Left Termination proof of /tmp/tmpialeel/sum-fbf.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 64 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 8 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).

Query: sum(a,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: sum(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | sum(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | sum(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX2 is free\nsum(T1, T2, T3) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: sum(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: sum(T13, T11, T14)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: sum(T13, T11, T14), SCOPE: 2, CLAUSE: 0 | sum(T13, T11, T14), SCOPE: 2, CLAUSE: 1\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: sum(T13, T11, T14), SCOPE: 2, CLAUSE: 0\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: sum(T13, T11, T14), SCOPE: 2, CLAUSE: 1\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: sum(T29, T27, T30)\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 40 [arrowhead = none ];
40 -> 4;

41 [label="EVAL with clause\nsum(X2, 0, X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> 0,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 41 [arrowhead = none ];
41 -> 9;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 42 [arrowhead = none ];
42 -> 13;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 43 [arrowhead = none ];
43 -> 16;

44 [label="EVAL with clause\nsum(X9, s(X10), s(X11)) :- sum(X9, X10, X11).\nand substitutionT1 -> T13,\nX9 -> T13,\nX10 -> T11,\nT2 -> s(T11),\nX11 -> T14,\nT3 -> s(T14),\nT10 -> T13,\nT12 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 44 [arrowhead = none ];
44 -> 20;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 45 [arrowhead = none ];
45 -> 21;

46 [label="BACKTRACK\nfor clause: sum(X, s(Y), s(Z)) :- sum(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 46 [arrowhead = none ];
46 -> 18;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 47 [arrowhead = none ];
47 -> 23;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 48 [arrowhead = none ];
48 -> 25;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 49 [arrowhead = none ];
49 -> 26;

50 [label="EVAL with clause\nsum(X16, 0, X16).\nand substitutionT13 -> T19,\nX16 -> T19,\nT11 -> 0,\nT14 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 50 [arrowhead = none ];
50 -> 31;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 51 [arrowhead = none ];
51 -> 33;

52 [label="EVAL with clause\nsum(X23, s(X24), s(X25)) :- sum(X23, X24, X25).\nand substitutionT13 -> T29,\nX23 -> T29,\nX24 -> T27,\nT11 -> s(T27),\nX25 -> T30,\nT14 -> s(T30),\nT26 -> T29,\nT28 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 52 [arrowhead = none ];
52 -> 38;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 53 [arrowhead = none ];
53 -> 39;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 54 [arrowhead = none ];
54 -> 34;

55 [label="INSTANCE with matching:\nT1 -> T29\nT2 -> T27\nT3 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 55 [arrowhead = none , style=dashed];
55 -> 1[style=dashed];

}

(2) Obligation:

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

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

(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:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → U1_AGA(T29, T27, T30, sumA_in_aga(T29, T27, T30))
SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

The argument filtering Pi contains the following mapping:
sumA_in_aga(x1, x2, x3) = sumA_in_aga(x2)
0 = 0
sumA_out_aga(x1, x2, x3) = sumA_out_aga(x2)
s(x1) = s(x1)
U1_aga(x1, x2, x3, x4) = U1_aga(x2, x4)
SUMA_IN_AGA(x1, x2, x3) = SUMA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4) = U1_AGA(x2, x4)

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

(4) Obligation:

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

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → U1_AGA(T29, T27, T30, sumA_in_aga(T29, T27, T30))
SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

(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:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

(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:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
SUMA_IN_AGA(x1, x2, x3) = SUMA_IN_AGA(x2)

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:

SUMA_IN_AGA(s(s(T27))) → SUMA_IN_AGA(T27)

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:

SUMA_IN_AGA(s(s(T27))) → SUMA_IN_AGA(T27)
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