Left Termination of the query pattern p_in_1(g) 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:

p(b).
p(a) :- p1(X).
p1(b) :- !.
p1(a) :- p1(X).

Queries:

p(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

p(b).
p(a) :- p1(X).
p1(b) :- true.
p1(a) :- p1(X).

Queries:

p(g).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

p(b).
p(a) :- p1(X).
p1(b) :- true.
p1(a) :- p1(X).
true.

Queries:

p(g).

(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:
p_in: (b)
p1_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_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:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_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:

P_IN_G(a) → U1_G(p1_in_a(X))
P_IN_G(a) → P1_IN_A(X)
P1_IN_A(b) → U2_A(true_in_)
P1_IN_A(b) → TRUE_IN_
P1_IN_A(a) → U3_A(p1_in_a(X))
P1_IN_A(a) → P1_IN_A(X)

The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_a(x1)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1)  =  U1_G'(x1)
P1_IN_A(x1)  =  P1_IN_A
U2_A(x1)  =  U2_A(x1)
TRUE_IN_  =  TRUE_IN_
U3_A(x1)  =  U3_A(x1)

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

(8) Obligation:

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

P_IN_G(a) → U1_G(p1_in_a(X))
P_IN_G(a) → P1_IN_A(X)
P1_IN_A(b) → U2_A(true_in_)
P1_IN_A(b) → TRUE_IN_
P1_IN_A(a) → U3_A(p1_in_a(X))
P1_IN_A(a) → P1_IN_A(X)

The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_a(x1)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
P1_IN_A(x1)  =  P1_IN_A
U2_A(x1)  =  U2_A(x1)
TRUE_IN_  =  TRUE_IN_
U3_A(x1)  =  U3_A(x1)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(10) Obligation:

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

P1_IN_A(a) → P1_IN_A(X)

The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_a(x1)
P1_IN_A(x1)  =  P1_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:

P1_IN_A(a) → P1_IN_A(X)

R is empty.
The argument filtering Pi contains the following mapping:
a  =  a
P1_IN_A(x1)  =  P1_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:

P1_IN_A → P1_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 = P1_IN_A evaluates to t =P1_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 P1_IN_A to P1_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:
p_in: (b)
p1_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g(x1)
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_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:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g(x1)
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_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:

P_IN_G(a) → U1_G(p1_in_a(X))
P_IN_G(a) → P1_IN_A(X)
P1_IN_A(b) → U2_A(true_in_)
P1_IN_A(b) → TRUE_IN_
P1_IN_A(a) → U3_A(p1_in_a(X))
P1_IN_A(a) → P1_IN_A(X)

The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g(x1)
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_a(x1)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
P1_IN_A(x1)  =  P1_IN_A
U2_A(x1)  =  U2_A(x1)
TRUE_IN_  =  TRUE_IN_
U3_A(x1)  =  U3_A(x1)

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

(20) Obligation:

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

P_IN_G(a) → U1_G(p1_in_a(X))
P_IN_G(a) → P1_IN_A(X)
P1_IN_A(b) → U2_A(true_in_)
P1_IN_A(b) → TRUE_IN_
P1_IN_A(a) → U3_A(p1_in_a(X))
P1_IN_A(a) → P1_IN_A(X)

The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g(x1)
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_a(x1)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
P1_IN_A(x1)  =  P1_IN_A
U2_A(x1)  =  U2_A(x1)
TRUE_IN_  =  TRUE_IN_
U3_A(x1)  =  U3_A(x1)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(22) Obligation:

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

P1_IN_A(a) → P1_IN_A(X)

The TRS R consists of the following rules:

p_in_g(b) → p_out_g(b)
p_in_g(a) → U1_g(p1_in_a(X))
p1_in_a(b) → U2_a(true_in_)
true_in_ → true_out_
U2_a(true_out_) → p1_out_a(b)
p1_in_a(a) → U3_a(p1_in_a(X))
U3_a(p1_out_a(X)) → p1_out_a(a)
U1_g(p1_out_a(X)) → p_out_g(a)

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
b  =  b
p_out_g(x1)  =  p_out_g(x1)
a  =  a
U1_g(x1)  =  U1_g(x1)
p1_in_a(x1)  =  p1_in_a
U2_a(x1)  =  U2_a(x1)
true_in_  =  true_in_
true_out_  =  true_out_
p1_out_a(x1)  =  p1_out_a(x1)
U3_a(x1)  =  U3_a(x1)
P1_IN_A(x1)  =  P1_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:

P1_IN_A(a) → P1_IN_A(X)

R is empty.
The argument filtering Pi contains the following mapping:
a  =  a
P1_IN_A(x1)  =  P1_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:

P1_IN_A → P1_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 = P1_IN_A evaluates to t =P1_IN_A

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P1_IN_A to P1_IN_A.



(28) FALSE