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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 171 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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)), 66 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 30 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms]
(20) CdtProblem
(21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) 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(a, y) -> f(y, g(y))
   g(a) -> b
   g(b) -> b

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(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b


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

(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(a, y) -> f(y, g(y))
   g(a) -> b
   g(b) -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b

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(a, y) -> f(y, g(y))
   g(a) -> b
   g(b) -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b

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(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_b -> b
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(a) -> c
   ENCARG(b) -> c1
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_A -> c5
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_B -> c7
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
S tuples:
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_a, encode_g_1, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_A, ENCODE_G_1, ENCODE_B, F_2, G_1

Compound Symbols:   c, c1, c2_3, c3_2, c4_3, c5, c6_2, c7, c8_2, c9, c10


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ENCODE_B -> c7
   ENCODE_A -> c5
   ENCARG(a) -> c
   ENCARG(b) -> c1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_b -> b
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
S tuples:
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_a, encode_g_1, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_G_1, F_2, G_1

Compound Symbols:   c2_3, c3_2, c4_3, c6_2, c8_2, c9, c10


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_b -> b
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1) -> c(ENCARG(z0))
   ENCODE_F(z0, z1) -> c(ENCARG(z1))
   ENCODE_G(z0) -> c(G(encArg(z0)))
   ENCODE_G(z0) -> c(ENCARG(z0))
S tuples:
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_a, encode_g_1, encode_b

Defined Pair Symbols:   ENCARG_1, F_2, G_1, ENCODE_F_2, ENCODE_G_1

Compound Symbols:   c2_3, c3_2, c8_2, c9, c10, c_1


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

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_b -> b
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_a, encode_g_1, encode_b

Defined Pair Symbols:   ENCARG_1, F_2, G_1, ENCODE_F_2, ENCODE_G_1

Compound Symbols:   c2_3, c3_2, c8_2, c9, c10, c_1


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

(13) 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
   encode_g(z0) -> g(encArg(z0))
   encode_b -> b

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
K tuples:none
Defined Rule Symbols:   encArg_1, f_2, g_1

Defined Pair Symbols:   ENCARG_1, F_2, G_1, ENCODE_F_2, ENCODE_G_1

Compound Symbols:   c2_3, c3_2, c8_2, c9, c10, c_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.
   G(a) -> c9
We considered the (Usable) Rules:
   g(b) -> b
   g(a) -> b
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   encArg(b) -> b
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = [1]
   POL(F(x_1, x_2)) = x_2
   POL(G(x_1)) = x_1
   POL(a) = [1]
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c10) = 0
   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(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1]
   POL(f(x_1, x_2)) = x_2
   POL(g(x_1)) = 0

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(b) -> c10
K tuples:
   G(a) -> c9
Defined Rule Symbols:   encArg_1, f_2, g_1

Defined Pair Symbols:   ENCARG_1, F_2, G_1, ENCODE_F_2, ENCODE_G_1

Compound Symbols:   c2_3, c3_2, c8_2, c9, c10, c_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(a, z0) -> c8(F(z0, g(z0)), G(z0))
We considered the (Usable) Rules:
   g(b) -> b
   g(a) -> b
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   encArg(b) -> b
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [2] + [2]x_1
   POL(ENCODE_F(x_1, x_2)) = [2]
   POL(ENCODE_G(x_1)) = [1] + x_1
   POL(F(x_1, x_2)) = x_1 + x_2
   POL(G(x_1)) = 0
   POL(a) = [1]
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c10) = 0
   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(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(cons_f(x_1, x_2)) = [3] + x_1 + x_2
   POL(cons_g(x_1)) = x_1
   POL(encArg(x_1)) = [1]
   POL(f(x_1, x_2)) = 0
   POL(g(x_1)) = 0

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   G(b) -> c10
K tuples:
   G(a) -> c9
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
Defined Rule Symbols:   encArg_1, f_2, g_1

Defined Pair Symbols:   ENCARG_1, F_2, G_1, ENCODE_F_2, ENCODE_G_1

Compound Symbols:   c2_3, c3_2, c8_2, c9, c10, c_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(b) -> c10
We considered the (Usable) Rules:
   g(b) -> b
   g(a) -> b
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   encArg(b) -> b
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [2]x_1
   POL(ENCODE_F(x_1, x_2)) = [3]
   POL(ENCODE_G(x_1)) = [1]
   POL(F(x_1, x_2)) = x_1 + x_2
   POL(G(x_1)) = [1]
   POL(a) = [1]
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c10) = 0
   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(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1]
   POL(f(x_1, x_2)) = 0
   POL(g(x_1)) = 0

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(a, z0) -> f(z0, g(z0))
   g(a) -> b
   g(b) -> b
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(a) -> c9
   G(b) -> c10
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:none
K tuples:
   G(a) -> c9
   F(a, z0) -> c8(F(z0, g(z0)), G(z0))
   G(b) -> c10
Defined Rule Symbols:   encArg_1, f_2, g_1

Defined Pair Symbols:   ENCARG_1, F_2, G_1, ENCODE_F_2, ENCODE_G_1

Compound Symbols:   c2_3, c3_2, c8_2, c9, c10, c_1


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

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

(22)
BOUNDS(1, 1)