Left Termination proof of /tmp/tmpLXKKU8/less-bf.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 58 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 16 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Query: less(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: less(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | less(0, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\nless(T1, T2) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(0, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T7, T9)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T7, T9), SCOPE: 2, CLAUSE: 0 | less(T7, T9), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T7, T9), SCOPE: 2, CLAUSE: 0\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(T7, T9), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T19, T21)\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 42 [arrowhead = none ];
42 -> 4;

43 [label="EVAL with clause\nless(0, s(X3)).\nand substitutionT1 -> 0,\nX3 -> T4,\nT2 -> s(T4)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 43 [arrowhead = none ];
43 -> 9;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 44 [arrowhead = none ];
44 -> 11;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 45 [arrowhead = none ];
45 -> 13;

46 [label="EVAL with clause\nless(s(X8), s(X9)) :- less(X8, X9).\nand substitutionX8 -> T7,\nT1 -> s(T7),\nX9 -> T9,\nT2 -> s(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 46 [arrowhead = none ];
46 -> 18;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 47 [arrowhead = none ];
47 -> 21;

48 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 48 [arrowhead = none ];
48 -> 14;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 49 [arrowhead = none ];
49 -> 24;

50 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 50 [arrowhead = none ];
50 -> 27;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 51 [arrowhead = none ];
51 -> 29;

52 [label="EVAL with clause\nless(0, s(X14)).\nand substitutionT7 -> 0,\nX14 -> T14,\nT9 -> s(T14)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 52 [arrowhead = none ];
52 -> 31;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 53 [arrowhead = none ];
53 -> 34;

54 [label="EVAL with clause\nless(s(X19), s(X20)) :- less(X19, X20).\nand substitutionX19 -> T19,\nT7 -> s(T19),\nX20 -> T21,\nT9 -> s(T21),\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 54 [arrowhead = none ];
54 -> 40;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 55 [arrowhead = none ];
55 -> 41;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 56 [arrowhead = none ];
56 -> 37;

57 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 57 [arrowhead = none , style=dashed];
57 -> 3[style=dashed];

}

(2) Obligation:

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

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(0), s(s(T14))) → lessA_out_ga(s(0), s(s(T14)))
lessA_in_ga(s(s(T19)), s(s(T21))) → U1_ga(T19, T21, lessA_in_ga(T19, T21))
U1_ga(T19, T21, lessA_out_ga(T19, T21)) → lessA_out_ga(s(s(T19)), s(s(T21)))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2) = lessA_in_ga(x1)
0 = 0
lessA_out_ga(x1, x2) = lessA_out_ga(x1)
s(x1) = s(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)

(3) 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:

LESSA_IN_GA(s(s(T19)), s(s(T21))) → U1_GA(T19, T21, lessA_in_ga(T19, T21))
LESSA_IN_GA(s(s(T19)), s(s(T21))) → LESSA_IN_GA(T19, T21)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(0), s(s(T14))) → lessA_out_ga(s(0), s(s(T14)))
lessA_in_ga(s(s(T19)), s(s(T21))) → U1_ga(T19, T21, lessA_in_ga(T19, T21))
U1_ga(T19, T21, lessA_out_ga(T19, T21)) → lessA_out_ga(s(s(T19)), s(s(T21)))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2) = lessA_in_ga(x1)
0 = 0
lessA_out_ga(x1, x2) = lessA_out_ga(x1)
s(x1) = s(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)

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

(4) Obligation:

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

LESSA_IN_GA(s(s(T19)), s(s(T21))) → U1_GA(T19, T21, lessA_in_ga(T19, T21))
LESSA_IN_GA(s(s(T19)), s(s(T21))) → LESSA_IN_GA(T19, T21)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(0), s(s(T14))) → lessA_out_ga(s(0), s(s(T14)))
lessA_in_ga(s(s(T19)), s(s(T21))) → U1_ga(T19, T21, lessA_in_ga(T19, T21))
U1_ga(T19, T21, lessA_out_ga(T19, T21)) → lessA_out_ga(s(s(T19)), s(s(T21)))

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

LESSA_IN_GA(s(s(T19)), s(s(T21))) → LESSA_IN_GA(T19, T21)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(0), s(s(T14))) → lessA_out_ga(s(0), s(s(T14)))
lessA_in_ga(s(s(T19)), s(s(T21))) → U1_ga(T19, T21, lessA_in_ga(T19, T21))
U1_ga(T19, T21, lessA_out_ga(T19, T21)) → lessA_out_ga(s(s(T19)), s(s(T21)))

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

LESSA_IN_GA(s(s(T19)), s(s(T21))) → LESSA_IN_GA(T19, T21)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

LESSA_IN_GA(s(s(T19))) → LESSA_IN_GA(T19)

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

(11) 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:

LESSA_IN_GA(s(s(T19))) → LESSA_IN_GA(T19)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES