Left Termination of the query pattern list_in_1(g) 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 Instantiation (⇔)
    ↳24 QDP
    ↳25 NonTerminationProof (⇔)
    ↳26 FALSE
  ↳27 PrologToPiTRSProof (⇐)
    ↳28 PiTRS
    ↳29 DependencyPairsProof (⇔)
    ↳30 PiDP
    ↳31 DependencyGraphProof (⇔)
    ↳32 PiDP
    ↳33 UsableRulesProof (⇔)
    ↳34 PiDP
    ↳35 PiDPToQDPProof (⇐)
    ↳36 QDP
    ↳37 Narrowing (⇐)
    ↳38 QDP
    ↳39 UsableRulesProof (⇔)
    ↳40 QDP
    ↳41 QReductionProof (⇔)
    ↳42 QDP
    ↳43 QDPOrderProof (⇔)
    ↳44 QDP
      ↳45 Instantiation (⇔)
        ↳46 QDP
      ↳47 Instantiation (⇔)
        ↳48 QDP
        ↳49 NonTerminationProof (⇔)
        ↳50 FALSE

(0) Obligation:

Clauses:

list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
tail([], []).
tail(.(X, Xs), Xs).

Queries:

list(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

list([]) :- true.
list(X) :- ','(tail(X, T), list(T)).
tail([], []).
tail(.(X, Xs), Xs).

Queries:

list(g).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

list([]) :- true.
list(X) :- ','(tail(X, T), list(T)).
tail([], []).
tail(.(X, Xs), Xs).
true.

Queries:

list(g).

(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:
list_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g
U2_g(x1, x2)  =  U2_g(x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g
U2_g(x1, x2)  =  U2_g(x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x2)

(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:

LIST_IN_G([]) → U1_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U2_G(X, tail_out_ga(X, T)) → U3_G(X, list_in_g(T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g
U2_g(x1, x2)  =  U2_g(x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
TRUE_IN_  =  TRUE_IN_
U2_G(x1, x2)  =  U2_G(x2)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_G(x1, x2)  =  U3_G(x2)

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

(8) Obligation:

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

LIST_IN_G([]) → U1_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U2_G(X, tail_out_ga(X, T)) → U3_G(X, list_in_g(T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g
U2_g(x1, x2)  =  U2_g(x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
TRUE_IN_  =  TRUE_IN_
U2_G(x1, x2)  =  U2_G(x2)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_G(x1, x2)  =  U3_G(x2)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(10) Obligation:

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

LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g
U2_g(x1, x2)  =  U2_g(x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)

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:

LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
[]  =  []
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)

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:

LIST_IN_G(X) → U2_G(tail_in_ga(X))
U2_G(tail_out_ga(T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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(.(X, Xs)) → tail_out_ga(Xs)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + 2·x2   
POL(LIST_IN_G(x1)) = x1   
POL(U2_G(x1)) = x1   
POL([]) = 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:

LIST_IN_G(X) → U2_G(tail_in_ga(X))
U2_G(tail_out_ga(T)) → LIST_IN_G(T)

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.

(17) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LIST_IN_G(X) → U2_G(tail_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

LIST_IN_G([]) → U2_G(tail_out_ga([]))

(18) Obligation:

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

U2_G(tail_out_ga(T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G(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.

(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_G(tail_out_ga(T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G(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.

(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].

tail_in_ga(x0)

(22) Obligation:

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

U2_G(tail_out_ga(T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G(tail_out_ga([]))

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

(23) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_G(tail_out_ga(T)) → LIST_IN_G(T) we obtained the following new rules [LPAR04]:

U2_G(tail_out_ga([])) → LIST_IN_G([])

(24) Obligation:

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

LIST_IN_G([]) → U2_G(tail_out_ga([]))
U2_G(tail_out_ga([])) → LIST_IN_G([])

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

(25) 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_G(tail_out_ga([])) evaluates to t =U2_G(tail_out_ga([]))

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



Rewriting sequence

U2_G(tail_out_ga([])) → LIST_IN_G([])
with rule U2_G(tail_out_ga([])) → LIST_IN_G([]) at position [] and matcher [ ]

LIST_IN_G([]) → U2_G(tail_out_ga([]))
with rule LIST_IN_G([]) → U2_G(tail_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.



(26) FALSE

(27) PrologToPiTRSProof (SOUND transformation)

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

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(28) Obligation:

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

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x1, x2)

(29) 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:

LIST_IN_G([]) → U1_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U2_G(X, tail_out_ga(X, T)) → U3_G(X, list_in_g(T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
TRUE_IN_  =  TRUE_IN_
U2_G(x1, x2)  =  U2_G(x1, x2)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)

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

(30) Obligation:

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

LIST_IN_G([]) → U1_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U2_G(X, tail_out_ga(X, T)) → U3_G(X, list_in_g(T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
TRUE_IN_  =  TRUE_IN_
U2_G(x1, x2)  =  U2_G(x1, x2)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)

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

(31) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(32) Obligation:

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

LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

list_in_g([]) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → list_out_g([])
list_in_g(X) → U2_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_g(X, tail_out_ga(X, T)) → U3_g(X, list_in_g(T))
U3_g(X, list_out_g(T)) → list_out_g(X)

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
U1_g(x1)  =  U1_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
list_out_g(x1)  =  list_out_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_g(x1, x2)  =  U3_g(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

LIST_IN_G(X) → U2_G(X, tail_in_ga(X, T))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
[]  =  []
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

LIST_IN_G(X) → U2_G(X, tail_in_ga(X))
U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

tail_in_ga(x0)

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

(37) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LIST_IN_G(X) → U2_G(X, tail_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
LIST_IN_G(.(x0, x1)) → U2_G(.(x0, x1), tail_out_ga(.(x0, x1), x1))

(38) Obligation:

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

U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
LIST_IN_G(.(x0, x1)) → U2_G(.(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(.(X, Xs)) → tail_out_ga(.(X, Xs), Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(39) 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.

(40) Obligation:

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

U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
LIST_IN_G(.(x0, x1)) → U2_G(.(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.

(41) 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)

(42) Obligation:

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

U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
LIST_IN_G(.(x0, x1)) → U2_G(.(x0, x1), tail_out_ga(.(x0, x1), x1))

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LIST_IN_G(.(x0, x1)) → U2_G(.(x0, x1), tail_out_ga(.(x0, x1), x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U2_G(x1, x2)) = 0 +
[0,0]
·x1 +
[1,0]
·x2

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

POL(LIST_IN_G(x1)) = 0 +
[0,1]
·x1

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

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

The following usable rules [FROCOS05] were oriented: none

(44) Obligation:

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

U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U2_G([], tail_out_ga([], []))

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

(45) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T) we obtained the following new rules [LPAR04]:

U2_G([], tail_out_ga([], [])) → LIST_IN_G([])

(46) Obligation:

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

LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
U2_G([], tail_out_ga([], [])) → LIST_IN_G([])

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

(47) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_G(X, tail_out_ga(X, T)) → LIST_IN_G(T) we obtained the following new rules [LPAR04]:

U2_G([], tail_out_ga([], [])) → LIST_IN_G([])

(48) Obligation:

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

LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
U2_G([], tail_out_ga([], [])) → LIST_IN_G([])

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

(49) 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_G([], tail_out_ga([], [])) evaluates to t =U2_G([], tail_out_ga([], []))

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



Rewriting sequence

U2_G([], tail_out_ga([], [])) → LIST_IN_G([])
with rule U2_G([], tail_out_ga([], [])) → LIST_IN_G([]) at position [] and matcher [ ]

LIST_IN_G([]) → U2_G([], tail_out_ga([], []))
with rule LIST_IN_G([]) → U2_G([], tail_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.



(50) FALSE