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


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


WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 40 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(18) CdtProblem
(19) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 68 ms]
(22) CdtProblem
(23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 57 ms]
(24) CdtProblem
(25) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   f(f(x)) -> f(x)
   g(0) -> g(f(0))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(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_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_0 -> 0


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

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


The TRS R consists of the following rules:

   f(f(x)) -> f(x)
   g(0) -> g(f(0))

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_0 -> 0

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(f(x)) -> f(x)
   g(0) -> g(f(0))

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_0 -> 0

Rewrite Strategy: FULL
----------------------------------------

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   g(0) -> g(f(0))
   f(c_f(x)) -> f(x)

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_0 -> 0
   f(x0) -> c_f(x0)

Rewrite Strategy: FULL
----------------------------------------

(7) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(8)
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:

   g(0) -> g(f(0))
   f(c_f(x)) -> f(x)

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_0 -> 0
   f(x0) -> c_f(x0)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(0) -> c
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_0 -> c5
   F(z0) -> c6
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)), F(0))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)), F(0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_f_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_G_1, ENCODE_0, F_1, G_1

Compound Symbols:   c, c1_2, c2_2, c3_2, c4_2, c5, c6, c7_1, c8_2


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

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   F(z0) -> c6
   ENCARG(0) -> c
   ENCODE_0 -> c5

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c4(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)), F(0))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)), F(0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_f_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_G_1, F_1, G_1

Compound Symbols:   c1_2, c2_2, c3_2, c4_2, c7_1, c8_2


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

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c4(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_f_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_G_1, F_1, G_1

Compound Symbols:   c1_2, c2_2, c3_2, c4_2, c7_1, c8_1


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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_F(z0) -> c(ENCARG(z0))
   ENCODE_G(z0) -> c(G(encArg(z0)))
   ENCODE_G(z0) -> c(ENCARG(z0))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_f_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c7_1, c8_1, c_1


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

(17) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0) -> c(ENCARG(z0))
   ENCODE_G(z0) -> c(ENCARG(z0))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_f_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c7_1, c8_1, c_1


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

(19) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c7_1, c8_1, c_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(0) -> c8(G(f(0)))
We considered the (Usable) Rules:
   f(z0) -> c_f(z0)
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   g(0) -> g(f(0))
   encArg(0) -> 0
   f(c_f(z0)) -> f(z0)
And the Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(ENCARG(x_1)) = [2]x_1
   POL(ENCODE_F(x_1)) = [1]
   POL(ENCODE_G(x_1)) = [3]
   POL(F(x_1)) = 0
   POL(G(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c_f(x_1)) = 0
   POL(cons_f(x_1)) = x_1
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [2]
   POL(f(x_1)) = 0
   POL(g(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(c_f(z0)) -> c7(F(z0))
K tuples:
   G(0) -> c8(G(f(0)))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c7_1, c8_1, c_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(c_f(z0)) -> c7(F(z0))
We considered the (Usable) Rules:
   f(z0) -> c_f(z0)
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   g(0) -> g(f(0))
   encArg(0) -> 0
   f(c_f(z0)) -> f(z0)
And the Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_F(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_G(x_1)) = [2] + x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(G(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c_f(x_1)) = [2] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_g(x_1)) = x_1
   POL(encArg(x_1)) = [2] + [2]x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = [2]

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0) -> c_f(z0)
   f(c_f(z0)) -> f(z0)
   g(0) -> g(f(0))
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   F(c_f(z0)) -> c7(F(z0))
   G(0) -> c8(G(f(0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:none
K tuples:
   G(0) -> c8(G(f(0)))
   F(c_f(z0)) -> c7(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c7_1, c8_1, c_1


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

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

(26)
BOUNDS(1, 1)