/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(?, O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 30 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 83 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 79 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 84 ms]
(22) CdtProblem
(23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) BOUNDS(1, 1)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   a(a(x1)) -> a(b(a(x1)))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   a(a(x1)) -> a(b(a(x1)))

The (relative) TRS S consists of the following rules:

   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   a(a(x1)) -> a(b(a(x1)))

The (relative) TRS S consists of the following rules:

   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   b(a(a(z0))) -> a(b(c(z0)))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c4(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   B(a(a(z0))) -> c7(A(b(c(z0))), B(c(z0)), C(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(B(a(z0)), A(z0))
   A(a(z0)) -> c10(A(b(a(z0))), B(a(z0)), A(z0))
S tuples:
   B(a(a(z0))) -> c7(A(b(c(z0))), B(c(z0)), C(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(B(a(z0)), A(z0))
   A(a(z0)) -> c10(A(b(a(z0))), B(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   b_1, c_1, a_1, encArg_1, encode_b_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_B_1, ENCODE_A_1, ENCODE_C_1, B_1, C_1, A_1

Compound Symbols:   c1_2, c2_2, c3_2, c4_2, c5_2, c6_2, c7_3, c8_2, c9_2, c10_3


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   B(a(a(z0))) -> c7(A(b(c(z0))), B(c(z0)), C(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   b(a(a(z0))) -> a(b(c(z0)))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c4(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(B(a(z0)), A(z0))
   A(a(z0)) -> c10(A(b(a(z0))), B(a(z0)), A(z0))
S tuples:
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(B(a(z0)), A(z0))
   A(a(z0)) -> c10(A(b(a(z0))), B(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   b_1, c_1, a_1, encArg_1, encode_b_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_B_1, ENCODE_A_1, ENCODE_C_1, C_1, A_1

Compound Symbols:   c1_2, c2_2, c3_2, c4_2, c5_2, c6_2, c8_2, c9_2, c10_3


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   b(a(a(z0))) -> a(b(c(z0)))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   ENCODE_B(z0) -> c4(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
S tuples:
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
K tuples:none
Defined Rule Symbols:   b_1, c_1, a_1, encArg_1, encode_b_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_C_1, C_1, ENCODE_B_1, A_1

Compound Symbols:   c2_2, c3_2, c5_2, c6_2, c8_2, c1_1, c4_1, c9_1, c10_1


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

(11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   b(a(a(z0))) -> a(b(c(z0)))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   ENCODE_B(z0) -> c4(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_A(z0) -> c7(ENCARG(z0))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
   ENCODE_C(z0) -> c7(ENCARG(z0))
S tuples:
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
K tuples:none
Defined Rule Symbols:   b_1, c_1, a_1, encArg_1, encode_b_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, C_1, ENCODE_B_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c3_2, c8_2, c1_1, c4_1, c9_1, c10_1, c7_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_B(z0) -> c4(ENCARG(z0))
   ENCODE_A(z0) -> c7(ENCARG(z0))
   ENCODE_C(z0) -> c7(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   b(a(a(z0))) -> a(b(c(z0)))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
K tuples:none
Defined Rule Symbols:   b_1, c_1, a_1, encArg_1, encode_b_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, C_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c3_2, c8_2, c1_1, c9_1, c10_1, c7_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_b(z0) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   b(a(a(z0))) -> a(b(c(z0)))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, c_1, a_1

Defined Pair Symbols:   ENCARG_1, C_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c3_2, c8_2, c1_1, c9_1, c10_1, c7_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   C(b(z0)) -> c9(A(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = 0
   POL(C(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = 0
   POL(ENCODE_C(x_1)) = [1]
   POL(a(x_1)) = x_1
   POL(b(x_1)) = [1] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   A(a(z0)) -> c10(A(z0))
K tuples:
   C(b(z0)) -> c9(A(z0))
Defined Rule Symbols:   encArg_1, c_1, a_1

Defined Pair Symbols:   ENCARG_1, C_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c3_2, c8_2, c1_1, c9_1, c10_1, c7_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.
   C(a(z0)) -> c8(A(c(z0)), C(z0))
We considered the (Usable) Rules:
   c(b(z0)) -> b(a(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(z0)) -> a(b(a(z0)))
   c(a(z0)) -> a(c(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
And the Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = 0
   POL(C(x_1)) = x_1
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_C(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(a(x_1)) = [2] + x_1
   POL(b(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = x_1
   POL(cons_c(x_1)) = [2] + x_1
   POL(encArg(x_1)) = [2]x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:
   A(a(z0)) -> c10(A(z0))
K tuples:
   C(b(z0)) -> c9(A(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
Defined Rule Symbols:   encArg_1, c_1, a_1

Defined Pair Symbols:   ENCARG_1, C_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c3_2, c8_2, c1_1, c9_1, c10_1, c7_1


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(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A(a(z0)) -> c10(A(z0))
We considered the (Usable) Rules:
   c(b(z0)) -> b(a(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(z0)) -> a(b(a(z0)))
   c(a(z0)) -> a(c(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
And the Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = x_1
   POL(C(x_1)) = [1] + x_1 + x_1^2
   POL(ENCARG(x_1)) = x_1^3
   POL(ENCODE_A(x_1)) = x_1
   POL(ENCODE_C(x_1)) = [1] + x_1 + x_1^2
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1

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(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   c(a(z0)) -> a(c(z0))
   c(b(z0)) -> b(a(z0))
   a(a(z0)) -> a(b(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   ENCARG(cons_b(z0)) -> c1(ENCARG(z0))
   C(b(z0)) -> c9(A(z0))
   A(a(z0)) -> c10(A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:none
K tuples:
   C(b(z0)) -> c9(A(z0))
   C(a(z0)) -> c8(A(c(z0)), C(z0))
   A(a(z0)) -> c10(A(z0))
Defined Rule Symbols:   encArg_1, c_1, a_1

Defined Pair Symbols:   ENCARG_1, C_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c3_2, c8_2, c1_1, c9_1, c10_1, c7_1


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(23) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(24)
BOUNDS(1, 1)