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


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WORST_CASE(?, O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 148 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^2)), 402 ms]
(18) CdtProblem
(19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) BOUNDS(1, 1)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   p(a(x0), p(a(a(a(x1))), x2)) -> p(a(x2), p(a(a(b(x0))), x2))

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(a(x_1)) -> a(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))


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

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


The TRS R consists of the following rules:

   p(a(x0), p(a(a(a(x1))), x2)) -> p(a(x2), p(a(a(b(x0))), x2))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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^2).


The TRS R consists of the following rules:

   p(a(x0), p(a(a(a(x1))), x2)) -> p(a(x2), p(a(a(b(x0))), x2))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   encode_p(z0, z1) -> p(encArg(z0), encArg(z1))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_P(z0, z1) -> c3(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_A(z0) -> c4(ENCARG(z0))
   ENCODE_B(z0) -> c5(ENCARG(z0))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(z2), p(a(a(b(z0))), z2)), P(a(a(b(z0))), z2))
S tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(z2), p(a(a(b(z0))), z2)), P(a(a(b(z0))), z2))
K tuples:none
Defined Rule Symbols:   p_2, encArg_1, encode_p_2, encode_a_1, encode_b_1

Defined Pair Symbols:   ENCARG_1, ENCODE_P_2, ENCODE_A_1, ENCODE_B_1, P_2

Compound Symbols:   c_1, c1_1, c2_3, c3_3, c4_1, c5_1, c6_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A(z0) -> c4(ENCARG(z0))
   ENCODE_B(z0) -> c5(ENCARG(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   encode_p(z0, z1) -> p(encArg(z0), encArg(z1))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_P(z0, z1) -> c3(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(z2), p(a(a(b(z0))), z2)), P(a(a(b(z0))), z2))
S tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(z2), p(a(a(b(z0))), z2)), P(a(a(b(z0))), z2))
K tuples:none
Defined Rule Symbols:   p_2, encArg_1, encode_p_2, encode_a_1, encode_b_1

Defined Pair Symbols:   ENCARG_1, ENCODE_P_2, P_2

Compound Symbols:   c_1, c1_1, c2_3, c3_3, c6_2


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   encode_p(z0, z1) -> p(encArg(z0), encArg(z1))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_P(z0, z1) -> c3(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
S tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
K tuples:none
Defined Rule Symbols:   p_2, encArg_1, encode_p_2, encode_a_1, encode_b_1

Defined Pair Symbols:   ENCARG_1, ENCODE_P_2, P_2

Compound Symbols:   c_1, c1_1, c2_3, c3_3, c6_1


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   encode_p(z0, z1) -> p(encArg(z0), encArg(z1))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
   ENCODE_P(z0, z1) -> c4(P(encArg(z0), encArg(z1)))
   ENCODE_P(z0, z1) -> c4(ENCARG(z0))
   ENCODE_P(z0, z1) -> c4(ENCARG(z1))
S tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
K tuples:none
Defined Rule Symbols:   p_2, encArg_1, encode_p_2, encode_a_1, encode_b_1

Defined Pair Symbols:   ENCARG_1, P_2, ENCODE_P_2

Compound Symbols:   c_1, c1_1, c2_3, c6_1, c4_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_P(z0, z1) -> c4(ENCARG(z0))
   ENCODE_P(z0, z1) -> c4(ENCARG(z1))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   encode_p(z0, z1) -> p(encArg(z0), encArg(z1))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
   ENCODE_P(z0, z1) -> c4(P(encArg(z0), encArg(z1)))
S tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
K tuples:none
Defined Rule Symbols:   p_2, encArg_1, encode_p_2, encode_a_1, encode_b_1

Defined Pair Symbols:   ENCARG_1, P_2, ENCODE_P_2

Compound Symbols:   c_1, c1_1, c2_3, c6_1, c4_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_p(z0, z1) -> p(encArg(z0), encArg(z1))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
   ENCODE_P(z0, z1) -> c4(P(encArg(z0), encArg(z1)))
S tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, p_2

Defined Pair Symbols:   ENCARG_1, P_2, ENCODE_P_2

Compound Symbols:   c_1, c1_1, c2_3, c6_1, c4_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
We considered the (Usable) Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   encArg(a(z0)) -> a(encArg(z0))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
And the Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
   ENCODE_P(z0, z1) -> c4(P(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [2] + x_1^2
   POL(ENCODE_P(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + x_1^2
   POL(P(x_1, x_2)) = [2] + [2]x_2
   POL(a(x_1)) = x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons_p(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(p(x_1, x_2)) = [2] + x_2

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a(z0)) -> a(encArg(z0))
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_p(z0, z1)) -> p(encArg(z0), encArg(z1))
   p(a(z0), p(a(a(a(z1))), z2)) -> p(a(z2), p(a(a(b(z0))), z2))
Tuples:
   ENCARG(a(z0)) -> c(ENCARG(z0))
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_p(z0, z1)) -> c2(P(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
   ENCODE_P(z0, z1) -> c4(P(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   P(a(z0), p(a(a(a(z1))), z2)) -> c6(P(a(a(b(z0))), z2))
Defined Rule Symbols:   encArg_1, p_2

Defined Pair Symbols:   ENCARG_1, P_2, ENCODE_P_2

Compound Symbols:   c_1, c1_1, c2_3, c6_1, c4_1


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

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

(20)
BOUNDS(1, 1)