WORST_CASE(?,O(n^1))
proof of input_QCitfDOhpv.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 97 ms]
(8) CdtProblem
(9) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 4 ms]
(12) CdtProblem
(13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms]
(14) CdtProblem
(15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
(16) CdtProblem
(17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) BOUNDS(1, 1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f_0(x) -> a
   f_1(x) -> g_1(x, x)
   g_1(s(x), y) -> b(f_0(y), g_1(x, y))
   f_2(x) -> g_2(x, x)
   g_2(s(x), y) -> b(f_1(y), g_2(x, y))
   f_3(x) -> g_3(x, x)
   g_3(s(x), y) -> b(f_2(y), g_3(x, y))
   f_4(x) -> g_4(x, x)
   g_4(s(x), y) -> b(f_3(y), g_4(x, y))
   f_5(x) -> g_5(x, x)
   g_5(s(x), y) -> b(f_4(y), g_5(x, y))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f_0(z0) -> a
   f_1(z0) -> g_1(z0, z0)
   g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1))
   f_2(z0) -> g_2(z0, z0)
   g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1))
   f_3(z0) -> g_3(z0, z0)
   g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1))
   f_4(z0) -> g_4(z0, z0)
   g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1))
   f_5(z0) -> g_5(z0, z0)
   g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1))
Tuples:
   F_0(z0) -> c
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c2(F_0(z1))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   F_0(z0) -> c
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c2(F_0(z1))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
K tuples:none
Defined Rule Symbols:   f_0_1, f_1_1, g_1_2, f_2_1, g_2_2, f_3_1, g_3_2, f_4_1, g_4_2, f_5_1, g_5_2

Defined Pair Symbols:   F_0_1, F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2

Compound Symbols:   c, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1


----------------------------------------

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F_5(z0) -> c13(G_5(z0, z0))
Removed 2 trailing nodes:
   G_1(s(z0), z1) -> c2(F_0(z1))
   F_0(z0) -> c

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f_0(z0) -> a
   f_1(z0) -> g_1(z0, z0)
   g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1))
   f_2(z0) -> g_2(z0, z0)
   g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1))
   f_3(z0) -> g_3(z0, z0)
   g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1))
   f_4(z0) -> g_4(z0, z0)
   g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1))
   f_5(z0) -> g_5(z0, z0)
   g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1))
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
K tuples:none
Defined Rule Symbols:   f_0_1, f_1_1, g_1_2, f_2_1, g_2_2, f_3_1, g_3_2, f_4_1, g_4_2, f_5_1, g_5_2

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


----------------------------------------

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f_0(z0) -> a
   f_1(z0) -> g_1(z0, z0)
   g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1))
   f_2(z0) -> g_2(z0, z0)
   g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1))
   f_3(z0) -> g_3(z0, z0)
   g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1))
   f_4(z0) -> g_4(z0, z0)
   g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1))
   f_5(z0) -> g_5(z0, z0)
   g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1))

----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


----------------------------------------

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F_1(x_1)) = 0
   POL(F_2(x_1)) = x_1
   POL(F_3(x_1)) = [1] + x_1
   POL(F_4(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = 0
   POL(G_2(x_1, x_2)) = x_1
   POL(G_3(x_1, x_2)) = x_2
   POL(G_4(x_1, x_2)) = [1] + x_2
   POL(G_5(x_1, x_2)) = [1] + x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
K tuples:
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


----------------------------------------

(9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F_1(z0) -> c1(G_1(z0, z0))
   F_4(z0) -> c10(G_4(z0, z0))

----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
K tuples:
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   F_4(z0) -> c10(G_4(z0, z0))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


----------------------------------------

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F_2(z0) -> c4(G_2(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F_1(x_1)) = 0
   POL(F_2(x_1)) = [1]
   POL(F_3(x_1)) = [1]
   POL(F_4(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = 0
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = [1]
   POL(G_4(x_1, x_2)) = [1] + x_1
   POL(G_5(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
K tuples:
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   F_4(z0) -> c10(G_4(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


----------------------------------------

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F_1(x_1)) = [1]
   POL(F_2(x_1)) = [1]
   POL(F_3(x_1)) = [1] + x_1
   POL(F_4(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = [1]
   POL(G_2(x_1, x_2)) = [1]
   POL(G_3(x_1, x_2)) = [1] + x_1
   POL(G_4(x_1, x_2)) = [1] + x_2
   POL(G_5(x_1, x_2)) = [1] + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
K tuples:
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   F_4(z0) -> c10(G_4(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


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(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F_1(x_1)) = [1] + x_1
   POL(F_2(x_1)) = [1] + x_1
   POL(F_3(x_1)) = [1] + x_1
   POL(F_4(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = [1] + x_1
   POL(G_2(x_1, x_2)) = [1] + x_2
   POL(G_3(x_1, x_2)) = [1] + x_2
   POL(G_4(x_1, x_2)) = [1] + x_2
   POL(G_5(x_1, x_2)) = [1] + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F_1(z0) -> c1(G_1(z0, z0))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
S tuples:none
K tuples:
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_3(z0) -> c7(G_3(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   F_4(z0) -> c10(G_4(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, G_5_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c14_1, c15_1


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(17) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(18)
BOUNDS(1, 1)