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


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


WORST_CASE(Omega(n^1), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 145 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), 3 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)), 225 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 153 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 533 ms]
    (24) CdtProblem
    (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 455 ms]
    (26) CdtProblem
    (27) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) BOUNDS(1, 1)
(29) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CpxRelTRS
    (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) typed CpxTrs
    (33) OrderProof [LOWER BOUND(ID), 0 ms]
    (34) typed CpxTrs
    (35) RewriteLemmaProof [LOWER BOUND(ID), 207 ms]
    (36) proven lower bound
    (37) LowerBoundPropagationProof [FINISHED, 0 ms]
    (38) BOUNDS(n^1, INF)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   i(i(x)) -> x
   +(i(x), x) -> 0
   +(x, i(x)) -> 0
   i(+(x, y)) -> +(i(x), i(y))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, i(y)), y) -> x
   +(+(x, y), i(y)) -> x

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_i(x_1)) -> i(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_0 -> 0
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   i(i(x)) -> x
   +(i(x), x) -> 0
   +(x, i(x)) -> 0
   i(+(x, y)) -> +(i(x), i(y))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, i(y)), y) -> x
   +(+(x, y), i(y)) -> x

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

   encArg(0) -> 0
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_0 -> 0
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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(n^1, n^3).


The TRS R consists of the following rules:

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   i(i(x)) -> x
   +(i(x), x) -> 0
   +(x, i(x)) -> 0
   i(+(x, y)) -> +(i(x), i(y))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, i(y)), y) -> x
   +(+(x, y), i(y)) -> x

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

   encArg(0) -> 0
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_0 -> 0
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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:

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   +(c_+(x, c_i(y)), y) -> x
   +(c_+(x, y), c_i(y)) -> x
   i(c_i(x)) -> x
   +(c_i(x), x) -> 0
   +(x, c_i(x)) -> 0
   +(x, c_+(y, z)) -> +(+(x, y), z)
   i(c_+(x, y)) -> +(i(x), i(y))

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

   encArg(0) -> 0
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_0 -> 0
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   +(x0, x1) -> c_+(x0, x1)
   i(x0) -> c_i(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:

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   +(c_+(x, c_i(y)), y) -> x
   +(c_+(x, y), c_i(y)) -> x
   i(c_i(x)) -> x
   +(c_i(x), x) -> 0
   +(x, c_i(x)) -> 0
   +(x, c_+(y, z)) -> +(+(x, y), z)
   i(c_+(x, y)) -> +(i(x), i(y))

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

   encArg(0) -> 0
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_0 -> 0
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   +(x0, x1) -> c_+(x0, x1)
   i(x0) -> c_i(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_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_0 -> 0
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
Tuples:
   ENCARG(0) -> c
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_I(z0) -> c3(I(encArg(z0)), ENCARG(z0))
   ENCODE_0 -> c4
   ENCODE_+(z0, z1) -> c5(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, z1) -> c6
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(z0) -> c14
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
K tuples:none
Defined Rule Symbols:   i_1, +_2, encArg_1, encode_i_1, encode_0, encode_+_2

Defined Pair Symbols:   ENCARG_1, ENCODE_I_1, ENCODE_0, ENCODE_+_2, +'_2, I_1

Compound Symbols:   c, c1_2, c2_3, c3_2, c4, c5_3, c6, c7, c8, c9, c10, c11, c12, c13_2, c14, c15, c16, c17_3


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

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ENCARG(0) -> c
   ENCODE_0 -> c4
   I(z0) -> c14
   +'(z0, z1) -> c6

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_0 -> 0
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_I(z0) -> c3(I(encArg(z0)), ENCARG(z0))
   ENCODE_+(z0, z1) -> c5(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
K tuples:none
Defined Rule Symbols:   i_1, +_2, encArg_1, encode_i_1, encode_0, encode_+_2

Defined Pair Symbols:   ENCARG_1, ENCODE_I_1, ENCODE_+_2, +'_2, I_1

Compound Symbols:   c1_2, c2_3, c3_2, c5_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_0 -> 0
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_I(z0) -> c(ENCARG(z0))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_+(z0, z1) -> c(ENCARG(z0))
   ENCODE_+(z0, z1) -> c(ENCARG(z1))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
K tuples:none
Defined Rule Symbols:   i_1, +_2, encArg_1, encode_i_1, encode_0, encode_+_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_1


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

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_I(z0) -> c(ENCARG(z0))
   ENCODE_+(z0, z1) -> c(ENCARG(z0))
   ENCODE_+(z0, z1) -> c(ENCARG(z1))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_0 -> 0
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
K tuples:none
Defined Rule Symbols:   i_1, +_2, encArg_1, encode_i_1, encode_0, encode_+_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_1


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

(17) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_i(z0) -> i(encArg(z0))
   encode_0 -> 0
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
K tuples:none
Defined Rule Symbols:   encArg_1, i_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_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.
   I(0) -> c15
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
We considered the (Usable) Rules:
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(0) -> 0
   i(c_i(z0)) -> z0
   i(0) -> 0
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   i(z0) -> c_i(z0)
And the Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(ENCARG(x_1)) = [2]x_1 + [2]x_1^2
   POL(ENCODE_+(x_1, x_2)) = [2] + x_1 + x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2
   POL(ENCODE_I(x_1)) = [1] + [2]x_1 + x_1^2
   POL(I(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c15) = 0
   POL(c16) = 0
   POL(c17(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c7) = 0
   POL(c8) = 0
   POL(c9) = 0
   POL(c_+(x_1, x_2)) = [2] + x_1 + x_2
   POL(c_i(x_1)) = x_1
   POL(cons_+(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_i(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + [2]x_1
   POL(i(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(c_i(z0)) -> c16
K tuples:
   I(0) -> c15
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
Defined Rule Symbols:   encArg_1, i_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_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.
   I(c_i(z0)) -> c16
We considered the (Usable) Rules:
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(0) -> 0
   i(c_i(z0)) -> z0
   i(0) -> 0
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   i(z0) -> c_i(z0)
And the Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = 0
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_+(x_1, x_2)) = [2] + [2]x_1 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(ENCODE_I(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(I(x_1)) = [2] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c15) = 0
   POL(c16) = 0
   POL(c17(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c7) = 0
   POL(c8) = 0
   POL(c9) = 0
   POL(c_+(x_1, x_2)) = [2] + x_1 + x_2
   POL(c_i(x_1)) = x_1
   POL(cons_+(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_i(x_1)) = [2] + x_1
   POL(encArg(x_1)) = [2]x_1
   POL(i(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
K tuples:
   I(0) -> c15
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   I(c_i(z0)) -> c16
Defined Rule Symbols:   encArg_1, i_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_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.
   +'(z0, 0) -> c8
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
We considered the (Usable) Rules:
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(0) -> 0
   i(c_i(z0)) -> z0
   i(0) -> 0
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   i(z0) -> c_i(z0)
And the Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(+'(x_1, x_2)) = x_2
   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1^2 + x_1^3
   POL(ENCODE_+(x_1, x_2)) = x_2
   POL(ENCODE_I(x_1)) = x_1^2
   POL(I(x_1)) = x_1^2
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c15) = 0
   POL(c16) = 0
   POL(c17(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c7) = 0
   POL(c8) = 0
   POL(c9) = 0
   POL(c_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(c_i(x_1)) = x_1
   POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_i(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(i(x_1)) = x_1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(0, z0) -> c7
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
K tuples:
   I(0) -> c15
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   I(c_i(z0)) -> c16
   +'(z0, 0) -> c8
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
Defined Rule Symbols:   encArg_1, i_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_1


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

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(0, z0) -> c7
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
We considered the (Usable) Rules:
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(0) -> 0
   i(c_i(z0)) -> z0
   i(0) -> 0
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   i(z0) -> c_i(z0)
And the Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(+'(x_1, x_2)) = [1] + x_2
   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1 + x_1^3
   POL(ENCODE_+(x_1, x_2)) = [1] + x_2
   POL(ENCODE_I(x_1)) = x_1 + x_1^2
   POL(I(x_1)) = x_1 + x_1^2
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c15) = 0
   POL(c16) = 0
   POL(c17(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c7) = 0
   POL(c8) = 0
   POL(c9) = 0
   POL(c_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(c_i(x_1)) = x_1
   POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_i(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(i(x_1)) = x_1

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_i(z0)) -> i(encArg(z0))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   i(z0) -> c_i(z0)
   i(0) -> 0
   i(c_i(z0)) -> z0
   i(c_+(z0, z1)) -> +(i(z0), i(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(0, z0) -> z0
   +(z0, 0) -> z0
   +(c_+(z0, c_i(z1)), z1) -> z0
   +(c_+(z0, z1), c_i(z1)) -> z0
   +(c_i(z0), z0) -> 0
   +(z0, c_i(z0)) -> 0
   +(z0, c_+(z1, z2)) -> +(+(z0, z1), z2)
Tuples:
   ENCARG(cons_i(z0)) -> c1(I(encArg(z0)), ENCARG(z0))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(0, z0) -> c7
   +'(z0, 0) -> c8
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   I(0) -> c15
   I(c_i(z0)) -> c16
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   I(0) -> c15
   I(c_+(z0, z1)) -> c17(+'(i(z0), i(z1)), I(z0), I(z1))
   I(c_i(z0)) -> c16
   +'(z0, 0) -> c8
   +'(z0, c_+(z1, z2)) -> c13(+'(+(z0, z1), z2), +'(z0, z1))
   +'(0, z0) -> c7
   +'(c_+(z0, c_i(z1)), z1) -> c9
   +'(c_+(z0, z1), c_i(z1)) -> c10
   +'(c_i(z0), z0) -> c11
   +'(z0, c_i(z0)) -> c12
Defined Rule Symbols:   encArg_1, i_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, I_1, ENCODE_I_1, ENCODE_+_2

Compound Symbols:   c1_2, c2_3, c7, c8, c9, c10, c11, c12, c13_2, c15, c16, c17_3, c_1


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

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

(28)
BOUNDS(1, 1)

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

(29) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(30)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   i(0') -> 0'
   +'(0', y) -> y
   +'(x, 0') -> x
   i(i(x)) -> x
   +'(i(x), x) -> 0'
   +'(x, i(x)) -> 0'
   i(+'(x, y)) -> +'(i(x), i(y))
   +'(x, +'(y, z)) -> +'(+'(x, y), z)
   +'(+'(x, i(y)), y) -> x
   +'(+'(x, y), i(y)) -> x

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

   encArg(0') -> 0'
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_0 -> 0'
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

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

(31) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(32)
Obligation:
TRS:
Rules:
i(0') -> 0'
+'(0', y) -> y
+'(x, 0') -> x
i(i(x)) -> x
+'(i(x), x) -> 0'
+'(x, i(x)) -> 0'
i(+'(x, y)) -> +'(i(x), i(y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
+'(+'(x, i(y)), y) -> x
+'(+'(x, y), i(y)) -> x
encArg(0') -> 0'
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_i(x_1) -> i(encArg(x_1))
encode_0 -> 0'
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
0' :: 0':cons_i:cons_+
+' :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
encArg :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
cons_i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
cons_+ :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
encode_i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
encode_0 :: 0':cons_i:cons_+
encode_+ :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
hole_0':cons_i:cons_+1_3 :: 0':cons_i:cons_+
gen_0':cons_i:cons_+2_3 :: Nat -> 0':cons_i:cons_+

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

(33) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
i, +', encArg

They will be analysed ascendingly in the following order:
+' < i
i < encArg
+' < encArg

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

(34)
Obligation:
TRS:
Rules:
i(0') -> 0'
+'(0', y) -> y
+'(x, 0') -> x
i(i(x)) -> x
+'(i(x), x) -> 0'
+'(x, i(x)) -> 0'
i(+'(x, y)) -> +'(i(x), i(y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
+'(+'(x, i(y)), y) -> x
+'(+'(x, y), i(y)) -> x
encArg(0') -> 0'
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_i(x_1) -> i(encArg(x_1))
encode_0 -> 0'
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
0' :: 0':cons_i:cons_+
+' :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
encArg :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
cons_i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
cons_+ :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
encode_i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
encode_0 :: 0':cons_i:cons_+
encode_+ :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
hole_0':cons_i:cons_+1_3 :: 0':cons_i:cons_+
gen_0':cons_i:cons_+2_3 :: Nat -> 0':cons_i:cons_+


Generator Equations:
gen_0':cons_i:cons_+2_3(0) <=> 0'
gen_0':cons_i:cons_+2_3(+(x, 1)) <=> cons_i(gen_0':cons_i:cons_+2_3(x))


The following defined symbols remain to be analysed:
+', i, encArg

They will be analysed ascendingly in the following order:
+' < i
i < encArg
+' < encArg

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':cons_i:cons_+2_3(n155_3)) -> gen_0':cons_i:cons_+2_3(0), rt in Omega(n155_3)

Induction Base:
encArg(gen_0':cons_i:cons_+2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':cons_i:cons_+2_3(+(n155_3, 1))) ->_R^Omega(0)
i(encArg(gen_0':cons_i:cons_+2_3(n155_3))) ->_IH
i(gen_0':cons_i:cons_+2_3(0)) ->_R^Omega(1)
0'

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(36)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
i(0') -> 0'
+'(0', y) -> y
+'(x, 0') -> x
i(i(x)) -> x
+'(i(x), x) -> 0'
+'(x, i(x)) -> 0'
i(+'(x, y)) -> +'(i(x), i(y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
+'(+'(x, i(y)), y) -> x
+'(+'(x, y), i(y)) -> x
encArg(0') -> 0'
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_i(x_1) -> i(encArg(x_1))
encode_0 -> 0'
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
0' :: 0':cons_i:cons_+
+' :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
encArg :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
cons_i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
cons_+ :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
encode_i :: 0':cons_i:cons_+ -> 0':cons_i:cons_+
encode_0 :: 0':cons_i:cons_+
encode_+ :: 0':cons_i:cons_+ -> 0':cons_i:cons_+ -> 0':cons_i:cons_+
hole_0':cons_i:cons_+1_3 :: 0':cons_i:cons_+
gen_0':cons_i:cons_+2_3 :: Nat -> 0':cons_i:cons_+


Generator Equations:
gen_0':cons_i:cons_+2_3(0) <=> 0'
gen_0':cons_i:cons_+2_3(+(x, 1)) <=> cons_i(gen_0':cons_i:cons_+2_3(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(37) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(38)
BOUNDS(n^1, INF)