Left Termination proof of /tmp/tmpyRVFSo/negationasfailure.pl

Left Termination of the query pattern q_in_1(g) w.r.t. the given Prolog program could successfully be proven:

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 QDPSizeChangeProof (⇔)

↳14 TRUE

(0) Obligation:

Clauses:

q(X) :- ','(not_zero(X), ','(p(X, Y), q(Y))).
p(0, 0).
p(s(X), X).
zero(0).
not_zero(X) :- ','(zero(X), ','(!, failure(a))).
not_zero(X1).
failure(b).

Queries:

q(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(not_zero(T2), ','(p(T2, X4), q(X4)))\n\nT2 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(not_zero(T2), ','(p(T2, X4), q(X4))), SCOPE: 2, CLAUSE: 4 | ','(not_zero(T2), ','(p(T2, X4), q(X4))), SCOPE: 2, CLAUSE: 5 | ?_2\n\nT2 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(zero(T3), ','(!_2, ','(failure(a), ','(p(T3, X4), q(X4))))) | ','(not_zero(T3), ','(p(T3, X4), q(X4))), SCOPE: 2, CLAUSE: 5 | ?_2\n\nT3 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(zero(T3), ','(!_2, ','(failure(a), ','(p(T3, X4), q(X4))))), SCOPE: 3, CLAUSE: 3 | ','(not_zero(T3), ','(p(T3, X4), q(X4))), SCOPE: 2, CLAUSE: 5 | ?_2\n\nT3 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, ','(failure(a), ','(p(0, X4), q(X4)))) | ','(not_zero(0), ','(p(0, X4), q(X4))), SCOPE: 2, CLAUSE: 5 | ?_2\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(not_zero(T3), ','(p(T3, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX4 is free; \nzero(T3) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(failure(a), ','(p(0, X4), q(X4)))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(failure(a), ','(p(0, X4), q(X4))), SCOPE: 4, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T4, X4), q(X4))\n\nT4 is ground\nX4 is free; \nzero(T4) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T4, X4), q(X4)), SCOPE: 5, CLAUSE: 1 | ','(p(T4, X4), q(X4)), SCOPE: 5, CLAUSE: 2\n\nT4 is ground\nX4 is free; \nzero(T4) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T4, X4), q(X4)), SCOPE: 5, CLAUSE: 2\n\nT4 is ground\nX4 is free; \nzero(T4) !=? zero(0)\np(T4, X4) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: q(T5)\n\nT5 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\nX4 is free\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 17 [arrowhead = none ];
17 -> 2;

18 [label="EVAL with clause\nq(X3) :- ','(not_zero(X3), ','(p(X3, X4), q(X4))).\nand substitution\nT1 -> T2\nX3 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 18 [arrowhead = none ];
18 -> 3;

19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 19 [arrowhead = none ];
19 -> 4;

20 [label="EVAL with clause\nnot_zero(X6) :- ','(zero(X6), ','(!, failure(a))).\nand substitution\nT2 -> T3\nX6 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 20 [arrowhead = none ];
20 -> 5;

21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 21 [arrowhead = none ];
21 -> 6;

22 [label="EVAL with clause\nzero(0).\nand substitution\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 22 [arrowhead = none ];
22 -> 7;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 23 [arrowhead = none ];
23 -> 8;

24 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 24 [arrowhead = none ];
24 -> 9;

25 [label="EVAL with clause\nnot_zero(X8).\nand substitution\nT3 -> T4\nX8 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 25 [arrowhead = none ];
25 -> 12;

26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 26 [arrowhead = none ];
26 -> 10;

27 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 27 [arrowhead = none ];
27 -> 11;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 28 [arrowhead = none ];
28 -> 13;

29 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T4), zero(0))", fontsize=14, style = filled, fillcolor = white];
13 -> 29 [arrowhead = none ];
29 -> 14;

30 [label="EVAL with clause\np(s(X10), X10).\nand substitution\nX10 -> T5\nT4 -> s(T5)\nX4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 30 [arrowhead = none ];
30 -> 15;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 31 [arrowhead = none ];
31 -> 16;

32 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 32 [arrowhead = none , style=dashed];
32 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

q1(s(T5)) :- q1(T5).

Queries:

q1(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:
q1_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

q1_in_g(s(T5)) → U1_g(T5, q1_in_g(T5))
U1_g(T5, q1_out_g(T5)) → q1_out_g(s(T5))

The argument filtering Pi contains the following mapping:
q1_in_g(x1) = q1_in_g(x1)
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
q1_out_g(x1) = q1_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

q1_in_g(s(T5)) → U1_g(T5, q1_in_g(T5))
U1_g(T5, q1_out_g(T5)) → q1_out_g(s(T5))

The argument filtering Pi contains the following mapping:
q1_in_g(x1) = q1_in_g(x1)
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
q1_out_g(x1) = q1_out_g

(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:

Q1_IN_G(s(T5)) → U1_G(T5, q1_in_g(T5))
Q1_IN_G(s(T5)) → Q1_IN_G(T5)

The TRS R consists of the following rules:

q1_in_g(s(T5)) → U1_g(T5, q1_in_g(T5))
U1_g(T5, q1_out_g(T5)) → q1_out_g(s(T5))

The argument filtering Pi contains the following mapping:
q1_in_g(x1) = q1_in_g(x1)
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
q1_out_g(x1) = q1_out_g
Q1_IN_G(x1) = Q1_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)

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

(6) Obligation:

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

Q1_IN_G(s(T5)) → U1_G(T5, q1_in_g(T5))
Q1_IN_G(s(T5)) → Q1_IN_G(T5)

The TRS R consists of the following rules:

q1_in_g(s(T5)) → U1_g(T5, q1_in_g(T5))
U1_g(T5, q1_out_g(T5)) → q1_out_g(s(T5))

(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:

Q1_IN_G(s(T5)) → Q1_IN_G(T5)

The TRS R consists of the following rules:

q1_in_g(s(T5)) → U1_g(T5, q1_in_g(T5))
U1_g(T5, q1_out_g(T5)) → q1_out_g(s(T5))

The argument filtering Pi contains the following mapping:
q1_in_g(x1) = q1_in_g(x1)
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
q1_out_g(x1) = q1_out_g
Q1_IN_G(x1) = Q1_IN_G(x1)

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:

Q1_IN_G(s(T5)) → Q1_IN_G(T5)

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

(11) PiDPToQDPProof (EQUIVALENT 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:

Q1_IN_G(s(T5)) → Q1_IN_G(T5)

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

(13) 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:

Q1_IN_G(s(T5)) → Q1_IN_G(T5)
The graph contains the following edges 1 > 1

(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 (EQUIVALENT transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) TRUE