Left Termination of the query pattern perm_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 NonTerminationProof (⇔)
        ↳20 FALSE
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔)
        ↳27 TRUE
  ↳28 PrologToPiTRSProof (⇐)
    ↳29 PiTRS
    ↳30 DependencyPairsProof (⇔)
    ↳31 PiDP
    ↳32 DependencyGraphProof (⇔)
    ↳33 AND
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔)
        ↳40 TRUE
      ↳41 PiDP
        ↳42 UsableRulesProof (⇔)
        ↳43 PiDP
        ↳44 PiDPToQDPProof (⇐)
        ↳45 QDP
        ↳46 NonTerminationProof (⇔)
        ↳47 FALSE
      ↳48 PiDP
        ↳49 UsableRulesProof (⇔)
        ↳50 PiDP
        ↳51 PiDPToQDPProof (⇐)
        ↳52 QDP
        ↳53 QDPSizeChangeProof (⇔)
        ↳54 TRUE

(0) Obligation:

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).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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

(2) Obligation:

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] 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)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
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)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)

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

(4) Obligation:

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)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
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)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APP_IN_GAA(.(Xs)) → APP_IN_GAA(Xs)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • APP_IN_GAA(.(Xs)) → APP_IN_GAA(Xs)
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APP_IN_AAA → APP_IN_AAA

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

(19) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

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



(20) FALSE

(21) Obligation:

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)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

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)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

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

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.

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • U2_AG(Ys, app_out_gaa) → PERM_IN_AG(Ys)
    The graph contains the following edges 1 >= 1

  • PERM_IN_AG(.(Ys)) → U1_AG(Ys, app_in_aaa)
    The graph contains the following edges 1 > 1

  • U1_AG(Ys, app_out_aaa(X1s)) → U2_AG(Ys, app_in_gaa(X1s))
    The graph contains the following edges 1 >= 1

(27) TRUE

(28) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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

(29) Obligation:

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)

(30) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] 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)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
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)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)

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

(31) Obligation:

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)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
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)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)

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

(32) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(33) Complex Obligation (AND)

(34) Obligation:

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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.

(39) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • APP_IN_GAA(.(Xs)) → APP_IN_GAA(Xs)
    The graph contains the following edges 1 > 1

(40) TRUE

(41) Obligation:

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

(42) UsableRulesProof (EQUIVALENT transformation)

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

(43) Obligation:

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

(44) PiDPToQDPProof (SOUND transformation)

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

(45) Obligation:

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.

(46) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

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



(47) FALSE

(48) Obligation:

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)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

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

(49) UsableRulesProof (EQUIVALENT transformation)

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

(50) Obligation:

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)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

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

(51) PiDPToQDPProof (SOUND transformation)

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

(52) Obligation:

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

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)
PERM_IN_AG(.(Ys)) → U1_AG(Ys, app_in_aaa)

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.

(53) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • U2_AG(Ys, app_out_gaa(X1s)) → PERM_IN_AG(Ys)
    The graph contains the following edges 1 >= 1

  • PERM_IN_AG(.(Ys)) → U1_AG(Ys, app_in_aaa)
    The graph contains the following edges 1 > 1

  • U1_AG(Ys, app_out_aaa(X1s)) → U2_AG(Ys, app_in_gaa(X1s))
    The graph contains the following edges 1 >= 1

(54) TRUE