(0) Obligation:
Clauses:
p(M, N, s(R), RES) :- p(M, R, N, RES).
p(M, s(N), R, RES) :- p(R, N, M, RES).
p(M, X1, X2, M).
Query: p(g,g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
pA(X1, s(X2), s(X3), X4) :- pA(X1, X2, X3, X4).
pA(X1, X2, s(s(X3)), X4) :- pA(X2, X3, X1, X4).
pA(X1, s(X2), s(X3), X4) :- pA(s(X3), X2, X1, X4).
pA(s(X1), s(X2), X3, X4) :- pA(X3, X1, X2, X4).
pA(X1, s(s(X2)), X3, X4) :- pA(X1, X2, X3, X4).
Clauses:
pcA(X1, s(X2), s(X3), X4) :- pcA(X1, X2, X3, X4).
pcA(X1, X2, s(s(X3)), X4) :- pcA(X2, X3, X1, X4).
pcA(X1, X2, s(X3), X1).
pcA(X1, s(X2), s(X3), X4) :- pcA(s(X3), X2, X1, X4).
pcA(X1, X2, s(X3), X1).
pcA(s(X1), s(X2), X3, X4) :- pcA(X3, X1, X2, X4).
pcA(X1, s(s(X2)), X3, X4) :- pcA(X1, X2, X3, X4).
pcA(X1, s(X2), X3, X3).
pcA(X1, s(X2), X3, X1).
pcA(X1, X2, X3, X1).
Afs:
pA(x1, x2, x3, x4) = pA(x1, x2, x3)
(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:
pA_in: (b,b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_GGGA(X1, s(X2), s(X3), X4) → U1_GGGA(X1, X2, X3, X4, pA_in_ggga(X1, X2, X3, X4))
PA_IN_GGGA(X1, s(X2), s(X3), X4) → PA_IN_GGGA(X1, X2, X3, X4)
PA_IN_GGGA(X1, X2, s(s(X3)), X4) → U2_GGGA(X1, X2, X3, X4, pA_in_ggga(X2, X3, X1, X4))
PA_IN_GGGA(X1, X2, s(s(X3)), X4) → PA_IN_GGGA(X2, X3, X1, X4)
PA_IN_GGGA(X1, s(X2), s(X3), X4) → U3_GGGA(X1, X2, X3, X4, pA_in_ggga(s(X3), X2, X1, X4))
PA_IN_GGGA(X1, s(X2), s(X3), X4) → PA_IN_GGGA(s(X3), X2, X1, X4)
PA_IN_GGGA(s(X1), s(X2), X3, X4) → U4_GGGA(X1, X2, X3, X4, pA_in_ggga(X3, X1, X2, X4))
PA_IN_GGGA(s(X1), s(X2), X3, X4) → PA_IN_GGGA(X3, X1, X2, X4)
PA_IN_GGGA(X1, s(s(X2)), X3, X4) → U5_GGGA(X1, X2, X3, X4, pA_in_ggga(X1, X2, X3, X4))
PA_IN_GGGA(X1, s(s(X2)), X3, X4) → PA_IN_GGGA(X1, X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
pA_in_ggga(
x1,
x2,
x3,
x4) =
pA_in_ggga(
x1,
x2,
x3)
s(
x1) =
s(
x1)
PA_IN_GGGA(
x1,
x2,
x3,
x4) =
PA_IN_GGGA(
x1,
x2,
x3)
U1_GGGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGGA(
x1,
x2,
x3,
x5)
U2_GGGA(
x1,
x2,
x3,
x4,
x5) =
U2_GGGA(
x1,
x2,
x3,
x5)
U3_GGGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGGA(
x1,
x2,
x3,
x5)
U4_GGGA(
x1,
x2,
x3,
x4,
x5) =
U4_GGGA(
x1,
x2,
x3,
x5)
U5_GGGA(
x1,
x2,
x3,
x4,
x5) =
U5_GGGA(
x1,
x2,
x3,
x5)
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:
PA_IN_GGGA(X1, s(X2), s(X3), X4) → U1_GGGA(X1, X2, X3, X4, pA_in_ggga(X1, X2, X3, X4))
PA_IN_GGGA(X1, s(X2), s(X3), X4) → PA_IN_GGGA(X1, X2, X3, X4)
PA_IN_GGGA(X1, X2, s(s(X3)), X4) → U2_GGGA(X1, X2, X3, X4, pA_in_ggga(X2, X3, X1, X4))
PA_IN_GGGA(X1, X2, s(s(X3)), X4) → PA_IN_GGGA(X2, X3, X1, X4)
PA_IN_GGGA(X1, s(X2), s(X3), X4) → U3_GGGA(X1, X2, X3, X4, pA_in_ggga(s(X3), X2, X1, X4))
PA_IN_GGGA(X1, s(X2), s(X3), X4) → PA_IN_GGGA(s(X3), X2, X1, X4)
PA_IN_GGGA(s(X1), s(X2), X3, X4) → U4_GGGA(X1, X2, X3, X4, pA_in_ggga(X3, X1, X2, X4))
PA_IN_GGGA(s(X1), s(X2), X3, X4) → PA_IN_GGGA(X3, X1, X2, X4)
PA_IN_GGGA(X1, s(s(X2)), X3, X4) → U5_GGGA(X1, X2, X3, X4, pA_in_ggga(X1, X2, X3, X4))
PA_IN_GGGA(X1, s(s(X2)), X3, X4) → PA_IN_GGGA(X1, X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
pA_in_ggga(
x1,
x2,
x3,
x4) =
pA_in_ggga(
x1,
x2,
x3)
s(
x1) =
s(
x1)
PA_IN_GGGA(
x1,
x2,
x3,
x4) =
PA_IN_GGGA(
x1,
x2,
x3)
U1_GGGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGGA(
x1,
x2,
x3,
x5)
U2_GGGA(
x1,
x2,
x3,
x4,
x5) =
U2_GGGA(
x1,
x2,
x3,
x5)
U3_GGGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGGA(
x1,
x2,
x3,
x5)
U4_GGGA(
x1,
x2,
x3,
x4,
x5) =
U4_GGGA(
x1,
x2,
x3,
x5)
U5_GGGA(
x1,
x2,
x3,
x4,
x5) =
U5_GGGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_GGGA(X1, X2, s(s(X3)), X4) → PA_IN_GGGA(X2, X3, X1, X4)
PA_IN_GGGA(X1, s(X2), s(X3), X4) → PA_IN_GGGA(X1, X2, X3, X4)
PA_IN_GGGA(X1, s(X2), s(X3), X4) → PA_IN_GGGA(s(X3), X2, X1, X4)
PA_IN_GGGA(s(X1), s(X2), X3, X4) → PA_IN_GGGA(X3, X1, X2, X4)
PA_IN_GGGA(X1, s(s(X2)), X3, X4) → PA_IN_GGGA(X1, X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
PA_IN_GGGA(
x1,
x2,
x3,
x4) =
PA_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:
PA_IN_GGGA(X1, X2, s(s(X3))) → PA_IN_GGGA(X2, X3, X1)
PA_IN_GGGA(X1, s(X2), s(X3)) → PA_IN_GGGA(X1, X2, X3)
PA_IN_GGGA(X1, s(X2), s(X3)) → PA_IN_GGGA(s(X3), X2, X1)
PA_IN_GGGA(s(X1), s(X2), X3) → PA_IN_GGGA(X3, X1, X2)
PA_IN_GGGA(X1, s(s(X2)), X3) → PA_IN_GGGA(X1, X2, X3)
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:
- PA_IN_GGGA(X1, X2, s(s(X3))) → PA_IN_GGGA(X2, X3, X1)
The graph contains the following edges 2 >= 1, 3 > 2, 1 >= 3
- PA_IN_GGGA(X1, s(X2), s(X3)) → PA_IN_GGGA(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3
- PA_IN_GGGA(X1, s(X2), s(X3)) → PA_IN_GGGA(s(X3), X2, X1)
The graph contains the following edges 3 >= 1, 2 > 2, 1 >= 3
- PA_IN_GGGA(s(X1), s(X2), X3) → PA_IN_GGGA(X3, X1, X2)
The graph contains the following edges 3 >= 1, 1 > 2, 2 > 3
- PA_IN_GGGA(X1, s(s(X2)), X3) → PA_IN_GGGA(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
(10) YES