y))
   U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y))
   U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X))
   U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y))
   U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y))
   U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y))
   U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X)))
   U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X)))

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

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

e_in_gg(x1, x2)  =  e_in_gg(x1, x2)

a  =  a

b  =  b

e_out_gg(x1, x2)  =  e_out_gg

q_out_gg(x1, x2)  =  q_out_gg

f(x1)  =  f(x1)

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

p_in_gg(x1, x2)  =  p_in_gg(x1, x2)

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

p_out_gg(x1, x2)  =  p_out_gg

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

r_in_gg(x1, x2)  =  r_in_gg(x1, x2)

U8_gg(x1, x2, x3)  =  U8_gg(x3)

r_out_gg(x1, x2)  =  r_out_gg

U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

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

U10_gg(x1, x2, x3)  =  U10_gg(x3)

U11_gg(x1, x2)  =  U11_gg(x2)

t_in_gg(x1, x2)  =  t_in_gg(x1, x2)

U12_gg(x1, x2, x3)  =  U12_gg(x3)

t_out_gg(x1, x2)  =  t_out_gg

U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)

U14_gg(x1, x2, x3)  =  U14_gg(x3)

U7_gg(x1, x2, x3)  =  U7_gg(x3)

U3_gg(x1, x2)  =  U3_gg(x2)

Q_IN_GG(x1, x2)  =  Q_IN_GG(x1, x2)

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

E_IN_GG(x1, x2)  =  E_IN_GG(x1, x2)

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

P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)

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

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

R_IN_GG(x1, x2)  =  R_IN_GG(x1, x2)

U8_GG(x1, x2, x3)  =  U8_GG(x3)

U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)

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

U10_GG(x1, x2, x3)  =  U10_GG(x3)

U11_GG(x1, x2)  =  U11_GG(x2)

T_IN_GG(x1, x2)  =  T_IN_GG(x1, x2)

U12_GG(x1, x2, x3)  =  U12_GG(x3)

U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)

U14_GG(x1, x2, x3)  =  U14_GG(x3)

U7_GG(x1, x2, x3)  =  U7_GG(x3)

U3_GG(x1, x2)  =  U3_GG(x2)


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

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

   Q_IN_GG(X, Y) -> U1_GG(X, Y, e_in_gg(X, Y))
   Q_IN_GG(X, Y) -> E_IN_GG(X, Y)
   Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X))))
   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, Y) -> U5_GG(X, Y, e_in_gg(X, Y))
   P_IN_GG(X, Y) -> E_IN_GG(X, Y)
   P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y)))
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(X, Y) -> U8_GG(X, Y, e_in_gg(X, Y))
   R_IN_GG(X, Y) -> E_IN_GG(X, Y)
   R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y))
   R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y)
   Q_IN_GG(X, f(f(Y))) -> U4_GG(X, Y, p_in_gg(X, f(Y)))
   Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y))
   U9_GG(X, Y, q_out_gg(X, Y)) -> U10_GG(X, Y, r_in_gg(X, Y))
   U9_GG(X, Y, q_out_gg(X, Y)) -> R_IN_GG(X, Y)
   R_IN_GG(f(X), f(X)) -> U11_GG(X, t_in_gg(f(X), f(X)))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(X, Y) -> U12_GG(X, Y, e_in_gg(X, Y))
   T_IN_GG(X, Y) -> E_IN_GG(X, Y)
   T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y)))
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))
   U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> U14_GG(X, Y, t_in_gg(X, Y))
   U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> T_IN_GG(X, Y)
   U6_GG(X, Y, r_out_gg(X, f(Y))) -> U7_GG(X, Y, p_in_gg(X, Y))
   U6_GG(X, Y, r_out_gg(X, f(Y))) -> P_IN_GG(X, Y)
   U2_GG(X, p_out_gg(X, f(f(X)))) -> U3_GG(X, q_in_gg(X, f(X)))
   U2_GG(X, p_out_gg(X, f(f(X)))) -> Q_IN_GG(X, f(X))

The TRS R consists of the following rules:

   q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg(a, b)
   U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y)
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y))
   U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y)
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y))
   U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y)
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y)))
   U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y)))
   U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y))
   U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y)
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y))
   U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y))
   U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X))
   U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y))
   U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y))
   U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y))
   U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X)))
   U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X)))

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

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

e_in_gg(x1, x2)  =  e_in_gg(x1, x2)

a  =  a

b  =  b

e_out_gg(x1, x2)  =  e_out_gg

q_out_gg(x1, x2)  =  q_out_gg

f(x1)  =  f(x1)

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

p_in_gg(x1, x2)  =  p_in_gg(x1, x2)

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

p_out_gg(x1, x2)  =  p_out_gg

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

r_in_gg(x1, x2)  =  r_in_gg(x1, x2)

U8_gg(x1, x2, x3)  =  U8_gg(x3)

r_out_gg(x1, x2)  =  r_out_gg

U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

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

U10_gg(x1, x2, x3)  =  U10_gg(x3)

U11_gg(x1, x2)  =  U11_gg(x2)

t_in_gg(x1, x2)  =  t_in_gg(x1, x2)

U12_gg(x1, x2, x3)  =  U12_gg(x3)

t_out_gg(x1, x2)  =  t_out_gg

U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)

U14_gg(x1, x2, x3)  =  U14_gg(x3)

U7_gg(x1, x2, x3)  =  U7_gg(x3)

U3_gg(x1, x2)  =  U3_gg(x2)

Q_IN_GG(x1, x2)  =  Q_IN_GG(x1, x2)

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

E_IN_GG(x1, x2)  =  E_IN_GG(x1, x2)

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

P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)

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

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

R_IN_GG(x1, x2)  =  R_IN_GG(x1, x2)

U8_GG(x1, x2, x3)  =  U8_GG(x3)

U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)

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

U10_GG(x1, x2, x3)  =  U10_GG(x3)

U11_GG(x1, x2)  =  U11_GG(x2)

T_IN_GG(x1, x2)  =  T_IN_GG(x1, x2)

U12_GG(x1, x2, x3)  =  U12_GG(x3)

U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)

U14_GG(x1, x2, x3)  =  U14_GG(x3)

U7_GG(x1, x2, x3)  =  U7_GG(x3)

U3_GG(x1, x2)  =  U3_GG(x2)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 14 less nodes.
----------------------------------------

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X))))
   U2_GG(X, p_out_gg(X, f(f(X)))) -> Q_IN_GG(X, f(X))
   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y)))
   U6_GG(X, Y, r_out_gg(X, f(Y))) -> P_IN_GG(X, Y)
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y))
   U9_GG(X, Y, q_out_gg(X, Y)) -> R_IN_GG(X, Y)
   R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y)
   Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y)))
   U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> T_IN_GG(X, Y)
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))

The TRS R consists of the following rules:

   q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg(a, b)
   U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y)
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y))
   U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y)
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y))
   U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y)
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y)))
   U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y)))
   U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y))
   U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y)
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y))
   U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y))
   U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X))
   U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y))
   U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y))
   U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y))
   U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X)))
   U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X)))

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

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

e_in_gg(x1, x2)  =  e_in_gg(x1, x2)

a  =  a

b  =  b

e_out_gg(x1, x2)  =  e_out_gg

q_out_gg(x1, x2)  =  q_out_gg

f(x1)  =  f(x1)

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

p_in_gg(x1, x2)  =  p_in_gg(x1, x2)

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

p_out_gg(x1, x2)  =  p_out_gg

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

r_in_gg(x1, x2)  =  r_in_gg(x1, x2)

U8_gg(x1, x2, x3)  =  U8_gg(x3)

r_out_gg(x1, x2)  =  r_out_gg

U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

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

U10_gg(x1, x2, x3)  =  U10_gg(x3)

U11_gg(x1, x2)  =  U11_gg(x2)

t_in_gg(x1, x2)  =  t_in_gg(x1, x2)

U12_gg(x1, x2, x3)  =  U12_gg(x3)

t_out_gg(x1, x2)  =  t_out_gg

U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)

U14_gg(x1, x2, x3)  =  U14_gg(x3)

U7_gg(x1, x2, x3)  =  U7_gg(x3)

U3_gg(x1, x2)  =  U3_gg(x2)

Q_IN_GG(x1, x2)  =  Q_IN_GG(x1, x2)

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

P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)

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

R_IN_GG(x1, x2)  =  R_IN_GG(x1, x2)

U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)

T_IN_GG(x1, x2)  =  T_IN_GG(x1, x2)

U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)


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

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X))))
   U2_GG(X, p_out_gg) -> Q_IN_GG(X, f(X))
   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y)))
   U6_GG(X, Y, r_out_gg) -> P_IN_GG(X, Y)
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y))
   U9_GG(X, Y, q_out_gg) -> R_IN_GG(X, Y)
   R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y)
   Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y)))
   U13_GG(X, Y, q_out_gg) -> T_IN_GG(X, Y)
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))

The TRS R consists of the following rules:

   q_in_gg(X, Y) -> U1_gg(e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg
   U1_gg(e_out_gg) -> q_out_gg
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y))
   U5_gg(e_out_gg) -> p_out_gg
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(e_in_gg(X, Y))
   U8_gg(e_out_gg) -> r_out_gg
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y)))
   U4_gg(p_out_gg) -> q_out_gg
   U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(e_in_gg(X, Y))
   U12_gg(e_out_gg) -> t_out_gg
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y))
   U14_gg(t_out_gg) -> t_out_gg
   U11_gg(t_out_gg) -> r_out_gg
   U10_gg(r_out_gg) -> r_out_gg
   U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y))
   U7_gg(p_out_gg) -> p_out_gg
   U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X)))
   U3_gg(q_out_gg) -> q_out_gg

The set Q consists of the following terms:

   q_in_gg(x0, x1)
   e_in_gg(x0, x1)
   U1_gg(x0)
   p_in_gg(x0, x1)
   U5_gg(x0)
   r_in_gg(x0, x1)
   U8_gg(x0)
   U4_gg(x0)
   U9_gg(x0, x1, x2)
   t_in_gg(x0, x1)
   U12_gg(x0)
   U13_gg(x0, x1, x2)
   U14_gg(x0)
   U11_gg(x0)
   U10_gg(x0)
   U6_gg(x0, x1, x2)
   U7_gg(x0)
   U2_gg(x0, x1)
   U3_gg(x0)

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

(9) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y)))
   U6_GG(X, Y, r_out_gg) -> P_IN_GG(X, Y)
   U9_GG(X, Y, q_out_gg) -> R_IN_GG(X, Y)
   R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y)
   Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y))
   T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y)))
   U13_GG(X, Y, q_out_gg) -> T_IN_GG(X, Y)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U2_GG_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U13_GG_3(x_1, ..., x_3) ) = 2x_2 + 1
POL( U2_gg_2(x_1, x_2) ) = 2
POL( U4_gg_1(x_1) ) = max{0, 2x_1 - 2}
POL( U7_gg_1(x_1) ) = max{0, -2}
POL( U6_GG_3(x_1, ..., x_3) ) = 2x_2 + 1
POL( U9_GG_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( p_in_gg_2(x_1, x_2) ) = 2
POL( U5_gg_1(x_1) ) = max{0, -2}
POL( e_in_gg_2(x_1, x_2) ) = 0
POL( f_1(x_1) ) = x_1 + 2
POL( U6_gg_3(x_1, ..., x_3) ) = 2
POL( r_in_gg_2(x_1, x_2) ) = max{0, -2}
POL( U10_gg_1(x_1) ) = max{0, -2}
POL( U8_gg_1(x_1) ) = 1
POL( U9_gg_3(x_1, ..., x_3) ) = max{0, -2}
POL( q_in_gg_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U11_gg_1(x_1) ) = max{0, -2}
POL( t_in_gg_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( U13_gg_3(x_1, ..., x_3) ) = 1
POL( U3_gg_1(x_1) ) = max{0, -2}
POL( U1_gg_1(x_1) ) = max{0, -2}
POL( p_out_gg ) = 0
POL( q_out_gg ) = 0
POL( r_out_gg ) = 0
POL( U14_gg_1(x_1) ) = max{0, -2}
POL( U12_gg_1(x_1) ) = max{0, -2}
POL( t_out_gg ) = 0
POL( a ) = 2
POL( b ) = 2
POL( e_out_gg ) = 0
POL( Q_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( P_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( R_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( T_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y))
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X)))
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y)))
   U4_gg(p_out_gg) -> q_out_gg
   U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y))
   U10_gg(r_out_gg) -> r_out_gg
   U11_gg(t_out_gg) -> r_out_gg
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y))
   U14_gg(t_out_gg) -> t_out_gg
   U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y))
   U7_gg(p_out_gg) -> p_out_gg
   U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X)))
   U3_gg(q_out_gg) -> q_out_gg
   U5_gg(e_out_gg) -> p_out_gg


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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X))))
   U2_GG(X, p_out_gg) -> Q_IN_GG(X, f(X))
   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))

The TRS R consists of the following rules:

   q_in_gg(X, Y) -> U1_gg(e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg
   U1_gg(e_out_gg) -> q_out_gg
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y))
   U5_gg(e_out_gg) -> p_out_gg
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(e_in_gg(X, Y))
   U8_gg(e_out_gg) -> r_out_gg
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y)))
   U4_gg(p_out_gg) -> q_out_gg
   U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(e_in_gg(X, Y))
   U12_gg(e_out_gg) -> t_out_gg
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y))
   U14_gg(t_out_gg) -> t_out_gg
   U11_gg(t_out_gg) -> r_out_gg
   U10_gg(r_out_gg) -> r_out_gg
   U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y))
   U7_gg(p_out_gg) -> p_out_gg
   U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X)))
   U3_gg(q_out_gg) -> q_out_gg

The set Q consists of the following terms:

   q_in_gg(x0, x1)
   e_in_gg(x0, x1)
   U1_gg(x0)
   p_in_gg(x0, x1)
   U5_gg(x0)
   r_in_gg(x0, x1)
   U8_gg(x0)
   U4_gg(x0)
   U9_gg(x0, x1, x2)
   t_in_gg(x0, x1)
   U12_gg(x0)
   U13_gg(x0, x1, x2)
   U14_gg(x0)
   U11_gg(x0)
   U10_gg(x0)
   U6_gg(x0, x1, x2)
   U7_gg(x0)
   U2_gg(x0, x1)
   U3_gg(x0)

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

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))

The TRS R consists of the following rules:

   q_in_gg(X, Y) -> U1_gg(e_in_gg(X, Y))
   e_in_gg(a, b) -> e_out_gg
   U1_gg(e_out_gg) -> q_out_gg
   q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X))))
   p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y))
   U5_gg(e_out_gg) -> p_out_gg
   p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y)))
   r_in_gg(X, Y) -> U8_gg(e_in_gg(X, Y))
   U8_gg(e_out_gg) -> r_out_gg
   r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y))
   q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y)))
   U4_gg(p_out_gg) -> q_out_gg
   U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y))
   r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X)))
   t_in_gg(X, Y) -> U12_gg(e_in_gg(X, Y))
   U12_gg(e_out_gg) -> t_out_gg
   t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y)))
   U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y))
   U14_gg(t_out_gg) -> t_out_gg
   U11_gg(t_out_gg) -> r_out_gg
   U10_gg(r_out_gg) -> r_out_gg
   U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y))
   U7_gg(p_out_gg) -> p_out_gg
   U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X)))
   U3_gg(q_out_gg) -> q_out_gg

The set Q consists of the following terms:

   q_in_gg(x0, x1)
   e_in_gg(x0, x1)
   U1_gg(x0)
   p_in_gg(x0, x1)
   U5_gg(x0)
   r_in_gg(x0, x1)
   U8_gg(x0)
   U4_gg(x0)
   U9_gg(x0, x1, x2)
   t_in_gg(x0, x1)
   U12_gg(x0)
   U13_gg(x0, x1, x2)
   U14_gg(x0)
   U11_gg(x0)
   U10_gg(x0)
   U6_gg(x0, x1, x2)
   U7_gg(x0)
   U2_gg(x0, x1)
   U3_gg(x0)

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

(13) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))

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

   q_in_gg(x0, x1)
   e_in_gg(x0, x1)
   U1_gg(x0)
   p_in_gg(x0, x1)
   U5_gg(x0)
   r_in_gg(x0, x1)
   U8_gg(x0)
   U4_gg(x0)
   U9_gg(x0, x1, x2)
   t_in_gg(x0, x1)
   U12_gg(x0)
   U13_gg(x0, x1, x2)
   U14_gg(x0)
   U11_gg(x0)
   U10_gg(x0)
   U6_gg(x0, x1, x2)
   U7_gg(x0)
   U2_gg(x0, x1)
   U3_gg(x0)

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

(15) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   q_in_gg(x0, x1)
   e_in_gg(x0, x1)
   U1_gg(x0)
   p_in_gg(x0, x1)
   U5_gg(x0)
   r_in_gg(x0, x1)
   U8_gg(x0)
   U4_gg(x0)
   U9_gg(x0, x1, x2)
   t_in_gg(x0, x1)
   U12_gg(x0)
   U13_gg(x0, x1, x2)
   U14_gg(x0)
   U11_gg(x0)
   U10_gg(x0)
   U6_gg(x0, x1, x2)
   U7_gg(x0)
   U2_gg(x0, x1)
   U3_gg(x0)


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

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

   Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
   P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
   R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
   T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))

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

(17) 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:
*P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y))
The graph contains the following edges 1 >= 1, 2 >= 2


*T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y))
The graph contains the following edges 1 >= 1, 2 >= 2


*R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X))
The graph contains the following edges 1 >= 1, 2 >= 1, 1 >= 2, 2 >= 2


*Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X)))
The graph contains the following edges 1 >= 1, 2 > 1, 2 >= 2


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

(18)
YES
