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


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


WORST_CASE(?, O(n^3))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 44 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 [ComplexityIfPolyImplication, 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^2)), 117 ms]
(22) CdtProblem
(23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 77 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^3).


The TRS R consists of the following rules:

   a(b(x1)) -> b(a(x1))
   b(a(x1)) -> a(c(b(x1)))

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(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   a(b(x1)) -> b(a(x1))
   b(a(x1)) -> a(c(b(x1)))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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^3).


The TRS R consists of the following rules:

   a(b(x1)) -> b(a(x1))
   b(a(x1)) -> a(c(b(x1)))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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^3).


The TRS R consists of the following rules:

   a(c_b(x1)) -> b(a(x1))
   b(c_a(x1)) -> a(c(b(x1)))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   a(x0) -> c_a(x0)
   b(x0) -> c_b(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^3).


The TRS R consists of the following rules:

   a(c_b(x1)) -> b(a(x1))
   b(c_a(x1)) -> a(c(b(x1)))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   a(x0) -> c_a(x0)
   b(x0) -> c_b(x0)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c6(ENCARG(z0))
   A(z0) -> c7
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(z0) -> c9
   B(c_a(z0)) -> c10(A(c(b(z0))), B(z0))
S tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(A(c(b(z0))), B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, A_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c6_1, c7, c8_2, c9, c10_2


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_C(z0) -> c6(ENCARG(z0))
Removed 2 trailing nodes:
   B(z0) -> c9
   A(z0) -> c7

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(A(c(b(z0))), B(z0))
S tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(A(c(b(z0))), B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_B_1, A_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c8_2, c10_2


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
S tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_B_1, A_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c8_2, c10_1


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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_A(z0) -> c6(ENCARG(z0))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
   ENCODE_B(z0) -> c6(ENCARG(z0))
S tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1

Compound Symbols:   c1_1, c2_2, c3_2, c8_2, c10_1, c6_1


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

(17) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A(z0) -> c6(ENCARG(z0))
   ENCODE_B(z0) -> c6(ENCARG(z0))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
S tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1

Compound Symbols:   c1_1, c2_2, c3_2, c8_2, c10_1, c6_1


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

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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
S tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1

Compound Symbols:   c1_1, c2_2, c3_2, c8_2, c10_1, c6_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.
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
We considered the (Usable) Rules:
   a(z0) -> c_a(z0)
   encArg(c(z0)) -> c(encArg(z0))
   b(z0) -> c_b(z0)
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(c_b(z0)) -> b(a(z0))
   b(c_a(z0)) -> a(c(b(z0)))
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [2]x_1
   POL(B(x_1)) = [1]
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_A(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_B(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(a(x_1)) = x_1
   POL(b(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_a(x_1)) = x_1
   POL(c_b(x_1)) = [1] + x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
S tuples:
   B(c_a(z0)) -> c10(B(z0))
K tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1

Compound Symbols:   c1_1, c2_2, c3_2, c8_2, c10_1, c6_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   B(c_a(z0)) -> c10(B(z0))
We considered the (Usable) Rules:
   a(z0) -> c_a(z0)
   encArg(c(z0)) -> c(encArg(z0))
   b(z0) -> c_b(z0)
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(c_b(z0)) -> b(a(z0))
   b(c_a(z0)) -> a(c(b(z0)))
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1] + x_1^2
   POL(B(x_1)) = x_1
   POL(ENCARG(x_1)) = x_1^3
   POL(ENCODE_A(x_1)) = [1] + x_1^2
   POL(ENCODE_B(x_1)) = x_1
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_a(x_1)) = [1] + x_1
   POL(c_b(x_1)) = [1] + x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1

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(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> c_a(z0)
   a(c_b(z0)) -> b(a(z0))
   b(z0) -> c_b(z0)
   b(c_a(z0)) -> a(c(b(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)))
   ENCODE_B(z0) -> c6(B(encArg(z0)))
S tuples:none
K tuples:
   A(c_b(z0)) -> c8(B(a(z0)), A(z0))
   B(c_a(z0)) -> c10(B(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1

Compound Symbols:   c1_1, c2_2, c3_2, c8_2, c10_1, c6_1


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(25) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(26)
BOUNDS(1, 1)