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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 85 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^1)), 107 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 34 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 52 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 43 ms]
    (22) CdtProblem
    (23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) BOUNDS(1, 1)
(25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(26) TRS for Loop Detection
    (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) 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:

   f(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   activate(X) -> X

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(g(x_1)) -> g(encArg(x_1))
   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(n__a) -> n__a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_n__a -> n__a
   encode_activate(x_1) -> activate(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:

   f(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   activate(X) -> X

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(n__a) -> n__a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_n__a -> n__a
   encode_activate(x_1) -> activate(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:

   f(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   activate(X) -> X

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(n__a) -> n__a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_n__a -> n__a
   encode_activate(x_1) -> activate(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(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_n__a -> n__a
   encode_activate(z0) -> activate(encArg(z0))
   f(f(a)) -> f(g(n__f(n__a)))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(n__a) -> c2
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c6(F(encArg(z0)), ENCARG(z0))
   ENCODE_A -> c7(A)
   ENCODE_G(z0) -> c8(ENCARG(z0))
   ENCODE_N__F(z0) -> c9(ENCARG(z0))
   ENCODE_N__A -> c10
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(f(a)) -> c12(F(g(n__f(n__a))))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
S tuples:
   F(f(a)) -> c12(F(g(n__f(n__a))))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
K tuples:none
Defined Rule Symbols:   f_1, a, activate_1, encArg_1, encode_f_1, encode_a, encode_g_1, encode_n__f_1, encode_n__a, encode_activate_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_A, ENCODE_G_1, ENCODE_N__F_1, ENCODE_N__A, ENCODE_ACTIVATE_1, F_1, A, ACTIVATE_1

Compound Symbols:   c_1, c1_1, c2, c3_2, c4_1, c5_2, c6_2, c7_1, c8_1, c9_1, c10, c11_2, c12_1, c13, c14, c15_2, c16_1, c17


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_G(z0) -> c8(ENCARG(z0))
   ENCODE_N__F(z0) -> c9(ENCARG(z0))
   ENCODE_A -> c7(A)
   F(f(a)) -> c12(F(g(n__f(n__a))))
Removed 2 trailing nodes:
   ENCODE_N__A -> c10
   ENCARG(n__a) -> c2

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_n__a -> n__a
   encode_activate(z0) -> activate(encArg(z0))
   f(f(a)) -> f(g(n__f(n__a)))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c6(F(encArg(z0)), ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
S tuples:
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
K tuples:none
Defined Rule Symbols:   f_1, a, activate_1, encArg_1, encode_f_1, encode_a, encode_g_1, encode_n__f_1, encode_n__a, encode_activate_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_ACTIVATE_1, F_1, A, ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c6_2, c11_2, c13, c14, c15_2, c16_1, c17


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_n__a -> n__a
   encode_activate(z0) -> activate(encArg(z0))
   f(f(a)) -> f(g(n__f(n__a)))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_F(z0) -> c2(ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ENCARG(z0))
S tuples:
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
K tuples:none
Defined Rule Symbols:   f_1, a, activate_1, encArg_1, encode_f_1, encode_a, encode_g_1, encode_n__f_1, encode_n__a, encode_activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0) -> c2(ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c2(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_n__a -> n__a
   encode_activate(z0) -> activate(encArg(z0))
   f(f(a)) -> f(g(n__f(n__a)))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
S tuples:
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
K tuples:none
Defined Rule Symbols:   f_1, a, activate_1, encArg_1, encode_f_1, encode_a, encode_g_1, encode_n__f_1, encode_n__a, encode_activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_n__a -> n__a
   encode_activate(z0) -> activate(encArg(z0))
   f(f(a)) -> f(g(n__f(n__a)))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
S tuples:
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = 0
   POL(ACTIVATE(x_1)) = [1]
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_ACTIVATE(x_1)) = [1] + x_1
   POL(ENCODE_F(x_1)) = 0
   POL(F(x_1)) = 0
   POL(a) = [1]
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c13) = 0
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1)) = x_1
   POL(c17) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a) = [1]
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = [1] + x_1
   POL(n__a) = [1]
   POL(n__f(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
S tuples:
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
K tuples:
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
Defined Rule Symbols:   encArg_1, f_1, a, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_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.
   A -> c14
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = [1]
   POL(ACTIVATE(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_ACTIVATE(x_1)) = [1]
   POL(ENCODE_F(x_1)) = 0
   POL(F(x_1)) = 0
   POL(a) = [1]
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c13) = 0
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1)) = x_1
   POL(c17) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a) = [1]
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = [1] + x_1
   POL(n__a) = [1]
   POL(n__f(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
S tuples:
   F(z0) -> c13
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
K tuples:
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   A -> c14
Defined Rule Symbols:   encArg_1, f_1, a, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_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.
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:
   encArg(n__a) -> n__a
   encArg(cons_a) -> a
   f(z0) -> n__f(z0)
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   activate(z0) -> z0
   activate(n__a) -> a
   activate(n__f(z0)) -> f(activate(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   a -> n__a
And the Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = 0
   POL(ACTIVATE(x_1)) = x_1
   POL(ENCARG(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_ACTIVATE(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_F(x_1)) = [2] + [2]x_1 + x_1^2
   POL(F(x_1)) = 0
   POL(a) = 0
   POL(activate(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c13) = 0
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1)) = x_1
   POL(c17) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a) = 0
   POL(cons_activate(x_1)) = [2] + x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(n__a) = 0
   POL(n__f(x_1)) = [2] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
S tuples:
   F(z0) -> c13
K tuples:
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:   encArg_1, f_1, a, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(z0) -> c13
We considered the (Usable) Rules:
   encArg(n__a) -> n__a
   encArg(cons_a) -> a
   f(z0) -> n__f(z0)
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   activate(z0) -> z0
   activate(n__a) -> a
   activate(n__f(z0)) -> f(activate(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   a -> n__a
And the Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = 0
   POL(ACTIVATE(x_1)) = x_1
   POL(ENCARG(x_1)) = [2] + [2]x_1^2
   POL(ENCODE_ACTIVATE(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_F(x_1)) = [1] + x_1^2
   POL(F(x_1)) = [1]
   POL(a) = 0
   POL(activate(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c13) = 0
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1)) = x_1
   POL(c17) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a) = 0
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = x_1
   POL(n__a) = 0
   POL(n__f(x_1)) = [1] + x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__a) -> n__a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> n__f(z0)
   a -> n__a
   activate(n__f(z0)) -> f(activate(z0))
   activate(n__a) -> a
   activate(z0) -> z0
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCARG(cons_activate(z0)) -> c5(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c13
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c2(ACTIVATE(encArg(z0)))
S tuples:none
K tuples:
   ACTIVATE(n__a) -> c16(A)
   ACTIVATE(z0) -> c17
   A -> c14
   ACTIVATE(n__f(z0)) -> c15(F(activate(z0)), ACTIVATE(z0))
   F(z0) -> c13
Defined Rule Symbols:   encArg_1, f_1, a, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, A, ACTIVATE_1, ENCODE_F_1, ENCODE_ACTIVATE_1

Compound Symbols:   c_1, c1_1, c3_2, c4_1, c5_2, c13, c14, c15_2, c16_1, c17, c2_1


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

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

(24)
BOUNDS(1, 1)

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

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

(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:

   f(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   activate(X) -> X

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(n__a) -> n__a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_n__a -> n__a
   encode_activate(x_1) -> activate(encArg(x_1))

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

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

The rewrite sequence

activate(n__f(X)) ->^+ f(activate(X))

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

The pumping substitution is [X / n__f(X)].

The result substitution is [ ].




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

(28)
Complex Obligation (BEST)

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

(29)
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:

   f(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   activate(X) -> X

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(n__a) -> n__a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_n__a -> n__a
   encode_activate(x_1) -> activate(encArg(x_1))

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

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

(31)
BOUNDS(n^1, INF)

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

(32)
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:

   f(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   activate(X) -> X

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(n__a) -> n__a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_n__a -> n__a
   encode_activate(x_1) -> activate(encArg(x_1))

Rewrite Strategy: INNERMOST