Left Termination proof of /tmp/tmpk1WGJk/psk03.pl

Left Termination of the query pattern p_in_1(a) 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 FALSE

    ↳15 PrologToPiTRSProof (⇐)

        ↳16 PiTRS

            ↳17 DependencyPairsProof (⇔)

                ↳18 PiDP

                    ↳19 DependencyGraphProof (⇔)

                        ↳20 PiDP

                            ↳21 UsableRulesProof (⇔)

                                ↳22 PiDP

                                    ↳23 PiDPToQDPProof (⇐)

                                        ↳24 QDP

                                            ↳25 NonTerminationProof (⇔)

                                                ↳26 FALSE

(0) Obligation:

Clauses:

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

Queries:

p(a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(q(a), !_1) | p(T1), SCOPE: 1, CLAUSE: 1 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(q(a), !_1), SCOPE: 2, CLAUSE: 2 | p(T1), SCOPE: 1, CLAUSE: 1 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(T4)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 7 [arrowhead = none ];
7 -> 2;

8 [label="EVAL with clause\np(X2) :- ','(q(a), !).\nand substitution\nT1 -> T2\nX2 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 8 [arrowhead = none ];
8 -> 3;

9 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 9 [arrowhead = none ];
9 -> 4;

10 [label="BACKTRACK\nfor clause: q(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
4 -> 10 [arrowhead = none ];
10 -> 5;

11 [label="EVAL with clause\np(X4) :- p(X4).\nand substitution\nT1 -> T4\nX4 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 11 [arrowhead = none ];
11 -> 6;

12 [label="INSTANCE with matching:\nT1 -> T4", fontsize=14, style = filled, fillcolor = lightgrey];
6 -> 12 [arrowhead = none , style=dashed];
12 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

p1(T4) :- p1(T4).

Queries:

p1(a).

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

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(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:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(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:

P1_IN_A(T4) → U1_A(T4, p1_in_a(T4))
P1_IN_A(T4) → P1_IN_A(T4)

The TRS R consists of the following rules:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(x1)
P1_IN_A(x1) = P1_IN_A
U1_A(x1, x2) = U1_A(x2)

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

(6) Obligation:

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

P1_IN_A(T4) → U1_A(T4, p1_in_a(T4))
P1_IN_A(T4) → P1_IN_A(T4)

The TRS R consists of the following rules:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

P1_IN_A(T4) → P1_IN_A(T4)

The TRS R consists of the following rules:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(x1)
P1_IN_A(x1) = P1_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:

P1_IN_A(T4) → P1_IN_A(T4)

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

P1_IN_A → P1_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 = 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.

(14) FALSE

(15) PrologToPiTRSProof (SOUND transformation)

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(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:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(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:

P1_IN_A(T4) → U1_A(T4, p1_in_a(T4))
P1_IN_A(T4) → P1_IN_A(T4)

The TRS R consists of the following rules:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

(18) Obligation:

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

P1_IN_A(T4) → U1_A(T4, p1_in_a(T4))
P1_IN_A(T4) → P1_IN_A(T4)

The TRS R consists of the following rules:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) Obligation:

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

P1_IN_A(T4) → P1_IN_A(T4)

The TRS R consists of the following rules:

p1_in_a(T4) → U1_a(T4, p1_in_a(T4))
U1_a(T4, p1_out_a(T4)) → p1_out_a(T4)

The argument filtering Pi contains the following mapping:
p1_in_a(x1) = p1_in_a
U1_a(x1, x2) = U1_a(x2)
p1_out_a(x1) = p1_out_a(x1)
P1_IN_A(x1) = P1_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:

P1_IN_A(T4) → P1_IN_A(T4)

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

P1_IN_A → P1_IN_A

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

(25) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) NonTerminationProof (EQUIVALENT transformation)

(14) FALSE

(15) PrologToPiTRSProof (SOUND transformation)

(16) Obligation:

(17) DependencyPairsProof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) Obligation:

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

(23) PiDPToQDPProof (SOUND transformation)

(24) Obligation:

(25) NonTerminationProof (EQUIVALENT transformation)

(26) FALSE