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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 154 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3731 ms]
(22) CdtProblem
(23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   f(f(a, f(x, a)), a) -> f(a, f(f(x, a), a))

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(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a


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

(2)
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:

   f(f(a, f(x, a)), a) -> f(a, f(f(x, a), a))

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

   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a

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^2).


The TRS R consists of the following rules:

   f(f(a, f(x, a)), a) -> f(a, f(f(x, a), a))

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

   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   ENCARG(a) -> c
   ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_A -> c3
   F(f(a, f(z0, a)), a) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a), F(z0, a))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a), F(z0, a))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_A, F_2

Compound Symbols:   c, c1_3, c2_3, c3, c4_3


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   ENCARG(a) -> c
   ENCODE_A -> c3

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(a, f(z0, a)), a) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a), F(z0, a))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a), F(z0, a))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, F_2

Compound Symbols:   c1_3, c2_3, c4_3


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, F_2

Compound Symbols:   c1_3, c2_3, c4_1


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1) -> c(ENCARG(z0))
   ENCODE_F(z0, z1) -> c(ENCARG(z1))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_a

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_3, c4_1, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0, z1) -> c(ENCARG(z0))
   ENCODE_F(z0, z1) -> c(ENCARG(z1))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_a

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_3, c4_1, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_3, c4_1, c_1


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

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by 
   ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0), ENCARG(a))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(a), ENCARG(x1))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0), ENCARG(a))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(a), ENCARG(x1))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCODE_F_2, ENCARG_1

Compound Symbols:   c4_1, c_1, c1_3


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

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0))
   ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1))
S tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCODE_F_2, ENCARG_1

Compound Symbols:   c4_1, c_1, c1_3, c1_2, c1_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(f(a, f(z0, a)), a) -> c4(F(z0, a))
We considered the (Usable) Rules:
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0))
   ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_F(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(F(x_1, x_2)) = [2]x_1 + [2]x_1*x_2
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(cons_f(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = [1] + x_1 + x_2

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(a, f(z0, a)), a) -> f(a, f(f(z0, a), a))
Tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0))
   ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1))
S tuples:none
K tuples:
   F(f(a, f(z0, a)), a) -> c4(F(z0, a))
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCODE_F_2, ENCARG_1

Compound Symbols:   c4_1, c_1, c1_3, c1_2, c1_1


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

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

(24)
BOUNDS(1, 1)