/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 44 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 9 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 20 ms]
(16) CdtProblem
(17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) BOUNDS(1, 1)


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(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:

   f(a) -> f(a)
   a -> b

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(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b


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

(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:

   f(a) -> f(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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:

   f(a) -> f(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   f(a) -> f(a)
   a -> b
Tuples:
   ENCARG(b) -> c
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_A -> c4(A)
   ENCODE_B -> c5
   F(a) -> c6(F(a), A)
   A -> c7
S tuples:
   F(a) -> c6(F(a), A)
   A -> c7
K tuples:none
Defined Rule Symbols:   f_1, a, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_A, ENCODE_B, F_1, A

Compound Symbols:   c, c1_2, c2_1, c3_2, c4_1, c5, c6_2, c7


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(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A -> c4(A)
   F(a) -> c6(F(a), A)
Removed 2 trailing nodes:
   ENCARG(b) -> c
   ENCODE_B -> c5

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   f(a) -> f(a)
   a -> b
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   A -> c7
S tuples:
   A -> c7
K tuples:none
Defined Rule Symbols:   f_1, a, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, A

Compound Symbols:   c1_2, c2_1, c3_2, c7


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   f(a) -> f(a)
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(z0)) -> c1(ENCARG(z0))
   ENCODE_F(z0) -> c3(ENCARG(z0))
S tuples:
   A -> c7
K tuples:none
Defined Rule Symbols:   f_1, a, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, A, ENCODE_F_1

Compound Symbols:   c2_1, c7, c1_1, c3_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_F(z0) -> c3(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   f(a) -> f(a)
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(z0)) -> c1(ENCARG(z0))
S tuples:
   A -> c7
K tuples:none
Defined Rule Symbols:   f_1, a, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, A

Compound Symbols:   c2_1, c7, c1_1


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(z0)) -> c1(ENCARG(z0))
S tuples:
   A -> c7
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   ENCARG_1, A

Compound Symbols:   c2_1, c7, c1_1


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(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A -> c7
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(z0)) -> c1(ENCARG(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = [2]
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c7) = 0
   POL(cons_a) = [2]
   POL(cons_f(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(z0)) -> c1(ENCARG(z0))
S tuples:none
K tuples:
   A -> c7
Defined Rule Symbols:none

Defined Pair Symbols:   ENCARG_1, A

Compound Symbols:   c2_1, c7, c1_1


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

(18)
BOUNDS(1, 1)