(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).

Query: rev(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

revA(.(X1, X2), X3, X4, X5) :- revA(X2, X1, .(X3, X4), X5).
revB(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10).

Clauses:

revcA([], X1, X2, .(X1, X2)).
revcA(.(X1, X2), X3, X4, X5) :- revcA(X2, X1, .(X3, X4), X5).

Afs:

revB(x1, x2)  =  revB(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
revB_in: (b,f)
revA_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → U2_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revA_in_ggga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10))
REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → REVA_IN_GGGA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)
REVA_IN_GGGA(.(X1, X2), X3, X4, X5) → U1_GGGA(X1, X2, X3, X4, X5, revA_in_ggga(X2, X1, .(X3, X4), X5))
REVA_IN_GGGA(.(X1, X2), X3, X4, X5) → REVA_IN_GGGA(X2, X1, .(X3, X4), X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revA_in_ggga(x1, x2, x3, x4)  =  revA_in_ggga(x1, x2, x3)
[]  =  []
REVB_IN_GA(x1, x2)  =  REVB_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)
REVA_IN_GGGA(x1, x2, x3, x4)  =  REVA_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:

REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → U2_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revA_in_ggga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10))
REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → REVA_IN_GGGA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)
REVA_IN_GGGA(.(X1, X2), X3, X4, X5) → U1_GGGA(X1, X2, X3, X4, X5, revA_in_ggga(X2, X1, .(X3, X4), X5))
REVA_IN_GGGA(.(X1, X2), X3, X4, X5) → REVA_IN_GGGA(X2, X1, .(X3, X4), X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revA_in_ggga(x1, x2, x3, x4)  =  revA_in_ggga(x1, x2, x3)
[]  =  []
REVB_IN_GA(x1, x2)  =  REVB_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)
REVA_IN_GGGA(x1, x2, x3, x4)  =  REVA_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:

REVA_IN_GGGA(.(X1, X2), X3, X4, X5) → REVA_IN_GGGA(X2, X1, .(X3, X4), X5)

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

REVA_IN_GGGA(.(X1, X2), X3, X4) → REVA_IN_GGGA(X2, X1, .(X3, X4))

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:

  • REVA_IN_GGGA(.(X1, X2), X3, X4) → REVA_IN_GGGA(X2, X1, .(X3, X4))
    The graph contains the following edges 1 > 1, 1 > 2

(10) YES