Left Termination of the query pattern p_in_1(g) 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 UsableRulesReductionPairsProof (⇔)
        ↳20 QDP
        ↳21 PisEmptyProof (⇔)
        ↳22 TRUE
  ↳23 PrologToPiTRSProof (⇐)
    ↳24 PiTRS
    ↳25 DependencyPairsProof (⇔)
    ↳26 PiDP
    ↳27 DependencyGraphProof (⇔)
    ↳28 AND
      ↳29 PiDP
        ↳30 UsableRulesProof (⇔)
        ↳31 PiDP
        ↳32 PiDPToQDPProof (⇐)
        ↳33 QDP
        ↳34 QDPSizeChangeProof (⇔)
        ↳35 TRUE
      ↳36 PiDP
        ↳37 UsableRulesProof (⇔)
        ↳38 PiDP
        ↳39 PiDPToQDPProof (⇐)
        ↳40 QDP

(0) Obligation:

Clauses:

p(0).
p(s(X)) :- ','(geq(X, Y), p(Y)).
geq(X, X).
geq(s(X), Y) :- geq(X, Y).

Queries:

p(g).

(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:
p_in: (b)
geq_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U2_g(x1, x2)  =  U2_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U2_g(x1, x2)  =  U2_g(x2)

(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:

P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))
P_IN_G(s(X)) → GEQ_IN_GA(X, Y)
GEQ_IN_GA(s(X), Y) → U3_GA(X, Y, geq_in_ga(X, Y))
GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)
U1_G(X, geq_out_ga(X, Y)) → U2_G(X, p_in_g(Y))
U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U2_g(x1, x2)  =  U2_g(x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U2_G(x1, x2)  =  U2_G(x2)

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

(4) Obligation:

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

P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))
P_IN_G(s(X)) → GEQ_IN_GA(X, Y)
GEQ_IN_GA(s(X), Y) → U3_GA(X, Y, geq_in_ga(X, Y))
GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)
U1_G(X, geq_out_ga(X, Y)) → U2_G(X, p_in_g(Y))
U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U2_g(x1, x2)  =  U2_g(x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U2_G(x1, x2)  =  U2_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 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U2_g(x1, x2)  =  U2_g(x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(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:

GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)

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

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

(10) PiDPToQDPProof (SOUND 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:

GEQ_IN_GA(s(X)) → GEQ_IN_GA(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:

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

(13) TRUE

(14) Obligation:

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

U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U2_g(x1, x2)  =  U2_g(x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(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:

U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))

The TRS R consists of the following rules:

geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(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:

U1_G(geq_out_ga(Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(geq_in_ga(X))

The TRS R consists of the following rules:

geq_in_ga(X) → geq_out_ga(X)
geq_in_ga(s(X)) → U3_ga(geq_in_ga(X))
U3_ga(geq_out_ga(Y)) → geq_out_ga(Y)

The set Q consists of the following terms:

geq_in_ga(x0)
U3_ga(x0)

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

(19) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

U1_G(geq_out_ga(Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(geq_in_ga(X))
The following rules are removed from R:

geq_in_ga(s(X)) → U3_ga(geq_in_ga(X))
U3_ga(geq_out_ga(Y)) → geq_out_ga(Y)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(P_IN_G(x1)) = 2 + x1   
POL(U1_G(x1)) = 1 + x1   
POL(U3_ga(x1)) = 1 + x1   
POL(geq_in_ga(x1)) = 2 + x1   
POL(geq_out_ga(x1)) = 2 + x1   
POL(s(x1)) = 2 + 2·x1   

(20) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

geq_in_ga(X) → geq_out_ga(X)

The set Q consists of the following terms:

geq_in_ga(x0)
U3_ga(x0)

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

(21) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(22) TRUE

(23) PrologToPiTRSProof (SOUND transformation)

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

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

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

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)

(25) 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:

P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))
P_IN_G(s(X)) → GEQ_IN_GA(X, Y)
GEQ_IN_GA(s(X), Y) → U3_GA(X, Y, geq_in_ga(X, Y))
GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)
U1_G(X, geq_out_ga(X, Y)) → U2_G(X, p_in_g(Y))
U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U2_G(x1, x2)  =  U2_G(x1, x2)

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

(26) Obligation:

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

P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))
P_IN_G(s(X)) → GEQ_IN_GA(X, Y)
GEQ_IN_GA(s(X), Y) → U3_GA(X, Y, geq_in_ga(X, Y))
GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)
U1_G(X, geq_out_ga(X, Y)) → U2_G(X, p_in_g(Y))
U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U2_G(x1, x2)  =  U2_G(x1, x2)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(28) Complex Obligation (AND)

(29) Obligation:

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

GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)

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

(30) UsableRulesProof (EQUIVALENT transformation)

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

(31) Obligation:

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

GEQ_IN_GA(s(X), Y) → GEQ_IN_GA(X, Y)

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

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

(32) PiDPToQDPProof (SOUND transformation)

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

(33) Obligation:

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

GEQ_IN_GA(s(X)) → GEQ_IN_GA(X)

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

(34) 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:

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

(35) TRUE

(36) Obligation:

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

U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))

The TRS R consists of the following rules:

p_in_g(0) → p_out_g(0)
p_in_g(s(X)) → U1_g(X, geq_in_ga(X, Y))
geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)
U1_g(X, geq_out_ga(X, Y)) → U2_g(X, p_in_g(Y))
U2_g(X, p_out_g(Y)) → p_out_g(s(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
0  =  0
p_out_g(x1)  =  p_out_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(X, geq_in_ga(X, Y))

The TRS R consists of the following rules:

geq_in_ga(X, X) → geq_out_ga(X, X)
geq_in_ga(s(X), Y) → U3_ga(X, Y, geq_in_ga(X, Y))
U3_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
geq_out_ga(x1, x2)  =  geq_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

U1_G(X, geq_out_ga(X, Y)) → P_IN_G(Y)
P_IN_G(s(X)) → U1_G(X, geq_in_ga(X))

The TRS R consists of the following rules:

geq_in_ga(X) → geq_out_ga(X, X)
geq_in_ga(s(X)) → U3_ga(X, geq_in_ga(X))
U3_ga(X, geq_out_ga(X, Y)) → geq_out_ga(s(X), Y)

The set Q consists of the following terms:

geq_in_ga(x0)
U3_ga(x0, x1)

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