Left Termination of the query pattern w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇔)
        ↳31 QDP
        ↳32 QDPSizeChangeProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP

(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) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
lte_in: (f,b)
even_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
even_out_g(x1)  =  even_out_g
goal_out_  =  goal_out_

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
even_out_g(x1)  =  even_out_g
goal_out_  =  goal_out_

(3) 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:

GOAL_IN_ → U3_1(lte_in_ag(X, s(s(s(s(0))))))
GOAL_IN_ → LTE_IN_AG(X, s(s(s(s(0)))))
LTE_IN_AG(s(X), s(Y)) → U2_AG(X, Y, lte_in_ag(X, Y))
LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → U4_1(even_in_g(X))
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → EVEN_IN_G(X)
EVEN_IN_G(s(s(X))) → U1_G(X, even_in_g(X))
EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
even_out_g(x1)  =  even_out_g
goal_out_  =  goal_out_
GOAL_IN_  =  GOAL_IN_
U3_1(x1)  =  U3_1(x1)
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U4_1(x1)  =  U4_1(x1)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_ → U3_1(lte_in_ag(X, s(s(s(s(0))))))
GOAL_IN_ → LTE_IN_AG(X, s(s(s(s(0)))))
LTE_IN_AG(s(X), s(Y)) → U2_AG(X, Y, lte_in_ag(X, Y))
LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → U4_1(even_in_g(X))
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → EVEN_IN_G(X)
EVEN_IN_G(s(s(X))) → U1_G(X, even_in_g(X))
EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
even_out_g(x1)  =  even_out_g
goal_out_  =  goal_out_
GOAL_IN_  =  GOAL_IN_
U3_1(x1)  =  U3_1(x1)
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U4_1(x1)  =  U4_1(x1)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)

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:

EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
even_out_g(x1)  =  even_out_g
goal_out_  =  goal_out_
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(10) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) 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:

  • EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
even_out_g(x1)  =  even_out_g
goal_out_  =  goal_out_
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LTE_IN_AG(s(Y)) → LTE_IN_AG(Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) 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:

  • LTE_IN_AG(s(Y)) → LTE_IN_AG(Y)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
lte_in: (f,b)
even_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1, x2)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
even_out_g(x1)  =  even_out_g(x1)
goal_out_  =  goal_out_

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1, x2)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
even_out_g(x1)  =  even_out_g(x1)
goal_out_  =  goal_out_

(23) 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:

GOAL_IN_ → U3_1(lte_in_ag(X, s(s(s(s(0))))))
GOAL_IN_ → LTE_IN_AG(X, s(s(s(s(0)))))
LTE_IN_AG(s(X), s(Y)) → U2_AG(X, Y, lte_in_ag(X, Y))
LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → U4_1(even_in_g(X))
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → EVEN_IN_G(X)
EVEN_IN_G(s(s(X))) → U1_G(X, even_in_g(X))
EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1, x2)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
even_out_g(x1)  =  even_out_g(x1)
goal_out_  =  goal_out_
GOAL_IN_  =  GOAL_IN_
U3_1(x1)  =  U3_1(x1)
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U4_1(x1)  =  U4_1(x1)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

We have to consider all (P,R,Pi)-chains

(24) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_ → U3_1(lte_in_ag(X, s(s(s(s(0))))))
GOAL_IN_ → LTE_IN_AG(X, s(s(s(s(0)))))
LTE_IN_AG(s(X), s(Y)) → U2_AG(X, Y, lte_in_ag(X, Y))
LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → U4_1(even_in_g(X))
U3_1(lte_out_ag(X, s(s(s(s(0)))))) → EVEN_IN_G(X)
EVEN_IN_G(s(s(X))) → U1_G(X, even_in_g(X))
EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1, x2)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
even_out_g(x1)  =  even_out_g(x1)
goal_out_  =  goal_out_
GOAL_IN_  =  GOAL_IN_
U3_1(x1)  =  U3_1(x1)
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U4_1(x1)  =  U4_1(x1)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

We have to consider all (P,R,Pi)-chains

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1, x2)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
even_out_g(x1)  =  even_out_g(x1)
goal_out_  =  goal_out_
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(28) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(29) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(30) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(32) 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:

  • EVEN_IN_G(s(s(X))) → EVEN_IN_G(X)
    The graph contains the following edges 1 > 1

(33) TRUE

(34) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)

The TRS R consists of the following rules:

goal_in_ → U3_(lte_in_ag(X, s(s(s(s(0))))))
lte_in_ag(s(X), s(Y)) → U2_ag(X, Y, lte_in_ag(X, Y))
lte_in_ag(0, Y) → lte_out_ag(0, Y)
U2_ag(X, Y, lte_out_ag(X, Y)) → lte_out_ag(s(X), s(Y))
U3_(lte_out_ag(X, s(s(s(s(0)))))) → U4_(even_in_g(X))
even_in_g(s(s(X))) → U1_g(X, even_in_g(X))
even_in_g(0) → even_out_g(0)
U1_g(X, even_out_g(X)) → even_out_g(s(s(X)))
U4_(even_out_g(X)) → goal_out_

The argument filtering Pi contains the following mapping:
goal_in_  =  goal_in_
U3_(x1)  =  U3_(x1)
lte_in_ag(x1, x2)  =  lte_in_ag(x2)
s(x1)  =  s(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
lte_out_ag(x1, x2)  =  lte_out_ag(x1, x2)
0  =  0
U4_(x1)  =  U4_(x1)
even_in_g(x1)  =  even_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
even_out_g(x1)  =  even_out_g(x1)
goal_out_  =  goal_out_
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(35) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(36) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LTE_IN_AG(s(X), s(Y)) → LTE_IN_AG(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LTE_IN_AG(x1, x2)  =  LTE_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(37) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LTE_IN_AG(s(Y)) → LTE_IN_AG(Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.