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


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WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 46 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(12) CdtProblem
(13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 48 ms]
(20) CdtProblem
(21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) 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:

   a(x1) -> x1
   a(a(b(x1))) -> b(a(b(a(a(x1)))))
   b(b(x1)) -> b(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_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   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:

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

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   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:

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

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   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(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   a(z0) -> z0
   a(a(b(z0))) -> b(a(b(a(a(z0)))))
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c3(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   A(a(b(z0))) -> c5(B(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
   B(b(z0)) -> c6(B(z0))
S tuples:
   A(z0) -> c4
   A(a(b(z0))) -> c5(B(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
   B(b(z0)) -> c6(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1

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

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4, c5_5, c6_1


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   A(a(b(z0))) -> c5(B(a(b(a(a(z0))))), A(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   a(z0) -> z0
   a(a(b(z0))) -> b(a(b(a(a(z0)))))
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c3(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
S tuples:
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1

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

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4, c6_1


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   a(z0) -> z0
   a(a(b(z0))) -> b(a(b(a(a(z0)))))
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_A(z0) -> c5(ENCARG(z0))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
   ENCODE_B(z0) -> c5(ENCARG(z0))
S tuples:
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1

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

Compound Symbols:   c_2, c1_2, c4, c6_1, c5_1


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

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   a(z0) -> z0
   a(a(b(z0))) -> b(a(b(a(a(z0)))))
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
S tuples:
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1

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

Compound Symbols:   c_2, c1_2, c4, c6_1, c5_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> z0
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
S tuples:
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, b_1

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

Compound Symbols:   c_2, c1_2, c4, c6_1, c5_1


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

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_b(cons_a(z0))) -> c1(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0)))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> z0
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c1(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0)))
S tuples:
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, b_1

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

Compound Symbols:   c_2, c4, c6_1, c5_1, c1_2


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A(z0) -> c4
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c1(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(B(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = [1]
   POL(ENCODE_B(x_1)) = [1]
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c4) = 0
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> z0
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c1(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0)))
S tuples:
   B(b(z0)) -> c6(B(z0))
K tuples:
   A(z0) -> c4
Defined Rule Symbols:   encArg_1, a_1, b_1

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

Compound Symbols:   c_2, c4, c6_1, c5_1, c1_2


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   B(b(z0)) -> c6(B(z0))
We considered the (Usable) Rules:
   a(z0) -> z0
   b(b(z0)) -> b(z0)
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
And the Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c1(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = 0
   POL(B(x_1)) = x_1
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_A(x_1)) = [2] + [2]x_1^2
   POL(ENCODE_B(x_1)) = [1] + x_1 + [2]x_1^2
   POL(a(x_1)) = x_1
   POL(b(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c4) = 0
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons_a(x_1)) = x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> z0
   b(b(z0)) -> b(z0)
Tuples:
   ENCARG(cons_a(z0)) -> c(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_B(z0) -> c5(B(encArg(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c1(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0)))
S tuples:none
K tuples:
   A(z0) -> c4
   B(b(z0)) -> c6(B(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1

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

Compound Symbols:   c_2, c4, c6_1, c5_1, c1_2


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

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

(22)
BOUNDS(1, 1)