(0) Obligation:

Clauses:

rev(L, R) :- rev(L, [], R).
rev([], Y, Y).
rev(L, S, R) :- ','(no(empty(L)), ','(head(L, X), ','(tail(L, T), rev(T, .(X, S), R)))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Queries:

rev(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

rev212(.(T1013, T1014), T965, T966, T968) :- rev212(T1014, T1013, .(T965, T966), T968).
rev1(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) :- rev212(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813).

Clauses:

revc212([], T949, T950, .(T949, T950)).
revc212(.(T1013, T1014), T965, T966, T968) :- revc212(T1014, T1013, .(T965, T966), T968).

Afs:

rev1(x1, x2)  =  rev1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev1_in: (b,f)
rev212_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → U2_GA(T811, T810, T809, T808, T807, T806, T805, T918, T919, T813, rev212_in_ggga(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813))
REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → REV212_IN_GGGA(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813)
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → U1_GGGA(T1013, T1014, T965, T966, T968, rev212_in_ggga(T1014, T1013, .(T965, T966), T968))
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → REV212_IN_GGGA(T1014, T1013, .(T965, T966), T968)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev212_in_ggga(x1, x2, x3, x4)  =  rev212_in_ggga(x1, x2, x3)
[]  =  []
REV1_IN_GA(x1, x2)  =  REV1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
REV212_IN_GGGA(x1, x2, x3, x4)  =  REV212_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → U2_GA(T811, T810, T809, T808, T807, T806, T805, T918, T919, T813, rev212_in_ggga(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813))
REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → REV212_IN_GGGA(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813)
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → U1_GGGA(T1013, T1014, T965, T966, T968, rev212_in_ggga(T1014, T1013, .(T965, T966), T968))
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → REV212_IN_GGGA(T1014, T1013, .(T965, T966), T968)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev212_in_ggga(x1, x2, x3, x4)  =  rev212_in_ggga(x1, x2, x3)
[]  =  []
REV1_IN_GA(x1, x2)  =  REV1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
REV212_IN_GGGA(x1, x2, x3, x4)  =  REV212_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → REV212_IN_GGGA(T1014, T1013, .(T965, T966), T968)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV212_IN_GGGA(x1, x2, x3, x4)  =  REV212_IN_GGGA(x1, x2, x3)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

REV212_IN_GGGA(.(T1013, T1014), T965, T966) → REV212_IN_GGGA(T1014, T1013, .(T965, T966))

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

(9) 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:

  • REV212_IN_GGGA(.(T1013, T1014), T965, T966) → REV212_IN_GGGA(T1014, T1013, .(T965, T966))
    The graph contains the following edges 1 > 1, 1 > 2

(10) YES