Left Termination of the query pattern len_in_2(g, a) w.r.t. the given Prolog program could not be shown:

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

(0) Obligation:

Clauses:

len([], X) :- ','(!, eq(X, 0)).
len(Xs, s(N)) :- ','(tail(Xs, Ys), len(Ys, N)).
tail([], []).
tail(.(X, Xs), Xs).
eq(X, X).

Queries:

len(g,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

len([], X) :- eq(X, 0).
len(Xs, s(N)) :- ','(tail(Xs, Ys), len(Ys, N)).
tail([], []).
tail(.(X, Xs), Xs).
eq(X, X).

Queries:

len(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

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

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
s(x1)  =  s(x1)

(5) 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:

LEN_IN_GA([], X) → U1_GA(X, eq_in_ag(X, 0))
LEN_IN_GA([], X) → EQ_IN_AG(X, 0)
LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
LEN_IN_GA(Xs, s(N)) → TAIL_IN_GA(Xs, Ys)
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → U3_GA(Xs, N, len_in_ga(Ys, N))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)

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

(6) Obligation:

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

LEN_IN_GA([], X) → U1_GA(X, eq_in_ag(X, 0))
LEN_IN_GA([], X) → EQ_IN_AG(X, 0)
LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
LEN_IN_GA(Xs, s(N)) → TAIL_IN_GA(Xs, Ys)
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → U3_GA(Xs, N, len_in_ga(Ys, N))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

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)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

LEN_IN_GA(Xs) → U2_GA(tail_in_ga(Xs))
U2_GA(tail_out_ga(Ys)) → LEN_IN_GA(Ys)

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.

(13) 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(LEN_IN_GA(x1)) = x1   
POL(U2_GA(x1)) = x1   
POL([]) = 0   
POL(tail_in_ga(x1)) = x1   
POL(tail_out_ga(x1)) = 2·x1   

(14) Obligation:

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

LEN_IN_GA(Xs) → U2_GA(tail_in_ga(Xs))
U2_GA(tail_out_ga(Ys)) → LEN_IN_GA(Ys)

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.

(15) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LEN_IN_GA(Xs) → U2_GA(tail_in_ga(Xs)) at position [0] we obtained the following new rules [LPAR04]:

LEN_IN_GA([]) → U2_GA(tail_out_ga([]))

(16) Obligation:

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

U2_GA(tail_out_ga(Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA(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.

(17) 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.

(18) Obligation:

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

U2_GA(tail_out_ga(Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA(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.

(19) 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)

(20) Obligation:

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

U2_GA(tail_out_ga(Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA(tail_out_ga([]))

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

(21) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GA(tail_out_ga(Ys)) → LEN_IN_GA(Ys) we obtained the following new rules [LPAR04]:

U2_GA(tail_out_ga([])) → LEN_IN_GA([])

(22) Obligation:

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

LEN_IN_GA([]) → U2_GA(tail_out_ga([]))
U2_GA(tail_out_ga([])) → LEN_IN_GA([])

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

(23) 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_GA(tail_out_ga([])) evaluates to t =U2_GA(tail_out_ga([]))

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



Rewriting sequence

U2_GA(tail_out_ga([])) → LEN_IN_GA([])
with rule U2_GA(tail_out_ga([])) → LEN_IN_GA([]) at position [] and matcher [ ]

LEN_IN_GA([]) → U2_GA(tail_out_ga([]))
with rule LEN_IN_GA([]) → U2_GA(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.



(24) FALSE

(25) PrologToPiTRSProof (SOUND transformation)

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

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(26) Obligation:

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

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
s(x1)  =  s(x1)

(27) 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:

LEN_IN_GA([], X) → U1_GA(X, eq_in_ag(X, 0))
LEN_IN_GA([], X) → EQ_IN_AG(X, 0)
LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
LEN_IN_GA(Xs, s(N)) → TAIL_IN_GA(Xs, Ys)
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → U3_GA(Xs, N, len_in_ga(Ys, N))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(28) Obligation:

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

LEN_IN_GA([], X) → U1_GA(X, eq_in_ag(X, 0))
LEN_IN_GA([], X) → EQ_IN_AG(X, 0)
LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
LEN_IN_GA(Xs, s(N)) → TAIL_IN_GA(Xs, Ys)
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → U3_GA(Xs, N, len_in_ga(Ys, N))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(29) DependencyGraphProof (EQUIVALENT transformation)

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

(30) Obligation:

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

LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

len_in_ga([], X) → U1_ga(X, eq_in_ag(X, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_ga(X, eq_out_ag(X, 0)) → len_out_ga([], X)
len_in_ga(Xs, s(N)) → U2_ga(Xs, N, tail_in_ga(Xs, Ys))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X, Xs), Xs) → tail_out_ga(.(X, Xs), Xs)
U2_ga(Xs, N, tail_out_ga(Xs, Ys)) → U3_ga(Xs, N, len_in_ga(Ys, N))
U3_ga(Xs, N, len_out_ga(Ys, N)) → len_out_ga(Xs, s(N))

The argument filtering Pi contains the following mapping:
len_in_ga(x1, x2)  =  len_in_ga(x1)
[]  =  []
U1_ga(x1, x2)  =  U1_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
0  =  0
len_out_ga(x1, x2)  =  len_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

LEN_IN_GA(Xs, s(N)) → U2_GA(Xs, N, tail_in_ga(Xs, Ys))
U2_GA(Xs, N, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys, N)

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)
s(x1)  =  s(x1)
LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

LEN_IN_GA(Xs) → U2_GA(Xs, tail_in_ga(Xs))
U2_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys)

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.

(35) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LEN_IN_GA(Xs) → U2_GA(Xs, tail_in_ga(Xs)) at position [1] we obtained the following new rules [LPAR04]:

LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
LEN_IN_GA(.(x0, x1)) → U2_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

(36) Obligation:

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

U2_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
LEN_IN_GA(.(x0, x1)) → U2_GA(.(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.

(37) 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.

(38) Obligation:

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

U2_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
LEN_IN_GA(.(x0, x1)) → U2_GA(.(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.

(39) 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)

(40) Obligation:

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

U2_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
LEN_IN_GA(.(x0, x1)) → U2_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LEN_IN_GA(.(x0, x1)) → U2_GA(.(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(LEN_IN_GA(x1)) = x1   
POL(U2_GA(x1, x2)) = x2   
POL([]) = 0   
POL(tail_out_ga(x1, x2)) = x2   

The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

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

U2_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys)
LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))

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

(43) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys) we obtained the following new rules [LPAR04]:

U2_GA([], tail_out_ga([], [])) → LEN_IN_GA([])

(44) Obligation:

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

LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
U2_GA([], tail_out_ga([], [])) → LEN_IN_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_GA(Xs, tail_out_ga(Xs, Ys)) → LEN_IN_GA(Ys) we obtained the following new rules [LPAR04]:

U2_GA([], tail_out_ga([], [])) → LEN_IN_GA([])

(46) Obligation:

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

LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
U2_GA([], tail_out_ga([], [])) → LEN_IN_GA([])

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

(47) 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_GA([], tail_out_ga([], [])) evaluates to t =U2_GA([], tail_out_ga([], []))

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



Rewriting sequence

U2_GA([], tail_out_ga([], [])) → LEN_IN_GA([])
with rule U2_GA([], tail_out_ga([], [])) → LEN_IN_GA([]) at position [] and matcher [ ]

LEN_IN_GA([]) → U2_GA([], tail_out_ga([], []))
with rule LEN_IN_GA([]) → U2_GA([], 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.



(48) FALSE