Left Termination of the query pattern f_in_1(a) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 CutEliminatorProof (⇐)
↳2 Prolog
↳3 UndefinedPredicateHandlerProof (⇐)
↳4 Prolog
  ↳5 PrologToPiTRSProof (⇐)
    ↳6 PiTRS
    ↳7 DependencyPairsProof (⇔)
    ↳8 PiDP
    ↳9 DependencyGraphProof (⇔)
    ↳10 PiDP
    ↳11 UsableRulesProof (⇔)
    ↳12 PiDP
    ↳13 PiDPToQDPProof (⇐)
    ↳14 QDP
    ↳15 NonTerminationProof (⇔)
    ↳16 FALSE
  ↳17 PrologToPiTRSProof (⇐)
    ↳18 PiTRS
    ↳19 DependencyPairsProof (⇔)
    ↳20 PiDP
    ↳21 DependencyGraphProof (⇔)
    ↳22 PiDP
    ↳23 UsableRulesProof (⇔)
    ↳24 PiDP
    ↳25 PiDPToQDPProof (⇐)
    ↳26 QDP
    ↳27 NonTerminationProof (⇔)
    ↳28 FALSE

(0) Obligation:

Clauses:

f(X) :- ','(p(X), q(X)).
p(a) :- !.
p(X) :- p(Y).
q(b).

Queries:

f(a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

f(X) :- ','(p(X), q(X)).
p(a) :- true.
p(X) :- p(Y).
q(b).

Queries:

f(a).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

f(X) :- ','(p(X), q(X)).
p(a) :- true.
p(X) :- p(Y).
q(b).
true.

Queries:

f(a).

(5) PrologToPiTRSProof (SOUND transformation)

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

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)

(7) 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:

F_IN_A(X) → U1_A(X, p_in_a(X))
F_IN_A(X) → P_IN_A(X)
P_IN_A(a) → U3_A(true_in_)
P_IN_A(a) → TRUE_IN_
P_IN_A(X) → U4_A(X, p_in_a(Y))
P_IN_A(X) → P_IN_A(Y)
U1_A(X, p_out_a(X)) → U2_A(X, q_in_a(X))
U1_A(X, p_out_a(X)) → Q_IN_A(X)

The TRS R consists of the following rules:

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)
F_IN_A(x1)  =  F_IN_A
U1_A(x1, x2)  =  U1_A(x2)
P_IN_A(x1)  =  P_IN_A
U3_A(x1)  =  U3_A(x1)
TRUE_IN_  =  TRUE_IN_
U4_A(x1, x2)  =  U4_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
Q_IN_A(x1)  =  Q_IN_A

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

(8) Obligation:

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

F_IN_A(X) → U1_A(X, p_in_a(X))
F_IN_A(X) → P_IN_A(X)
P_IN_A(a) → U3_A(true_in_)
P_IN_A(a) → TRUE_IN_
P_IN_A(X) → U4_A(X, p_in_a(Y))
P_IN_A(X) → P_IN_A(Y)
U1_A(X, p_out_a(X)) → U2_A(X, q_in_a(X))
U1_A(X, p_out_a(X)) → Q_IN_A(X)

The TRS R consists of the following rules:

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)
F_IN_A(x1)  =  F_IN_A
U1_A(x1, x2)  =  U1_A(x2)
P_IN_A(x1)  =  P_IN_A
U3_A(x1)  =  U3_A(x1)
TRUE_IN_  =  TRUE_IN_
U4_A(x1, x2)  =  U4_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
Q_IN_A(x1)  =  Q_IN_A

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(10) Obligation:

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

P_IN_A(X) → P_IN_A(Y)

The TRS R consists of the following rules:

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)
P_IN_A(x1)  =  P_IN_A

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

(11) UsableRulesProof (EQUIVALENT transformation)

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

(12) Obligation:

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

P_IN_A(X) → P_IN_A(Y)

R is empty.
The argument filtering Pi contains the following mapping:
P_IN_A(x1)  =  P_IN_A

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

P_IN_A → P_IN_A

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

(15) 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 = P_IN_A evaluates to t =P_IN_A

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 P_IN_A to P_IN_A.



(16) FALSE

(17) PrologToPiTRSProof (SOUND transformation)

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

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(18) Obligation:

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

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)

(19) 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:

F_IN_A(X) → U1_A(X, p_in_a(X))
F_IN_A(X) → P_IN_A(X)
P_IN_A(a) → U3_A(true_in_)
P_IN_A(a) → TRUE_IN_
P_IN_A(X) → U4_A(X, p_in_a(Y))
P_IN_A(X) → P_IN_A(Y)
U1_A(X, p_out_a(X)) → U2_A(X, q_in_a(X))
U1_A(X, p_out_a(X)) → Q_IN_A(X)

The TRS R consists of the following rules:

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)
F_IN_A(x1)  =  F_IN_A
U1_A(x1, x2)  =  U1_A(x2)
P_IN_A(x1)  =  P_IN_A
U3_A(x1)  =  U3_A(x1)
TRUE_IN_  =  TRUE_IN_
U4_A(x1, x2)  =  U4_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
Q_IN_A(x1)  =  Q_IN_A

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

(20) Obligation:

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

F_IN_A(X) → U1_A(X, p_in_a(X))
F_IN_A(X) → P_IN_A(X)
P_IN_A(a) → U3_A(true_in_)
P_IN_A(a) → TRUE_IN_
P_IN_A(X) → U4_A(X, p_in_a(Y))
P_IN_A(X) → P_IN_A(Y)
U1_A(X, p_out_a(X)) → U2_A(X, q_in_a(X))
U1_A(X, p_out_a(X)) → Q_IN_A(X)

The TRS R consists of the following rules:

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)
F_IN_A(x1)  =  F_IN_A
U1_A(x1, x2)  =  U1_A(x2)
P_IN_A(x1)  =  P_IN_A
U3_A(x1)  =  U3_A(x1)
TRUE_IN_  =  TRUE_IN_
U4_A(x1, x2)  =  U4_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
Q_IN_A(x1)  =  Q_IN_A

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(22) Obligation:

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

P_IN_A(X) → P_IN_A(Y)

The TRS R consists of the following rules:

f_in_a(X) → U1_a(X, p_in_a(X))
p_in_a(a) → U3_a(true_in_)
true_in_ → true_out_
U3_a(true_out_) → p_out_a(a)
p_in_a(X) → U4_a(X, p_in_a(Y))
U4_a(X, p_out_a(Y)) → p_out_a(X)
U1_a(X, p_out_a(X)) → U2_a(X, q_in_a(X))
q_in_a(b) → q_out_a(b)
U2_a(X, q_out_a(X)) → f_out_a(X)

The argument filtering Pi contains the following mapping:
f_in_a(x1)  =  f_in_a
U1_a(x1, x2)  =  U1_a(x2)
p_in_a(x1)  =  p_in_a
U3_a(x1)  =  U3_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p_out_a(x1)  =  p_out_a
U4_a(x1, x2)  =  U4_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a(x1)
f_out_a(x1)  =  f_out_a(x1)
P_IN_A(x1)  =  P_IN_A

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

(23) UsableRulesProof (EQUIVALENT transformation)

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

(24) Obligation:

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

P_IN_A(X) → P_IN_A(Y)

R is empty.
The argument filtering Pi contains the following mapping:
P_IN_A(x1)  =  P_IN_A

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

(25) PiDPToQDPProof (SOUND transformation)

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

(26) Obligation:

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

P_IN_A → P_IN_A

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

(27) 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 = P_IN_A evaluates to t =P_IN_A

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 P_IN_A to P_IN_A.



(28) FALSE