(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)).
Queries:
goal().
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:even35(s(s(T13))) :- even35(T13).
even35(0).
lte8(s(s(s(0)))).
lte8(s(s(0))).
lte8(s(0)).
lte8(0).
goal1 :- lte8(X22).
goal1 :- ','(lte8(s(T7)), even35(T7)).
goal1.
Queries:
goal1().
(3) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:even35_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goal1_in_ → U2_(lte8_in_a(X22))
lte8_in_a(s(s(s(0)))) → lte8_out_a(s(s(s(0))))
lte8_in_a(s(s(0))) → lte8_out_a(s(s(0)))
lte8_in_a(s(0)) → lte8_out_a(s(0))
lte8_in_a(0) → lte8_out_a(0)
U2_(lte8_out_a(X22)) → goal1_out_
goal1_in_ → U3_(lte8_in_a(s(T7)))
U3_(lte8_out_a(s(T7))) → U4_(even35_in_g(T7))
even35_in_g(s(s(T13))) → U1_g(T13, even35_in_g(T13))
even35_in_g(0) → even35_out_g(0)
U1_g(T13, even35_out_g(T13)) → even35_out_g(s(s(T13)))
U4_(even35_out_g(T7)) → goal1_out_
goal1_in_ → goal1_out_
The argument filtering Pi contains the following mapping:
goal1_in_ = goal1_in_
U2_(x1) = U2_(x1)
lte8_in_a(x1) = lte8_in_a
lte8_out_a(x1) = lte8_out_a(x1)
goal1_out_ = goal1_out_
U3_(x1) = U3_(x1)
s(x1) = s(x1)
U4_(x1) = U4_(x1)
even35_in_g(x1) = even35_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
0 = 0
even35_out_g(x1) = even35_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:
goal1_in_ → U2_(lte8_in_a(X22))
lte8_in_a(s(s(s(0)))) → lte8_out_a(s(s(s(0))))
lte8_in_a(s(s(0))) → lte8_out_a(s(s(0)))
lte8_in_a(s(0)) → lte8_out_a(s(0))
lte8_in_a(0) → lte8_out_a(0)
U2_(lte8_out_a(X22)) → goal1_out_
goal1_in_ → U3_(lte8_in_a(s(T7)))
U3_(lte8_out_a(s(T7))) → U4_(even35_in_g(T7))
even35_in_g(s(s(T13))) → U1_g(T13, even35_in_g(T13))
even35_in_g(0) → even35_out_g(0)
U1_g(T13, even35_out_g(T13)) → even35_out_g(s(s(T13)))
U4_(even35_out_g(T7)) → goal1_out_
goal1_in_ → goal1_out_
The argument filtering Pi contains the following mapping:
goal1_in_ = goal1_in_
U2_(x1) = U2_(x1)
lte8_in_a(x1) = lte8_in_a
lte8_out_a(x1) = lte8_out_a(x1)
goal1_out_ = goal1_out_
U3_(x1) = U3_(x1)
s(x1) = s(x1)
U4_(x1) = U4_(x1)
even35_in_g(x1) = even35_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
0 = 0
even35_out_g(x1) = even35_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:
GOAL1_IN_ → U2_1(lte8_in_a(X22))
GOAL1_IN_ → LTE8_IN_A(X22)
GOAL1_IN_ → U3_1(lte8_in_a(s(T7)))
GOAL1_IN_ → LTE8_IN_A(s(T7))
U3_1(lte8_out_a(s(T7))) → U4_1(even35_in_g(T7))
U3_1(lte8_out_a(s(T7))) → EVEN35_IN_G(T7)
EVEN35_IN_G(s(s(T13))) → U1_G(T13, even35_in_g(T13))
EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)
The TRS R consists of the following rules:
goal1_in_ → U2_(lte8_in_a(X22))
lte8_in_a(s(s(s(0)))) → lte8_out_a(s(s(s(0))))
lte8_in_a(s(s(0))) → lte8_out_a(s(s(0)))
lte8_in_a(s(0)) → lte8_out_a(s(0))
lte8_in_a(0) → lte8_out_a(0)
U2_(lte8_out_a(X22)) → goal1_out_
goal1_in_ → U3_(lte8_in_a(s(T7)))
U3_(lte8_out_a(s(T7))) → U4_(even35_in_g(T7))
even35_in_g(s(s(T13))) → U1_g(T13, even35_in_g(T13))
even35_in_g(0) → even35_out_g(0)
U1_g(T13, even35_out_g(T13)) → even35_out_g(s(s(T13)))
U4_(even35_out_g(T7)) → goal1_out_
goal1_in_ → goal1_out_
The argument filtering Pi contains the following mapping:
goal1_in_ = goal1_in_
U2_(x1) = U2_(x1)
lte8_in_a(x1) = lte8_in_a
lte8_out_a(x1) = lte8_out_a(x1)
goal1_out_ = goal1_out_
U3_(x1) = U3_(x1)
s(x1) = s(x1)
U4_(x1) = U4_(x1)
even35_in_g(x1) = even35_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
0 = 0
even35_out_g(x1) = even35_out_g
GOAL1_IN_ = GOAL1_IN_
U2_1(x1) = U2_1(x1)
LTE8_IN_A(x1) = LTE8_IN_A
U3_1(x1) = U3_1(x1)
U4_1(x1) = U4_1(x1)
EVEN35_IN_G(x1) = EVEN35_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:
GOAL1_IN_ → U2_1(lte8_in_a(X22))
GOAL1_IN_ → LTE8_IN_A(X22)
GOAL1_IN_ → U3_1(lte8_in_a(s(T7)))
GOAL1_IN_ → LTE8_IN_A(s(T7))
U3_1(lte8_out_a(s(T7))) → U4_1(even35_in_g(T7))
U3_1(lte8_out_a(s(T7))) → EVEN35_IN_G(T7)
EVEN35_IN_G(s(s(T13))) → U1_G(T13, even35_in_g(T13))
EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)
The TRS R consists of the following rules:
goal1_in_ → U2_(lte8_in_a(X22))
lte8_in_a(s(s(s(0)))) → lte8_out_a(s(s(s(0))))
lte8_in_a(s(s(0))) → lte8_out_a(s(s(0)))
lte8_in_a(s(0)) → lte8_out_a(s(0))
lte8_in_a(0) → lte8_out_a(0)
U2_(lte8_out_a(X22)) → goal1_out_
goal1_in_ → U3_(lte8_in_a(s(T7)))
U3_(lte8_out_a(s(T7))) → U4_(even35_in_g(T7))
even35_in_g(s(s(T13))) → U1_g(T13, even35_in_g(T13))
even35_in_g(0) → even35_out_g(0)
U1_g(T13, even35_out_g(T13)) → even35_out_g(s(s(T13)))
U4_(even35_out_g(T7)) → goal1_out_
goal1_in_ → goal1_out_
The argument filtering Pi contains the following mapping:
goal1_in_ = goal1_in_
U2_(x1) = U2_(x1)
lte8_in_a(x1) = lte8_in_a
lte8_out_a(x1) = lte8_out_a(x1)
goal1_out_ = goal1_out_
U3_(x1) = U3_(x1)
s(x1) = s(x1)
U4_(x1) = U4_(x1)
even35_in_g(x1) = even35_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
0 = 0
even35_out_g(x1) = even35_out_g
GOAL1_IN_ = GOAL1_IN_
U2_1(x1) = U2_1(x1)
LTE8_IN_A(x1) = LTE8_IN_A
U3_1(x1) = U3_1(x1)
U4_1(x1) = U4_1(x1)
EVEN35_IN_G(x1) = EVEN35_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:
EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)
The TRS R consists of the following rules:
goal1_in_ → U2_(lte8_in_a(X22))
lte8_in_a(s(s(s(0)))) → lte8_out_a(s(s(s(0))))
lte8_in_a(s(s(0))) → lte8_out_a(s(s(0)))
lte8_in_a(s(0)) → lte8_out_a(s(0))
lte8_in_a(0) → lte8_out_a(0)
U2_(lte8_out_a(X22)) → goal1_out_
goal1_in_ → U3_(lte8_in_a(s(T7)))
U3_(lte8_out_a(s(T7))) → U4_(even35_in_g(T7))
even35_in_g(s(s(T13))) → U1_g(T13, even35_in_g(T13))
even35_in_g(0) → even35_out_g(0)
U1_g(T13, even35_out_g(T13)) → even35_out_g(s(s(T13)))
U4_(even35_out_g(T7)) → goal1_out_
goal1_in_ → goal1_out_
The argument filtering Pi contains the following mapping:
goal1_in_ = goal1_in_
U2_(x1) = U2_(x1)
lte8_in_a(x1) = lte8_in_a
lte8_out_a(x1) = lte8_out_a(x1)
goal1_out_ = goal1_out_
U3_(x1) = U3_(x1)
s(x1) = s(x1)
U4_(x1) = U4_(x1)
even35_in_g(x1) = even35_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
0 = 0
even35_out_g(x1) = even35_out_g
EVEN35_IN_G(x1) = EVEN35_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:
EVEN35_IN_G(s(s(T13))) → EVEN35_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:
EVEN35_IN_G(s(s(T13))) → EVEN35_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:
- EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)
The graph contains the following edges 1 > 1