(0) Obligation:
Clauses:
max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).
Query: max(g,a,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
lessA(s(X1), s(X2)) :- lessA(X1, X2).
lessB(s(X1), s(X2)) :- lessB(X1, X2).
maxC(s(X1), s(X2), s(X1)) :- lessA(X2, X1).
maxC(s(X1), X2, X2) :- lessB(X1, X2).
maxC(s(X1), X2, X2) :- lessB(X1, X2).
Clauses:
lesscA(0, s(X1)).
lesscA(s(X1), s(X2)) :- lesscA(X1, X2).
lesscB(0, s(X1)).
lesscB(s(X1), s(X2)) :- lesscB(X1, X2).
Afs:
maxC(x1, x2, x3) = maxC(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:
maxC_in: (b,f,f)
lessA_in: (f,b)
lessB_in: (b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
MAXC_IN_GAA(s(X1), s(X2), s(X1)) → U3_GAA(X1, X2, lessA_in_ag(X2, X1))
MAXC_IN_GAA(s(X1), s(X2), s(X1)) → LESSA_IN_AG(X2, X1)
LESSA_IN_AG(s(X1), s(X2)) → U1_AG(X1, X2, lessA_in_ag(X1, X2))
LESSA_IN_AG(s(X1), s(X2)) → LESSA_IN_AG(X1, X2)
MAXC_IN_GAA(s(X1), X2, X2) → U4_GAA(X1, X2, lessB_in_ga(X1, X2))
MAXC_IN_GAA(s(X1), X2, X2) → LESSB_IN_GA(X1, X2)
LESSB_IN_GA(s(X1), s(X2)) → U2_GA(X1, X2, lessB_in_ga(X1, X2))
LESSB_IN_GA(s(X1), s(X2)) → LESSB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
lessA_in_ag(
x1,
x2) =
lessA_in_ag(
x2)
lessB_in_ga(
x1,
x2) =
lessB_in_ga(
x1)
MAXC_IN_GAA(
x1,
x2,
x3) =
MAXC_IN_GAA(
x1)
U3_GAA(
x1,
x2,
x3) =
U3_GAA(
x1,
x3)
LESSA_IN_AG(
x1,
x2) =
LESSA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3) =
U1_AG(
x2,
x3)
U4_GAA(
x1,
x2,
x3) =
U4_GAA(
x1,
x3)
LESSB_IN_GA(
x1,
x2) =
LESSB_IN_GA(
x1)
U2_GA(
x1,
x2,
x3) =
U2_GA(
x1,
x3)
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:
MAXC_IN_GAA(s(X1), s(X2), s(X1)) → U3_GAA(X1, X2, lessA_in_ag(X2, X1))
MAXC_IN_GAA(s(X1), s(X2), s(X1)) → LESSA_IN_AG(X2, X1)
LESSA_IN_AG(s(X1), s(X2)) → U1_AG(X1, X2, lessA_in_ag(X1, X2))
LESSA_IN_AG(s(X1), s(X2)) → LESSA_IN_AG(X1, X2)
MAXC_IN_GAA(s(X1), X2, X2) → U4_GAA(X1, X2, lessB_in_ga(X1, X2))
MAXC_IN_GAA(s(X1), X2, X2) → LESSB_IN_GA(X1, X2)
LESSB_IN_GA(s(X1), s(X2)) → U2_GA(X1, X2, lessB_in_ga(X1, X2))
LESSB_IN_GA(s(X1), s(X2)) → LESSB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
lessA_in_ag(
x1,
x2) =
lessA_in_ag(
x2)
lessB_in_ga(
x1,
x2) =
lessB_in_ga(
x1)
MAXC_IN_GAA(
x1,
x2,
x3) =
MAXC_IN_GAA(
x1)
U3_GAA(
x1,
x2,
x3) =
U3_GAA(
x1,
x3)
LESSA_IN_AG(
x1,
x2) =
LESSA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3) =
U1_AG(
x2,
x3)
U4_GAA(
x1,
x2,
x3) =
U4_GAA(
x1,
x3)
LESSB_IN_GA(
x1,
x2) =
LESSB_IN_GA(
x1)
U2_GA(
x1,
x2,
x3) =
U2_GA(
x1,
x3)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LESSB_IN_GA(s(X1), s(X2)) → LESSB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
LESSB_IN_GA(
x1,
x2) =
LESSB_IN_GA(
x1)
We have to consider all (P,R,Pi)-chains
(8) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LESSB_IN_GA(s(X1)) → LESSB_IN_GA(X1)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(10) 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:
- LESSB_IN_GA(s(X1)) → LESSB_IN_GA(X1)
The graph contains the following edges 1 > 1
(11) YES
(12) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LESSA_IN_AG(s(X1), s(X2)) → LESSA_IN_AG(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
LESSA_IN_AG(
x1,
x2) =
LESSA_IN_AG(
x2)
We have to consider all (P,R,Pi)-chains
(13) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LESSA_IN_AG(s(X2)) → LESSA_IN_AG(X2)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(15) 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:
- LESSA_IN_AG(s(X2)) → LESSA_IN_AG(X2)
The graph contains the following edges 1 > 1
(16) YES