Left Termination of the query pattern app_in_3(g, a, a) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 CutEliminatorProof (⇐)
↳2 Prolog
↳3 UndefinedPredicateHandlerProof (⇐)
↳4 Prolog
  ↳5 PrologToPiTRSProof (⇐)
    ↳6 PiTRS
    ↳7 DependencyPairsProof (⇔)
    ↳8 PiDP
    ↳9 DependencyGraphProof (⇔)
    ↳10 PiDP
    ↳11 UsableRulesProof (⇔)
    ↳12 PiDP
    ↳13 PiDPToQDPProof (⇐)
    ↳14 QDP
    ↳15 UsableRulesReductionPairsProof (⇔)
    ↳16 QDP
    ↳17 Narrowing (⇐)
    ↳18 QDP
    ↳19 UsableRulesProof (⇔)
    ↳20 QDP
    ↳21 QReductionProof (⇔)
    ↳22 QDP
    ↳23 Narrowing (⇐)
    ↳24 QDP
    ↳25 UsableRulesProof (⇔)
    ↳26 QDP
    ↳27 QReductionProof (⇔)
    ↳28 QDP
    ↳29 Instantiation (⇔)
    ↳30 QDP
    ↳31 NonTerminationProof (⇔)
    ↳32 FALSE
  ↳33 PrologToPiTRSProof (⇐)
    ↳34 PiTRS
    ↳35 DependencyPairsProof (⇔)
    ↳36 PiDP
    ↳37 DependencyGraphProof (⇔)
    ↳38 PiDP
    ↳39 UsableRulesProof (⇔)
    ↳40 PiDP
    ↳41 PiDPToQDPProof (⇐)
    ↳42 QDP
    ↳43 Narrowing (⇐)
    ↳44 QDP
    ↳45 UsableRulesProof (⇔)
    ↳46 QDP
    ↳47 QReductionProof (⇔)
    ↳48 QDP
    ↳49 Narrowing (⇐)
    ↳50 QDP
    ↳51 UsableRulesProof (⇔)
    ↳52 QDP
    ↳53 QReductionProof (⇔)
    ↳54 QDP
      ↳55 QDPOrderProof (⇔)
        ↳56 QDP
        ↳57 DependencyGraphProof (⇔)
        ↳58 QDP
          ↳59 Instantiation (⇔)
            ↳60 QDP
          ↳61 Instantiation (⇔)
            ↳62 QDP
            ↳63 NonTerminationProof (⇔)
            ↳64 FALSE
      ↳65 QDPOrderProof (⇔)
        ↳66 QDP

(0) Obligation:

Clauses:

app([], L, L) :- !.
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).

Queries:

app(g,a,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

app([], L, L) :- true.
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).

Queries:

app(g,a,a).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

app([], L, L) :- true.
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
true.

Queries:

app(g,a,a).

(5) PrologToPiTRSProof (SOUND transformation)

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

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)

(7) 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:

APP_IN_GAA([], L, L) → U1_GAA(L, true_in_)
APP_IN_GAA([], L, L) → TRUE_IN_
APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U4_GAA(X, Y, H, Z, app_in_gaa(T, Y, Z))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_GAA(x1, x2)  =  U1_GAA(x2)
TRUE_IN_  =  TRUE_IN_
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)

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

(8) Obligation:

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

APP_IN_GAA([], L, L) → U1_GAA(L, true_in_)
APP_IN_GAA([], L, L) → TRUE_IN_
APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U4_GAA(X, Y, H, Z, app_in_gaa(T, Y, Z))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_GAA(x1, x2)  =  U1_GAA(x2)
TRUE_IN_  =  TRUE_IN_
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(10) Obligation:

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

APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)

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

(11) UsableRulesProof (EQUIVALENT transformation)

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

(12) Obligation:

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

APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

APP_IN_GAA(X) → U2_GAA(X, head_in_ga(X))
U2_GAA(X, head_out_ga) → U3_GAA(tail_in_ga(X))
U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(X, X2)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(X3, Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(15) UsableRulesReductionPairsProof (EQUIVALENT transformation)

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

No dependency pairs are removed.

The following rules are removed from R:

tail_in_ga(.(X3, Xs)) → tail_out_ga(Xs)
head_in_ga(.(X, X2)) → head_out_ga
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + 2·x2   
POL(APP_IN_GAA(x1)) = 2·x1   
POL(U2_GAA(x1, x2)) = x1 + x2   
POL(U3_GAA(x1)) = x1   
POL([]) = 0   
POL(head_in_ga(x1)) = x1   
POL(head_out_ga) = 0   
POL(tail_in_ga(x1)) = x1   
POL(tail_out_ga(x1)) = 2·x1   

(16) Obligation:

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

APP_IN_GAA(X) → U2_GAA(X, head_in_ga(X))
U2_GAA(X, head_out_ga) → U3_GAA(tail_in_ga(X))
U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
head_in_ga([]) → head_out_ga

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(17) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule APP_IN_GAA(X) → U2_GAA(X, head_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

APP_IN_GAA([]) → U2_GAA([], head_out_ga)

(18) Obligation:

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

U2_GAA(X, head_out_ga) → U3_GAA(tail_in_ga(X))
U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
head_in_ga([]) → head_out_ga

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

U2_GAA(X, head_out_ga) → U3_GAA(tail_in_ga(X))
U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(21) QReductionProof (EQUIVALENT transformation)

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

head_in_ga(x0)

(22) Obligation:

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

U2_GAA(X, head_out_ga) → U3_GAA(tail_in_ga(X))
U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])

The set Q consists of the following terms:

tail_in_ga(x0)

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

(23) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U2_GAA(X, head_out_ga) → U3_GAA(tail_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([]))

(24) Obligation:

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

U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga)
U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([]))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])

The set Q consists of the following terms:

tail_in_ga(x0)

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

(25) UsableRulesProof (EQUIVALENT transformation)

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

(26) Obligation:

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

U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga)
U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([]))

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

tail_in_ga(x0)

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

(27) QReductionProof (EQUIVALENT transformation)

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

tail_in_ga(x0)

(28) Obligation:

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

U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga)
U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([]))

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

(29) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAA(tail_out_ga(T)) → APP_IN_GAA(T) we obtained the following new rules [LPAR04]:

U3_GAA(tail_out_ga([])) → APP_IN_GAA([])

(30) Obligation:

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

APP_IN_GAA([]) → U2_GAA([], head_out_ga)
U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([]))
U3_GAA(tail_out_ga([])) → APP_IN_GAA([])

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

(31) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U2_GAA([], head_out_ga) evaluates to t =U2_GAA([], head_out_ga)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([]))
with rule U2_GAA([], head_out_ga) → U3_GAA(tail_out_ga([])) at position [] and matcher [ ]

U3_GAA(tail_out_ga([])) → APP_IN_GAA([])
with rule U3_GAA(tail_out_ga([])) → APP_IN_GAA([]) at position [] and matcher [ ]

APP_IN_GAA([]) → U2_GAA([], head_out_ga)
with rule APP_IN_GAA([]) → U2_GAA([], head_out_ga)

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.



(32) FALSE

(33) PrologToPiTRSProof (SOUND transformation)

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

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(34) Obligation:

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

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x1, x5)

(35) 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:

APP_IN_GAA([], L, L) → U1_GAA(L, true_in_)
APP_IN_GAA([], L, L) → TRUE_IN_
APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U4_GAA(X, Y, H, Z, app_in_gaa(T, Y, Z))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x1, x5)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_GAA(x1, x2)  =  U1_GAA(x2)
TRUE_IN_  =  TRUE_IN_
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x1, x5)

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

(36) Obligation:

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

APP_IN_GAA([], L, L) → U1_GAA(L, true_in_)
APP_IN_GAA([], L, L) → TRUE_IN_
APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U4_GAA(X, Y, H, Z, app_in_gaa(T, Y, Z))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x1, x5)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_GAA(x1, x2)  =  U1_GAA(x2)
TRUE_IN_  =  TRUE_IN_
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x1, x5)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(38) Obligation:

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

APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

app_in_gaa([], L, L) → U1_gaa(L, true_in_)
true_in_ → true_out_
U1_gaa(L, true_out_) → app_out_gaa([], L, L)
app_in_gaa(X, Y, .(H, Z)) → U2_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U2_gaa(X, Y, H, Z, head_out_ga(X, H)) → U3_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U3_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U4_gaa(X, Y, H, Z, app_in_gaa(T, Y, Z))
U4_gaa(X, Y, H, Z, app_out_gaa(T, Y, Z)) → app_out_gaa(X, Y, .(H, Z))

The argument filtering Pi contains the following mapping:
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
[]  =  []
U1_gaa(x1, x2)  =  U1_gaa(x2)
true_in_  =  true_in_
true_out_  =  true_out_
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x1, x5)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)

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

(39) UsableRulesProof (EQUIVALENT transformation)

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

(40) Obligation:

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

APP_IN_GAA(X, Y, .(H, Z)) → U2_GAA(X, Y, H, Z, head_in_ga(X, H))
U2_GAA(X, Y, H, Z, head_out_ga(X, H)) → U3_GAA(X, Y, H, Z, tail_in_ga(X, T))
U3_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)

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

(41) PiDPToQDPProof (SOUND transformation)

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

(42) Obligation:

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

APP_IN_GAA(X) → U2_GAA(X, head_in_ga(X))
U2_GAA(X, head_out_ga(X)) → U3_GAA(X, tail_in_ga(X))
U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X, X2)) → head_out_ga(.(X, X2))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs)) → tail_out_ga(.(X3, Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(43) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule APP_IN_GAA(X) → U2_GAA(X, head_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))

(44) Obligation:

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

U2_GAA(X, head_out_ga(X)) → U3_GAA(X, tail_in_ga(X))
U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X, X2)) → head_out_ga(.(X, X2))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs)) → tail_out_ga(.(X3, Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

U2_GAA(X, head_out_ga(X)) → U3_GAA(X, tail_in_ga(X))
U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs)) → tail_out_ga(.(X3, Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(47) QReductionProof (EQUIVALENT transformation)

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

head_in_ga(x0)

(48) Obligation:

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

U2_GAA(X, head_out_ga(X)) → U3_GAA(X, tail_in_ga(X))
U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs)) → tail_out_ga(.(X3, Xs), Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U2_GAA(X, head_out_ga(X)) → U3_GAA(X, tail_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U2_GAA(.(x0, x1), head_out_ga(.(x0, x1))) → U3_GAA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

(50) Obligation:

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

U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U2_GAA(.(x0, x1), head_out_ga(.(x0, x1))) → U3_GAA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs)) → tail_out_ga(.(X3, Xs), Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U2_GAA(.(x0, x1), head_out_ga(.(x0, x1))) → U3_GAA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

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

tail_in_ga(x0)

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

(53) QReductionProof (EQUIVALENT transformation)

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

tail_in_ga(x0)

(54) Obligation:

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

U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U2_GAA(.(x0, x1), head_out_ga(.(x0, x1))) → U3_GAA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

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

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U3_GAA(x1, x2)) = 0 +
[0,0]
·x1 +
[1,0]
·x2

POL(tail_out_ga(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/01\
\00/
·x2

POL(APP_IN_GAA(x1)) = 0 +
[0,1]
·x1

POL([]) =
/0\
\0/

POL(U2_GAA(x1, x2)) = 0 +
[0,0]
·x1 +
[0,1]
·x2

POL(head_out_ga(x1)) =
/0\
\0/
 +
/00\
\10/
·x1

POL(.(x1, x2)) =
/0\
\1/
 +
/00\
\11/
·x1 +
/01\
\11/
·x2

The following usable rules [FROCOS05] were oriented: none

(56) Obligation:

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

U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U2_GAA(.(x0, x1), head_out_ga(.(x0, x1))) → U3_GAA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

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

(57) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(58) Obligation:

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

APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)

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

(59) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T) we obtained the following new rules [LPAR04]:

U3_GAA([], tail_out_ga([], [])) → APP_IN_GAA([])

(60) Obligation:

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

APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U3_GAA([], tail_out_ga([], [])) → APP_IN_GAA([])

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

(61) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T) we obtained the following new rules [LPAR04]:

U3_GAA([], tail_out_ga([], [])) → APP_IN_GAA([])

(62) Obligation:

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

APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
U3_GAA([], tail_out_ga([], [])) → APP_IN_GAA([])

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

(63) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U2_GAA([], head_out_ga([])) evaluates to t =U2_GAA([], head_out_ga([]))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))
with rule U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], [])) at position [] and matcher [ ]

U3_GAA([], tail_out_ga([], [])) → APP_IN_GAA([])
with rule U3_GAA([], tail_out_ga([], [])) → APP_IN_GAA([]) at position [] and matcher [ ]

APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
with rule APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))

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.



(64) FALSE

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2_GAA(.(x0, x1), head_out_ga(.(x0, x1))) → U3_GAA(.(x0, x1), tail_out_ga(.(x0, x1), x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(APP_IN_GAA(x1)) = x1   
POL(U2_GAA(x1, x2)) = x2   
POL(U3_GAA(x1, x2)) = x2   
POL([]) = 0   
POL(head_out_ga(x1)) = x1   
POL(tail_out_ga(x1, x2)) = x2   

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U3_GAA(X, tail_out_ga(X, T)) → APP_IN_GAA(T)
APP_IN_GAA([]) → U2_GAA([], head_out_ga([]))
APP_IN_GAA(.(x0, x1)) → U2_GAA(.(x0, x1), head_out_ga(.(x0, x1)))
U2_GAA([], head_out_ga([])) → U3_GAA([], tail_out_ga([], []))

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