MAYBE Left Termination proof of ../tpdb/LP/BCGGV05/permutation-fb.pl
Left Termination of the query pattern perm_in_2(a, g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

perm(a,g).

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

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)

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:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)


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

PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_GAA(X1s, X2s, Zs)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U4_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

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:

PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_GAA(X1s, X2s, Zs)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U4_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 5 less nodes.

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

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_GAA(.(Xs)) → APP_IN_GAA(Xs)

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
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_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
  ↳ PrologToPiTRSProof

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

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

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

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

APP_IN_AAA → APP_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:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none


s = APP_IN_AAA evaluates to t =APP_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 APP_IN_AAA to APP_IN_AAA.





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

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

U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

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
  ↳ PrologToPiTRSProof

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

U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

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
  ↳ PrologToPiTRSProof

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

PERM_IN_AG(.(Ys)) → U1_AG(Ys, app_in_aaa)
U2_AG(Ys, app_out_gaa) → PERM_IN_AG(Ys)
U1_AG(Ys, app_out_aaa(X1s)) → U2_AG(Ys, app_in_gaa(X1s))

The TRS R consists of the following rules:

app_in_gaa([]) → app_out_gaa
app_in_gaa(.(Xs)) → U4_gaa(app_in_gaa(Xs))
app_in_aaa → app_out_aaa([])
app_in_aaa → U4_aaa(app_in_aaa)
U4_gaa(app_out_gaa) → app_out_gaa
U4_aaa(app_out_aaa(Xs)) → app_out_aaa(.(Xs))

The set Q consists of the following terms:

app_in_gaa(x0)
app_in_aaa
U4_gaa(x0)
U4_aaa(x0)

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:


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

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)

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:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)


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

PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_GAA(X1s, X2s, Zs)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U4_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

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:

PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_GAA(X1s, X2s, Zs)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U4_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 5 less nodes.

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

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GAA(.(Xs)) → APP_IN_GAA(Xs)

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
                ↳ UsableRulesProof
              ↳ PiDP

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

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_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

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

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

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

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

APP_IN_AAA → APP_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:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none


s = APP_IN_AAA evaluates to t =APP_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 APP_IN_AAA to APP_IN_AAA.





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

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

U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U1_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_ag(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U2_ag(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → U3_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U3_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

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

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

U1_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U2_AG(Xs, X, Ys, app_in_gaa(X1s, X2s, Zs))
U2_AG(Xs, X, Ys, app_out_gaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U1_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in_gaa([], X, X) → app_out_gaa([], X, X)
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U4_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x2, x5)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

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

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

PERM_IN_AG(.(Ys)) → U1_AG(Ys, app_in_aaa)
U1_AG(Ys, app_out_aaa(X1s)) → U2_AG(Ys, app_in_gaa(X1s))
U2_AG(Ys, app_out_gaa(X1s)) → PERM_IN_AG(Ys)

The TRS R consists of the following rules:

app_in_gaa([]) → app_out_gaa([])
app_in_gaa(.(Xs)) → U4_gaa(Xs, app_in_gaa(Xs))
app_in_aaa → app_out_aaa([])
app_in_aaa → U4_aaa(app_in_aaa)
U4_gaa(Xs, app_out_gaa(Xs)) → app_out_gaa(.(Xs))
U4_aaa(app_out_aaa(Xs)) → app_out_aaa(.(Xs))

The set Q consists of the following terms:

app_in_gaa(x0)
app_in_aaa
U4_gaa(x0, x1)
U4_aaa(x0)

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: