Runtime Complexity of the query pattern
p(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

p(X, X).
p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W))).

Query: p(g,a)

(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\np(T1, T2) !=? p(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(f(T10), f(X12)), p(X12, g(X13)))\n\nT10 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(f(T10), f(X12))\n\nT10 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T13, g(X13))\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(f(T10), f(X12)), SCOPE: 2, CLAUSE: 0 | p(f(T10), f(X12)), SCOPE: 2, CLAUSE: 1\n\nT10 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | p(f(T17), f(X12)), SCOPE: 2, CLAUSE: 1\n\nT17 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(f(T17), f(X12)), SCOPE: 2, CLAUSE: 1\n\nT17 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T13, g(X13)), SCOPE: 3, CLAUSE: 0 | p(T13, g(X13)), SCOPE: 3, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true | p(T13, g(X13)), SCOPE: 3, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(T13, g(X13)), SCOPE: 3, CLAUSE: 1\n\nX13 is free\nX28 is free\np(T13, g(X13)) !=? p(X28, X28)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: p(T13, g(X13)), SCOPE: 3, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(f(T32), f(X39)), p(X39, g(X40)))\n\nX39 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(f(T32), f(X39))\n\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: p(T34, g(X40))\n\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: p(f(T32), f(X39)), SCOPE: 4, CLAUSE: 0 | p(f(T32), f(X39)), SCOPE: 4, CLAUSE: 1\n\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true | p(f(T32), f(X39)), SCOPE: 4, CLAUSE: 1\n\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(f(T32), f(X39)), SCOPE: 4, CLAUSE: 1\n\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(f(T47), f(X64)), p(X64, g(X65)))\n\nX64 is free\nX65 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(f(T53), f(X79)), p(X79, g(X80)))\n\nT53 is ground\nX3 is free\nX79 is free\nX80 is free\np(f(T53), T2) !=? p(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 56 [arrowhead = none ];
56 -> 3;

57 [label="EVAL with clause\np(X3, X3).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 57 [arrowhead = none ];
57 -> 5;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 58 [arrowhead = none ];
58 -> 7;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 59 [arrowhead = none ];
59 -> 9;

60 [label="EVAL with clause\np(f(X77), g(X78)) :- ','(p(f(X77), f(X79)), p(X79, g(X80))).\nand substitutionX77 -> T53,\nT1 -> f(T53),\nX78 -> T54,\nT2 -> g(T54)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 60 [arrowhead = none ];
60 -> 54;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 61 [arrowhead = none ];
61 -> 55;

62 [label="EVAL with clause\np(f(X10), g(X11)) :- ','(p(f(X10), f(X12)), p(X12, g(X13))).\nand substitutionX10 -> T10,\nT5 -> f(T10),\nX11 -> T11,\nT2 -> g(T11)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 64 [arrowhead = none ];
64 -> 17;

65 [label="SPLIT 2\nnew knowledge:\nT10 is ground\nreplacements:X12 -> T13", fontsize=14, style = filled, fillcolor = violet];
13 -> 65 [arrowhead = none ];
65 -> 18;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 66 [arrowhead = none ];
66 -> 20;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 67 [arrowhead = none ];
67 -> 26;

68 [label="ONLY EVAL with clause\np(X21, X21).\nand substitutionT10 -> T17,\nX21 -> f(T17),\nX12 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 68 [arrowhead = none ];
68 -> 22;

69 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 69 [arrowhead = none ];
69 -> 23;

70 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 70 [arrowhead = none ];
70 -> 24;

71 [label="EVAL with clause\np(X28, X28).\nand substitutionT13 -> g(T28),\nX28 -> g(T28),\nX13 -> T28,\nT27 -> g(T28)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 71 [arrowhead = none ];
71 -> 31;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 72 [arrowhead = none ];
72 -> 32;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 73 [arrowhead = none ];
73 -> 33;

74 [label="EVAL with clause\np(f(X62), g(X63)) :- ','(p(f(X62), f(X64)), p(X64, g(X65))).\nand substitutionX62 -> T47,\nT13 -> f(T47),\nX13 -> X66,\nX63 -> X66,\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 74 [arrowhead = none ];
74 -> 51;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 75 [arrowhead = none ];
75 -> 52;

76 [label="EVAL with clause\np(f(X37), g(X38)) :- ','(p(f(X37), f(X39)), p(X39, g(X40))).\nand substitutionX37 -> T32,\nT13 -> f(T32),\nX13 -> X41,\nX38 -> X41,\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 76 [arrowhead = none ];
76 -> 36;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 77 [arrowhead = none ];
77 -> 37;

78 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 78 [arrowhead = none ];
78 -> 38;

79 [label="SPLIT 2\nreplacements:X39 -> T34", fontsize=14, style = filled, fillcolor = violet];
36 -> 79 [arrowhead = none ];
79 -> 39;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 80 [arrowhead = none ];
80 -> 41;

81 [label="INSTANCE with matching:\nT13 -> T34\nX13 -> X40", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 81 [arrowhead = none , style=dashed];
81 -> 18[style=dashed];

82 [label="ONLY EVAL with clause\np(X49, X49).\nand substitutionT32 -> T38,\nX49 -> f(T38),\nX39 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 82 [arrowhead = none ];
82 -> 43;

83 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 83 [arrowhead = none ];
83 -> 44;

84 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 84 [arrowhead = none ];
84 -> 46;

85 [label="INSTANCE with matching:\nT32 -> T47\nX39 -> X64\nX40 -> X65", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 85 [arrowhead = none , style=dashed];
85 -> 36[style=dashed];

86 [label="INSTANCE with matching:\nT10 -> T53\nX12 -> X79\nX13 -> X80", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 86 [arrowhead = none , style=dashed];
86 -> 13[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1, f(z0)) → f1_out1
U2(f13_out1, f(z0)) → f1_out1
f18_in → f18_out1
f18_in → U3(f36_in)
f18_in → U4(f36_in)
U3(f36_out1) → f18_out1
U4(f36_out1) → f18_out1
f17_in(z0) → f17_out1
f38_in → f38_out1
f13_in(z0) → U5(f17_in(z0), z0)
U5(f17_out1, z0) → U6(f18_in, z0)
U6(f18_out1, z0) → f13_out1
f36_in → U7(f38_in)
U7(f38_out1) → U8(f18_in)
U8(f18_out1) → f36_out1

Tuples:

F1_IN(f(z0)) → c1(U1'(f13_in(z0), f(z0)), F13_IN(z0))
F1_IN(f(z0)) → c2(U2'(f13_in(z0), f(z0)), F13_IN(z0))
F18_IN → c6(U3'(f36_in), F36_IN)
F18_IN → c7(U4'(f36_in), F36_IN)
F13_IN(z0) → c12(U5'(f17_in(z0), z0), F17_IN(z0))
U5'(f17_out1, z0) → c13(U6'(f18_in, z0), F18_IN)
F36_IN → c15(U7'(f38_in), F38_IN)
U7'(f38_out1) → c16(U8'(f18_in), F18_IN)

S tuples:

F1_IN(f(z0)) → c1(U1'(f13_in(z0), f(z0)), F13_IN(z0))
F1_IN(f(z0)) → c2(U2'(f13_in(z0), f(z0)), F13_IN(z0))
F18_IN → c6(U3'(f36_in), F36_IN)
F18_IN → c7(U4'(f36_in), F36_IN)
F13_IN(z0) → c12(U5'(f17_in(z0), z0), F17_IN(z0))
U5'(f17_out1, z0) → c13(U6'(f18_in, z0), F18_IN)
F36_IN → c15(U7'(f38_in), F38_IN)
U7'(f38_out1) → c16(U8'(f18_in), F18_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f18_in, U3, U4, f17_in, f38_in, f13_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F1_IN, F18_IN, F13_IN, U5', F36_IN, U7'

Compound Symbols:

c1, c2, c6, c7, c12, c13, c15, c16

(3) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1, f(z0)) → f1_out1
U2(f13_out1, f(z0)) → f1_out1
f18_in → f18_out1
f18_in → U3(f36_in)
f18_in → U4(f36_in)
U3(f36_out1) → f18_out1
U4(f36_out1) → f18_out1
f17_in(z0) → f17_out1
f38_in → f38_out1
f13_in(z0) → U5(f17_in(z0), z0)
U5(f17_out1, z0) → U6(f18_in, z0)
U6(f18_out1, z0) → f13_out1
f36_in → U7(f38_in)
U7(f38_out1) → U8(f18_in)
U8(f18_out1) → f36_out1

Tuples:

F18_IN → c6(U3'(f36_in), F36_IN)
F18_IN → c7(U4'(f36_in), F36_IN)
F36_IN → c15(U7'(f38_in), F38_IN)
U7'(f38_out1) → c16(U8'(f18_in), F18_IN)
F1_IN(f(z0)) → c(U1'(f13_in(z0), f(z0)))
F1_IN(f(z0)) → c(F13_IN(z0))
F1_IN(f(z0)) → c(U2'(f13_in(z0), f(z0)))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
F13_IN(z0) → c(F17_IN(z0))
U5'(f17_out1, z0) → c(U6'(f18_in, z0))
U5'(f17_out1, z0) → c(F18_IN)

S tuples:

F18_IN → c6(U3'(f36_in), F36_IN)
F18_IN → c7(U4'(f36_in), F36_IN)
F36_IN → c15(U7'(f38_in), F38_IN)
U7'(f38_out1) → c16(U8'(f18_in), F18_IN)
F1_IN(f(z0)) → c(U1'(f13_in(z0), f(z0)))
F1_IN(f(z0)) → c(F13_IN(z0))
F1_IN(f(z0)) → c(U2'(f13_in(z0), f(z0)))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
F13_IN(z0) → c(F17_IN(z0))
U5'(f17_out1, z0) → c(U6'(f18_in, z0))
U5'(f17_out1, z0) → c(F18_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f18_in, U3, U4, f17_in, f38_in, f13_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F18_IN, F36_IN, U7', F1_IN, F13_IN, U5'

Compound Symbols:

c6, c7, c15, c16, c

(5) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 8 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1, f(z0)) → f1_out1
U2(f13_out1, f(z0)) → f1_out1
f18_in → f18_out1
f18_in → U3(f36_in)
f18_in → U4(f36_in)
U3(f36_out1) → f18_out1
U4(f36_out1) → f18_out1
f17_in(z0) → f17_out1
f38_in → f38_out1
f13_in(z0) → U5(f17_in(z0), z0)
U5(f17_out1, z0) → U6(f18_in, z0)
U6(f18_out1, z0) → f13_out1
f36_in → U7(f38_in)
U7(f38_out1) → U8(f18_in)
U8(f18_out1) → f36_out1

Tuples:

F1_IN(f(z0)) → c(F13_IN(z0))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
U5'(f17_out1, z0) → c(F18_IN)
F18_IN → c6(F36_IN)
F18_IN → c7(F36_IN)
F36_IN → c15(U7'(f38_in))
U7'(f38_out1) → c16(F18_IN)
F1_IN(f(z0)) → c
F13_IN(z0) → c
U5'(f17_out1, z0) → c

S tuples:

F1_IN(f(z0)) → c(F13_IN(z0))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
U5'(f17_out1, z0) → c(F18_IN)
F18_IN → c6(F36_IN)
F18_IN → c7(F36_IN)
F36_IN → c15(U7'(f38_in))
U7'(f38_out1) → c16(F18_IN)
F1_IN(f(z0)) → c
F13_IN(z0) → c
U5'(f17_out1, z0) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f18_in, U3, U4, f17_in, f38_in, f13_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F1_IN, F13_IN, U5', F18_IN, F36_IN, U7'

Compound Symbols:

c, c6, c7, c15, c16, c

(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(f(z0)) → c(F13_IN(z0))
F1_IN(f(z0)) → c(F13_IN(z0))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
U5'(f17_out1, z0) → c(F18_IN)
F1_IN(f(z0)) → c
F1_IN(f(z0)) → c
F13_IN(z0) → c
U5'(f17_out1, z0) → c
F13_IN(z0) → c(U5'(f17_in(z0), z0))
F13_IN(z0) → c
U5'(f17_out1, z0) → c(F18_IN)
U5'(f17_out1, z0) → c

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1, f(z0)) → f1_out1
U2(f13_out1, f(z0)) → f1_out1
f18_in → f18_out1
f18_in → U3(f36_in)
f18_in → U4(f36_in)
U3(f36_out1) → f18_out1
U4(f36_out1) → f18_out1
f17_in(z0) → f17_out1
f38_in → f38_out1
f13_in(z0) → U5(f17_in(z0), z0)
U5(f17_out1, z0) → U6(f18_in, z0)
U6(f18_out1, z0) → f13_out1
f36_in → U7(f38_in)
U7(f38_out1) → U8(f18_in)
U8(f18_out1) → f36_out1

Tuples:

F1_IN(f(z0)) → c(F13_IN(z0))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
U5'(f17_out1, z0) → c(F18_IN)
F18_IN → c6(F36_IN)
F18_IN → c7(F36_IN)
F36_IN → c15(U7'(f38_in))
U7'(f38_out1) → c16(F18_IN)
F1_IN(f(z0)) → c
F13_IN(z0) → c
U5'(f17_out1, z0) → c

S tuples:

F18_IN → c6(F36_IN)
F18_IN → c7(F36_IN)
F36_IN → c15(U7'(f38_in))
U7'(f38_out1) → c16(F18_IN)

K tuples:

F1_IN(f(z0)) → c(F13_IN(z0))
F13_IN(z0) → c(U5'(f17_in(z0), z0))
U5'(f17_out1, z0) → c(F18_IN)
F1_IN(f(z0)) → c
F13_IN(z0) → c
U5'(f17_out1, z0) → c

Defined Rule Symbols:

f1_in, U1, U2, f18_in, U3, U4, f17_in, f38_in, f13_in, U5, U6, f36_in, U7, U8

Defined Pair Symbols:

F1_IN, F13_IN, U5', F18_IN, F36_IN, U7'

Compound Symbols:

c, c6, c7, c15, c16, c

(9) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX2 is free\np(T1, T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(f(T7), f(X7)), p(X7, g(X8)))\n\nT7 is ground\nX7 is free\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(f(T7), f(X7)), p(X7, g(X8))), SCOPE: 2, CLAUSE: 0 | ','(p(f(T7), f(X7)), p(X7, g(X8))), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX7 is free\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(T15, g(X8)) | ','(p(f(T15), f(X7)), p(X7, g(X8))), SCOPE: 2, CLAUSE: 1\n\nT15 is ground\nX7 is free\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(T15, g(X8))\n\nT15 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(f(T15), f(X7)), p(X7, g(X8))), SCOPE: 2, CLAUSE: 1\n\nT15 is ground\nX7 is free\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: p(T15, g(X8)), SCOPE: 3, CLAUSE: 0 | p(T15, g(X8)), SCOPE: 3, CLAUSE: 1\n\nT15 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true | p(g(T29), g(X8)), SCOPE: 3, CLAUSE: 1\n\nT29 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(T15, g(X8)), SCOPE: 3, CLAUSE: 1\n\nT15 is ground\nX8 is free\nX20 is free\np(T15, g(X8)) !=? p(X20, X20)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(g(T29), g(X8)), SCOPE: 3, CLAUSE: 1\n\nT29 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(f(T32), f(X33)), p(X33, g(X34)))\n\nT32 is ground\nX33 is free\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(p(f(T37), f(X46)), p(X46, g(X47)))\n\nT37 is ground\nX2 is free\nX46 is free\nX47 is free\np(f(T37), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(f(T37), f(X46)), p(X46, g(X47))), SCOPE: 4, CLAUSE: 0 | ','(p(f(T37), f(X46)), p(X46, g(X47))), SCOPE: 4, CLAUSE: 1\n\nT37 is ground\nX2 is free\nX46 is free\nX47 is free\np(f(T37), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: p(T45, g(X47)) | ','(p(f(T45), f(X46)), p(X46, g(X47))), SCOPE: 4, CLAUSE: 1\n\nT45 is ground\nX2 is free\nX46 is free\nX47 is free\np(f(T45), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: p(T45, g(X47))\n\nT45 is ground\nX2 is free\nX47 is free\np(f(T45), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(p(f(T45), f(X46)), p(X46, g(X47))), SCOPE: 4, CLAUSE: 1\n\nT45 is ground\nX2 is free\nX46 is free\nX47 is free\np(f(T45), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="EVAL with clause\np(X2, X2).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 6;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 56 [arrowhead = none ];
56 -> 8;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 57 [arrowhead = none ];
57 -> 10;

58 [label="EVAL with clause\np(f(X44), g(X45)) :- ','(p(f(X44), f(X46)), p(X46, g(X47))).\nand substitutionX44 -> T37,\nT1 -> f(T37),\nX45 -> T38,\nT2 -> g(T38)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 58 [arrowhead = none ];
58 -> 42;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 59 [arrowhead = none ];
59 -> 45;

60 [label="EVAL with clause\np(f(X5), g(X6)) :- ','(p(f(X5), f(X7)), p(X7, g(X8))).\nand substitutionX5 -> T7,\nT4 -> f(T7),\nX6 -> T8,\nT2 -> g(T8)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 60 [arrowhead = none ];
60 -> 11;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 61 [arrowhead = none ];
61 -> 12;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 62 [arrowhead = none ];
62 -> 15;

63 [label="ONLY EVAL with clause\np(X13, X13).\nand substitutionT7 -> T15,\nX13 -> f(T15),\nX7 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 63 [arrowhead = none ];
63 -> 16;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 64 [arrowhead = none ];
64 -> 19;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 66 [arrowhead = none ];
66 -> 25;

67 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 67 [arrowhead = none ];
67 -> 40;

68 [label="EVAL with clause\np(X20, X20).\nand substitutionT15 -> g(T29),\nX20 -> g(T29),\nX8 -> T29,\nT28 -> g(T29)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 68 [arrowhead = none ];
68 -> 27;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 69 [arrowhead = none ];
69 -> 28;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 70 [arrowhead = none ];
70 -> 29;

71 [label="EVAL with clause\np(f(X31), g(X32)) :- ','(p(f(X31), f(X33)), p(X33, g(X34))).\nand substitutionX31 -> T32,\nT15 -> f(T32),\nX8 -> X35,\nX32 -> X35", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 71 [arrowhead = none ];
71 -> 34;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 72 [arrowhead = none ];
72 -> 35;

73 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 73 [arrowhead = none ];
73 -> 30;

74 [label="INSTANCE with matching:\nT7 -> T32\nX7 -> X33\nX8 -> X34", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 74 [arrowhead = none , style=dashed];
74 -> 11[style=dashed];

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 75 [arrowhead = none ];
75 -> 47;

76 [label="ONLY EVAL with clause\np(X52, X52).\nand substitutionT37 -> T45,\nX52 -> f(T45),\nX46 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 76 [arrowhead = none ];
76 -> 48;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 77 [arrowhead = none ];
77 -> 49;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 78 [arrowhead = none ];
78 -> 50;

79 [label="INSTANCE with matching:\nT15 -> T45\nX8 -> X47", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 79 [arrowhead = none , style=dashed];
79 -> 19[style=dashed];

80 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 80 [arrowhead = none ];
80 -> 53;

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1
f2_in(f(z0)) → U1(f11_in(z0), f(z0))
f2_in(f(z0)) → U2(f48_in(z0), f(z0))
U1(f11_out1, f(z0)) → f2_out1
U2(f48_out1, f(z0)) → f2_out1
U2(f48_out2, f(z0)) → f2_out1
f19_in(g(z0)) → f19_out1
f19_in(f(z0)) → U3(f11_in(z0), f(z0))
U3(f11_out1, f(z0)) → f19_out1
f11_in(z0) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f11_out1
U4(f16_out2, z0) → f11_out1
f16_in(z0) → U5(f19_in(z0), f21_in(z0), z0)
U5(f19_out1, z0, z1) → f16_out1
U5(z0, f21_out1, z1) → f16_out2
f48_in(z0) → U6(f19_in(z0), f50_in(z0), z0)
U6(f19_out1, z0, z1) → f48_out1
U6(z0, f50_out1, z1) → f48_out2

Tuples:

F2_IN(f(z0)) → c1(U1'(f11_in(z0), f(z0)), F11_IN(z0))
F2_IN(f(z0)) → c2(U2'(f48_in(z0), f(z0)), F48_IN(z0))
F19_IN(f(z0)) → c7(U3'(f11_in(z0), f(z0)), F11_IN(z0))
F11_IN(z0) → c9(U4'(f16_in(z0), z0), F16_IN(z0))
F16_IN(z0) → c12(U5'(f19_in(z0), f21_in(z0), z0), F19_IN(z0))
F48_IN(z0) → c15(U6'(f19_in(z0), f50_in(z0), z0), F19_IN(z0))

S tuples:

F2_IN(f(z0)) → c1(U1'(f11_in(z0), f(z0)), F11_IN(z0))
F2_IN(f(z0)) → c2(U2'(f48_in(z0), f(z0)), F48_IN(z0))
F19_IN(f(z0)) → c7(U3'(f11_in(z0), f(z0)), F11_IN(z0))
F11_IN(z0) → c9(U4'(f16_in(z0), z0), F16_IN(z0))
F16_IN(z0) → c12(U5'(f19_in(z0), f21_in(z0), z0), F19_IN(z0))
F48_IN(z0) → c15(U6'(f19_in(z0), f50_in(z0), z0), F19_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f19_in, U3, f11_in, U4, f16_in, U5, f48_in, U6

Defined Pair Symbols:

F2_IN, F19_IN, F11_IN, F16_IN, F48_IN

Compound Symbols:

c1, c2, c7, c9, c12, c15

(11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1
f2_in(f(z0)) → U1(f11_in(z0), f(z0))
f2_in(f(z0)) → U2(f48_in(z0), f(z0))
U1(f11_out1, f(z0)) → f2_out1
U2(f48_out1, f(z0)) → f2_out1
U2(f48_out2, f(z0)) → f2_out1
f19_in(g(z0)) → f19_out1
f19_in(f(z0)) → U3(f11_in(z0), f(z0))
U3(f11_out1, f(z0)) → f19_out1
f11_in(z0) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f11_out1
U4(f16_out2, z0) → f11_out1
f16_in(z0) → U5(f19_in(z0), f21_in(z0), z0)
U5(f19_out1, z0, z1) → f16_out1
U5(z0, f21_out1, z1) → f16_out2
f48_in(z0) → U6(f19_in(z0), f50_in(z0), z0)
U6(f19_out1, z0, z1) → f48_out1
U6(z0, f50_out1, z1) → f48_out2

Tuples:

F19_IN(f(z0)) → c7(U3'(f11_in(z0), f(z0)), F11_IN(z0))
F11_IN(z0) → c9(U4'(f16_in(z0), z0), F16_IN(z0))
F16_IN(z0) → c12(U5'(f19_in(z0), f21_in(z0), z0), F19_IN(z0))
F2_IN(f(z0)) → c(U1'(f11_in(z0), f(z0)))
F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(U2'(f48_in(z0), f(z0)))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(U6'(f19_in(z0), f50_in(z0), z0))
F48_IN(z0) → c(F19_IN(z0))

S tuples:

F19_IN(f(z0)) → c7(U3'(f11_in(z0), f(z0)), F11_IN(z0))
F11_IN(z0) → c9(U4'(f16_in(z0), z0), F16_IN(z0))
F16_IN(z0) → c12(U5'(f19_in(z0), f21_in(z0), z0), F19_IN(z0))
F2_IN(f(z0)) → c(U1'(f11_in(z0), f(z0)))
F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(U2'(f48_in(z0), f(z0)))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(U6'(f19_in(z0), f50_in(z0), z0))
F48_IN(z0) → c(F19_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f19_in, U3, f11_in, U4, f16_in, U5, f48_in, U6

Defined Pair Symbols:

F19_IN, F11_IN, F16_IN, F2_IN, F48_IN

Compound Symbols:

c7, c9, c12, c

(13) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1
f2_in(f(z0)) → U1(f11_in(z0), f(z0))
f2_in(f(z0)) → U2(f48_in(z0), f(z0))
U1(f11_out1, f(z0)) → f2_out1
U2(f48_out1, f(z0)) → f2_out1
U2(f48_out2, f(z0)) → f2_out1
f19_in(g(z0)) → f19_out1
f19_in(f(z0)) → U3(f11_in(z0), f(z0))
U3(f11_out1, f(z0)) → f19_out1
f11_in(z0) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f11_out1
U4(f16_out2, z0) → f11_out1
f16_in(z0) → U5(f19_in(z0), f21_in(z0), z0)
U5(f19_out1, z0, z1) → f16_out1
U5(z0, f21_out1, z1) → f16_out2
f48_in(z0) → U6(f19_in(z0), f50_in(z0), z0)
U6(f19_out1, z0, z1) → f48_out1
U6(z0, f50_out1, z1) → f48_out2

Tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c

S tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f19_in, U3, f11_in, U4, f16_in, U5, f48_in, U6

Defined Pair Symbols:

F2_IN, F48_IN, F19_IN, F11_IN, F16_IN

Compound Symbols:

c, c7, c9, c12, c

(15) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F2_IN(f(z0)) → c
F2_IN(f(z0)) → c
F48_IN(z0) → c
F48_IN(z0) → c(F19_IN(z0))
F48_IN(z0) → c

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1
f2_in(f(z0)) → U1(f11_in(z0), f(z0))
f2_in(f(z0)) → U2(f48_in(z0), f(z0))
U1(f11_out1, f(z0)) → f2_out1
U2(f48_out1, f(z0)) → f2_out1
U2(f48_out2, f(z0)) → f2_out1
f19_in(g(z0)) → f19_out1
f19_in(f(z0)) → U3(f11_in(z0), f(z0))
U3(f11_out1, f(z0)) → f19_out1
f11_in(z0) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f11_out1
U4(f16_out2, z0) → f11_out1
f16_in(z0) → U5(f19_in(z0), f21_in(z0), z0)
U5(f19_out1, z0, z1) → f16_out1
U5(z0, f21_out1, z1) → f16_out2
f48_in(z0) → U6(f19_in(z0), f50_in(z0), z0)
U6(f19_out1, z0, z1) → f48_out1
U6(z0, f50_out1, z1) → f48_out2

Tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c

S tuples:

F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))

K tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c

Defined Rule Symbols:

f2_in, U1, U2, f19_in, U3, f11_in, U4, f16_in, U5, f48_in, U6

Defined Pair Symbols:

F2_IN, F48_IN, F19_IN, F11_IN, F16_IN

Compound Symbols:

c, c7, c9, c12, c

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F11_IN(z0) → c9(F16_IN(z0))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F11_IN(x₁)) = [2] + [2]x₁   
POL(F16_IN(x₁)) = [2]x₁   
POL(F19_IN(x₁)) = [2]x₁   
POL(F2_IN(x₁)) = [3]x₁   
POL(F48_IN(x₁)) = [3] + [3]x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [1] + x₁   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1
f2_in(f(z0)) → U1(f11_in(z0), f(z0))
f2_in(f(z0)) → U2(f48_in(z0), f(z0))
U1(f11_out1, f(z0)) → f2_out1
U2(f48_out1, f(z0)) → f2_out1
U2(f48_out2, f(z0)) → f2_out1
f19_in(g(z0)) → f19_out1
f19_in(f(z0)) → U3(f11_in(z0), f(z0))
U3(f11_out1, f(z0)) → f19_out1
f11_in(z0) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f11_out1
U4(f16_out2, z0) → f11_out1
f16_in(z0) → U5(f19_in(z0), f21_in(z0), z0)
U5(f19_out1, z0, z1) → f16_out1
U5(z0, f21_out1, z1) → f16_out2
f48_in(z0) → U6(f19_in(z0), f50_in(z0), z0)
U6(f19_out1, z0, z1) → f48_out1
U6(z0, f50_out1, z1) → f48_out2

Tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c

S tuples:

F19_IN(f(z0)) → c7(F11_IN(z0))
F16_IN(z0) → c12(F19_IN(z0))

K tuples:

F2_IN(f(z0)) → c(F11_IN(z0))
F2_IN(f(z0)) → c(F48_IN(z0))
F48_IN(z0) → c(F19_IN(z0))
F2_IN(f(z0)) → c
F48_IN(z0) → c
F11_IN(z0) → c9(F16_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, f19_in, U3, f11_in, U4, f16_in, U5, f48_in, U6

Defined Pair Symbols:

F2_IN, F48_IN, F19_IN, F11_IN, F16_IN

Compound Symbols:

c, c7, c9, c12, c

(19) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F16_IN(z0) → c12(F19_IN(z0))
F19_IN(f(z0)) → c7(F11_IN(z0))
F11_IN(z0) → c9(F16_IN(z0))

Now S is empty

(20) BOUNDS(O(1), O(1))