Left Termination of the query pattern select_in_3(g, a, a) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
    ↳3 PrologToPiTRSProof (⇐)
        ↳4 PiTRS
        ↳5 DependencyPairsProof (⇔)
        ↳6 PiDP
        ↳7 DependencyGraphProof (⇔)
        ↳8 PiDP
        ↳9 UsableRulesProof (⇔)
        ↳10 PiDP
        ↳11 PiDPToQDPProof (⇐)
        ↳12 QDP
        ↳13 NonTerminationProof (⇔)
        ↳14 NO
    ↳15 PrologToPiTRSProof (⇐)
        ↳16 PiTRS
        ↳17 DependencyPairsProof (⇔)
        ↳18 PiDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 PiDP
        ↳21 UsableRulesProof (⇔)
        ↳22 PiDP
        ↳23 PiDPToQDPProof (⇐)
        ↳24 QDP
        ↳25 NonTerminationProof (⇔)
        ↳26 NO

(0) Obligation:

Clauses:

select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).

Queries:

select(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

select1(T6, .(T6, T7), T7).
select1(T26, .(T13, .(T26, T27)), .(T13, T27)).
select1(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) :- select1(T36, T40, T41).
select1(T64, .(T51, .(T64, T65)), .(T51, T65)).
select1(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) :- select1(T74, T78, T79).

Queries:

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

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x6)

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_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x6)

(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_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_GAA(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
SELECT1_IN_GAA(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_GAA(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))

The TRS R consists of the following rules:

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x6)
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x6)

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

(6) Obligation:

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

SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_GAA(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
SELECT1_IN_GAA(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_GAA(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))

The TRS R consists of the following rules:

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x6)
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)

The TRS R consists of the following rules:

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x6)
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)

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:

SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)

R is empty.
The argument filtering Pi contains the following mapping:
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)

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:

SELECT1_IN_GAA(T36) → SELECT1_IN_GAA(T36)

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

(13) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = SELECT1_IN_GAA(T36) evaluates to t =SELECT1_IN_GAA(T36)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SELECT1_IN_GAA(T36) to SELECT1_IN_GAA(T36).



(14) NO

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

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)

(17) 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_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_GAA(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
SELECT1_IN_GAA(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_GAA(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))

The TRS R consists of the following rules:

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x1, x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x1, x6)

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

(18) Obligation:

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

SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_GAA(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
SELECT1_IN_GAA(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_GAA(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))

The TRS R consists of the following rules:

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x1, x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x1, x6)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(20) Obligation:

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

SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)

The TRS R consists of the following rules:

select1_in_gaa(T6, .(T6, T7), T7) → select1_out_gaa(T6, .(T6, T7), T7)
select1_in_gaa(T26, .(T13, .(T26, T27)), .(T13, T27)) → select1_out_gaa(T26, .(T13, .(T26, T27)), .(T13, T27))
select1_in_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_gaa(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
select1_in_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_gaa(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
U2_gaa(T74, T51, T75, T78, T79, select1_out_gaa(T74, T78, T79)) → select1_out_gaa(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79)))
U1_gaa(T36, T13, T37, T40, T41, select1_out_gaa(T36, T40, T41)) → select1_out_gaa(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3)  =  select1_in_gaa(x1)
select1_out_gaa(x1, x2, x3)  =  select1_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)

R is empty.
The argument filtering Pi contains the following mapping:
SELECT1_IN_GAA(x1, x2, x3)  =  SELECT1_IN_GAA(x1)

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

SELECT1_IN_GAA(T36) → SELECT1_IN_GAA(T36)

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 semiunifying a rule from P directly.

s = SELECT1_IN_GAA(T36) evaluates to t =SELECT1_IN_GAA(T36)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SELECT1_IN_GAA(T36) to SELECT1_IN_GAA(T36).



(26) NO