/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

e(g,a)

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

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 28 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) PiDP
(7) PiDPToQDPProof [SOUND, 13 ms]
(8) QDP
(9) UsableRulesReductionPairsProof [EQUIVALENT, 56 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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

(0)
Obligation:
Clauses:

e(L, T) :- t(L, T).
e(L, T) :- ','(t(L, .(plus, C)), e(C, T)).
t(L, T) :- n(L, T).
t(L, T) :- ','(n(L, .(star, C)), t(C, T)).
n(.(L, T), T) :- z(L).
n(.(lbrace, A), B) :- e(A, .(rbrace, B)).
z(a).
z(b).
z(c).


Query: e(g,a)
----------------------------------------

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

e_in_2: (b,f)

t_in_2: (b,f)

n_in_2: (b,f)

Transforming Prolog into the following Term Rewriting System:

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

   e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T))
   t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T))
   n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L))
   z_in_g(a) -> z_out_g(a)
   z_in_g(b) -> z_out_g(b)
   z_in_g(c) -> z_out_g(c)
   U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T)
   n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B)))
   e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C)))
   t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C)))
   U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T))
   U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T)
   U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T))
   U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T)
   U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B)
   U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T)
   U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

t_in_ga(x1, x2)  =  t_in_ga(x1)

U4_ga(x1, x2, x3)  =  U4_ga(x3)

n_in_ga(x1, x2)  =  n_in_ga(x1)

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

U7_ga(x1, x2, x3)  =  U7_ga(x2, x3)

z_in_g(x1)  =  z_in_g(x1)

a  =  a

z_out_g(x1)  =  z_out_g

b  =  b

c  =  c

n_out_ga(x1, x2)  =  n_out_ga(x2)

lbrace  =  lbrace

U8_ga(x1, x2, x3)  =  U8_ga(x3)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

U5_ga(x1, x2, x3)  =  U5_ga(x3)

star  =  star

U6_ga(x1, x2, x3)  =  U6_ga(x3)

t_out_ga(x1, x2)  =  t_out_ga(x2)

plus  =  plus

U3_ga(x1, x2, x3)  =  U3_ga(x3)

e_out_ga(x1, x2)  =  e_out_ga(x2)

rbrace  =  rbrace





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T))
   t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T))
   n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L))
   z_in_g(a) -> z_out_g(a)
   z_in_g(b) -> z_out_g(b)
   z_in_g(c) -> z_out_g(c)
   U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T)
   n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B)))
   e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C)))
   t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C)))
   U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T))
   U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T)
   U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T))
   U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T)
   U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B)
   U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T)
   U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

t_in_ga(x1, x2)  =  t_in_ga(x1)

U4_ga(x1, x2, x3)  =  U4_ga(x3)

n_in_ga(x1, x2)  =  n_in_ga(x1)

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

U7_ga(x1, x2, x3)  =  U7_ga(x2, x3)

z_in_g(x1)  =  z_in_g(x1)

a  =  a

z_out_g(x1)  =  z_out_g

b  =  b

c  =  c

n_out_ga(x1, x2)  =  n_out_ga(x2)

lbrace  =  lbrace

U8_ga(x1, x2, x3)  =  U8_ga(x3)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

U5_ga(x1, x2, x3)  =  U5_ga(x3)

star  =  star

U6_ga(x1, x2, x3)  =  U6_ga(x3)

t_out_ga(x1, x2)  =  t_out_ga(x2)

plus  =  plus

U3_ga(x1, x2, x3)  =  U3_ga(x3)

e_out_ga(x1, x2)  =  e_out_ga(x2)

rbrace  =  rbrace



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

(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:

   E_IN_GA(L, T) -> U1_GA(L, T, t_in_ga(L, T))
   E_IN_GA(L, T) -> T_IN_GA(L, T)
   T_IN_GA(L, T) -> U4_GA(L, T, n_in_ga(L, T))
   T_IN_GA(L, T) -> N_IN_GA(L, T)
   N_IN_GA(.(L, T), T) -> U7_GA(L, T, z_in_g(L))
   N_IN_GA(.(L, T), T) -> Z_IN_G(L)
   N_IN_GA(.(lbrace, A), B) -> U8_GA(A, B, e_in_ga(A, .(rbrace, B)))
   N_IN_GA(.(lbrace, A), B) -> E_IN_GA(A, .(rbrace, B))
   E_IN_GA(L, T) -> U2_GA(L, T, t_in_ga(L, .(plus, C)))
   E_IN_GA(L, T) -> T_IN_GA(L, .(plus, C))
   T_IN_GA(L, T) -> U5_GA(L, T, n_in_ga(L, .(star, C)))
   T_IN_GA(L, T) -> N_IN_GA(L, .(star, C))
   U5_GA(L, T, n_out_ga(L, .(star, C))) -> U6_GA(L, T, t_in_ga(C, T))
   U5_GA(L, T, n_out_ga(L, .(star, C))) -> T_IN_GA(C, T)
   U2_GA(L, T, t_out_ga(L, .(plus, C))) -> U3_GA(L, T, e_in_ga(C, T))
   U2_GA(L, T, t_out_ga(L, .(plus, C))) -> E_IN_GA(C, T)

The TRS R consists of the following rules:

   e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T))
   t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T))
   n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L))
   z_in_g(a) -> z_out_g(a)
   z_in_g(b) -> z_out_g(b)
   z_in_g(c) -> z_out_g(c)
   U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T)
   n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B)))
   e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C)))
   t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C)))
   U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T))
   U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T)
   U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T))
   U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T)
   U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B)
   U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T)
   U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

t_in_ga(x1, x2)  =  t_in_ga(x1)

U4_ga(x1, x2, x3)  =  U4_ga(x3)

n_in_ga(x1, x2)  =  n_in_ga(x1)

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

U7_ga(x1, x2, x3)  =  U7_ga(x2, x3)

z_in_g(x1)  =  z_in_g(x1)

a  =  a

z_out_g(x1)  =  z_out_g

b  =  b

c  =  c

n_out_ga(x1, x2)  =  n_out_ga(x2)

lbrace  =  lbrace

U8_ga(x1, x2, x3)  =  U8_ga(x3)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

U5_ga(x1, x2, x3)  =  U5_ga(x3)

star  =  star

U6_ga(x1, x2, x3)  =  U6_ga(x3)

t_out_ga(x1, x2)  =  t_out_ga(x2)

plus  =  plus

U3_ga(x1, x2, x3)  =  U3_ga(x3)

e_out_ga(x1, x2)  =  e_out_ga(x2)

rbrace  =  rbrace

E_IN_GA(x1, x2)  =  E_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)

T_IN_GA(x1, x2)  =  T_IN_GA(x1)

U4_GA(x1, x2, x3)  =  U4_GA(x3)

N_IN_GA(x1, x2)  =  N_IN_GA(x1)

U7_GA(x1, x2, x3)  =  U7_GA(x2, x3)

Z_IN_G(x1)  =  Z_IN_G(x1)

U8_GA(x1, x2, x3)  =  U8_GA(x3)

U2_GA(x1, x2, x3)  =  U2_GA(x3)

U5_GA(x1, x2, x3)  =  U5_GA(x3)

U6_GA(x1, x2, x3)  =  U6_GA(x3)

U3_GA(x1, x2, x3)  =  U3_GA(x3)


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

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

   E_IN_GA(L, T) -> U1_GA(L, T, t_in_ga(L, T))
   E_IN_GA(L, T) -> T_IN_GA(L, T)
   T_IN_GA(L, T) -> U4_GA(L, T, n_in_ga(L, T))
   T_IN_GA(L, T) -> N_IN_GA(L, T)
   N_IN_GA(.(L, T), T) -> U7_GA(L, T, z_in_g(L))
   N_IN_GA(.(L, T), T) -> Z_IN_G(L)
   N_IN_GA(.(lbrace, A), B) -> U8_GA(A, B, e_in_ga(A, .(rbrace, B)))
   N_IN_GA(.(lbrace, A), B) -> E_IN_GA(A, .(rbrace, B))
   E_IN_GA(L, T) -> U2_GA(L, T, t_in_ga(L, .(plus, C)))
   E_IN_GA(L, T) -> T_IN_GA(L, .(plus, C))
   T_IN_GA(L, T) -> U5_GA(L, T, n_in_ga(L, .(star, C)))
   T_IN_GA(L, T) -> N_IN_GA(L, .(star, C))
   U5_GA(L, T, n_out_ga(L, .(star, C))) -> U6_GA(L, T, t_in_ga(C, T))
   U5_GA(L, T, n_out_ga(L, .(star, C))) -> T_IN_GA(C, T)
   U2_GA(L, T, t_out_ga(L, .(plus, C))) -> U3_GA(L, T, e_in_ga(C, T))
   U2_GA(L, T, t_out_ga(L, .(plus, C))) -> E_IN_GA(C, T)

The TRS R consists of the following rules:

   e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T))
   t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T))
   n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L))
   z_in_g(a) -> z_out_g(a)
   z_in_g(b) -> z_out_g(b)
   z_in_g(c) -> z_out_g(c)
   U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T)
   n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B)))
   e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C)))
   t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C)))
   U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T))
   U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T)
   U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T))
   U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T)
   U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B)
   U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T)
   U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

t_in_ga(x1, x2)  =  t_in_ga(x1)

U4_ga(x1, x2, x3)  =  U4_ga(x3)

n_in_ga(x1, x2)  =  n_in_ga(x1)

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

U7_ga(x1, x2, x3)  =  U7_ga(x2, x3)

z_in_g(x1)  =  z_in_g(x1)

a  =  a

z_out_g(x1)  =  z_out_g

b  =  b

c  =  c

n_out_ga(x1, x2)  =  n_out_ga(x2)

lbrace  =  lbrace

U8_ga(x1, x2, x3)  =  U8_ga(x3)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

U5_ga(x1, x2, x3)  =  U5_ga(x3)

star  =  star

U6_ga(x1, x2, x3)  =  U6_ga(x3)

t_out_ga(x1, x2)  =  t_out_ga(x2)

plus  =  plus

U3_ga(x1, x2, x3)  =  U3_ga(x3)

e_out_ga(x1, x2)  =  e_out_ga(x2)

rbrace  =  rbrace

E_IN_GA(x1, x2)  =  E_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)

T_IN_GA(x1, x2)  =  T_IN_GA(x1)

U4_GA(x1, x2, x3)  =  U4_GA(x3)

N_IN_GA(x1, x2)  =  N_IN_GA(x1)

U7_GA(x1, x2, x3)  =  U7_GA(x2, x3)

Z_IN_G(x1)  =  Z_IN_G(x1)

U8_GA(x1, x2, x3)  =  U8_GA(x3)

U2_GA(x1, x2, x3)  =  U2_GA(x3)

U5_GA(x1, x2, x3)  =  U5_GA(x3)

U6_GA(x1, x2, x3)  =  U6_GA(x3)

U3_GA(x1, x2, x3)  =  U3_GA(x3)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.
----------------------------------------

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

   E_IN_GA(L, T) -> T_IN_GA(L, T)
   T_IN_GA(L, T) -> N_IN_GA(L, T)
   N_IN_GA(.(lbrace, A), B) -> E_IN_GA(A, .(rbrace, B))
   E_IN_GA(L, T) -> U2_GA(L, T, t_in_ga(L, .(plus, C)))
   U2_GA(L, T, t_out_ga(L, .(plus, C))) -> E_IN_GA(C, T)
   E_IN_GA(L, T) -> T_IN_GA(L, .(plus, C))
   T_IN_GA(L, T) -> U5_GA(L, T, n_in_ga(L, .(star, C)))
   U5_GA(L, T, n_out_ga(L, .(star, C))) -> T_IN_GA(C, T)
   T_IN_GA(L, T) -> N_IN_GA(L, .(star, C))

The TRS R consists of the following rules:

   e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T))
   t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T))
   n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L))
   z_in_g(a) -> z_out_g(a)
   z_in_g(b) -> z_out_g(b)
   z_in_g(c) -> z_out_g(c)
   U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T)
   n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B)))
   e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C)))
   t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C)))
   U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T))
   U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T)
   U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T))
   U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T)
   U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B)
   U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T)
   U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

t_in_ga(x1, x2)  =  t_in_ga(x1)

U4_ga(x1, x2, x3)  =  U4_ga(x3)

n_in_ga(x1, x2)  =  n_in_ga(x1)

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

U7_ga(x1, x2, x3)  =  U7_ga(x2, x3)

z_in_g(x1)  =  z_in_g(x1)

a  =  a

z_out_g(x1)  =  z_out_g

b  =  b

c  =  c

n_out_ga(x1, x2)  =  n_out_ga(x2)

lbrace  =  lbrace

U8_ga(x1, x2, x3)  =  U8_ga(x3)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

U5_ga(x1, x2, x3)  =  U5_ga(x3)

star  =  star

U6_ga(x1, x2, x3)  =  U6_ga(x3)

t_out_ga(x1, x2)  =  t_out_ga(x2)

plus  =  plus

U3_ga(x1, x2, x3)  =  U3_ga(x3)

e_out_ga(x1, x2)  =  e_out_ga(x2)

rbrace  =  rbrace

E_IN_GA(x1, x2)  =  E_IN_GA(x1)

T_IN_GA(x1, x2)  =  T_IN_GA(x1)

N_IN_GA(x1, x2)  =  N_IN_GA(x1)

U2_GA(x1, x2, x3)  =  U2_GA(x3)

U5_GA(x1, x2, x3)  =  U5_GA(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:

   E_IN_GA(L) -> T_IN_GA(L)
   T_IN_GA(L) -> N_IN_GA(L)
   N_IN_GA(.(lbrace, A)) -> E_IN_GA(A)
   E_IN_GA(L) -> U2_GA(t_in_ga(L))
   U2_GA(t_out_ga(.(plus, C))) -> E_IN_GA(C)
   T_IN_GA(L) -> U5_GA(n_in_ga(L))
   U5_GA(n_out_ga(.(star, C))) -> T_IN_GA(C)

The TRS R consists of the following rules:

   e_in_ga(L) -> U1_ga(t_in_ga(L))
   t_in_ga(L) -> U4_ga(n_in_ga(L))
   n_in_ga(.(L, T)) -> U7_ga(T, z_in_g(L))
   z_in_g(a) -> z_out_g
   z_in_g(b) -> z_out_g
   z_in_g(c) -> z_out_g
   U7_ga(T, z_out_g) -> n_out_ga(T)
   n_in_ga(.(lbrace, A)) -> U8_ga(e_in_ga(A))
   e_in_ga(L) -> U2_ga(t_in_ga(L))
   t_in_ga(L) -> U5_ga(n_in_ga(L))
   U5_ga(n_out_ga(.(star, C))) -> U6_ga(t_in_ga(C))
   U6_ga(t_out_ga(T)) -> t_out_ga(T)
   U2_ga(t_out_ga(.(plus, C))) -> U3_ga(e_in_ga(C))
   U3_ga(e_out_ga(T)) -> e_out_ga(T)
   U8_ga(e_out_ga(.(rbrace, B))) -> n_out_ga(B)
   U4_ga(n_out_ga(T)) -> t_out_ga(T)
   U1_ga(t_out_ga(T)) -> e_out_ga(T)

The set Q consists of the following terms:

   e_in_ga(x0)
   t_in_ga(x0)
   n_in_ga(x0)
   z_in_g(x0)
   U7_ga(x0, x1)
   U5_ga(x0)
   U6_ga(x0)
   U2_ga(x0)
   U3_ga(x0)
   U8_ga(x0)
   U4_ga(x0)
   U1_ga(x0)

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

(9) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   N_IN_GA(.(lbrace, A)) -> E_IN_GA(A)
   U2_GA(t_out_ga(.(plus, C))) -> E_IN_GA(C)
   U5_GA(n_out_ga(.(star, C))) -> T_IN_GA(C)
The following rules are removed from R:

   n_in_ga(.(L, T)) -> U7_ga(T, z_in_g(L))
   z_in_g(a) -> z_out_g
   z_in_g(b) -> z_out_g
   z_in_g(c) -> z_out_g
   n_in_ga(.(lbrace, A)) -> U8_ga(e_in_ga(A))
   U5_ga(n_out_ga(.(star, C))) -> U6_ga(t_in_ga(C))
   U6_ga(t_out_ga(T)) -> t_out_ga(T)
   U2_ga(t_out_ga(.(plus, C))) -> U3_ga(e_in_ga(C))
   U8_ga(e_out_ga(.(rbrace, B))) -> n_out_ga(B)
Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 2*x_1 + x_2
   POL(E_IN_GA(x_1)) = x_1
   POL(N_IN_GA(x_1)) = x_1
   POL(T_IN_GA(x_1)) = x_1
   POL(U1_ga(x_1)) = x_1
   POL(U2_GA(x_1)) = x_1
   POL(U2_ga(x_1)) = x_1
   POL(U3_ga(x_1)) = x_1
   POL(U4_ga(x_1)) = x_1
   POL(U5_GA(x_1)) = x_1
   POL(U5_ga(x_1)) = x_1
   POL(U6_ga(x_1)) = 2 + x_1
   POL(U7_ga(x_1, x_2)) = x_1 + x_2
   POL(U8_ga(x_1)) = x_1
   POL(a) = 0
   POL(b) = 0
   POL(c) = 0
   POL(e_in_ga(x_1)) = x_1
   POL(e_out_ga(x_1)) = x_1
   POL(lbrace) = 0
   POL(n_in_ga(x_1)) = x_1
   POL(n_out_ga(x_1)) = x_1
   POL(plus) = 1
   POL(rbrace) = 0
   POL(star) = 2
   POL(t_in_ga(x_1)) = x_1
   POL(t_out_ga(x_1)) = x_1
   POL(z_in_g(x_1)) = x_1
   POL(z_out_g) = 0


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

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

   E_IN_GA(L) -> T_IN_GA(L)
   T_IN_GA(L) -> N_IN_GA(L)
   E_IN_GA(L) -> U2_GA(t_in_ga(L))
   T_IN_GA(L) -> U5_GA(n_in_ga(L))

The TRS R consists of the following rules:

   e_in_ga(L) -> U1_ga(t_in_ga(L))
   e_in_ga(L) -> U2_ga(t_in_ga(L))
   t_in_ga(L) -> U4_ga(n_in_ga(L))
   t_in_ga(L) -> U5_ga(n_in_ga(L))
   U3_ga(e_out_ga(T)) -> e_out_ga(T)
   U4_ga(n_out_ga(T)) -> t_out_ga(T)
   U1_ga(t_out_ga(T)) -> e_out_ga(T)
   U7_ga(T, z_out_g) -> n_out_ga(T)

The set Q consists of the following terms:

   e_in_ga(x0)
   t_in_ga(x0)
   n_in_ga(x0)
   z_in_g(x0)
   U7_ga(x0, x1)
   U5_ga(x0)
   U6_ga(x0)
   U2_ga(x0)
   U3_ga(x0)
   U8_ga(x0)
   U4_ga(x0)
   U1_ga(x0)

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

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
----------------------------------------

(12)
TRUE