(0) Obligation:

Clauses:

select(X, Y, Zs) :- ','(no(empty(Y)), ','(head(Y, X), tail(Y, Zs))).
select(X, Y, Z) :- ','(no(empty(Y)), ','(head(Y, H), ','(head(Z, H), ','(tail(Y, T), ','(tail(Z, Zs), select(X, T, Zs)))))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Queries:

select(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

select46(T31, .(T31, [])).
select46(T33, .(T40, T41)) :- select46(T33, T41).
select1(T12, .(T12, T13), T13).
select1(T31, .(T23, .(T31, [])), []).
select1(T33, .(T23, .(T40, T41)), []) :- select46(T33, T41).
select1(T14, .(T47, T46), .(T47, T49)) :- select1(T14, T46, T49).
select1(T14, .(T55, T54), .(T55, T57)) :- select1(T14, T54, T57).

Queries:

select1(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:
select1_in: (b,b,f)
select46_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x3)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
select46_in_gg(x1, x2)  =  select46_in_gg(x1, x2)
select46_out_gg(x1, x2)  =  select46_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, 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:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x3)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
select46_in_gg(x1, x2)  =  select46_in_gg(x1, x2)
select46_out_gg(x1, x2)  =  select46_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, 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:

SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → U2_GGA(T33, T23, T40, T41, select46_in_gg(T33, T41))
SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → SELECT46_IN_GG(T33, T41)
SELECT46_IN_GG(T33, .(T40, T41)) → U1_GG(T33, T40, T41, select46_in_gg(T33, T41))
SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → U3_GGA(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)
SELECT1_IN_GGA(T14, .(T55, T54), .(T55, T57)) → U4_GGA(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x3)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
select46_in_gg(x1, x2)  =  select46_in_gg(x1, x2)
select46_out_gg(x1, x2)  =  select46_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
SELECT46_IN_GG(x1, x2)  =  SELECT46_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x2, x5)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x5)

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

(6) Obligation:

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

SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → U2_GGA(T33, T23, T40, T41, select46_in_gg(T33, T41))
SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → SELECT46_IN_GG(T33, T41)
SELECT46_IN_GG(T33, .(T40, T41)) → U1_GG(T33, T40, T41, select46_in_gg(T33, T41))
SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → U3_GGA(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)
SELECT1_IN_GGA(T14, .(T55, T54), .(T55, T57)) → U4_GGA(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x3)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
select46_in_gg(x1, x2)  =  select46_in_gg(x1, x2)
select46_out_gg(x1, x2)  =  select46_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
SELECT46_IN_GG(x1, x2)  =  SELECT46_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x2, x5)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x3)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
select46_in_gg(x1, x2)  =  select46_in_gg(x1, x2)
select46_out_gg(x1, x2)  =  select46_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
SELECT46_IN_GG(x1, x2)  =  SELECT46_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)
    The graph contains the following edges 1 >= 1, 2 > 2

(15) TRUE

(16) Obligation:

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

SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x3)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
select46_in_gg(x1, x2)  =  select46_in_gg(x1, x2)
select46_out_gg(x1, x2)  =  select46_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

SELECT1_IN_GGA(T14, .(T47, T46)) → SELECT1_IN_GGA(T14, T46)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • SELECT1_IN_GGA(T14, .(T47, T46)) → SELECT1_IN_GGA(T14, T46)
    The graph contains the following edges 1 >= 1, 2 > 2

(22) TRUE