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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 174 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^3)), 698 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 86 ms]
    (24) CdtProblem
    (25) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) BOUNDS(1, 1)
(27) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CpxRelTRS
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) typed CpxTrs
    (31) OrderProof [LOWER BOUND(ID), 0 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 564 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
            (40) BOUNDS(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:

   +(*(x, y), *(x, z)) -> *(x, +(y, z))
   +(+(x, y), z) -> +(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_u -> u


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

(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:

   +(*(x, y), *(x, z)) -> *(x, +(y, z))
   +(+(x, y), z) -> +(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_u -> u

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:

   +(*(x, y), *(x, z)) -> *(x, +(y, z))
   +(+(x, y), z) -> +(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_u -> u

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:

   +(*(x, y), *(x, z)) -> *(x, +(y, z))
   +(c_+(x, y), z) -> +(x, +(y, z))
   +(*(x, y), c_+(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_u -> u
   +(x0, x1) -> c_+(x0, x1)

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:

   +(*(x, y), *(x, z)) -> *(x, +(y, z))
   +(c_+(x, y), z) -> +(x, +(y, z))
   +(*(x, y), c_+(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_u -> u
   +(x0, x1) -> c_+(x0, x1)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_u -> u
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(u) -> c1
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_*(z0, z1) -> c4(ENCARG(z0), ENCARG(z1))
   ENCODE_U -> c5
   +'(z0, z1) -> c6
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
S tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2, encode_u

Defined Pair Symbols:   ENCARG_1, ENCODE_+_2, ENCODE_*_2, ENCODE_U, +'_2

Compound Symbols:   c_2, c1, c2_3, c3_3, c4_2, c5, c6, c7_1, c8_2, c9_2


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_*(z0, z1) -> c4(ENCARG(z0), ENCARG(z1))
Removed 3 trailing nodes:
   ENCODE_U -> c5
   ENCARG(u) -> c1
   +'(z0, z1) -> c6

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_u -> u
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
S tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2, encode_u

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

Compound Symbols:   c_2, c2_3, c3_3, c7_1, c8_2, c9_2


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_u -> u
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
S tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2, encode_u

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

Compound Symbols:   c_2, c2_3, c3_3, c7_1, c8_2, c9_1


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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_u -> u
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1)))
   ENCODE_+(z0, z1) -> c1(ENCARG(z0))
   ENCODE_+(z0, z1) -> c1(ENCARG(z1))
S tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2, encode_u

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

Compound Symbols:   c_2, c2_3, c7_1, c8_2, c9_1, c1_1


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

(17) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_+(z0, z1) -> c1(ENCARG(z0))
   ENCODE_+(z0, z1) -> c1(ENCARG(z1))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_u -> u
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2, encode_u

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

Compound Symbols:   c_2, c2_3, c7_1, c8_2, c9_1, c1_1


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

(19) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_u -> u

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
K tuples:none
Defined Rule Symbols:   encArg_1, +_2

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

Compound Symbols:   c_2, c2_3, c7_1, c8_2, c9_1, c1_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
We considered the (Usable) Rules:
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
   encArg(u) -> u
And the Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(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_1 + x_2
   POL(+'(x_1, x_2)) = x_1^2
   POL(ENCARG(x_1)) = x_1^3
   POL(ENCODE_+(x_1, x_2)) = x_1^2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(c_+(x_1, x_2)) = x_1 + x_2
   POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(u) = 0

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
K tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
Defined Rule Symbols:   encArg_1, +_2

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

Compound Symbols:   c_2, c2_3, c7_1, c8_2, c9_1, c1_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
We considered the (Usable) Rules:
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
   encArg(u) -> u
And the Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(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)) = [1] + x_1 + x_2
   POL(+'(x_1, x_2)) = x_1
   POL(ENCARG(x_1)) = [2] + 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(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(c_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_+(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = [2] + [2]x_1
   POL(u) = [1]

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(u) -> u
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(z0, z1) -> c_+(z0, z1)
   +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2))
   +(c_+(z0, z1), z2) -> +(z0, +(z1, z2))
   +(*(z0, z1), c_+(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   +'(*(z0, z1), *(z0, z2)) -> c7(+'(z1, z2))
   +'(*(z0, z1), c_+(*(z0, z2), u)) -> c9(+'(z1, z2))
   +'(c_+(z0, z1), z2) -> c8(+'(z0, +(z1, z2)), +'(z1, z2))
Defined Rule Symbols:   encArg_1, +_2

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

Compound Symbols:   c_2, c2_3, c7_1, c8_2, c9_1, c1_1


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

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

(26)
BOUNDS(1, 1)

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

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

(28)
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:

   +'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
   +'(+'(x, y), z) -> +'(x, +'(y, z))
   +'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)

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

   encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
   encode_u -> u

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

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

(30)
Obligation:
TRS:
Rules:
+'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
+'(+'(x, y), z) -> +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(u) -> u
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_u -> u

Types:
+' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
*' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
u :: *':u:cons_+
encArg :: *':u:cons_+ -> *':u:cons_+
cons_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_* :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_u :: *':u:cons_+
hole_*':u:cons_+1_0 :: *':u:cons_+
gen_*':u:cons_+2_0 :: Nat -> *':u:cons_+

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

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

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

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

(32)
Obligation:
TRS:
Rules:
+'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
+'(+'(x, y), z) -> +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(u) -> u
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_u -> u

Types:
+' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
*' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
u :: *':u:cons_+
encArg :: *':u:cons_+ -> *':u:cons_+
cons_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_* :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_u :: *':u:cons_+
hole_*':u:cons_+1_0 :: *':u:cons_+
gen_*':u:cons_+2_0 :: Nat -> *':u:cons_+


Generator Equations:
gen_*':u:cons_+2_0(0) <=> u
gen_*':u:cons_+2_0(+(x, 1)) <=> *'(u, gen_*':u:cons_+2_0(x))


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

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

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_*':u:cons_+2_0(+(1, n4_0)), gen_*':u:cons_+2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
+'(gen_*':u:cons_+2_0(+(1, 0)), gen_*':u:cons_+2_0(+(1, 0)))

Induction Step:
+'(gen_*':u:cons_+2_0(+(1, +(n4_0, 1))), gen_*':u:cons_+2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
*'(u, +'(gen_*':u:cons_+2_0(+(1, n4_0)), gen_*':u:cons_+2_0(+(1, n4_0)))) ->_IH
*'(u, *3_0)

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

(34)
Complex Obligation (BEST)

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

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
+'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
+'(+'(x, y), z) -> +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(u) -> u
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_u -> u

Types:
+' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
*' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
u :: *':u:cons_+
encArg :: *':u:cons_+ -> *':u:cons_+
cons_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_* :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_u :: *':u:cons_+
hole_*':u:cons_+1_0 :: *':u:cons_+
gen_*':u:cons_+2_0 :: Nat -> *':u:cons_+


Generator Equations:
gen_*':u:cons_+2_0(0) <=> u
gen_*':u:cons_+2_0(+(x, 1)) <=> *'(u, gen_*':u:cons_+2_0(x))


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

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

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
TRS:
Rules:
+'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
+'(+'(x, y), z) -> +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(u) -> u
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_u -> u

Types:
+' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
*' :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
u :: *':u:cons_+
encArg :: *':u:cons_+ -> *':u:cons_+
cons_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_+ :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_* :: *':u:cons_+ -> *':u:cons_+ -> *':u:cons_+
encode_u :: *':u:cons_+
hole_*':u:cons_+1_0 :: *':u:cons_+
gen_*':u:cons_+2_0 :: Nat -> *':u:cons_+


Lemmas:
+'(gen_*':u:cons_+2_0(+(1, n4_0)), gen_*':u:cons_+2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_*':u:cons_+2_0(0) <=> u
gen_*':u:cons_+2_0(+(x, 1)) <=> *'(u, gen_*':u:cons_+2_0(x))


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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_*':u:cons_+2_0(n1547_0)) -> gen_*':u:cons_+2_0(n1547_0), rt in Omega(0)

Induction Base:
encArg(gen_*':u:cons_+2_0(0)) ->_R^Omega(0)
u

Induction Step:
encArg(gen_*':u:cons_+2_0(+(n1547_0, 1))) ->_R^Omega(0)
*'(encArg(u), encArg(gen_*':u:cons_+2_0(n1547_0))) ->_R^Omega(0)
*'(u, encArg(gen_*':u:cons_+2_0(n1547_0))) ->_IH
*'(u, gen_*':u:cons_+2_0(c1548_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(40)
BOUNDS(1, INF)