Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
    ↳3 PrologToPiTRSProof (⇐)
        ↳4 PiTRS
        ↳5 DependencyPairsProof (⇔)
        ↳6 PiDP
        ↳7 DependencyGraphProof (⇔)
        ↳8 PiDP
        ↳9 UsableRulesProof (⇔)
        ↳10 PiDP
        ↳11 PiDPToQDPProof (⇐)
        ↳12 QDP
        ↳13 NonTerminationProof (⇔)
        ↳14 NO
    ↳15 PrologToPiTRSProof (⇐)
        ↳16 PiTRS
        ↳17 DependencyPairsProof (⇔)
        ↳18 PiDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 PiDP
        ↳21 UsableRulesProof (⇔)
        ↳22 PiDP
        ↳23 PiDPToQDPProof (⇐)
        ↳24 QDP
        ↳25 NonTerminationProof (⇔)
        ↳26 NO

(0) Obligation:

Clauses:

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

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

p12(a).
p12(X15) :- p12(X14).
goal1(T6) :- p12(X7).
goal1(b) :- p12(T7).

Queries:

goal1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
p12_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g
b  =  b
U3_g(x1)  =  U3_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g
b  =  b
U3_g(x1)  =  U3_g(x1)

(5) 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:

GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)
GOAL1_IN_G(b) → U3_G(p12_in_a(T7))
GOAL1_IN_G(b) → P12_IN_A(T7)

The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g
b  =  b
U3_g(x1)  =  U3_g(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)
P12_IN_A(x1)  =  P12_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_G(x1)  =  U3_G(x1)

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

(6) Obligation:

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

GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)
GOAL1_IN_G(b) → U3_G(p12_in_a(T7))
GOAL1_IN_G(b) → P12_IN_A(T7)

The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g
b  =  b
U3_g(x1)  =  U3_g(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)
P12_IN_A(x1)  =  P12_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_G(x1)  =  U3_G(x1)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

P12_IN_A(X15) → P12_IN_A(X14)

The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g
b  =  b
U3_g(x1)  =  U3_g(x1)
P12_IN_A(x1)  =  P12_IN_A

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

P12_IN_A(X15) → P12_IN_A(X14)

R is empty.
The argument filtering Pi contains the following mapping:
P12_IN_A(x1)  =  P12_IN_A

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

P12_IN_A → P12_IN_A

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

(13) 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 = P12_IN_A evaluates to t =P12_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 P12_IN_A to P12_IN_A.



(14) NO

(15) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
p12_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g(x1)
b  =  b
U3_g(x1)  =  U3_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g(x1)
b  =  b
U3_g(x1)  =  U3_g(x1)

(17) 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:

GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)
GOAL1_IN_G(b) → U3_G(p12_in_a(T7))
GOAL1_IN_G(b) → P12_IN_A(T7)

The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g(x1)
b  =  b
U3_g(x1)  =  U3_g(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
P12_IN_A(x1)  =  P12_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_G(x1)  =  U3_G(x1)

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

(18) Obligation:

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

GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)
GOAL1_IN_G(b) → U3_G(p12_in_a(T7))
GOAL1_IN_G(b) → P12_IN_A(T7)

The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g(x1)
b  =  b
U3_g(x1)  =  U3_g(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
P12_IN_A(x1)  =  P12_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_G(x1)  =  U3_G(x1)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) Obligation:

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

P12_IN_A(X15) → P12_IN_A(X14)

The TRS R consists of the following rules:

goal1_in_g(T6) → U2_g(T6, p12_in_a(X7))
p12_in_a(a) → p12_out_a(a)
p12_in_a(X15) → U1_a(X15, p12_in_a(X14))
U1_a(X15, p12_out_a(X14)) → p12_out_a(X15)
U2_g(T6, p12_out_a(X7)) → goal1_out_g(T6)
goal1_in_g(b) → U3_g(p12_in_a(T7))
U3_g(p12_out_a(T7)) → goal1_out_g(b)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
p12_in_a(x1)  =  p12_in_a
p12_out_a(x1)  =  p12_out_a
U1_a(x1, x2)  =  U1_a(x2)
goal1_out_g(x1)  =  goal1_out_g(x1)
b  =  b
U3_g(x1)  =  U3_g(x1)
P12_IN_A(x1)  =  P12_IN_A

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

P12_IN_A(X15) → P12_IN_A(X14)

R is empty.
The argument filtering Pi contains the following mapping:
P12_IN_A(x1)  =  P12_IN_A

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

P12_IN_A → P12_IN_A

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

(25) 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 = P12_IN_A evaluates to t =P12_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 P12_IN_A to P12_IN_A.



(26) NO