Left Termination of the query pattern p_in_2(g, g) w.r.t. the given Prolog program could not be shown:

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 NonTerminationProof (⇔)
        ↳13 FALSE
      ↳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 NonTerminationProof (⇔)
        ↳33 FALSE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇔)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔)
        ↳40 TRUE

(0) Obligation:

Clauses:

p(X, Y) :- ','(q(X, Y), r(X)).
q(a, 0).
q(X, s(Y)) :- q(X, Y).
r(b) :- r(b).

Queries:

p(g,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,b)
q_in: (b,b)
r_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
q_in_gg(x1, x2)  =  q_in_gg(x1, x2)
a  =  a
0  =  0
q_out_gg(x1, x2)  =  q_out_gg
s(x1)  =  s(x1)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
r_in_g(x1)  =  r_in_g(x1)
b  =  b
U4_g(x1)  =  U4_g(x1)
r_out_g(x1)  =  r_out_g
p_out_gg(x1, x2)  =  p_out_gg

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_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
q_in_gg(x1, x2)  =  q_in_gg(x1, x2)
a  =  a
0  =  0
q_out_gg(x1, x2)  =  q_out_gg
s(x1)  =  s(x1)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
r_in_g(x1)  =  r_in_g(x1)
b  =  b
U4_g(x1)  =  U4_g(x1)
r_out_g(x1)  =  r_out_g
p_out_gg(x1, x2)  =  p_out_gg

(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_GG(X, Y) → U1_GG(X, Y, q_in_gg(X, Y))
P_IN_GG(X, Y) → Q_IN_GG(X, Y)
Q_IN_GG(X, s(Y)) → U3_GG(X, Y, q_in_gg(X, Y))
Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)
U1_GG(X, Y, q_out_gg(X, Y)) → U2_GG(X, Y, r_in_g(X))
U1_GG(X, Y, q_out_gg(X, Y)) → R_IN_G(X)
R_IN_G(b) → U4_G(r_in_g(b))
R_IN_G(b) → R_IN_G(b)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
q_in_gg(x1, x2)  =  q_in_gg(x1, x2)
a  =  a
0  =  0
q_out_gg(x1, x2)  =  q_out_gg
s(x1)  =  s(x1)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
r_in_g(x1)  =  r_in_g(x1)
b  =  b
U4_g(x1)  =  U4_g(x1)
r_out_g(x1)  =  r_out_g
p_out_gg(x1, x2)  =  p_out_gg
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
Q_IN_GG(x1, x2)  =  Q_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
R_IN_G(x1)  =  R_IN_G(x1)
U4_G(x1)  =  U4_G(x1)

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

(4) Obligation:

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

P_IN_GG(X, Y) → U1_GG(X, Y, q_in_gg(X, Y))
P_IN_GG(X, Y) → Q_IN_GG(X, Y)
Q_IN_GG(X, s(Y)) → U3_GG(X, Y, q_in_gg(X, Y))
Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)
U1_GG(X, Y, q_out_gg(X, Y)) → U2_GG(X, Y, r_in_g(X))
U1_GG(X, Y, q_out_gg(X, Y)) → R_IN_G(X)
R_IN_G(b) → U4_G(r_in_g(b))
R_IN_G(b) → R_IN_G(b)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
q_in_gg(x1, x2)  =  q_in_gg(x1, x2)
a  =  a
0  =  0
q_out_gg(x1, x2)  =  q_out_gg
s(x1)  =  s(x1)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
r_in_g(x1)  =  r_in_g(x1)
b  =  b
U4_g(x1)  =  U4_g(x1)
r_out_g(x1)  =  r_out_g
p_out_gg(x1, x2)  =  p_out_gg
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
Q_IN_GG(x1, x2)  =  Q_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
R_IN_G(x1)  =  R_IN_G(x1)
U4_G(x1)  =  U4_G(x1)

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:

R_IN_G(b) → R_IN_G(b)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
q_in_gg(x1, x2)  =  q_in_gg(x1, x2)
a  =  a
0  =  0
q_out_gg(x1, x2)  =  q_out_gg
s(x1)  =  s(x1)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
r_in_g(x1)  =  r_in_g(x1)
b  =  b
U4_g(x1)  =  U4_g(x1)
r_out_g(x1)  =  r_out_g
p_out_gg(x1, x2)  =  p_out_gg
R_IN_G(x1)  =  R_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:

R_IN_G(b) → R_IN_G(b)

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:

R_IN_G(b) → R_IN_G(b)

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

(12) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = R_IN_G(b) evaluates to t =R_IN_G(b)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from R_IN_G(b) to R_IN_G(b).



(13) FALSE

(14) Obligation:

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

Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
q_in_gg(x1, x2)  =  q_in_gg(x1, x2)
a  =  a
0  =  0
q_out_gg(x1, x2)  =  q_out_gg
s(x1)  =  s(x1)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
r_in_g(x1)  =  r_in_g(x1)
b  =  b
U4_g(x1)  =  U4_g(x1)
r_out_g(x1)  =  r_out_g
p_out_gg(x1, x2)  =  p_out_gg
Q_IN_GG(x1, x2)  =  Q_IN_GG(x1, 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:

Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)

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

(17) PiDPToQDPProof (EQUIVALENT 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:

Q_IN_GG(X, s(Y)) → Q_IN_GG(X, 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:

  • Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)
    The graph contains the following edges 1 >= 1, 2 > 2

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

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

Pi is empty.

(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:

P_IN_GG(X, Y) → U1_GG(X, Y, q_in_gg(X, Y))
P_IN_GG(X, Y) → Q_IN_GG(X, Y)
Q_IN_GG(X, s(Y)) → U3_GG(X, Y, q_in_gg(X, Y))
Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)
U1_GG(X, Y, q_out_gg(X, Y)) → U2_GG(X, Y, r_in_g(X))
U1_GG(X, Y, q_out_gg(X, Y)) → R_IN_G(X)
R_IN_G(b) → U4_G(r_in_g(b))
R_IN_G(b) → R_IN_G(b)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

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

(24) Obligation:

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

P_IN_GG(X, Y) → U1_GG(X, Y, q_in_gg(X, Y))
P_IN_GG(X, Y) → Q_IN_GG(X, Y)
Q_IN_GG(X, s(Y)) → U3_GG(X, Y, q_in_gg(X, Y))
Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)
U1_GG(X, Y, q_out_gg(X, Y)) → U2_GG(X, Y, r_in_g(X))
U1_GG(X, Y, q_out_gg(X, Y)) → R_IN_G(X)
R_IN_G(b) → U4_G(r_in_g(b))
R_IN_G(b) → R_IN_G(b)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

Pi is empty.
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:

R_IN_G(b) → R_IN_G(b)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

Pi is empty.
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:

R_IN_G(b) → R_IN_G(b)

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:

R_IN_G(b) → R_IN_G(b)

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

(32) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = R_IN_G(b) evaluates to t =R_IN_G(b)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from R_IN_G(b) to R_IN_G(b).



(33) FALSE

(34) Obligation:

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

Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)

The TRS R consists of the following rules:

p_in_gg(X, Y) → U1_gg(X, Y, q_in_gg(X, Y))
q_in_gg(a, 0) → q_out_gg(a, 0)
q_in_gg(X, s(Y)) → U3_gg(X, Y, q_in_gg(X, Y))
U3_gg(X, Y, q_out_gg(X, Y)) → q_out_gg(X, s(Y))
U1_gg(X, Y, q_out_gg(X, Y)) → U2_gg(X, Y, r_in_g(X))
r_in_g(b) → U4_g(r_in_g(b))
U4_g(r_out_g(b)) → r_out_g(b)
U2_gg(X, Y, r_out_g(X)) → p_out_gg(X, Y)

Pi is empty.
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:

Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)

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

(37) PiDPToQDPProof (EQUIVALENT 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:

Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)

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

(39) 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:

  • Q_IN_GG(X, s(Y)) → Q_IN_GG(X, Y)
    The graph contains the following edges 1 >= 1, 2 > 2

(40) TRUE