WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 58 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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 86 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(20) TRS for Loop Detection
    (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   a(x1) -> x1
   a(a(b(x1))) -> c(c(b(a(x1))))
   b(c(x1)) -> a(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(c(x_1)) -> c(encArg(x_1))
   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))
   encode_c(x_1) -> c(encArg(x_1))


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

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


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   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))
   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(n^1, n^2).


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   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))
   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(c(z0)) -> c(encArg(z0))
   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))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> z0
   a(a(b(z0))) -> c(c(b(a(z0))))
   b(c(z0)) -> a(b(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c6(ENCARG(z0))
   A(z0) -> c7
   A(a(b(z0))) -> c8(B(a(z0)), A(z0))
   B(c(z0)) -> c9(A(b(z0)), B(z0))
S tuples:
   A(z0) -> c7
   A(a(b(z0))) -> c8(B(a(z0)), A(z0))
   B(c(z0)) -> c9(A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

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

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c6_1, c7, c8_2, c9_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_C(z0) -> c6(ENCARG(z0))
   A(a(b(z0))) -> c8(B(a(z0)), A(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

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

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

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c7, c9_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

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

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

Compound Symbols:   c1_1, c2_2, c3_2, c7, c9_2, c6_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A(z0) -> c6(ENCARG(z0))
   ENCODE_B(z0) -> c6(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

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

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

Compound Symbols:   c1_1, c2_2, c3_2, c7, c9_2, c6_1


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(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))
   encode_c(z0) -> c(encArg(z0))
   a(a(b(z0))) -> c(c(b(a(z0))))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> z0
   b(c(z0)) -> a(b(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   B(c(z0)) -> c9(A(b(z0)), B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
S tuples:
   A(z0) -> c7
   B(c(z0)) -> c9(A(b(z0)), 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:   c1_1, c2_2, c3_2, c7, c9_2, c6_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.
   A(z0) -> c7
   B(c(z0)) -> c9(A(b(z0)), B(z0))
We considered the (Usable) Rules:
   a(z0) -> z0
   encArg(c(z0)) -> c(encArg(z0))
   b(c(z0)) -> a(b(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   B(c(z0)) -> c9(A(b(z0)), B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(B(x_1)) = [2]x_1
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A(x_1)) = [1] + [2]x_1 + x_1^2
   POL(ENCODE_B(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(a(x_1)) = x_1
   POL(b(x_1)) = [2] + x_1
   POL(c(x_1)) = [2] + x_1
   POL(c1(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(c6(x_1)) = x_1
   POL(c7) = 0
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a(x_1)) = [2] + x_1
   POL(cons_b(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> z0
   b(c(z0)) -> a(b(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   B(c(z0)) -> c9(A(b(z0)), B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
S tuples:none
K tuples:
   A(z0) -> c7
   B(c(z0)) -> c9(A(b(z0)), 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:   c1_1, c2_2, c3_2, c7, c9_2, c6_1


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

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

(18)
BOUNDS(1, 1)

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(19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(20)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   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))
   encode_c(x_1) -> c(encArg(x_1))

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

(21) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

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

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x1 / c(x1)].

The result substitution is [ ].




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(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   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))
   encode_c(x_1) -> c(encArg(x_1))

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

(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(25)
BOUNDS(n^1, INF)

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(26)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   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))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST