WORST_CASE(?,O(n^3))
proof of input_U6acDrFo5o.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^3).

(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^3)), 166 ms]
(8) CdtProblem
(9) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms]
(12) CdtProblem
(13) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 48 ms]
(16) CdtProblem
(17) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 39 ms]
(20) CdtProblem
(21) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
(23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 35 ms]
(24) CdtProblem
(25) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms]
(28) CdtProblem
(29) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 12 ms]
(30) CdtProblem
(31) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms]
(32) CdtProblem
(33) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms]
(34) CdtProblem
(35) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) BOUNDS(1, 1)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


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))
   f_6(x) -> g_6(x, x)
   g_6(s(x), y) -> b(f_5(y), g_6(x, y))
   f_7(x) -> g_7(x, x)
   g_7(s(x), y) -> b(f_6(y), g_7(x, y))
   f_8(x) -> g_8(x, x)
   g_8(s(x), y) -> b(f_7(y), g_8(x, y))
   f_9(x) -> g_9(x, x)
   g_9(s(x), y) -> b(f_8(y), g_9(x, y))
   f_10(x) -> g_10(x, x)
   g_10(s(x), y) -> b(f_9(y), g_10(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))
   f_6(z0) -> g_6(z0, z0)
   g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1))
   f_7(z0) -> g_7(z0, z0)
   g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1))
   f_8(z0) -> g_8(z0, z0)
   g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1))
   f_9(z0) -> g_9(z0, z0)
   g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1))
   f_10(z0) -> g_10(z0, z0)
   g_10(s(z0), z1) -> b(f_9(z1), g_10(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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   F_10(z0) -> c28(G_10(z0, z0))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   F_10(z0) -> c28(G_10(z0, z0))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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, f_6_1, g_6_2, f_7_1, g_7_2, f_8_1, g_8_2, f_9_1, g_9_2, f_10_1, g_10_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, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, F_10_1, G_10_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, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F_10(z0) -> c28(G_10(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))
   f_6(z0) -> g_6(z0, z0)
   g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1))
   f_7(z0) -> g_7(z0, z0)
   g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1))
   f_8(z0) -> g_8(z0, z0)
   g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1))
   f_9(z0) -> g_9(z0, z0)
   g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1))
   f_10(z0) -> g_10(z0, z0)
   g_10(s(z0), z1) -> b(f_9(z1), g_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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, f_6_1, g_6_2, f_7_1, g_7_2, f_8_1, g_8_2, f_9_1, g_9_2, f_10_1, g_10_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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_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))
   f_6(z0) -> g_6(z0, z0)
   g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1))
   f_7(z0) -> g_7(z0, z0)
   g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1))
   f_8(z0) -> g_8(z0, z0)
   g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1))
   f_9(z0) -> g_9(z0, z0)
   g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1))
   f_10(z0) -> g_10(z0, z0)
   g_10(s(z0), z1) -> b(f_9(z1), g_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = 0
   POL(F_3(x_1)) = 0
   POL(F_4(x_1)) = 0
   POL(F_5(x_1)) = 0
   POL(F_6(x_1)) = 0
   POL(F_7(x_1)) = 0
   POL(F_8(x_1)) = x_1 + x_1^2
   POL(F_9(x_1)) = [1] + x_1 + x_1^2 + x_1^3
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = x_1 + x_1*x_2 + x_1^2 + x_1^3 + x_1^2*x_2 + x_1*x_2^2 + x_2^3
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = 0
   POL(G_4(x_1, x_2)) = 0
   POL(G_5(x_1, x_2)) = 0
   POL(G_6(x_1, x_2)) = 0
   POL(G_7(x_1, x_2)) = 0
   POL(G_8(x_1, x_2)) = 0
   POL(G_9(x_1, x_2)) = x_2 + x_2^2 + x_1^3
   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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F_8(z0) -> c22(G_8(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_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_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = 0
   POL(F_3(x_1)) = 0
   POL(F_4(x_1)) = 0
   POL(F_5(x_1)) = 0
   POL(F_6(x_1)) = [2]
   POL(F_7(x_1)) = [2] + x_1
   POL(F_8(x_1)) = [3] + [3]x_1
   POL(F_9(x_1)) = [3] + [3]x_1
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = 0
   POL(G_4(x_1, x_2)) = 0
   POL(G_5(x_1, x_2)) = 0
   POL(G_6(x_1, x_2)) = [1]
   POL(G_7(x_1, x_2)) = [2] + x_1
   POL(G_8(x_1, x_2)) = [1] + [2]x_1 + x_2
   POL(G_9(x_1, x_2)) = [3] + [3]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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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)) = [2] + 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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))

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

(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = 0
   POL(F_3(x_1)) = 0
   POL(F_4(x_1)) = 0
   POL(F_5(x_1)) = x_1
   POL(F_6(x_1)) = x_1
   POL(F_7(x_1)) = x_1
   POL(F_8(x_1)) = x_1
   POL(F_9(x_1)) = [2]x_1 + [2]x_1^2
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = [2]x_2 + [2]x_2^2 + [2]x_1*x_2
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = 0
   POL(G_4(x_1, x_2)) = 0
   POL(G_5(x_1, x_2)) = x_1
   POL(G_6(x_1, x_2)) = x_2
   POL(G_7(x_1, x_2)) = x_2
   POL(G_8(x_1, x_2)) = x_2
   POL(G_9(x_1, x_2)) = 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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

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

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

(18)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
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))
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = x_1
   POL(F_3(x_1)) = x_1
   POL(F_4(x_1)) = x_1
   POL(F_5(x_1)) = x_1
   POL(F_6(x_1)) = x_1
   POL(F_7(x_1)) = x_1
   POL(F_8(x_1)) = x_1
   POL(F_9(x_1)) = [2]x_1 + [2]x_1^2
   POL(G_1(x_1, x_2)) = [1]
   POL(G_10(x_1, x_2)) = [2]x_2 + [2]x_2^2 + x_1*x_2
   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)) = x_2
   POL(G_5(x_1, x_2)) = x_2
   POL(G_6(x_1, x_2)) = x_2
   POL(G_7(x_1, x_2)) = x_2
   POL(G_8(x_1, x_2)) = x_2
   POL(G_9(x_1, x_2)) = 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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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)) = [2] + x_1

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

(20)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   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))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(21) 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))

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

(22)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
S tuples:
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   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))
   G_4(s(z0), z1) -> c11(F_3(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G_4(s(z0), z1) -> c11(F_3(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = 0
   POL(F_3(x_1)) = 0
   POL(F_4(x_1)) = [1]
   POL(F_5(x_1)) = [1]
   POL(F_6(x_1)) = [1]
   POL(F_7(x_1)) = [1]
   POL(F_8(x_1)) = [2]
   POL(F_9(x_1)) = [2] + [2]x_1^2
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = [2] + [2]x_1 + x_2 + [2]x_2^2 + [2]x_1*x_2 + x_1^2
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = 0
   POL(G_4(x_1, x_2)) = [1]
   POL(G_5(x_1, x_2)) = [1]
   POL(G_6(x_1, x_2)) = [1]
   POL(G_7(x_1, x_2)) = [1]
   POL(G_8(x_1, x_2)) = [1]
   POL(G_9(x_1, x_2)) = [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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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)) = x_1

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

(24)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
S tuples:
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   F_2(z0) -> c4(G_2(z0, z0))
   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))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(25) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F_3(z0) -> c7(G_3(z0, z0))

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

(26)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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) -> c12(G_4(z0, z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   F_3(z0) -> c7(G_3(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(27) 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_6(s(z0), z1) -> c18(G_6(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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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]
   POL(F_5(x_1)) = [1]
   POL(F_6(x_1)) = [1] + x_1
   POL(F_7(x_1)) = [1] + x_1
   POL(F_8(x_1)) = [1] + x_1
   POL(F_9(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = [1] + x_2
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = [1]
   POL(G_4(x_1, x_2)) = [1]
   POL(G_5(x_1, x_2)) = [1]
   POL(G_6(x_1, x_2)) = x_1
   POL(G_7(x_1, x_2)) = [1] + x_2
   POL(G_8(x_1, x_2)) = [1] + x_2
   POL(G_9(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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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

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

(28)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   F_3(z0) -> c7(G_3(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_6(s(z0), z1) -> c18(G_6(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(29) 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_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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = 0
   POL(F_3(x_1)) = [1]
   POL(F_4(x_1)) = [1] + x_1
   POL(F_5(x_1)) = [1] + x_1
   POL(F_6(x_1)) = [1] + x_1
   POL(F_7(x_1)) = [1] + x_1
   POL(F_8(x_1)) = [1] + x_1
   POL(F_9(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = x_1 + x_2
   POL(G_2(x_1, x_2)) = 0
   POL(G_3(x_1, x_2)) = [1]
   POL(G_4(x_1, x_2)) = x_1
   POL(G_5(x_1, x_2)) = [1] + x_2
   POL(G_6(x_1, x_2)) = [1] + x_2
   POL(G_7(x_1, x_2)) = [1] + x_2
   POL(G_8(x_1, x_2)) = [1] + x_2
   POL(G_9(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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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

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

(30)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
S tuples:
   G_1(s(z0), z1) -> c3(G_1(z0, z1))
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   F_3(z0) -> c7(G_3(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   G_3(s(z0), z1) -> c8(F_2(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(31) 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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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(F_5(x_1)) = [1] + x_1
   POL(F_6(x_1)) = [1] + x_1
   POL(F_7(x_1)) = [1] + x_1
   POL(F_8(x_1)) = [1] + x_1
   POL(F_9(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = [1] + x_1
   POL(G_10(x_1, x_2)) = [1] + x_2
   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(G_6(x_1, x_2)) = [1] + x_2
   POL(G_7(x_1, x_2)) = [1] + x_2
   POL(G_8(x_1, x_2)) = [1] + x_2
   POL(G_9(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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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

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

(32)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
S tuples:
   G_3(s(z0), z1) -> c9(G_3(z0, z1))
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   F_3(z0) -> c7(G_3(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_4(s(z0), z1) -> c12(G_4(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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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

(33) 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) -> 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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(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)) = 0
   POL(F_3(x_1)) = [1] + x_1
   POL(F_4(x_1)) = [1] + x_1
   POL(F_5(x_1)) = [1] + x_1
   POL(F_6(x_1)) = [1] + x_1
   POL(F_7(x_1)) = [1] + x_1
   POL(F_8(x_1)) = [1] + x_1
   POL(F_9(x_1)) = [1] + x_1
   POL(G_1(x_1, x_2)) = 0
   POL(G_10(x_1, x_2)) = x_1 + x_2
   POL(G_2(x_1, x_2)) = 0
   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(G_6(x_1, x_2)) = [1] + x_2
   POL(G_7(x_1, x_2)) = [1] + x_2
   POL(G_8(x_1, x_2)) = [1] + x_2
   POL(G_9(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(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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

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

(34)
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))
   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))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   F_7(z0) -> c19(G_7(z0, z0))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
S tuples:none
K tuples:
   F_9(z0) -> c25(G_9(z0, z0))
   G_9(s(z0), z1) -> c26(F_8(z1))
   G_9(s(z0), z1) -> c27(G_9(z0, z1))
   G_10(s(z0), z1) -> c29(F_9(z1))
   G_10(s(z0), z1) -> c30(G_10(z0, z1))
   F_8(z0) -> c22(G_8(z0, z0))
   F_6(z0) -> c16(G_6(z0, z0))
   G_6(s(z0), z1) -> c17(F_5(z1))
   G_7(s(z0), z1) -> c20(F_6(z1))
   G_7(s(z0), z1) -> c21(G_7(z0, z1))
   G_8(s(z0), z1) -> c23(F_7(z1))
   G_8(s(z0), z1) -> c24(G_8(z0, z1))
   F_5(z0) -> c13(G_5(z0, z0))
   F_7(z0) -> c19(G_7(z0, z0))
   G_5(s(z0), z1) -> c14(F_4(z1))
   G_5(s(z0), z1) -> c15(G_5(z0, z1))
   F_4(z0) -> c10(G_4(z0, z0))
   G_2(s(z0), z1) -> c5(F_1(z1))
   G_2(s(z0), z1) -> c6(G_2(z0, z1))
   F_1(z0) -> c1(G_1(z0, z0))
   G_4(s(z0), z1) -> c11(F_3(z1))
   F_3(z0) -> c7(G_3(z0, z0))
   F_2(z0) -> c4(G_2(z0, z0))
   G_6(s(z0), z1) -> c18(G_6(z0, z1))
   G_3(s(z0), z1) -> c8(F_2(z1))
   G_4(s(z0), z1) -> c12(G_4(z0, z1))
   G_1(s(z0), z1) -> c3(G_1(z0, 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, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_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, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1


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(35) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(36)
BOUNDS(1, 1)