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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 215 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)), 46 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 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^1).


The TRS R consists of the following rules:

   f(c(a, z, x)) -> b(a, z)
   b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
   b(y, z) -> z

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, x_2, x_3)) -> c(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(a) -> a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1, x_2, x_3) -> c(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   f(c(a, z, x)) -> b(a, z)
   b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
   b(y, z) -> z

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

   encArg(c(x_1, x_2, x_3)) -> c(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(a) -> a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1, x_2, x_3) -> c(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2))

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^1).


The TRS R consists of the following rules:

   f(c(a, z, x)) -> b(a, z)
   b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
   b(y, z) -> z

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

   encArg(c(x_1, x_2, x_3)) -> c(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(a) -> a
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1, x_2, x_3) -> c(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0, z1, z2) -> c(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_b(z0, z1) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, b(z1, z2)) -> f(b(f(f(z1)), c(z0, z1, z2)))
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(a) -> c2
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0) -> c5(F(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0, z1, z2) -> c6(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_A -> c7
   ENCODE_B(z0, z1) -> c8(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, b(z1, z2)) -> c10(F(b(f(f(z1)), c(z0, z1, z2))), B(f(f(z1)), c(z0, z1, z2)), F(f(z1)), F(z1))
   B(z0, z1) -> c11
S tuples:
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, b(z1, z2)) -> c10(F(b(f(f(z1)), c(z0, z1, z2))), B(f(f(z1)), c(z0, z1, z2)), F(f(z1)), F(z1))
   B(z0, z1) -> c11
K tuples:none
Defined Rule Symbols:   f_1, b_2, encArg_1, encode_f_1, encode_c_3, encode_a, encode_b_2

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_C_3, ENCODE_A, ENCODE_B_2, F_1, B_2

Compound Symbols:   c1_3, c2, c3_2, c4_3, c5_2, c6_3, c7, c8_3, c9_1, c10_4, c11


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_C(z0, z1, z2) -> c6(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   B(z0, b(z1, z2)) -> c10(F(b(f(f(z1)), c(z0, z1, z2))), B(f(f(z1)), c(z0, z1, z2)), F(f(z1)), F(z1))
Removed 2 trailing nodes:
   ENCODE_A -> c7
   ENCARG(a) -> c2

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0, z1, z2) -> c(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_b(z0, z1) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, b(z1, z2)) -> f(b(f(f(z1)), c(z0, z1, z2)))
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0) -> c5(F(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0, z1) -> c8(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
S tuples:
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
K tuples:none
Defined Rule Symbols:   f_1, b_2, encArg_1, encode_f_1, encode_c_3, encode_a, encode_b_2

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_B_2, F_1, B_2

Compound Symbols:   c1_3, c3_2, c4_3, c5_2, c8_3, c9_1, c11


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

(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, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0, z1, z2) -> c(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_b(z0, z1) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, b(z1, z2)) -> f(b(f(f(z1)), c(z0, z1, z2)))
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_F(z0) -> c2(ENCARG(z0))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
   ENCODE_B(z0, z1) -> c2(ENCARG(z0))
   ENCODE_B(z0, z1) -> c2(ENCARG(z1))
S tuples:
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
K tuples:none
Defined Rule Symbols:   f_1, b_2, encArg_1, encode_f_1, encode_c_3, encode_a, encode_b_2

Defined Pair Symbols:   ENCARG_1, F_1, B_2, ENCODE_F_1, ENCODE_B_2

Compound Symbols:   c1_3, c3_2, c4_3, c9_1, c11, c2_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_F(z0) -> c2(ENCARG(z0))
   ENCODE_B(z0, z1) -> c2(ENCARG(z0))
   ENCODE_B(z0, z1) -> c2(ENCARG(z1))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0, z1, z2) -> c(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_b(z0, z1) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, b(z1, z2)) -> f(b(f(f(z1)), c(z0, z1, z2)))
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
S tuples:
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
K tuples:none
Defined Rule Symbols:   f_1, b_2, encArg_1, encode_f_1, encode_c_3, encode_a, encode_b_2

Defined Pair Symbols:   ENCARG_1, F_1, B_2, ENCODE_F_1, ENCODE_B_2

Compound Symbols:   c1_3, c3_2, c4_3, c9_1, c11, c2_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0, z1, z2) -> c(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_b(z0, z1) -> b(encArg(z0), encArg(z1))
   b(z0, b(z1, z2)) -> f(b(f(f(z1)), c(z0, z1, z2)))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
S tuples:
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, b_2

Defined Pair Symbols:   ENCARG_1, F_1, B_2, ENCODE_F_1, ENCODE_B_2

Compound Symbols:   c1_3, c3_2, c4_3, c9_1, c11, 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.
   B(z0, z1) -> c11
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(B(x_1, x_2)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_B(x_1, x_2)) = [1] + x_2
   POL(ENCODE_F(x_1)) = [1] + x_1
   POL(F(x_1)) = [1]
   POL(a) = [1]
   POL(b(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons_b(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_f(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
S tuples:
   F(c(a, z0, z1)) -> c9(B(a, z0))
K tuples:
   B(z0, z1) -> c11
Defined Rule Symbols:   encArg_1, f_1, b_2

Defined Pair Symbols:   ENCARG_1, F_1, B_2, ENCODE_F_1, ENCODE_B_2

Compound Symbols:   c1_3, c3_2, c4_3, c9_1, c11, 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.
   F(c(a, z0, z1)) -> c9(B(a, z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(B(x_1, x_2)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_B(x_1, x_2)) = 0
   POL(ENCODE_F(x_1)) = [1] + x_1
   POL(F(x_1)) = [1]
   POL(a) = [1]
   POL(b(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons_b(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_f(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1, z2)) -> c(encArg(z0), encArg(z1), encArg(z2))
   encArg(a) -> a
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1))
   f(c(a, z0, z1)) -> b(a, z0)
   b(z0, z1) -> z1
Tuples:
   ENCARG(c(z0, z1, z2)) -> c1(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0, z1)) -> c4(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(c(a, z0, z1)) -> c9(B(a, z0))
   B(z0, z1) -> c11
   ENCODE_F(z0) -> c2(F(encArg(z0)))
   ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   B(z0, z1) -> c11
   F(c(a, z0, z1)) -> c9(B(a, z0))
Defined Rule Symbols:   encArg_1, f_1, b_2

Defined Pair Symbols:   ENCARG_1, F_1, B_2, ENCODE_F_1, ENCODE_B_2

Compound Symbols:   c1_3, c3_2, c4_3, c9_1, c11, c2_1


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(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(20)
BOUNDS(1, 1)