/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 170 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), 15 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(16) CdtProblem
(17) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 257 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 209 ms]
(22) CdtProblem
(23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) 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:

   or(x, x) -> x
   and(x, x) -> x
   not(not(x)) -> x
   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))

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(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))


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

(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:

   or(x, x) -> x
   and(x, x) -> x
   not(not(x)) -> x
   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))

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

   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(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^2).


The TRS R consists of the following rules:

   or(x, x) -> x
   and(x, x) -> x
   not(not(x)) -> x
   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))

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

   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(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^2).


The TRS R consists of the following rules:

   or(x, x) -> x
   and(x, x) -> x
   not(c_and(x, y)) -> or(not(x), not(y))
   not(c_not(x)) -> x
   not(c_or(x, y)) -> and(not(x), not(y))

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

   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   or(x0, x1) -> c_or(x0, x1)
   and(x0, x1) -> c_and(x0, x1)
   not(x0) -> c_not(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:

   or(x, x) -> x
   and(x, x) -> x
   not(c_and(x, y)) -> or(not(x), not(y))
   not(c_not(x)) -> x
   not(c_or(x, y)) -> and(not(x), not(y))

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

   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   or(x0, x1) -> c_or(x0, x1)
   and(x0, x1) -> c_and(x0, x1)
   not(x0) -> c_not(x0)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCODE_OR(z0, z1) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AND(z0, z1) -> c4(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_NOT(z0) -> c5(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z1) -> c6
   OR(z0, z0) -> c7
   AND(z0, z1) -> c8
   AND(z0, z0) -> c9
   NOT(z0) -> c10
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
S tuples:
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
K tuples:none
Defined Rule Symbols:   or_2, and_2, not_1, encArg_1, encode_or_2, encode_and_2, encode_not_1

Defined Pair Symbols:   ENCARG_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1, OR_2, AND_2, NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c3_3, c4_3, c5_2, c6, c7, c8, c9, c10, c11_3, c12, c13_3


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

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   NOT(z0) -> c10
   AND(z0, z1) -> c8
   OR(z0, z1) -> c6

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCODE_OR(z0, z1) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AND(z0, z1) -> c4(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_NOT(z0) -> c5(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
S tuples:
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
K tuples:none
Defined Rule Symbols:   or_2, and_2, not_1, encArg_1, encode_or_2, encode_and_2, encode_not_1

Defined Pair Symbols:   ENCARG_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1, OR_2, AND_2, NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c3_3, c4_3, c5_2, c7, c9, c11_3, c12, c13_3


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c6(ENCARG(z0))
   ENCODE_OR(z0, z1) -> c6(ENCARG(z1))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(ENCARG(z0))
   ENCODE_AND(z0, z1) -> c6(ENCARG(z1))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
   ENCODE_NOT(z0) -> c6(ENCARG(z0))
S tuples:
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
K tuples:none
Defined Rule Symbols:   or_2, and_2, not_1, encArg_1, encode_or_2, encode_and_2, encode_not_1

Defined Pair Symbols:   ENCARG_1, OR_2, AND_2, NOT_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c7, c9, c11_3, c12, c13_3, c6_1


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

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 5 leading nodes:
   ENCODE_OR(z0, z1) -> c6(ENCARG(z0))
   ENCODE_OR(z0, z1) -> c6(ENCARG(z1))
   ENCODE_AND(z0, z1) -> c6(ENCARG(z0))
   ENCODE_AND(z0, z1) -> c6(ENCARG(z1))
   ENCODE_NOT(z0) -> c6(ENCARG(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
S tuples:
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
K tuples:none
Defined Rule Symbols:   or_2, and_2, not_1, encArg_1, encode_or_2, encode_and_2, encode_not_1

Defined Pair Symbols:   ENCARG_1, OR_2, AND_2, NOT_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c7, c9, c11_3, c12, c13_3, c6_1


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

(17) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
S tuples:
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
K tuples:none
Defined Rule Symbols:   encArg_1, or_2, and_2, not_1

Defined Pair Symbols:   ENCARG_1, OR_2, AND_2, NOT_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c7, c9, c11_3, c12, c13_3, c6_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
We considered the (Usable) Rules:
   encArg(cons_not(z0)) -> not(encArg(z0))
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
   or(z0, z1) -> c_or(z0, z1)
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   or(z0, z0) -> z0
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_not(z0)) -> z0
   and(z0, z1) -> c_and(z0, z1)
And the Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AND(x_1, x_2)) = 0
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_AND(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + x_1^2
   POL(ENCODE_NOT(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_OR(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + x_1^2
   POL(NOT(x_1)) = [2]x_1
   POL(OR(x_1, x_2)) = 0
   POL(and(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12) = 0
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7) = 0
   POL(c9) = 0
   POL(c_and(x_1, x_2)) = [1] + x_1 + x_2
   POL(c_not(x_1)) = x_1
   POL(c_or(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_and(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_not(x_1)) = [2] + x_1
   POL(cons_or(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(not(x_1)) = x_1
   POL(or(x_1, x_2)) = [1] + x_1 + x_2

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
S tuples:
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_not(z0)) -> c12
K tuples:
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
Defined Rule Symbols:   encArg_1, or_2, and_2, not_1

Defined Pair Symbols:   ENCARG_1, OR_2, AND_2, NOT_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c7, c9, c11_3, c12, c13_3, 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.
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_not(z0)) -> c12
We considered the (Usable) Rules:
   encArg(cons_not(z0)) -> not(encArg(z0))
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
   or(z0, z1) -> c_or(z0, z1)
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   or(z0, z0) -> z0
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_not(z0)) -> z0
   and(z0, z1) -> c_and(z0, z1)
And the Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AND(x_1, x_2)) = [2]
   POL(ENCARG(x_1)) = [2] + [2]x_1^2
   POL(ENCODE_AND(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2
   POL(ENCODE_NOT(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_OR(x_1, x_2)) = [2] + [2]x_1 + [2]x_2^2 + x_1*x_2 + [2]x_1^2
   POL(NOT(x_1)) = [2] + [2]x_1
   POL(OR(x_1, x_2)) = [2]
   POL(and(x_1, x_2)) = [2] + x_1 + x_2
   POL(c(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12) = 0
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7) = 0
   POL(c9) = 0
   POL(c_and(x_1, x_2)) = [2] + x_1 + x_2
   POL(c_not(x_1)) = x_1
   POL(c_or(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_and(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_not(x_1)) = [2] + x_1
   POL(cons_or(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(not(x_1)) = x_1
   POL(or(x_1, x_2)) = [2] + x_1 + x_2

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_not(z0)) -> not(encArg(z0))
   or(z0, z1) -> c_or(z0, z1)
   or(z0, z0) -> z0
   and(z0, z1) -> c_and(z0, z1)
   and(z0, z0) -> z0
   not(z0) -> c_not(z0)
   not(c_and(z0, z1)) -> or(not(z0), not(z1))
   not(c_not(z0)) -> z0
   not(c_or(z0, z1)) -> and(not(z0), not(z1))
Tuples:
   ENCARG(cons_or(z0, z1)) -> c(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c1(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_not(z0)) -> c12
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   ENCODE_OR(z0, z1) -> c6(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c6(AND(encArg(z0), encArg(z1)))
   ENCODE_NOT(z0) -> c6(NOT(encArg(z0)))
S tuples:none
K tuples:
   NOT(c_and(z0, z1)) -> c11(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
   NOT(c_or(z0, z1)) -> c13(AND(not(z0), not(z1)), NOT(z0), NOT(z1))
   OR(z0, z0) -> c7
   AND(z0, z0) -> c9
   NOT(c_not(z0)) -> c12
Defined Rule Symbols:   encArg_1, or_2, and_2, not_1

Defined Pair Symbols:   ENCARG_1, OR_2, AND_2, NOT_1, ENCODE_OR_2, ENCODE_AND_2, ENCODE_NOT_1

Compound Symbols:   c_3, c1_3, c2_2, c7, c9, c11_3, c12, c13_3, c6_1


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

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

(24)
BOUNDS(1, 1)