Left Termination proof of /tmp/tmpLQeKcs/transitive

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 88 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 TRUE

(0) Obligation:

Clauses:

p(a, b).
p(b, c).
tc(X, X).
tc(X, Y) :- ','(p(X, Z), tc(Z, Y)).

Query: tc(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: tc(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: tc(T1, T2), SCOPE: 1, CLAUSE: 2 | tc(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | tc(T4, T2), SCOPE: 1, CLAUSE: 3\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: tc(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX2 is free\ntc(T1, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: tc(T4, T2), SCOPE: 1, CLAUSE: 3\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(p(T7, X7), tc(X7, T9))\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 0 | ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: tc(b, T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: tc(b, T9), SCOPE: 3, CLAUSE: 2 | tc(b, T9), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: tc(b, T9), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: tc(b, T9), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(b, X24), tc(X24, T15))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(b, X24), tc(X24, T15)), SCOPE: 4, CLAUSE: 0 | ','(p(b, X24), tc(X24, T15)), SCOPE: 4, CLAUSE: 1\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(p(b, X24), tc(X24, T15)), SCOPE: 4, CLAUSE: 1\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: tc(c, T15)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: tc(c, T15), SCOPE: 5, CLAUSE: 2 | tc(c, T15), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: tc(c, T15), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: tc(c, T15), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(p(c, X45), tc(X45, T22))\n\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(c, X45), tc(X45, T22)), SCOPE: 6, CLAUSE: 0 | ','(p(c, X45), tc(X45, T22)), SCOPE: 6, CLAUSE: 1\n\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(c, X45), tc(X45, T22)), SCOPE: 6, CLAUSE: 1\n\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: tc(c, T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T26, X54), tc(X54, T28))\n\nT26 is ground\nX2 is free\nX54 is free\ntc(T26, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T26, X54), tc(X54, T28)), SCOPE: 7, CLAUSE: 0 | ','(p(T26, X54), tc(X54, T28)), SCOPE: 7, CLAUSE: 1\n\nT26 is ground\nX2 is free\nX54 is free\ntc(T26, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T26, X54), tc(X54, T28)), SCOPE: 7, CLAUSE: 0\n\nT26 is ground\nX2 is free\nX54 is free\ntc(T26, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T26, X54), tc(X54, T28)), SCOPE: 7, CLAUSE: 1\n\nT26 is ground\nX2 is free\nX54 is free\ntc(T26, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tc(b, T28)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: tc(c, T28)\n\n", 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];
1 -> 42 [arrowhead = none ];
42 -> 2;

43 [label="EVAL with clause\ntc(X2, X2).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 43 [arrowhead = none ];
43 -> 3;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 44 [arrowhead = none ];
44 -> 4;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 45 [arrowhead = none ];
45 -> 5;

46 [label="ONLY EVAL with clause\ntc(X52, X53) :- ','(p(X52, X54), tc(X54, X53)).\nand substitutionT1 -> T26,\nX52 -> T26,\nT2 -> T28,\nX53 -> T28,\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 46 [arrowhead = none ];
46 -> 34;

47 [label="ONLY EVAL with clause\ntc(X5, X6) :- ','(p(X5, X7), tc(X7, X6)).\nand substitutionT4 -> T7,\nX5 -> T7,\nT2 -> T9,\nX6 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 47 [arrowhead = none ];
47 -> 6;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 48 [arrowhead = none ];
48 -> 7;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 49 [arrowhead = none ];
49 -> 8;

50 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 50 [arrowhead = none ];
50 -> 9;

51 [label="EVAL with clause\np(a, b).\nand substitutionT7 -> a,\nX7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 51 [arrowhead = none ];
51 -> 10;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 52 [arrowhead = none ];
52 -> 11;

53 [label="EVAL with clause\np(b, c).\nand substitutionT7 -> b,\nX7 -> c", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 53 [arrowhead = none ];
53 -> 32;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 54 [arrowhead = none ];
54 -> 33;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 55 [arrowhead = none ];
55 -> 12;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 56 [arrowhead = none ];
56 -> 13;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 57 [arrowhead = none ];
57 -> 14;

58 [label="EVAL with clause\ntc(X14, X14).\nand substitutionX14 -> b,\nT9 -> b", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 58 [arrowhead = none ];
58 -> 15;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 59 [arrowhead = none ];
59 -> 16;

60 [label="ONLY EVAL with clause\ntc(X22, X23) :- ','(p(X22, X24), tc(X24, X23)).\nand substitutionX22 -> b,\nT9 -> T15,\nX23 -> T15,\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 60 [arrowhead = none ];
60 -> 18;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 62 [arrowhead = none ];
62 -> 19;

63 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 63 [arrowhead = none ];
63 -> 20;

64 [label="ONLY EVAL with clause\np(b, c).\nand substitutionX24 -> c", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 64 [arrowhead = none ];
64 -> 21;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 65 [arrowhead = none ];
65 -> 22;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 66 [arrowhead = none ];
66 -> 23;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 67 [arrowhead = none ];
67 -> 24;

68 [label="EVAL with clause\ntc(X35, X35).\nand substitutionX35 -> c,\nT15 -> c", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 68 [arrowhead = none ];
68 -> 25;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="ONLY EVAL with clause\ntc(X43, X44) :- ','(p(X43, X45), tc(X45, X44)).\nand substitutionX43 -> c,\nT15 -> T22,\nX44 -> T22,\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 71 [arrowhead = none ];
71 -> 27;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 72 [arrowhead = none ];
72 -> 29;

73 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 73 [arrowhead = none ];
73 -> 30;

74 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 74 [arrowhead = none ];
74 -> 31;

75 [label="INSTANCE with matching:\nT15 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 75 [arrowhead = none , style=dashed];
75 -> 21[style=dashed];

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 76 [arrowhead = none ];
76 -> 35;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 77 [arrowhead = none ];
77 -> 36;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 78 [arrowhead = none ];
78 -> 37;

79 [label="EVAL with clause\np(a, b).\nand substitutionT26 -> a,\nX54 -> b", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 79 [arrowhead = none ];
79 -> 38;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 80 [arrowhead = none ];
80 -> 39;

81 [label="EVAL with clause\np(b, c).\nand substitutionT26 -> b,\nX54 -> c", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 81 [arrowhead = none ];
81 -> 40;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 82 [arrowhead = none ];
82 -> 41;

83 [label="INSTANCE with matching:\nT9 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 83 [arrowhead = none , style=dashed];
83 -> 10[style=dashed];

84 [label="INSTANCE with matching:\nT15 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 84 [arrowhead = none , style=dashed];
84 -> 21[style=dashed];

}

(2) Obligation:

Clauses:

tcA(c).
tcB(b).
tcB(T15) :- tcA(T15).
tcC(T4, T4).
tcC(a, T9) :- tcB(T9).
tcC(b, T9) :- tcA(T9).
tcC(a, T28) :- tcB(T28).
tcC(b, T28) :- tcA(T28).

Query: tcC(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
tcC_in: (b,f)
tcB_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

tcC_in_ga(T4, T4) → tcC_out_ga(T4, T4)
tcC_in_ga(a, T9) → U2_ga(T9, tcB_in_a(T9))
tcB_in_a(b) → tcB_out_a(b)
tcB_in_a(T15) → U1_a(T15, tcA_in_a(T15))
tcA_in_a(c) → tcA_out_a(c)
U1_a(T15, tcA_out_a(T15)) → tcB_out_a(T15)
U2_ga(T9, tcB_out_a(T9)) → tcC_out_ga(a, T9)
tcC_in_ga(b, T9) → U3_ga(T9, tcA_in_a(T9))
U3_ga(T9, tcA_out_a(T9)) → tcC_out_ga(b, T9)
tcC_in_ga(a, T28) → U4_ga(T28, tcB_in_a(T28))
U4_ga(T28, tcB_out_a(T28)) → tcC_out_ga(a, T28)
tcC_in_ga(b, T28) → U5_ga(T28, tcA_in_a(T28))
U5_ga(T28, tcA_out_a(T28)) → tcC_out_ga(b, T28)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

tcC_in_ga(T4, T4) → tcC_out_ga(T4, T4)
tcC_in_ga(a, T9) → U2_ga(T9, tcB_in_a(T9))
tcB_in_a(b) → tcB_out_a(b)
tcB_in_a(T15) → U1_a(T15, tcA_in_a(T15))
tcA_in_a(c) → tcA_out_a(c)
U1_a(T15, tcA_out_a(T15)) → tcB_out_a(T15)
U2_ga(T9, tcB_out_a(T9)) → tcC_out_ga(a, T9)
tcC_in_ga(b, T9) → U3_ga(T9, tcA_in_a(T9))
U3_ga(T9, tcA_out_a(T9)) → tcC_out_ga(b, T9)
tcC_in_ga(a, T28) → U4_ga(T28, tcB_in_a(T28))
U4_ga(T28, tcB_out_a(T28)) → tcC_out_ga(a, T28)
tcC_in_ga(b, T28) → U5_ga(T28, tcA_in_a(T28))
U5_ga(T28, tcA_out_a(T28)) → tcC_out_ga(b, T28)

(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:

TCC_IN_GA(a, T9) → U2_GA(T9, tcB_in_a(T9))
TCC_IN_GA(a, T9) → TCB_IN_A(T9)
TCB_IN_A(T15) → U1_A(T15, tcA_in_a(T15))
TCB_IN_A(T15) → TCA_IN_A(T15)
TCC_IN_GA(b, T9) → U3_GA(T9, tcA_in_a(T9))
TCC_IN_GA(b, T9) → TCA_IN_A(T9)
TCC_IN_GA(a, T28) → U4_GA(T28, tcB_in_a(T28))
TCC_IN_GA(b, T28) → U5_GA(T28, tcA_in_a(T28))

The TRS R consists of the following rules:

tcC_in_ga(T4, T4) → tcC_out_ga(T4, T4)
tcC_in_ga(a, T9) → U2_ga(T9, tcB_in_a(T9))
tcB_in_a(b) → tcB_out_a(b)
tcB_in_a(T15) → U1_a(T15, tcA_in_a(T15))
tcA_in_a(c) → tcA_out_a(c)
U1_a(T15, tcA_out_a(T15)) → tcB_out_a(T15)
U2_ga(T9, tcB_out_a(T9)) → tcC_out_ga(a, T9)
tcC_in_ga(b, T9) → U3_ga(T9, tcA_in_a(T9))
U3_ga(T9, tcA_out_a(T9)) → tcC_out_ga(b, T9)
tcC_in_ga(a, T28) → U4_ga(T28, tcB_in_a(T28))
U4_ga(T28, tcB_out_a(T28)) → tcC_out_ga(a, T28)
tcC_in_ga(b, T28) → U5_ga(T28, tcA_in_a(T28))
U5_ga(T28, tcA_out_a(T28)) → tcC_out_ga(b, T28)

The argument filtering Pi contains the following mapping:
tcC_in_ga(x1, x2) = tcC_in_ga(x1)
tcC_out_ga(x1, x2) = tcC_out_ga(x2)
a = a
U2_ga(x1, x2) = U2_ga(x2)
tcB_in_a(x1) = tcB_in_a
tcB_out_a(x1) = tcB_out_a(x1)
U1_a(x1, x2) = U1_a(x2)
tcA_in_a(x1) = tcA_in_a
tcA_out_a(x1) = tcA_out_a(x1)
b = b
U3_ga(x1, x2) = U3_ga(x2)
U4_ga(x1, x2) = U4_ga(x2)
U5_ga(x1, x2) = U5_ga(x2)
TCC_IN_GA(x1, x2) = TCC_IN_GA(x1)
U2_GA(x1, x2) = U2_GA(x2)
TCB_IN_A(x1) = TCB_IN_A
U1_A(x1, x2) = U1_A(x2)
TCA_IN_A(x1) = TCA_IN_A
U3_GA(x1, x2) = U3_GA(x2)
U4_GA(x1, x2) = U4_GA(x2)
U5_GA(x1, x2) = U5_GA(x2)

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

(6) Obligation:

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