(0) Obligation:
Clauses:
even(s(s(X))) :- even(X).
even(0).
lte(s(X), s(Y)) :- lte(X, Y).
lte(0, Y).
goal :- ','(lte(X, s(s(s(s(0))))), even(X)).
Query: goal()
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
evenA(s(s(T13))) :- evenA(T13).
evenA(0).
lteB(s(s(s(0)))).
lteB(s(s(0))).
lteB(s(0)).
lteB(0).
goalC :- lteB(X22).
goalC :- ','(lteB(s(T7)), evenA(T7)).
goalC.
Query: goalC()
(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:
evenA_in: (b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goalC_in_ → U2_(lteB_in_a(X22))
lteB_in_a(s(s(s(0)))) → lteB_out_a(s(s(s(0))))
lteB_in_a(s(s(0))) → lteB_out_a(s(s(0)))
lteB_in_a(s(0)) → lteB_out_a(s(0))
lteB_in_a(0) → lteB_out_a(0)
U2_(lteB_out_a(X22)) → goalC_out_
goalC_in_ → U3_(lteB_in_a(s(T7)))
U3_(lteB_out_a(s(T7))) → U4_(evenA_in_g(T7))
evenA_in_g(s(s(T13))) → U1_g(T13, evenA_in_g(T13))
evenA_in_g(0) → evenA_out_g(0)
U1_g(T13, evenA_out_g(T13)) → evenA_out_g(s(s(T13)))
U4_(evenA_out_g(T7)) → goalC_out_
goalC_in_ → goalC_out_
The argument filtering Pi contains the following mapping:
goalC_in_ =
goalC_in_
U2_(
x1) =
U2_(
x1)
lteB_in_a(
x1) =
lteB_in_a
lteB_out_a(
x1) =
lteB_out_a(
x1)
goalC_out_ =
goalC_out_
U3_(
x1) =
U3_(
x1)
s(
x1) =
s(
x1)
U4_(
x1) =
U4_(
x1)
evenA_in_g(
x1) =
evenA_in_g(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
0 =
0
evenA_out_g(
x1) =
evenA_out_g
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goalC_in_ → U2_(lteB_in_a(X22))
lteB_in_a(s(s(s(0)))) → lteB_out_a(s(s(s(0))))
lteB_in_a(s(s(0))) → lteB_out_a(s(s(0)))
lteB_in_a(s(0)) → lteB_out_a(s(0))
lteB_in_a(0) → lteB_out_a(0)
U2_(lteB_out_a(X22)) → goalC_out_
goalC_in_ → U3_(lteB_in_a(s(T7)))
U3_(lteB_out_a(s(T7))) → U4_(evenA_in_g(T7))
evenA_in_g(s(s(T13))) → U1_g(T13, evenA_in_g(T13))
evenA_in_g(0) → evenA_out_g(0)
U1_g(T13, evenA_out_g(T13)) → evenA_out_g(s(s(T13)))
U4_(evenA_out_g(T7)) → goalC_out_
goalC_in_ → goalC_out_
The argument filtering Pi contains the following mapping:
goalC_in_ =
goalC_in_
U2_(
x1) =
U2_(
x1)
lteB_in_a(
x1) =
lteB_in_a
lteB_out_a(
x1) =
lteB_out_a(
x1)
goalC_out_ =
goalC_out_
U3_(
x1) =
U3_(
x1)
s(
x1) =
s(
x1)
U4_(
x1) =
U4_(
x1)
evenA_in_g(
x1) =
evenA_in_g(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
0 =
0
evenA_out_g(
x1) =
evenA_out_g
(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:
GOALC_IN_ → U2_1(lteB_in_a(X22))
GOALC_IN_ → LTEB_IN_A(X22)
GOALC_IN_ → U3_1(lteB_in_a(s(T7)))
GOALC_IN_ → LTEB_IN_A(s(T7))
U3_1(lteB_out_a(s(T7))) → U4_1(evenA_in_g(T7))
U3_1(lteB_out_a(s(T7))) → EVENA_IN_G(T7)
EVENA_IN_G(s(s(T13))) → U1_G(T13, evenA_in_g(T13))
EVENA_IN_G(s(s(T13))) → EVENA_IN_G(T13)
The TRS R consists of the following rules:
goalC_in_ → U2_(lteB_in_a(X22))
lteB_in_a(s(s(s(0)))) → lteB_out_a(s(s(s(0))))
lteB_in_a(s(s(0))) → lteB_out_a(s(s(0)))
lteB_in_a(s(0)) → lteB_out_a(s(0))
lteB_in_a(0) → lteB_out_a(0)
U2_(lteB_out_a(X22)) → goalC_out_
goalC_in_ → U3_(lteB_in_a(s(T7)))
U3_(lteB_out_a(s(T7))) → U4_(evenA_in_g(T7))
evenA_in_g(s(s(T13))) → U1_g(T13, evenA_in_g(T13))
evenA_in_g(0) → evenA_out_g(0)
U1_g(T13, evenA_out_g(T13)) → evenA_out_g(s(s(T13)))
U4_(evenA_out_g(T7)) → goalC_out_
goalC_in_ → goalC_out_
The argument filtering Pi contains the following mapping:
goalC_in_ =
goalC_in_
U2_(
x1) =
U2_(
x1)
lteB_in_a(
x1) =
lteB_in_a
lteB_out_a(
x1) =
lteB_out_a(
x1)
goalC_out_ =
goalC_out_
U3_(
x1) =
U3_(
x1)
s(
x1) =
s(
x1)
U4_(
x1) =
U4_(
x1)
evenA_in_g(
x1) =
evenA_in_g(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
0 =
0
evenA_out_g(
x1) =
evenA_out_g
GOALC_IN_ =
GOALC_IN_
U2_1(
x1) =
U2_1(
x1)
LTEB_IN_A(
x1) =
LTEB_IN_A
U3_1(
x1) =
U3_1(
x1)
U4_1(
x1) =
U4_1(
x1)
EVENA_IN_G(
x1) =
EVENA_IN_G(
x1)
U1_G(
x1,
x2) =
U1_G(
x2)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
GOALC_IN_ → U2_1(lteB_in_a(X22))
GOALC_IN_ → LTEB_IN_A(X22)
GOALC_IN_ → U3_1(lteB_in_a(s(T7)))
GOALC_IN_ → LTEB_IN_A(s(T7))
U3_1(lteB_out_a(s(T7))) → U4_1(evenA_in_g(T7))
U3_1(lteB_out_a(s(T7))) → EVENA_IN_G(T7)
EVENA_IN_G(s(s(T13))) → U1_G(T13, evenA_in_g(T13))
EVENA_IN_G(s(s(T13))) → EVENA_IN_G(T13)
The TRS R consists of the following rules:
goalC_in_ → U2_(lteB_in_a(X22))
lteB_in_a(s(s(s(0)))) → lteB_out_a(s(s(s(0))))
lteB_in_a(s(s(0))) → lteB_out_a(s(s(0)))
lteB_in_a(s(0)) → lteB_out_a(s(0))
lteB_in_a(0) → lteB_out_a(0)
U2_(lteB_out_a(X22)) → goalC_out_
goalC_in_ → U3_(lteB_in_a(s(T7)))
U3_(lteB_out_a(s(T7))) → U4_(evenA_in_g(T7))
evenA_in_g(s(s(T13))) → U1_g(T13, evenA_in_g(T13))
evenA_in_g(0) → evenA_out_g(0)
U1_g(T13, evenA_out_g(T13)) → evenA_out_g(s(s(T13)))
U4_(evenA_out_g(T7)) → goalC_out_
goalC_in_ → goalC_out_
The argument filtering Pi contains the following mapping:
goalC_in_ =
goalC_in_
U2_(
x1) =
U2_(
x1)
lteB_in_a(
x1) =
lteB_in_a
lteB_out_a(
x1) =
lteB_out_a(
x1)
goalC_out_ =
goalC_out_
U3_(
x1) =
U3_(
x1)
s(
x1) =
s(
x1)
U4_(
x1) =
U4_(
x1)
evenA_in_g(
x1) =
evenA_in_g(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
0 =
0
evenA_out_g(
x1) =
evenA_out_g
GOALC_IN_ =
GOALC_IN_
U2_1(
x1) =
U2_1(
x1)
LTEB_IN_A(
x1) =
LTEB_IN_A
U3_1(
x1) =
U3_1(
x1)
U4_1(
x1) =
U4_1(
x1)
EVENA_IN_G(
x1) =
EVENA_IN_G(
x1)
U1_G(
x1,
x2) =
U1_G(
x2)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
EVENA_IN_G(s(s(T13))) → EVENA_IN_G(T13)
The TRS R consists of the following rules:
goalC_in_ → U2_(lteB_in_a(X22))
lteB_in_a(s(s(s(0)))) → lteB_out_a(s(s(s(0))))
lteB_in_a(s(s(0))) → lteB_out_a(s(s(0)))
lteB_in_a(s(0)) → lteB_out_a(s(0))
lteB_in_a(0) → lteB_out_a(0)
U2_(lteB_out_a(X22)) → goalC_out_
goalC_in_ → U3_(lteB_in_a(s(T7)))
U3_(lteB_out_a(s(T7))) → U4_(evenA_in_g(T7))
evenA_in_g(s(s(T13))) → U1_g(T13, evenA_in_g(T13))
evenA_in_g(0) → evenA_out_g(0)
U1_g(T13, evenA_out_g(T13)) → evenA_out_g(s(s(T13)))
U4_(evenA_out_g(T7)) → goalC_out_
goalC_in_ → goalC_out_
The argument filtering Pi contains the following mapping:
goalC_in_ =
goalC_in_
U2_(
x1) =
U2_(
x1)
lteB_in_a(
x1) =
lteB_in_a
lteB_out_a(
x1) =
lteB_out_a(
x1)
goalC_out_ =
goalC_out_
U3_(
x1) =
U3_(
x1)
s(
x1) =
s(
x1)
U4_(
x1) =
U4_(
x1)
evenA_in_g(
x1) =
evenA_in_g(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
0 =
0
evenA_out_g(
x1) =
evenA_out_g
EVENA_IN_G(
x1) =
EVENA_IN_G(
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:
EVENA_IN_G(s(s(T13))) → EVENA_IN_G(T13)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (EQUIVALENT 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:
EVENA_IN_G(s(s(T13))) → EVENA_IN_G(T13)
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:
- EVENA_IN_G(s(s(T13))) → EVENA_IN_G(T13)
The graph contains the following edges 1 > 1
(14) YES