Runtime Complexity proof of /tmp/tmp1G8G4O/log.pl

Runtime Complexity of the query pattern
log(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^2)).

0 Prolog

↳1 LPReorderTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 69 ms)

↳4 CdtProblem

    ↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

    ↳7 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳8 CdtProblem

    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 240 ms)

    ↳10 CdtProblem

    ↳11 CdtKnowledgeProof (⇔, 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 122 ms)

    ↳14 CdtProblem

    ↳15 SIsEmptyProof (⇔, 0 ms)

    ↳16 BOUNDS(O(1), O(1))

↳17 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 60 ms)

↳18 CdtProblem

    ↳19 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳20 CdtProblem

    ↳21 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳22 CdtProblem

    ↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 270 ms)

    ↳24 CdtProblem

    ↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 60 ms)

    ↳26 CdtProblem

    ↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 30 ms)

    ↳28 CdtProblem

    ↳29 CdtKnowledgeProof (⇔, 0 ms)

    ↳30 CdtProblem

(0) Obligation:

Clauses:

half(0, 0).
half(s(0), 0).
half(s(s(X)), s(Y)) :- half(X, Y).
log(0, s(0)).
log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y)).

Query: log(g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

half(0, 0).
half(s(0), 0).
log(0, s(0)).
half(s(s(X)), s(Y)) :- half(X, Y).
log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y)).

Query: log(g,a)

(3) 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: log(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: log(T1, T2), SCOPE: 1, CLAUSE: 2 | log(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | log(0, T2), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: log(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nlog(T1, T2) !=? log(0, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: log(0, T2), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(half(s(T7), X10), log(X10, T9))\n\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: half(s(T7), X10)\n\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: log(T10, T9)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: half(s(T7), X10), SCOPE: 2, CLAUSE: 0 | half(s(T7), X10), SCOPE: 2, CLAUSE: 1 | half(s(T7), X10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: half(s(T7), X10), SCOPE: 2, CLAUSE: 1 | half(s(T7), X10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true | half(s(0), X10), SCOPE: 2, CLAUSE: 3\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: half(s(T7), X10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX10 is free\nhalf(s(T7), X10) !=? half(s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: half(s(0), X10), SCOPE: 2, CLAUSE: 3\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: half(T13, X21)\n\nT13 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: half(T13, X21), SCOPE: 3, CLAUSE: 0 | half(T13, X21), SCOPE: 3, CLAUSE: 1 | half(T13, X21), SCOPE: 3, CLAUSE: 3\n\nT13 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | half(0, X21), SCOPE: 3, CLAUSE: 1 | half(0, X21), SCOPE: 3, CLAUSE: 3\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: half(T13, X21), SCOPE: 3, CLAUSE: 1 | half(T13, X21), SCOPE: 3, CLAUSE: 3\n\nT13 is ground\nX21 is free\nhalf(T13, X21) !=? half(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: half(0, X21), SCOPE: 3, CLAUSE: 1 | half(0, X21), SCOPE: 3, CLAUSE: 3\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: half(0, X21), SCOPE: 3, CLAUSE: 3\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true | half(s(0), X21), SCOPE: 3, CLAUSE: 3\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: half(T13, X21), SCOPE: 3, CLAUSE: 3\n\nT13 is ground\nX21 is free\nhalf(T13, X21) !=? half(0, 0)\nhalf(T13, X21) !=? half(s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: half(s(0), X21), SCOPE: 3, CLAUSE: 3\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: half(T16, X34)\n\nT16 is ground\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 68 [arrowhead = none ];
68 -> 4;

69 [label="EVAL with clause\nlog(0, s(0)).\nand substitutionT1 -> 0,\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 69 [arrowhead = none ];
69 -> 5;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 70 [arrowhead = none ];
70 -> 7;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 71 [arrowhead = none ];
71 -> 9;

72 [label="EVAL with clause\nlog(s(X8), s(X9)) :- ','(half(s(X8), X10), log(X10, X9)).\nand substitutionX8 -> T7,\nT1 -> s(T7),\nX9 -> T9,\nT2 -> s(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 72 [arrowhead = none ];
72 -> 13;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="BACKTRACK\nfor clause: log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 74 [arrowhead = none ];
74 -> 11;

75 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 75 [arrowhead = none ];
75 -> 21;

76 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nT10 is ground\nreplacements:X10 -> T10", fontsize=14, style = filled, fillcolor = violet];
13 -> 76 [arrowhead = none ];
76 -> 22;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 77 [arrowhead = none ];
77 -> 26;

78 [label="INSTANCE with matching:\nT1 -> T10\nT2 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 78 [arrowhead = none , style=dashed];
78 -> 1[style=dashed];

79 [label="BACKTRACK\nfor clause: half(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 79 [arrowhead = none ];
79 -> 28;

80 [label="EVAL with clause\nhalf(s(0), 0).\nand substitutionT7 -> 0,\nX10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 80 [arrowhead = none ];
80 -> 32;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 81 [arrowhead = none ];
81 -> 34;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 82 [arrowhead = none ];
82 -> 36;

83 [label="EVAL with clause\nhalf(s(s(X19)), s(X20)) :- half(X19, X20).\nand substitutionX19 -> T13,\nT7 -> s(T13),\nX20 -> X21,\nX10 -> s(X21)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 83 [arrowhead = none ];
83 -> 39;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 84 [arrowhead = none ];
84 -> 41;

85 [label="BACKTRACK\nfor clause: half(s(s(X)), s(Y)) :- half(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 85 [arrowhead = none ];
85 -> 37;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 86 [arrowhead = none ];
86 -> 53;

87 [label="EVAL with clause\nhalf(0, 0).\nand substitutionT13 -> 0,\nX21 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 87 [arrowhead = none ];
87 -> 55;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 88 [arrowhead = none ];
88 -> 56;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 89 [arrowhead = none ];
89 -> 57;

90 [label="EVAL with clause\nhalf(s(0), 0).\nand substitutionT13 -> s(0),\nX21 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 90 [arrowhead = none ];
90 -> 60;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 91 [arrowhead = none ];
91 -> 62;

92 [label="BACKTRACK\nfor clause: half(s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 92 [arrowhead = none ];
92 -> 58;

93 [label="BACKTRACK\nfor clause: half(s(s(X)), s(Y)) :- half(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 93 [arrowhead = none ];
93 -> 59;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 94 [arrowhead = none ];
94 -> 63;

95 [label="EVAL with clause\nhalf(s(s(X32)), s(X33)) :- half(X32, X33).\nand substitutionX32 -> T16,\nT13 -> s(s(T16)),\nX33 -> X34,\nX21 -> s(X34)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 95 [arrowhead = none ];
95 -> 66;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 96 [arrowhead = none ];
96 -> 67;

97 [label="BACKTRACK\nfor clause: half(s(s(X)), s(Y)) :- half(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 97 [arrowhead = none ];
97 -> 65;

98 [label="INSTANCE with matching:\nT13 -> T16\nX21 -> X34", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 98 [arrowhead = none , style=dashed];
98 -> 39[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1(s(0))
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
U1(f13_out1(z0, z1), s(z2)) → f1_out1(s(z1))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))
f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
f13_in(z0) → U4(f21_in(z0), z0)
U4(f21_out1(z0), z1) → U5(f1_in(z0), z1, z0)
U5(f1_out1(z0), z1, z2) → f13_out1(z2, z0)

Tuples:

F1_IN(s(z0)) → c1(U1'(f13_in(z0), s(z0)), F13_IN(z0))
F39_IN(s(s(z0))) → c5(U2'(f39_in(z0), s(s(z0))), F39_IN(z0))
F21_IN(s(z0)) → c8(U3'(f39_in(z0), s(z0)), F39_IN(z0))
F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
U4'(f21_out1(z0), z1) → c11(U5'(f1_in(z0), z1, z0), F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(U1'(f13_in(z0), s(z0)), F13_IN(z0))
F39_IN(s(s(z0))) → c5(U2'(f39_in(z0), s(s(z0))), F39_IN(z0))
F21_IN(s(z0)) → c8(U3'(f39_in(z0), s(z0)), F39_IN(z0))
F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
U4'(f21_out1(z0), z1) → c11(U5'(f1_in(z0), z1, z0), F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f39_in, U2, f21_in, U3, f13_in, U4, U5

Defined Pair Symbols:

F1_IN, F39_IN, F21_IN, F13_IN, U4'

Compound Symbols:

c1, c5, c8, c10, c11

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1(s(0))
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
U1(f13_out1(z0, z1), s(z2)) → f1_out1(s(z1))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))
f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
f13_in(z0) → U4(f21_in(z0), z0)
U4(f21_out1(z0), z1) → U5(f1_in(z0), z1, z0)
U5(f1_out1(z0), z1, z2) → f13_out1(z2, z0)

Tuples:

F1_IN(s(z0)) → c1(U1'(f13_in(z0), s(z0)), F13_IN(z0))
F39_IN(s(s(z0))) → c5(U2'(f39_in(z0), s(s(z0))), F39_IN(z0))
F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
U4'(f21_out1(z0), z1) → c11(U5'(f1_in(z0), z1, z0), F1_IN(z0))
F21_IN(s(z0)) → c(U3'(f39_in(z0), s(z0)))
F21_IN(s(z0)) → c(F39_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(U1'(f13_in(z0), s(z0)), F13_IN(z0))
F39_IN(s(s(z0))) → c5(U2'(f39_in(z0), s(s(z0))), F39_IN(z0))
F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
U4'(f21_out1(z0), z1) → c11(U5'(f1_in(z0), z1, z0), F1_IN(z0))
F21_IN(s(z0)) → c(U3'(f39_in(z0), s(z0)))
F21_IN(s(z0)) → c(F39_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f39_in, U2, f21_in, U3, f13_in, U4, U5

Defined Pair Symbols:

F1_IN, F39_IN, F13_IN, U4', F21_IN

Compound Symbols:

c1, c5, c10, c11, c

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1(s(0))
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
U1(f13_out1(z0, z1), s(z2)) → f1_out1(s(z1))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))
f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
f13_in(z0) → U4(f21_in(z0), z0)
U4(f21_out1(z0), z1) → U5(f1_in(z0), z1, z0)
U5(f1_out1(z0), z1, z2) → f13_out1(z2, z0)

Tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

S tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f39_in, U2, f21_in, U3, f13_in, U4, U5

Defined Pair Symbols:

F13_IN, F21_IN, F1_IN, F39_IN, U4'

Compound Symbols:

c10, c, c1, c5, c11, c

(9) 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.

F1_IN(s(z0)) → c1(F13_IN(z0))

We considered the (Usable) Rules:

f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))

And the Tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

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

POL(0) = 0
POL(F13_IN(x₁)) = x₁
POL(F1_IN(x₁)) = x₁
POL(F21_IN(x₁)) = 0
POL(F39_IN(x₁)) = 0
POL(U2(x₁, x₂)) = [1] + x₁
POL(U3(x₁, x₂)) = [1] + x₁
POL(U4'(x₁, x₂)) = x₁
POL(c) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(f21_in(x₁)) = x₁
POL(f21_out1(x₁)) = x₁
POL(f39_in(x₁)) = x₁
POL(f39_out1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1(s(0))
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
U1(f13_out1(z0, z1), s(z2)) → f1_out1(s(z1))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))
f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
f13_in(z0) → U4(f21_in(z0), z0)
U4(f21_out1(z0), z1) → U5(f1_in(z0), z1, z0)
U5(f1_out1(z0), z1, z2) → f13_out1(z2, z0)

Tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

S tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

K tuples:

F1_IN(s(z0)) → c1(F13_IN(z0))

Defined Rule Symbols:

f1_in, U1, f39_in, U2, f21_in, U3, f13_in, U4, U5

Defined Pair Symbols:

F13_IN, F21_IN, F1_IN, F39_IN, U4'

Compound Symbols:

c10, c, c1, c5, c11, c

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c
F21_IN(s(z0)) → c(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c
F1_IN(s(z0)) → c1(F13_IN(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1(s(0))
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
U1(f13_out1(z0, z1), s(z2)) → f1_out1(s(z1))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))
f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
f13_in(z0) → U4(f21_in(z0), z0)
U4(f21_out1(z0), z1) → U5(f1_in(z0), z1, z0)
U5(f1_out1(z0), z1, z2) → f13_out1(z2, z0)

Tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

S tuples:

F39_IN(s(s(z0))) → c5(F39_IN(z0))

K tuples:

F1_IN(s(z0)) → c1(F13_IN(z0))
F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

Defined Rule Symbols:

f1_in, U1, f39_in, U2, f21_in, U3, f13_in, U4, U5

Defined Pair Symbols:

F13_IN, F21_IN, F1_IN, F39_IN, U4'

Compound Symbols:

c10, c, c1, c5, c11, c

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F39_IN(s(s(z0))) → c5(F39_IN(z0))

We considered the (Usable) Rules:

f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))

And the Tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

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

POL(0) = 0
POL(F13_IN(x₁)) = [1] + x₁ + x₁²
POL(F1_IN(x₁)) = [1] + x₁²
POL(F21_IN(x₁)) = x₁
POL(F39_IN(x₁)) = x₁
POL(U2(x₁, x₂)) = [1] + x₁
POL(U3(x₁, x₂)) = [1] + x₁
POL(U4'(x₁, x₂)) = [1] + x₁²
POL(c) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(f21_in(x₁)) = x₁
POL(f21_out1(x₁)) = x₁
POL(f39_in(x₁)) = x₁
POL(f39_out1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1(s(0))
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
U1(f13_out1(z0, z1), s(z2)) → f1_out1(s(z1))
f39_in(0) → f39_out1(0)
f39_in(s(0)) → f39_out1(0)
f39_in(s(s(z0))) → U2(f39_in(z0), s(s(z0)))
U2(f39_out1(z0), s(s(z1))) → f39_out1(s(z0))
f21_in(0) → f21_out1(0)
f21_in(s(z0)) → U3(f39_in(z0), s(z0))
U3(f39_out1(z0), s(z1)) → f21_out1(s(z0))
f13_in(z0) → U4(f21_in(z0), z0)
U4(f21_out1(z0), z1) → U5(f1_in(z0), z1, z0)
U5(f1_out1(z0), z1, z2) → f13_out1(z2, z0)

Tuples:

F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
F1_IN(s(z0)) → c1(F13_IN(z0))
F39_IN(s(s(z0))) → c5(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c

S tuples:none
K tuples:

F1_IN(s(z0)) → c1(F13_IN(z0))
F13_IN(z0) → c10(U4'(f21_in(z0), z0), F21_IN(z0))
F21_IN(s(z0)) → c(F39_IN(z0))
U4'(f21_out1(z0), z1) → c11(F1_IN(z0))
F21_IN(s(z0)) → c
F39_IN(s(s(z0))) → c5(F39_IN(z0))

Defined Rule Symbols:

f1_in, U1, f39_in, U2, f21_in, U3, f13_in, U4, U5

Defined Pair Symbols:

F13_IN, F21_IN, F1_IN, F39_IN, U4'

Compound Symbols:

c10, c, c1, c5, c11, c

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))

(17) 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: log(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: log(T1, T2), SCOPE: 1, CLAUSE: 2 | log(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | log(0, T2), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: log(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nlog(T1, T2) !=? log(0, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: log(0, T2), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(half(s(T5), X7), log(X7, T7))\n\nT5 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(half(s(T5), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 0 | ','(half(s(T5), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 1 | ','(half(s(T5), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(half(s(T5), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 1 | ','(half(s(T5), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: log(0, T7) | ','(half(s(0), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 3\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(half(s(T5), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX7 is free\nhalf(s(T5), X7) !=? half(s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: log(0, T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(half(s(0), X7), log(X7, T7)), SCOPE: 2, CLAUSE: 3\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: log(0, T7), SCOPE: 3, CLAUSE: 2 | log(0, T7), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true | log(0, T7), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: log(0, T7), SCOPE: 3, CLAUSE: 4\n\nlog(0, T7) !=? log(0, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: log(0, T7), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(half(T10, X22), log(s(X22), T7))\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: half(T10, X22)\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: log(s(T11), T7)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: half(T10, X22), SCOPE: 4, CLAUSE: 0 | half(T10, X22), SCOPE: 4, CLAUSE: 1 | half(T10, X22), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true | half(0, X22), SCOPE: 4, CLAUSE: 1 | half(0, X22), SCOPE: 4, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: half(T10, X22), SCOPE: 4, CLAUSE: 1 | half(T10, X22), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX22 is free\nhalf(T10, X22) !=? half(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: half(0, X22), SCOPE: 4, CLAUSE: 1 | half(0, X22), SCOPE: 4, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: half(0, X22), SCOPE: 4, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true | half(s(0), X22), SCOPE: 4, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: half(T10, X22), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX22 is free\nhalf(T10, X22) !=? half(0, 0)\nhalf(T10, X22) !=? half(s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: half(s(0), X22), SCOPE: 4, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: half(T14, X35)\n\nT14 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 65 [arrowhead = none ];
65 -> 3;

66 [label="EVAL with clause\nlog(0, s(0)).\nand substitutionT1 -> 0,\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 66 [arrowhead = none ];
66 -> 6;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 67 [arrowhead = none ];
67 -> 8;

68 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 68 [arrowhead = none ];
68 -> 10;

69 [label="EVAL with clause\nlog(s(X5), s(X6)) :- ','(half(s(X5), X7), log(X7, X6)).\nand substitutionX5 -> T5,\nT1 -> s(T5),\nX6 -> T7,\nT2 -> s(T7),\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 69 [arrowhead = none ];
69 -> 15;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 70 [arrowhead = none ];
70 -> 16;

71 [label="BACKTRACK\nfor clause: log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 71 [arrowhead = none ];
71 -> 12;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="BACKTRACK\nfor clause: half(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 73 [arrowhead = none ];
73 -> 18;

74 [label="EVAL with clause\nhalf(s(0), 0).\nand substitutionT5 -> 0,\nX7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 74 [arrowhead = none ];
74 -> 19;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 75 [arrowhead = none ];
75 -> 20;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 76 [arrowhead = none ];
76 -> 23;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 77 [arrowhead = none ];
77 -> 24;

78 [label="EVAL with clause\nhalf(s(s(X20)), s(X21)) :- half(X20, X21).\nand substitutionX20 -> T10,\nT5 -> s(T10),\nX21 -> X22,\nX7 -> s(X22)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 78 [arrowhead = none ];
78 -> 38;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 79 [arrowhead = none ];
79 -> 40;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 80 [arrowhead = none ];
80 -> 25;

81 [label="BACKTRACK\nfor clause: half(s(s(X)), s(Y)) :- half(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 81 [arrowhead = none ];
81 -> 35;

82 [label="EVAL with clause\nlog(0, s(0)).\nand substitutionT7 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 82 [arrowhead = none ];
82 -> 27;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="BACKTRACK\nfor clause: log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 85 [arrowhead = none ];
85 -> 33;

86 [label="BACKTRACK\nfor clause: log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 86 [arrowhead = none ];
86 -> 31;

87 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
38 -> 87 [arrowhead = none ];
87 -> 42;

88 [label="SPLIT 2\nnew knowledge:\nT10 is ground\nT11 is ground\nreplacements:X22 -> T11", fontsize=14, style = filled, fillcolor = violet];
38 -> 88 [arrowhead = none ];
88 -> 43;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 89 [arrowhead = none ];
89 -> 44;

90 [label="INSTANCE with matching:\nT1 -> s(T11)\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 90 [arrowhead = none , style=dashed];
90 -> 2[style=dashed];

91 [label="EVAL with clause\nhalf(0, 0).\nand substitutionT10 -> 0,\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 91 [arrowhead = none ];
91 -> 45;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 92 [arrowhead = none ];
92 -> 46;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 93 [arrowhead = none ];
93 -> 47;

94 [label="EVAL with clause\nhalf(s(0), 0).\nand substitutionT10 -> s(0),\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 94 [arrowhead = none ];
94 -> 50;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 95 [arrowhead = none ];
95 -> 51;

96 [label="BACKTRACK\nfor clause: half(s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 96 [arrowhead = none ];
96 -> 48;

97 [label="BACKTRACK\nfor clause: half(s(s(X)), s(Y)) :- half(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 97 [arrowhead = none ];
97 -> 49;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 98 [arrowhead = none ];
98 -> 52;

99 [label="EVAL with clause\nhalf(s(s(X33)), s(X34)) :- half(X33, X34).\nand substitutionX33 -> T14,\nT10 -> s(s(T14)),\nX34 -> X35,\nX22 -> s(X35)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 99 [arrowhead = none ];
99 -> 61;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 100 [arrowhead = none ];
100 -> 64;

101 [label="BACKTRACK\nfor clause: half(s(s(X)), s(Y)) :- half(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 101 [arrowhead = none ];
101 -> 54;

102 [label="INSTANCE with matching:\nT10 -> T14\nX22 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 102 [arrowhead = none , style=dashed];
102 -> 42[style=dashed];

}

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F2_IN(s(0)) → c1(U1'(f19_in, s(0)), F19_IN)
F2_IN(s(s(z0))) → c2(U2'(f38_in(z0), s(s(z0))), F38_IN(z0))
F42_IN(s(s(z0))) → c8(U3'(f42_in(z0), s(s(z0))), F42_IN(z0))
F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(U5'(f2_in(s(z0)), z1, z0), F2_IN(s(z0)))
F19_IN → c14(U6'(f23_in, f24_in), F23_IN)

S tuples:

F2_IN(s(0)) → c1(U1'(f19_in, s(0)), F19_IN)
F2_IN(s(s(z0))) → c2(U2'(f38_in(z0), s(s(z0))), F38_IN(z0))
F42_IN(s(s(z0))) → c8(U3'(f42_in(z0), s(s(z0))), F42_IN(z0))
F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(U5'(f2_in(s(z0)), z1, z0), F2_IN(s(z0)))
F19_IN → c14(U6'(f23_in, f24_in), F23_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F2_IN, F42_IN, F38_IN, U4', F19_IN

Compound Symbols:

c1, c2, c8, c11, c12, c14

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

Split RHS of tuples not part of any SCC

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F2_IN(s(s(z0))) → c2(U2'(f38_in(z0), s(s(z0))), F38_IN(z0))
F42_IN(s(s(z0))) → c8(U3'(f42_in(z0), s(s(z0))), F42_IN(z0))
F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(U5'(f2_in(s(z0)), z1, z0), F2_IN(s(z0)))
F2_IN(s(0)) → c(U1'(f19_in, s(0)))
F2_IN(s(0)) → c(F19_IN)
F19_IN → c(U6'(f23_in, f24_in))
F19_IN → c(F23_IN)

S tuples:

F2_IN(s(s(z0))) → c2(U2'(f38_in(z0), s(s(z0))), F38_IN(z0))
F42_IN(s(s(z0))) → c8(U3'(f42_in(z0), s(s(z0))), F42_IN(z0))
F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(U5'(f2_in(s(z0)), z1, z0), F2_IN(s(z0)))
F2_IN(s(0)) → c(U1'(f19_in, s(0)))
F2_IN(s(0)) → c(F19_IN)
F19_IN → c(U6'(f23_in, f24_in))
F19_IN → c(F23_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F2_IN, F42_IN, F38_IN, U4', F19_IN

Compound Symbols:

c2, c8, c11, c12, c

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

Removed 6 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

S tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F38_IN, F2_IN, F42_IN, U4', F19_IN

Compound Symbols:

c11, c, c2, c8, c12, c

(23) 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.

F2_IN(s(0)) → c
F19_IN → c

We considered the (Usable) Rules:

f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))

And the Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

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

POL(0) = 0
POL(F19_IN) = [1]
POL(F2_IN(x₁)) = [1]
POL(F38_IN(x₁)) = [1]
POL(F42_IN(x₁)) = 0
POL(U3(x₁, x₂)) = 0
POL(U4'(x₁, x₂)) = [1]
POL(c) = 0
POL(c(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(f42_in(x₁)) = 0
POL(f42_out1(x₁)) = 0
POL(s(x₁)) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

S tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))

K tuples:

F2_IN(s(0)) → c
F19_IN → c

Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F38_IN, F2_IN, F42_IN, U4', F19_IN

Compound Symbols:

c11, c, c2, c8, c12, c

(25) 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.

F2_IN(s(0)) → c(F19_IN)

We considered the (Usable) Rules:

f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))

And the Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

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

POL(0) = 0
POL(F19_IN) = 0
POL(F2_IN(x₁)) = [1]
POL(F38_IN(x₁)) = [1]
POL(F42_IN(x₁)) = 0
POL(U3(x₁, x₂)) = 0
POL(U4'(x₁, x₂)) = [1]
POL(c) = 0
POL(c(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(f42_in(x₁)) = 0
POL(f42_out1(x₁)) = 0
POL(s(x₁)) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

S tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))

K tuples:

F2_IN(s(0)) → c
F19_IN → c
F2_IN(s(0)) → c(F19_IN)

Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F38_IN, F2_IN, F42_IN, U4', F19_IN

Compound Symbols:

c11, c, c2, c8, c12, c

(27) 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.

F2_IN(s(s(z0))) → c2(F38_IN(z0))

We considered the (Usable) Rules:

f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))

And the Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

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

POL(0) = 0
POL(F19_IN) = [2]
POL(F2_IN(x₁)) = [2]x₁
POL(F38_IN(x₁)) = [2] + [2]x₁
POL(F42_IN(x₁)) = 0
POL(U3(x₁, x₂)) = [1] + x₁
POL(U4'(x₁, x₂)) = [2] + [2]x₁
POL(c) = 0
POL(c(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(f42_in(x₁)) = x₁
POL(f42_out1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

S tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))

K tuples:

F2_IN(s(0)) → c
F19_IN → c
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F38_IN, F2_IN, F42_IN, U4', F19_IN

Compound Symbols:

c11, c, c2, c8, c12, c

(29) CdtKnowledgeProof (EQUIVALENT transformation)

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

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F2_IN(s(0)) → c

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1(s(0))
f2_in(s(0)) → U1(f19_in, s(0))
f2_in(s(s(z0))) → U2(f38_in(z0), s(s(z0)))
U1(f19_out1(z0), s(0)) → f2_out1(s(z0))
U1(f19_out2(z0, z1), s(0)) → f2_out1(s(z1))
U2(f38_out1(z0, z1), s(s(z2))) → f2_out1(s(z1))
f42_in(0) → f42_out1(0)
f42_in(s(0)) → f42_out1(0)
f42_in(s(s(z0))) → U3(f42_in(z0), s(s(z0)))
U3(f42_out1(z0), s(s(z1))) → f42_out1(s(z0))
f23_in → f23_out1(s(0))
f38_in(z0) → U4(f42_in(z0), z0)
U4(f42_out1(z0), z1) → U5(f2_in(s(z0)), z1, z0)
U5(f2_out1(z0), z1, z2) → f38_out1(z2, z0)
f19_in → U6(f23_in, f24_in)
U6(f23_out1(z0), z1) → f19_out1(z0)
U6(z0, f24_out1(z1, z2)) → f19_out2(z1, z2)

Tuples:

F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F42_IN(s(s(z0))) → c8(F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))
F2_IN(s(0)) → c
F19_IN → c

S tuples:

F42_IN(s(s(z0))) → c8(F42_IN(z0))

K tuples:

F2_IN(s(0)) → c
F19_IN → c
F2_IN(s(0)) → c(F19_IN)
F2_IN(s(s(z0))) → c2(F38_IN(z0))
F38_IN(z0) → c11(U4'(f42_in(z0), z0), F42_IN(z0))
U4'(f42_out1(z0), z1) → c12(F2_IN(s(z0)))

Defined Rule Symbols:

f2_in, U1, U2, f42_in, U3, f23_in, f38_in, U4, U5, f19_in, U6

Defined Pair Symbols:

F38_IN, F2_IN, F42_IN, U4', F19_IN

Compound Symbols:

c11, c, c2, c8, c12, c