Left Termination of the query pattern select_in_3(g, 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 Narrowing (⇐)
    ↳22 QDP
    ↳23 UsableRulesProof (⇔)
    ↳24 QDP
    ↳25 QReductionProof (⇔)
    ↳26 QDP
    ↳27 Instantiation (⇔)
    ↳28 QDP
    ↳29 NonTerminationProof (⇔)
    ↳30 FALSE
  ↳31 PrologToPiTRSProof (⇐)
    ↳32 PiTRS
    ↳33 DependencyPairsProof (⇔)
    ↳34 PiDP
    ↳35 DependencyGraphProof (⇔)
    ↳36 PiDP
    ↳37 UsableRulesProof (⇔)
    ↳38 PiDP
    ↳39 PiDPToQDPProof (⇐)
    ↳40 QDP
    ↳41 Narrowing (⇐)
    ↳42 QDP
    ↳43 UsableRulesProof (⇔)
    ↳44 QDP
    ↳45 QReductionProof (⇔)
    ↳46 QDP
    ↳47 Narrowing (⇐)
    ↳48 QDP
    ↳49 UsableRulesProof (⇔)
    ↳50 QDP
    ↳51 QReductionProof (⇔)
    ↳52 QDP
    ↳53 QDPOrderProof (⇔)
    ↳54 QDP
    ↳55 DependencyGraphProof (⇔)
    ↳56 QDP
      ↳57 Instantiation (⇔)
        ↳58 QDP
      ↳59 Instantiation (⇔)
        ↳60 QDP
        ↳61 NonTerminationProof (⇔)
        ↳62 FALSE

(0) Obligation:

Clauses:

select(X1, [], X2) :- ','(!, failure(a)).
select(X, Y, Zs) :- ','(head(Y, X), tail(Y, Zs)).
select(X, Y, .(H, Zs)) :- ','(head(Y, H), ','(tail(Y, T), select(X, T, Zs))).
head([], X3).
head(.(H, X4), H).
tail([], []).
tail(.(X5, T), T).
failure(b).

Queries:

select(g,g,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

select(X1, [], X2) :- failure(a).
select(X, Y, Zs) :- ','(head(Y, X), tail(Y, Zs)).
select(X, Y, .(H, Zs)) :- ','(head(Y, H), ','(tail(Y, T), select(X, T, Zs))).
head([], X3).
head(.(H, X4), H).
tail([], []).
tail(.(X5, T), T).
failure(b).

Queries:

select(g,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:
select_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)

(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:

SELECT_IN_GGA(X1, [], X2) → U1_GGA(X1, X2, failure_in_g(a))
SELECT_IN_GGA(X1, [], X2) → FAILURE_IN_G(a)
SELECT_IN_GGA(X, Y, Zs) → U2_GGA(X, Y, Zs, head_in_gg(Y, X))
SELECT_IN_GGA(X, Y, Zs) → HEAD_IN_GG(Y, X)
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → U3_GGA(X, Y, Zs, tail_in_ga(Y, Zs))
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → TAIL_IN_GA(Y, Zs)
SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
SELECT_IN_GGA(X, Y, .(H, Zs)) → HEAD_IN_GA(Y, H)
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_GGA(X, Y, H, Zs, select_in_gga(X, T, Zs))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)
HEAD_IN_GG(x1, x2)  =  HEAD_IN_GG(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x5)

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

(6) Obligation:

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

SELECT_IN_GGA(X1, [], X2) → U1_GGA(X1, X2, failure_in_g(a))
SELECT_IN_GGA(X1, [], X2) → FAILURE_IN_G(a)
SELECT_IN_GGA(X, Y, Zs) → U2_GGA(X, Y, Zs, head_in_gg(Y, X))
SELECT_IN_GGA(X, Y, Zs) → HEAD_IN_GG(Y, X)
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → U3_GGA(X, Y, Zs, tail_in_ga(Y, Zs))
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → TAIL_IN_GA(Y, Zs)
SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
SELECT_IN_GGA(X, Y, .(H, Zs)) → HEAD_IN_GA(Y, H)
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_GGA(X, Y, H, Zs, select_in_gga(X, T, Zs))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)
HEAD_IN_GG(x1, x2)  =  HEAD_IN_GG(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 9 less nodes.

(8) Obligation:

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

SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)

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:

SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)

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:

SELECT_IN_GGA(X, Y) → U4_GGA(X, Y, head_in_ga(Y))
U4_GGA(X, Y, head_out_ga) → U5_GGA(X, tail_in_ga(Y))
U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(H, X4)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(X5, T)) → tail_out_ga(T)

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.

(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(.(X5, T)) → tail_out_ga(T)
head_in_ga(.(H, X4)) → head_out_ga
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + 2·x2   
POL(SELECT_IN_GGA(x1, x2)) = x1 + 2·x2   
POL(U4_GGA(x1, x2, x3)) = x1 + x2 + x3   
POL(U5_GGA(x1, x2)) = x1 + x2   
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   

(14) Obligation:

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

SELECT_IN_GGA(X, Y) → U4_GGA(X, Y, head_in_ga(Y))
U4_GGA(X, Y, head_out_ga) → U5_GGA(X, tail_in_ga(Y))
U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, 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.

(15) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SELECT_IN_GGA(X, Y) → U4_GGA(X, Y, head_in_ga(Y)) at position [2] we obtained the following new rules [LPAR04]:

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga)

(16) Obligation:

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

U4_GGA(X, Y, head_out_ga) → U5_GGA(X, tail_in_ga(Y))
U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], 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.

(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:

U4_GGA(X, Y, head_out_ga) → U5_GGA(X, tail_in_ga(Y))
U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], 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.

(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].

head_in_ga(x0)

(20) Obligation:

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

U4_GGA(X, Y, head_out_ga) → U5_GGA(X, tail_in_ga(Y))
U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], 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.

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U4_GGA(X, Y, head_out_ga) → U5_GGA(X, tail_in_ga(Y)) at position [1] we obtained the following new rules [LPAR04]:

U4_GGA(y0, [], head_out_ga) → U5_GGA(y0, tail_out_ga([]))

(22) Obligation:

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

U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga)
U4_GGA(y0, [], head_out_ga) → U5_GGA(y0, 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.

(23) 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.

(24) Obligation:

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

U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga)
U4_GGA(y0, [], head_out_ga) → U5_GGA(y0, 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.

(25) 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)

(26) Obligation:

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

U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga)
U4_GGA(y0, [], head_out_ga) → U5_GGA(y0, tail_out_ga([]))

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

(27) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GGA(X, tail_out_ga(T)) → SELECT_IN_GGA(X, T) we obtained the following new rules [LPAR04]:

U5_GGA(z0, tail_out_ga([])) → SELECT_IN_GGA(z0, [])

(28) Obligation:

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

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga)
U4_GGA(y0, [], head_out_ga) → U5_GGA(y0, tail_out_ga([]))
U5_GGA(z0, tail_out_ga([])) → SELECT_IN_GGA(z0, [])

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

(29) 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 = U4_GGA(y0, [], head_out_ga) evaluates to t =U4_GGA(y0, [], head_out_ga)

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



Rewriting sequence

U4_GGA(y0, [], head_out_ga) → U5_GGA(y0, tail_out_ga([]))
with rule U4_GGA(y0', [], head_out_ga) → U5_GGA(y0', tail_out_ga([])) at position [] and matcher [y0' / y0]

U5_GGA(y0, tail_out_ga([])) → SELECT_IN_GGA(y0, [])
with rule U5_GGA(z0, tail_out_ga([])) → SELECT_IN_GGA(z0, []) at position [] and matcher [z0 / y0]

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga)
with rule SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], 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.



(30) FALSE

(31) PrologToPiTRSProof (SOUND transformation)

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

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga(x1, x2)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(32) Obligation:

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

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga(x1, x2)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)

(33) 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:

SELECT_IN_GGA(X1, [], X2) → U1_GGA(X1, X2, failure_in_g(a))
SELECT_IN_GGA(X1, [], X2) → FAILURE_IN_G(a)
SELECT_IN_GGA(X, Y, Zs) → U2_GGA(X, Y, Zs, head_in_gg(Y, X))
SELECT_IN_GGA(X, Y, Zs) → HEAD_IN_GG(Y, X)
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → U3_GGA(X, Y, Zs, tail_in_ga(Y, Zs))
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → TAIL_IN_GA(Y, Zs)
SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
SELECT_IN_GGA(X, Y, .(H, Zs)) → HEAD_IN_GA(Y, H)
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_GGA(X, Y, H, Zs, select_in_gga(X, T, Zs))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga(x1, x2)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
HEAD_IN_GG(x1, x2)  =  HEAD_IN_GG(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x5)

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

(34) Obligation:

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

SELECT_IN_GGA(X1, [], X2) → U1_GGA(X1, X2, failure_in_g(a))
SELECT_IN_GGA(X1, [], X2) → FAILURE_IN_G(a)
SELECT_IN_GGA(X, Y, Zs) → U2_GGA(X, Y, Zs, head_in_gg(Y, X))
SELECT_IN_GGA(X, Y, Zs) → HEAD_IN_GG(Y, X)
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → U3_GGA(X, Y, Zs, tail_in_ga(Y, Zs))
U2_GGA(X, Y, Zs, head_out_gg(Y, X)) → TAIL_IN_GA(Y, Zs)
SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
SELECT_IN_GGA(X, Y, .(H, Zs)) → HEAD_IN_GA(Y, H)
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_GGA(X, Y, H, Zs, select_in_gga(X, T, Zs))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga(x1, x2)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
HEAD_IN_GG(x1, x2)  =  HEAD_IN_GG(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x5)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 9 less nodes.

(36) Obligation:

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

SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

select_in_gga(X1, [], X2) → U1_gga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U1_gga(X1, X2, failure_out_g(a)) → select_out_gga(X1, [], X2)
select_in_gga(X, Y, Zs) → U2_gga(X, Y, Zs, head_in_gg(Y, X))
head_in_gg([], X3) → head_out_gg([], X3)
head_in_gg(.(H, X4), H) → head_out_gg(.(H, X4), H)
U2_gga(X, Y, Zs, head_out_gg(Y, X)) → U3_gga(X, Y, Zs, tail_in_ga(Y, Zs))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)
U3_gga(X, Y, Zs, tail_out_ga(Y, Zs)) → select_out_gga(X, Y, Zs)
select_in_gga(X, Y, .(H, Zs)) → U4_gga(X, Y, H, Zs, head_in_ga(Y, H))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
U4_gga(X, Y, H, Zs, head_out_ga(Y, H)) → U5_gga(X, Y, H, Zs, tail_in_ga(Y, T))
U5_gga(X, Y, H, Zs, tail_out_ga(Y, T)) → U6_gga(X, Y, H, Zs, select_in_gga(X, T, Zs))
U6_gga(X, Y, H, Zs, select_out_gga(X, T, Zs)) → select_out_gga(X, Y, .(H, Zs))

The argument filtering Pi contains the following mapping:
select_in_gga(x1, x2, x3)  =  select_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
select_out_gga(x1, x2, x3)  =  select_out_gga(x1, x2)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_gg(x1, x2)  =  head_in_gg(x1, x2)
head_out_gg(x1, x2)  =  head_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

SELECT_IN_GGA(X, Y, .(H, Zs)) → U4_GGA(X, Y, H, Zs, head_in_ga(Y, H))
U4_GGA(X, Y, H, Zs, head_out_ga(Y, H)) → U5_GGA(X, Y, H, Zs, tail_in_ga(Y, T))
U5_GGA(X, Y, H, Zs, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T, Zs)

The TRS R consists of the following rules:

head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(H, X4), H) → head_out_ga(.(H, X4), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, T), T) → tail_out_ga(.(X5, T), T)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
SELECT_IN_GGA(x1, x2, x3)  =  SELECT_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

SELECT_IN_GGA(X, Y) → U4_GGA(X, Y, head_in_ga(Y))
U4_GGA(X, Y, head_out_ga(Y)) → U5_GGA(X, Y, tail_in_ga(Y))
U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(H, X4)) → head_out_ga(.(H, X4))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, T)) → tail_out_ga(.(X5, T), T)

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.

(41) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SELECT_IN_GGA(X, Y) → U4_GGA(X, Y, head_in_ga(Y)) at position [2] we obtained the following new rules [LPAR04]:

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))

(42) Obligation:

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

U4_GGA(X, Y, head_out_ga(Y)) → U5_GGA(X, Y, tail_in_ga(Y))
U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(H, X4)) → head_out_ga(.(H, X4))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, T)) → tail_out_ga(.(X5, T), T)

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) 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.

(44) Obligation:

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

U4_GGA(X, Y, head_out_ga(Y)) → U5_GGA(X, Y, tail_in_ga(Y))
U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, T)) → tail_out_ga(.(X5, T), T)

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) 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)

(46) Obligation:

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

U4_GGA(X, Y, head_out_ga(Y)) → U5_GGA(X, Y, tail_in_ga(Y))
U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, T)) → tail_out_ga(.(X5, T), T)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(47) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U4_GGA(X, Y, head_out_ga(Y)) → U5_GGA(X, Y, tail_in_ga(Y)) at position [2] we obtained the following new rules [LPAR04]:

U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1))) → U5_GGA(y0, .(x0, x1), tail_out_ga(.(x0, x1), x1))

(48) Obligation:

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

U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1))) → U5_GGA(y0, .(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(.(X5, T)) → tail_out_ga(.(X5, T), T)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(49) 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.

(50) Obligation:

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

U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1))) → U5_GGA(y0, .(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.

(51) 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)

(52) Obligation:

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

U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1))) → U5_GGA(y0, .(x0, x1), tail_out_ga(.(x0, x1), x1))

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1))) → U5_GGA(y0, .(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(SELECT_IN_GGA(x1, x2)) = x2   
POL(U4_GGA(x1, x2, x3)) = x3   
POL(U5_GGA(x1, x2, x3)) = x3   
POL([]) = 0   
POL(head_out_ga(x1)) = x1   
POL(tail_out_ga(x1, x2)) = x2   

The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)
SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
SELECT_IN_GGA(y0, .(x0, x1)) → U4_GGA(y0, .(x0, x1), head_out_ga(.(x0, x1)))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))

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

(55) DependencyGraphProof (EQUIVALENT transformation)

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

(56) Obligation:

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

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T)

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

(57) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T) we obtained the following new rules [LPAR04]:

U5_GGA(z0, [], tail_out_ga([], [])) → SELECT_IN_GGA(z0, [])

(58) Obligation:

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

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U5_GGA(z0, [], tail_out_ga([], [])) → SELECT_IN_GGA(z0, [])

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

(59) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GGA(X, Y, tail_out_ga(Y, T)) → SELECT_IN_GGA(X, T) we obtained the following new rules [LPAR04]:

U5_GGA(z0, [], tail_out_ga([], [])) → SELECT_IN_GGA(z0, [])

(60) Obligation:

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

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
U5_GGA(z0, [], tail_out_ga([], [])) → SELECT_IN_GGA(z0, [])

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

(61) 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 = U4_GGA(y0, [], head_out_ga([])) evaluates to t =U4_GGA(y0, [], head_out_ga([]))

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



Rewriting sequence

U4_GGA(y0, [], head_out_ga([])) → U5_GGA(y0, [], tail_out_ga([], []))
with rule U4_GGA(y0', [], head_out_ga([])) → U5_GGA(y0', [], tail_out_ga([], [])) at position [] and matcher [y0' / y0]

U5_GGA(y0, [], tail_out_ga([], [])) → SELECT_IN_GGA(y0, [])
with rule U5_GGA(z0, [], tail_out_ga([], [])) → SELECT_IN_GGA(z0, []) at position [] and matcher [z0 / y0]

SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], head_out_ga([]))
with rule SELECT_IN_GGA(y0, []) → U4_GGA(y0, [], 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.



(62) FALSE