(0) Obligation:

Clauses:

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

Query: app(g,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

(2) Obligation:

Clauses:

appA([], T5, T5).
appA(.(T57, T58), T47, .(T57, T48)) :- appA(T58, T47, T48).

Query: appA(g,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
appA_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(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:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(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:

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → U1_GAA(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)

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

(6) Obligation:

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

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → U1_GAA(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(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 1 less node.

(8) Obligation:

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

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
APPA_IN_GAA(x1, x2, x3)  =  APPA_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:

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

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

APPA_IN_GAA(.(T57, T58)) → APPA_IN_GAA(T58)

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

(13) 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:

  • APPA_IN_GAA(.(T57, T58)) → APPA_IN_GAA(T58)
    The graph contains the following edges 1 > 1

(14) YES