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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 174 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)), 57 ms]
(16) CdtProblem
(17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) 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:

   a(a(y, 0), 0) -> y
   c(c(y)) -> y
   c(a(c(c(y)), x)) -> a(c(c(c(a(x, 0)))), y)

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(0) -> 0
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_c(x_1) -> c(encArg(x_1))


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

(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:

   a(a(y, 0), 0) -> y
   c(c(y)) -> y
   c(a(c(c(y)), x)) -> a(c(c(c(a(x, 0)))), y)

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

   encArg(0) -> 0
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   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(1, n^1).


The TRS R consists of the following rules:

   a(a(y, 0), 0) -> y
   c(c(y)) -> y
   c(a(c(c(y)), x)) -> a(c(c(c(a(x, 0)))), y)

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

   encArg(0) -> 0
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   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(0) -> 0
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_c(z0) -> c(encArg(z0))
   a(a(z0, 0), 0) -> z0
   c(c(z0)) -> z0
   c(a(c(c(z0)), z1)) -> a(c(c(c(a(z1, 0)))), z0)
Tuples:
   ENCARG(0) -> c1
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0, z1) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_0 -> c5
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   C(a(c(c(z0)), z1)) -> c9(A(c(c(c(a(z1, 0)))), z0), C(c(c(a(z1, 0)))), C(c(a(z1, 0))), C(a(z1, 0)), A(z1, 0))
S tuples:
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   C(a(c(c(z0)), z1)) -> c9(A(c(c(c(a(z1, 0)))), z0), C(c(c(a(z1, 0)))), C(c(a(z1, 0))), C(a(z1, 0)), A(z1, 0))
K tuples:none
Defined Rule Symbols:   a_2, c_1, encArg_1, encode_a_2, encode_0, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_2, ENCODE_0, ENCODE_C_1, A_2, C_1

Compound Symbols:   c1, c2_3, c3_2, c4_3, c5, c6_2, c7, c8, c9_5


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   C(a(c(c(z0)), z1)) -> c9(A(c(c(c(a(z1, 0)))), z0), C(c(c(a(z1, 0)))), C(c(a(z1, 0))), C(a(z1, 0)), A(z1, 0))
Removed 2 trailing nodes:
   ENCARG(0) -> c1
   ENCODE_0 -> c5

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_c(z0) -> c(encArg(z0))
   a(a(z0, 0), 0) -> z0
   c(c(z0)) -> z0
   c(a(c(c(z0)), z1)) -> a(c(c(c(a(z1, 0)))), z0)
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0, z1) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
S tuples:
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
K tuples:none
Defined Rule Symbols:   a_2, c_1, encArg_1, encode_a_2, encode_0, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_2, ENCODE_C_1, A_2, C_1

Compound Symbols:   c2_3, c3_2, c4_3, c6_2, c7, c8


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_c(z0) -> c(encArg(z0))
   a(a(z0, 0), 0) -> z0
   c(c(z0)) -> z0
   c(a(c(c(z0)), z1)) -> a(c(c(c(a(z1, 0)))), z0)
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   ENCODE_A(z0, z1) -> c1(A(encArg(z0), encArg(z1)))
   ENCODE_A(z0, z1) -> c1(ENCARG(z0))
   ENCODE_A(z0, z1) -> c1(ENCARG(z1))
   ENCODE_C(z0) -> c1(C(encArg(z0)))
   ENCODE_C(z0) -> c1(ENCARG(z0))
S tuples:
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
K tuples:none
Defined Rule Symbols:   a_2, c_1, encArg_1, encode_a_2, encode_0, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_2, C_1, ENCODE_A_2, ENCODE_C_1

Compound Symbols:   c2_3, c3_2, c7, c8, c1_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_A(z0, z1) -> c1(ENCARG(z0))
   ENCODE_A(z0, z1) -> c1(ENCARG(z1))
   ENCODE_C(z0) -> c1(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_c(z0) -> c(encArg(z0))
   a(a(z0, 0), 0) -> z0
   c(c(z0)) -> z0
   c(a(c(c(z0)), z1)) -> a(c(c(c(a(z1, 0)))), z0)
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   ENCODE_A(z0, z1) -> c1(A(encArg(z0), encArg(z1)))
   ENCODE_C(z0) -> c1(C(encArg(z0)))
S tuples:
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
K tuples:none
Defined Rule Symbols:   a_2, c_1, encArg_1, encode_a_2, encode_0, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_2, C_1, ENCODE_A_2, ENCODE_C_1

Compound Symbols:   c2_3, c3_2, c7, c8, c1_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_c(z0) -> c(encArg(z0))
   c(a(c(c(z0)), z1)) -> a(c(c(c(a(z1, 0)))), z0)

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(z0, 0), 0) -> z0
   c(c(z0)) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   ENCODE_A(z0, z1) -> c1(A(encArg(z0), encArg(z1)))
   ENCODE_C(z0) -> c1(C(encArg(z0)))
S tuples:
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
K tuples:none
Defined Rule Symbols:   encArg_1, a_2, c_1

Defined Pair Symbols:   ENCARG_1, A_2, C_1, ENCODE_A_2, ENCODE_C_1

Compound Symbols:   c2_3, c3_2, c7, c8, c1_1


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(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   ENCODE_A(z0, z1) -> c1(A(encArg(z0), encArg(z1)))
   ENCODE_C(z0) -> c1(C(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(A(x_1, x_2)) = [1]
   POL(C(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1, x_2)) = [1]
   POL(ENCODE_C(x_1)) = [1]
   POL(a(x_1, x_2)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0
   POL(c8) = 0
   POL(cons_a(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_c(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(z0, 0), 0) -> z0
   c(c(z0)) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
   ENCODE_A(z0, z1) -> c1(A(encArg(z0), encArg(z1)))
   ENCODE_C(z0) -> c1(C(encArg(z0)))
S tuples:none
K tuples:
   A(a(z0, 0), 0) -> c7
   C(c(z0)) -> c8
Defined Rule Symbols:   encArg_1, a_2, c_1

Defined Pair Symbols:   ENCARG_1, A_2, C_1, ENCODE_A_2, ENCODE_C_1

Compound Symbols:   c2_3, c3_2, c7, c8, c1_1


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

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

(18)
BOUNDS(1, 1)