Left Termination of the query pattern t_in_1(g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

t(N) :- ','(ll(N, Xs), ','(select(X, Xs, Xs1), ','(ll(M, Xs1), t(M)))).
t(0).
ll(s(N), .(X, Xs)) :- ll(N, Xs).
ll(0, []).
select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys).
select(X, .(X, Xs), Xs).

Queries:

t(g).

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

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))
T_IN_G(N) → LL_IN_GA(N, Xs)
LL_IN_GA(s(N), .(X, Xs)) → U5_GA(N, X, Xs, ll_in_ga(N, Xs))
LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
U1_G(N, ll_out_ga(N, Xs)) → SELECT_IN_AAA(X, Xs, Xs1)
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → U6_AAA(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)
U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U2_G(N, select_out_aaa(X, Xs, Xs1)) → LL_IN_AA(M, Xs1)
LL_IN_AA(s(N), .(X, Xs)) → U5_AA(N, X, Xs, ll_in_aa(N, Xs))
LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)
U3_G(N, ll_out_aa(M, Xs1)) → U4_G(N, t_in_g(M))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g
U2_G(x1, x2)  =  U2_G(x2)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U4_G(x1, x2)  =  U4_G(x2)
LL_IN_AA(x1, x2)  =  LL_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)
LL_IN_GA(x1, x2)  =  LL_IN_GA(x1)
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))
T_IN_G(N) → LL_IN_GA(N, Xs)
LL_IN_GA(s(N), .(X, Xs)) → U5_GA(N, X, Xs, ll_in_ga(N, Xs))
LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
U1_G(N, ll_out_ga(N, Xs)) → SELECT_IN_AAA(X, Xs, Xs1)
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → U6_AAA(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)
U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U2_G(N, select_out_aaa(X, Xs, Xs1)) → LL_IN_AA(M, Xs1)
LL_IN_AA(s(N), .(X, Xs)) → U5_AA(N, X, Xs, ll_in_aa(N, Xs))
LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)
U3_G(N, ll_out_aa(M, Xs1)) → U4_G(N, t_in_g(M))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g
U2_G(x1, x2)  =  U2_G(x2)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U4_G(x1, x2)  =  U4_G(x2)
LL_IN_AA(x1, x2)  =  LL_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)
LL_IN_GA(x1, x2)  =  LL_IN_GA(x1)
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 7 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g
LL_IN_AA(x1, x2)  =  LL_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)

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

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LL_IN_AA → LL_IN_AA

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

The TRS P consists of the following rules:

LL_IN_AA → LL_IN_AA

The TRS R consists of the following rules:none


s = LL_IN_AA evaluates to t =LL_IN_AA

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LL_IN_AA to LL_IN_AA.





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)

R is empty.
The argument filtering Pi contains the following mapping:
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SELECT_IN_AAA → SELECT_IN_AAA

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

The TRS P consists of the following rules:

SELECT_IN_AAA → SELECT_IN_AAA

The TRS R consists of the following rules:none


s = SELECT_IN_AAA evaluates to t =SELECT_IN_AAA

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SELECT_IN_AAA to SELECT_IN_AAA.





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g
LL_IN_GA(x1, x2)  =  LL_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)

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

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LL_IN_GA(s(N)) → LL_IN_GA(N)

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

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
U2_g(x1, x2)  =  U2_g(x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x2)
t_out_g(x1)  =  t_out_g
U2_G(x1, x2)  =  U2_G(x2)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))

The TRS R consists of the following rules:

ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))

The argument filtering Pi contains the following mapping:
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U2_G(x1, x2)  =  U2_G(x2)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

T_IN_G(N) → U1_G(ll_in_ga(N))
U3_G(ll_out_aa(M)) → T_IN_G(M)
U2_G(select_out_aaa) → U3_G(ll_in_aa)
U1_G(ll_out_ga) → U2_G(select_in_aaa)

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(ll_in_ga(N))
ll_in_ga(0) → ll_out_ga
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(ll_out_ga) → ll_out_ga

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U2_G(select_out_aaa) → U3_G(ll_in_aa) at position [0] we obtained the following new rules:

U2_G(select_out_aaa) → U3_G(U5_aa(ll_in_aa))
U2_G(select_out_aaa) → U3_G(ll_out_aa(0))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U2_G(select_out_aaa) → U3_G(U5_aa(ll_in_aa))
T_IN_G(N) → U1_G(ll_in_ga(N))
U3_G(ll_out_aa(M)) → T_IN_G(M)
U1_G(ll_out_ga) → U2_G(select_in_aaa)
U2_G(select_out_aaa) → U3_G(ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(ll_in_ga(N))
ll_in_ga(0) → ll_out_ga
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(ll_out_ga) → ll_out_ga

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_G(ll_out_ga) → U2_G(select_in_aaa) at position [0] we obtained the following new rules:

U1_G(ll_out_ga) → U2_G(U6_aaa(select_in_aaa))
U1_G(ll_out_ga) → U2_G(select_out_aaa)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U2_G(select_out_aaa) → U3_G(U5_aa(ll_in_aa))
T_IN_G(N) → U1_G(ll_in_ga(N))
U1_G(ll_out_ga) → U2_G(U6_aaa(select_in_aaa))
U3_G(ll_out_aa(M)) → T_IN_G(M)
U1_G(ll_out_ga) → U2_G(select_out_aaa)
U2_G(select_out_aaa) → U3_G(ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(ll_in_ga(N))
ll_in_ga(0) → ll_out_ga
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(ll_out_ga) → ll_out_ga

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule T_IN_G(N) → U1_G(ll_in_ga(N)) at position [0] we obtained the following new rules:

T_IN_G(0) → U1_G(ll_out_ga)
T_IN_G(s(x0)) → U1_G(U5_ga(ll_in_ga(x0)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ ForwardInstantiation
  ↳ PrologToPiTRSProof

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

U2_G(select_out_aaa) → U3_G(U5_aa(ll_in_aa))
T_IN_G(0) → U1_G(ll_out_ga)
T_IN_G(s(x0)) → U1_G(U5_ga(ll_in_ga(x0)))
U3_G(ll_out_aa(M)) → T_IN_G(M)
U1_G(ll_out_ga) → U2_G(U6_aaa(select_in_aaa))
U1_G(ll_out_ga) → U2_G(select_out_aaa)
U2_G(select_out_aaa) → U3_G(ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(ll_in_ga(N))
ll_in_ga(0) → ll_out_ga
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(ll_out_ga) → ll_out_ga

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U3_G(ll_out_aa(M)) → T_IN_G(M) we obtained the following new rules:

U3_G(ll_out_aa(0)) → T_IN_G(0)
U3_G(ll_out_aa(s(y_0))) → T_IN_G(s(y_0))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ ForwardInstantiation
                                      ↳ QDP
                                        ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

U3_G(ll_out_aa(0)) → T_IN_G(0)
U2_G(select_out_aaa) → U3_G(U5_aa(ll_in_aa))
U3_G(ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
T_IN_G(0) → U1_G(ll_out_ga)
T_IN_G(s(x0)) → U1_G(U5_ga(ll_in_ga(x0)))
U1_G(ll_out_ga) → U2_G(U6_aaa(select_in_aaa))
U1_G(ll_out_ga) → U2_G(select_out_aaa)
U2_G(select_out_aaa) → U3_G(ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(ll_in_ga(N))
ll_in_ga(0) → ll_out_ga
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(ll_out_ga) → ll_out_ga

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0)

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

The TRS P consists of the following rules:

U3_G(ll_out_aa(0)) → T_IN_G(0)
U2_G(select_out_aaa) → U3_G(U5_aa(ll_in_aa))
U3_G(ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
T_IN_G(0) → U1_G(ll_out_ga)
T_IN_G(s(x0)) → U1_G(U5_ga(ll_in_ga(x0)))
U1_G(ll_out_ga) → U2_G(U6_aaa(select_in_aaa))
U1_G(ll_out_ga) → U2_G(select_out_aaa)
U2_G(select_out_aaa) → U3_G(ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(ll_in_ga(N))
ll_in_ga(0) → ll_out_ga
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(ll_out_ga) → ll_out_ga


s = T_IN_G(0) evaluates to t =T_IN_G(0)

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



Rewriting sequence

T_IN_G(0) → U1_G(ll_out_ga)
with rule T_IN_G(0) → U1_G(ll_out_ga) at position [] and matcher [ ]

U1_G(ll_out_ga) → U2_G(select_out_aaa)
with rule U1_G(ll_out_ga) → U2_G(select_out_aaa) at position [] and matcher [ ]

U2_G(select_out_aaa) → U3_G(ll_out_aa(0))
with rule U2_G(select_out_aaa) → U3_G(ll_out_aa(0)) at position [] and matcher [ ]

U3_G(ll_out_aa(0)) → T_IN_G(0)
with rule U3_G(ll_out_aa(0)) → T_IN_G(0)

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


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




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

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))
T_IN_G(N) → LL_IN_GA(N, Xs)
LL_IN_GA(s(N), .(X, Xs)) → U5_GA(N, X, Xs, ll_in_ga(N, Xs))
LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
U1_G(N, ll_out_ga(N, Xs)) → SELECT_IN_AAA(X, Xs, Xs1)
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → U6_AAA(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)
U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U2_G(N, select_out_aaa(X, Xs, Xs1)) → LL_IN_AA(M, Xs1)
LL_IN_AA(s(N), .(X, Xs)) → U5_AA(N, X, Xs, ll_in_aa(N, Xs))
LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)
U3_G(N, ll_out_aa(M, Xs1)) → U4_G(N, t_in_g(M))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
LL_IN_AA(x1, x2)  =  LL_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_G(x1, x2)  =  U3_G(x1, x2)
LL_IN_GA(x1, x2)  =  LL_IN_GA(x1)
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))
T_IN_G(N) → LL_IN_GA(N, Xs)
LL_IN_GA(s(N), .(X, Xs)) → U5_GA(N, X, Xs, ll_in_ga(N, Xs))
LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
U1_G(N, ll_out_ga(N, Xs)) → SELECT_IN_AAA(X, Xs, Xs1)
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → U6_AAA(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)
U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U2_G(N, select_out_aaa(X, Xs, Xs1)) → LL_IN_AA(M, Xs1)
LL_IN_AA(s(N), .(X, Xs)) → U5_AA(N, X, Xs, ll_in_aa(N, Xs))
LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)
U3_G(N, ll_out_aa(M, Xs1)) → U4_G(N, t_in_g(M))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
LL_IN_AA(x1, x2)  =  LL_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_G(x1, x2)  =  U3_G(x1, x2)
LL_IN_GA(x1, x2)  =  LL_IN_GA(x1)
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 7 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)
LL_IN_AA(x1, x2)  =  LL_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LL_IN_AA(s(N), .(X, Xs)) → LL_IN_AA(N, Xs)

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

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LL_IN_AA → LL_IN_AA

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

The TRS P consists of the following rules:

LL_IN_AA → LL_IN_AA

The TRS R consists of the following rules:none


s = LL_IN_AA evaluates to t =LL_IN_AA

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LL_IN_AA to LL_IN_AA.





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

SELECT_IN_AAA(X, .(Y, Xs), .(Y, Ys)) → SELECT_IN_AAA(X, Xs, Ys)

R is empty.
The argument filtering Pi contains the following mapping:
SELECT_IN_AAA(x1, x2, x3)  =  SELECT_IN_AAA

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP

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

SELECT_IN_AAA → SELECT_IN_AAA

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

The TRS P consists of the following rules:

SELECT_IN_AAA → SELECT_IN_AAA

The TRS R consists of the following rules:none


s = SELECT_IN_AAA evaluates to t =SELECT_IN_AAA

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SELECT_IN_AAA to SELECT_IN_AAA.





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)
LL_IN_GA(x1, x2)  =  LL_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

LL_IN_GA(s(N), .(X, Xs)) → LL_IN_GA(N, Xs)

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

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

LL_IN_GA(s(N)) → LL_IN_GA(N)

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

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



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))

The TRS R consists of the following rules:

t_in_g(N) → U1_g(N, ll_in_ga(N, Xs))
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))
U1_g(N, ll_out_ga(N, Xs)) → U2_g(N, select_in_aaa(X, Xs, Xs1))
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U2_g(N, select_out_aaa(X, Xs, Xs1)) → U3_g(N, ll_in_aa(M, Xs1))
ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U3_g(N, ll_out_aa(M, Xs1)) → U4_g(N, t_in_g(M))
t_in_g(0) → t_out_g(0)
U4_g(N, t_out_g(M)) → t_out_g(N)

The argument filtering Pi contains the following mapping:
t_in_g(x1)  =  t_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
U3_g(x1, x2)  =  U3_g(x1, x2)
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
t_out_g(x1)  =  t_out_g(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_G(x1, x2)  =  U3_G(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

U2_G(N, select_out_aaa(X, Xs, Xs1)) → U3_G(N, ll_in_aa(M, Xs1))
U3_G(N, ll_out_aa(M, Xs1)) → T_IN_G(M)
U1_G(N, ll_out_ga(N, Xs)) → U2_G(N, select_in_aaa(X, Xs, Xs1))
T_IN_G(N) → U1_G(N, ll_in_ga(N, Xs))

The TRS R consists of the following rules:

ll_in_aa(s(N), .(X, Xs)) → U5_aa(N, X, Xs, ll_in_aa(N, Xs))
ll_in_aa(0, []) → ll_out_aa(0, [])
select_in_aaa(X, .(Y, Xs), .(Y, Ys)) → U6_aaa(X, Y, Xs, Ys, select_in_aaa(X, Xs, Ys))
select_in_aaa(X, .(X, Xs), Xs) → select_out_aaa(X, .(X, Xs), Xs)
ll_in_ga(s(N), .(X, Xs)) → U5_ga(N, X, Xs, ll_in_ga(N, Xs))
ll_in_ga(0, []) → ll_out_ga(0, [])
U5_aa(N, X, Xs, ll_out_aa(N, Xs)) → ll_out_aa(s(N), .(X, Xs))
U6_aaa(X, Y, Xs, Ys, select_out_aaa(X, Xs, Ys)) → select_out_aaa(X, .(Y, Xs), .(Y, Ys))
U5_ga(N, X, Xs, ll_out_ga(N, Xs)) → ll_out_ga(s(N), .(X, Xs))

The argument filtering Pi contains the following mapping:
ll_in_ga(x1, x2)  =  ll_in_ga(x1)
s(x1)  =  s(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
0  =  0
ll_out_ga(x1, x2)  =  ll_out_ga(x1)
select_in_aaa(x1, x2, x3)  =  select_in_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
select_out_aaa(x1, x2, x3)  =  select_out_aaa
ll_in_aa(x1, x2)  =  ll_in_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
ll_out_aa(x1, x2)  =  ll_out_aa(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
T_IN_G(x1)  =  T_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_G(x1, x2)  =  U3_G(x1, x2)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(N, ll_out_ga(N)) → U2_G(N, select_in_aaa)
U2_G(N, select_out_aaa) → U3_G(N, ll_in_aa)
T_IN_G(N) → U1_G(N, ll_in_ga(N))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U2_G(N, select_out_aaa) → U3_G(N, ll_in_aa) at position [1] we obtained the following new rules:

U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(N, ll_out_ga(N)) → U2_G(N, select_in_aaa)
U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))
T_IN_G(N) → U1_G(N, ll_in_ga(N))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_G(N, ll_out_ga(N)) → U2_G(N, select_in_aaa) at position [1] we obtained the following new rules:

U1_G(y0, ll_out_ga(y0)) → U2_G(y0, select_out_aaa)
U1_G(y0, ll_out_ga(y0)) → U2_G(y0, U6_aaa(select_in_aaa))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing

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

U1_G(y0, ll_out_ga(y0)) → U2_G(y0, select_out_aaa)
U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))
U1_G(y0, ll_out_ga(y0)) → U2_G(y0, U6_aaa(select_in_aaa))
T_IN_G(N) → U1_G(N, ll_in_ga(N))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule T_IN_G(N) → U1_G(N, ll_in_ga(N)) at position [1] we obtained the following new rules:

T_IN_G(0) → U1_G(0, ll_out_ga(0))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(y0, ll_out_ga(y0)) → U2_G(y0, select_out_aaa)
U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
U1_G(y0, ll_out_ga(y0)) → U2_G(y0, U6_aaa(select_in_aaa))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U1_G(y0, ll_out_ga(y0)) → U2_G(y0, select_out_aaa) we obtained the following new rules:

U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
U1_G(y0, ll_out_ga(y0)) → U2_G(y0, U6_aaa(select_in_aaa))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U1_G(y0, ll_out_ga(y0)) → U2_G(y0, U6_aaa(select_in_aaa)) we obtained the following new rules:

U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation
                                          ↳ QDP
                                            ↳ Instantiation

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_G(y0, select_out_aaa) → U3_G(y0, ll_out_aa(0)) we obtained the following new rules:

U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ Instantiation

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
T_IN_G(0) → U1_G(0, ll_out_ga(0))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_G(y0, select_out_aaa) → U3_G(y0, U5_aa(ll_in_aa)) we obtained the following new rules:

U2_G(s(z0), select_out_aaa) → U3_G(s(z0), U5_aa(ll_in_aa))
U2_G(0, select_out_aaa) → U3_G(0, U5_aa(ll_in_aa))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation

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

U3_G(N, ll_out_aa(M)) → T_IN_G(M)
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), U5_aa(ll_in_aa))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U2_G(0, select_out_aaa) → U3_G(0, U5_aa(ll_in_aa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_G(N, ll_out_aa(M)) → T_IN_G(M) we obtained the following new rules:

U3_G(0, ll_out_aa(x1)) → T_IN_G(x1)
U3_G(s(z0), ll_out_aa(x1)) → T_IN_G(x1)
U3_G(s(z0), ll_out_aa(0)) → T_IN_G(0)
U3_G(0, ll_out_aa(0)) → T_IN_G(0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ ForwardInstantiation

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

U2_G(s(z0), select_out_aaa) → U3_G(s(z0), U5_aa(ll_in_aa))
U3_G(0, ll_out_aa(x1)) → T_IN_G(x1)
U3_G(s(z0), ll_out_aa(x1)) → T_IN_G(x1)
U3_G(0, ll_out_aa(0)) → T_IN_G(0)
U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U2_G(0, select_out_aaa) → U3_G(0, U5_aa(ll_in_aa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
U3_G(s(z0), ll_out_aa(0)) → T_IN_G(0)
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U3_G(0, ll_out_aa(x1)) → T_IN_G(x1) we obtained the following new rules:

U3_G(0, ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
U3_G(0, ll_out_aa(0)) → T_IN_G(0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ ForwardInstantiation
                                                          ↳ QDP
                                                            ↳ ForwardInstantiation

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

U2_G(s(z0), select_out_aaa) → U3_G(s(z0), U5_aa(ll_in_aa))
U3_G(s(z0), ll_out_aa(x1)) → T_IN_G(x1)
U3_G(0, ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U3_G(0, ll_out_aa(0)) → T_IN_G(0)
U2_G(0, select_out_aaa) → U3_G(0, U5_aa(ll_in_aa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
T_IN_G(0) → U1_G(0, ll_out_ga(0))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
U3_G(s(z0), ll_out_aa(0)) → T_IN_G(0)
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U3_G(s(z0), ll_out_aa(x1)) → T_IN_G(x1) we obtained the following new rules:

U3_G(s(x0), ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
U3_G(s(x0), ll_out_aa(0)) → T_IN_G(0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ Instantiation
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ ForwardInstantiation
                                                          ↳ QDP
                                                            ↳ ForwardInstantiation
                                                              ↳ QDP
                                                                ↳ NonTerminationProof

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

U2_G(s(z0), select_out_aaa) → U3_G(s(z0), U5_aa(ll_in_aa))
U3_G(0, ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
U3_G(0, ll_out_aa(0)) → T_IN_G(0)
U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U2_G(0, select_out_aaa) → U3_G(0, U5_aa(ll_in_aa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U3_G(s(x0), ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
U3_G(s(z0), ll_out_aa(0)) → T_IN_G(0)
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))

The set Q consists of the following terms:

ll_in_aa
select_in_aaa
ll_in_ga(x0)
U5_aa(x0)
U6_aaa(x0)
U5_ga(x0, x1)

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

The TRS P consists of the following rules:

U2_G(s(z0), select_out_aaa) → U3_G(s(z0), U5_aa(ll_in_aa))
U3_G(0, ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
U3_G(0, ll_out_aa(0)) → T_IN_G(0)
U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
U2_G(0, select_out_aaa) → U3_G(0, U5_aa(ll_in_aa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), U6_aaa(select_in_aaa))
U1_G(s(z0), ll_out_ga(s(z0))) → U2_G(s(z0), select_out_aaa)
U1_G(0, ll_out_ga(0)) → U2_G(0, U6_aaa(select_in_aaa))
U3_G(s(x0), ll_out_aa(s(y_0))) → T_IN_G(s(y_0))
T_IN_G(s(x0)) → U1_G(s(x0), U5_ga(x0, ll_in_ga(x0)))
T_IN_G(0) → U1_G(0, ll_out_ga(0))
U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
U3_G(s(z0), ll_out_aa(0)) → T_IN_G(0)
U2_G(s(z0), select_out_aaa) → U3_G(s(z0), ll_out_aa(0))

The TRS R consists of the following rules:

ll_in_aa → U5_aa(ll_in_aa)
ll_in_aa → ll_out_aa(0)
select_in_aaa → U6_aaa(select_in_aaa)
select_in_aaa → select_out_aaa
ll_in_ga(s(N)) → U5_ga(N, ll_in_ga(N))
ll_in_ga(0) → ll_out_ga(0)
U5_aa(ll_out_aa(N)) → ll_out_aa(s(N))
U6_aaa(select_out_aaa) → select_out_aaa
U5_ga(N, ll_out_ga(N)) → ll_out_ga(s(N))


s = T_IN_G(0) evaluates to t =T_IN_G(0)

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



Rewriting sequence

T_IN_G(0) → U1_G(0, ll_out_ga(0))
with rule T_IN_G(0) → U1_G(0, ll_out_ga(0)) at position [] and matcher [ ]

U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa)
with rule U1_G(0, ll_out_ga(0)) → U2_G(0, select_out_aaa) at position [] and matcher [ ]

U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0))
with rule U2_G(0, select_out_aaa) → U3_G(0, ll_out_aa(0)) at position [] and matcher [ ]

U3_G(0, ll_out_aa(0)) → T_IN_G(0)
with rule U3_G(0, ll_out_aa(0)) → T_IN_G(0)

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


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