WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1560 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 11 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 517 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 7261 ms]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 2007 ms]
(14) BEST
    (15) proven lower bound
        (16) LowerBoundPropagationProof [FINISHED, 0 ms]
        (17) BOUNDS(n^1, INF)
    (18) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
   A__U11(z0, z1) -> c1
   A__U12(tt) -> c2
   A__U12(z0) -> c3
   A__U21(tt) -> c4
   A__U21(z0) -> c5
   A__U31(tt, z0) -> c6(MARK(z0))
   A__U31(z0, z1) -> c7
   A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
   A__U41(z0, z1, z2) -> c9
   A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U42(z0, z1, z2) -> c12
   A__ISNAT(0) -> c13
   A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
   A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
   A__ISNAT(z0) -> c16
   A__PLUS(z0, 0) -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
   A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
   A__PLUS(z0, z1) -> c19
   MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
   MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
   MARK(isNat(z0)) -> c22(A__ISNAT(z0))
   MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
   MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
   MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
   MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c29
   MARK(s(z0)) -> c30(MARK(z0))
   MARK(0) -> c31

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

   a__U11(tt, z0) -> a__U12(a__isNat(z0))
   a__U11(z0, z1) -> U11(z0, z1)
   a__U12(tt) -> tt
   a__U12(z0) -> U12(z0)
   a__U21(tt) -> tt
   a__U21(z0) -> U21(z0)
   a__U31(tt, z0) -> mark(z0)
   a__U31(z0, z1) -> U31(z0, z1)
   a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
   a__U41(z0, z1, z2) -> U41(z0, z1, z2)
   a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U42(z0, z1, z2) -> U42(z0, z1, z2)
   a__isNat(0) -> tt
   a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
   a__isNat(s(z0)) -> a__U21(a__isNat(z0))
   a__isNat(z0) -> isNat(z0)
   a__plus(z0, 0) -> a__U31(a__isNat(z0), z0)
   a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
   mark(U12(z0)) -> a__U12(mark(z0))
   mark(isNat(z0)) -> a__isNat(z0)
   mark(U21(z0)) -> a__U21(mark(z0))
   mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
   mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
   mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
   A__U11(z0, z1) -> c1
   A__U12(tt) -> c2
   A__U12(z0) -> c3
   A__U21(tt) -> c4
   A__U21(z0) -> c5
   A__U31(tt, z0) -> c6(MARK(z0))
   A__U31(z0, z1) -> c7
   A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
   A__U41(z0, z1, z2) -> c9
   A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U42(z0, z1, z2) -> c12
   A__ISNAT(0) -> c13
   A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
   A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
   A__ISNAT(z0) -> c16
   A__PLUS(z0, 0) -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
   A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
   A__PLUS(z0, z1) -> c19
   MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
   MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
   MARK(isNat(z0)) -> c22(A__ISNAT(z0))
   MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
   MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
   MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
   MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c29
   MARK(s(z0)) -> c30(MARK(z0))
   MARK(0) -> c31

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

   a__U11(tt, z0) -> a__U12(a__isNat(z0))
   a__U11(z0, z1) -> U11(z0, z1)
   a__U12(tt) -> tt
   a__U12(z0) -> U12(z0)
   a__U21(tt) -> tt
   a__U21(z0) -> U21(z0)
   a__U31(tt, z0) -> mark(z0)
   a__U31(z0, z1) -> U31(z0, z1)
   a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
   a__U41(z0, z1, z2) -> U41(z0, z1, z2)
   a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U42(z0, z1, z2) -> U42(z0, z1, z2)
   a__isNat(0) -> tt
   a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
   a__isNat(s(z0)) -> a__U21(a__isNat(z0))
   a__isNat(z0) -> isNat(z0)
   a__plus(z0, 0) -> a__U31(a__isNat(z0), z0)
   a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
   mark(U12(z0)) -> a__U12(mark(z0))
   mark(isNat(z0)) -> a__isNat(z0)
   mark(U21(z0)) -> a__U21(mark(z0))
   mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
   mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
   mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
   A__U11(z0, z1) -> c1
   A__U12(tt) -> c2
   A__U12(z0) -> c3
   A__U21(tt) -> c4
   A__U21(z0) -> c5
   A__U31(tt, z0) -> c6(MARK(z0))
   A__U31(z0, z1) -> c7
   A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
   A__U41(z0, z1, z2) -> c9
   A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U42(z0, z1, z2) -> c12
   A__ISNAT(0') -> c13
   A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
   A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
   A__ISNAT(z0) -> c16
   A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
   A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
   A__PLUS(z0, z1) -> c19
   MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
   MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
   MARK(isNat(z0)) -> c22(A__ISNAT(z0))
   MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
   MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
   MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
   MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c29
   MARK(s(z0)) -> c30(MARK(z0))
   MARK(0') -> c31

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

   a__U11(tt, z0) -> a__U12(a__isNat(z0))
   a__U11(z0, z1) -> U11(z0, z1)
   a__U12(tt) -> tt
   a__U12(z0) -> U12(z0)
   a__U21(tt) -> tt
   a__U21(z0) -> U21(z0)
   a__U31(tt, z0) -> mark(z0)
   a__U31(z0, z1) -> U31(z0, z1)
   a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
   a__U41(z0, z1, z2) -> U41(z0, z1, z2)
   a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U42(z0, z1, z2) -> U42(z0, z1, z2)
   a__isNat(0') -> tt
   a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
   a__isNat(s(z0)) -> a__U21(a__isNat(z0))
   a__isNat(z0) -> isNat(z0)
   a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
   a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
   mark(U12(z0)) -> a__U12(mark(z0))
   mark(isNat(z0)) -> a__isNat(z0)
   mark(U21(z0)) -> a__U21(mark(z0))
   mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
   mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
   mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0') -> 0'

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

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

(6)
Obligation:
Innermost TRS:
Rules:
A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
A__U11(z0, z1) -> c1
A__U12(tt) -> c2
A__U12(z0) -> c3
A__U21(tt) -> c4
A__U21(z0) -> c5
A__U31(tt, z0) -> c6(MARK(z0))
A__U31(z0, z1) -> c7
A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
A__U41(z0, z1, z2) -> c9
A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
A__U42(z0, z1, z2) -> c12
A__ISNAT(0') -> c13
A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
A__ISNAT(z0) -> c16
A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
A__PLUS(z0, z1) -> c19
MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
MARK(isNat(z0)) -> c22(A__ISNAT(z0))
MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
MARK(tt) -> c29
MARK(s(z0)) -> c30(MARK(z0))
MARK(0') -> c31
a__U11(tt, z0) -> a__U12(a__isNat(z0))
a__U11(z0, z1) -> U11(z0, z1)
a__U12(tt) -> tt
a__U12(z0) -> U12(z0)
a__U21(tt) -> tt
a__U21(z0) -> U21(z0)
a__U31(tt, z0) -> mark(z0)
a__U31(z0, z1) -> U31(z0, z1)
a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
a__U41(z0, z1, z2) -> U41(z0, z1, z2)
a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
a__U42(z0, z1, z2) -> U42(z0, z1, z2)
a__isNat(0') -> tt
a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
a__isNat(s(z0)) -> a__U21(a__isNat(z0))
a__isNat(z0) -> isNat(z0)
a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
a__plus(z0, z1) -> plus(z0, z1)
mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
mark(U12(z0)) -> a__U12(mark(z0))
mark(isNat(z0)) -> a__isNat(z0)
mark(U21(z0)) -> a__U21(mark(z0))
mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
mark(tt) -> tt
mark(s(z0)) -> s(mark(z0))
mark(0') -> 0'

Types:
A__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c:c1
tt :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c :: c2:c3 -> c13:c14:c15:c16 -> c:c1
A__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c2:c3
a__isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
A__ISNAT :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c13:c14:c15:c16
c1 :: c:c1
c2 :: c2:c3
c3 :: c2:c3
A__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
A__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c6:c7
c6 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c6:c7
MARK :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c7 :: c6:c7
A__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c8:c9
c8 :: c10:c11:c12 -> c13:c14:c15:c16 -> c8:c9
A__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c10:c11:c12
c9 :: c8:c9
c10 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
A__PLUS :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c17:c18:c19
mark :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c11 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
c12 :: c10:c11:c12
0' :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c13 :: c13:c14:c15:c16
plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c14 :: c:c1 -> c13:c14:c15:c16 -> c13:c14:c15:c16
s :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c15 :: c4:c5 -> c13:c14:c15:c16 -> c13:c14:c15:c16
c16 :: c13:c14:c15:c16
c17 :: c6:c7 -> c13:c14:c15:c16 -> c17:c18:c19
c18 :: c8:c9 -> c13:c14:c15:c16 -> c17:c18:c19
c19 :: c17:c18:c19
U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c20 :: c:c1 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c21 :: c2:c3 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c22 :: c13:c14:c15:c16 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c23 :: c4:c5 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c24 :: c6:c7 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c25 :: c8:c9 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c26 :: c10:c11:c12 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c27 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c28 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c29 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c30 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c31 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
a__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c:c11_32 :: c:c1
hole_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U422_32 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c2:c33_32 :: c2:c3
hole_c13:c14:c15:c164_32 :: c13:c14:c15:c16
hole_c4:c55_32 :: c4:c5
hole_c6:c76_32 :: c6:c7
hole_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c317_32 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
hole_c8:c98_32 :: c8:c9
hole_c10:c11:c129_32 :: c10:c11:c12
hole_c17:c18:c1910_32 :: c17:c18:c19
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32 :: Nat -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
gen_c13:c14:c15:c1612_32 :: Nat -> c13:c14:c15:c16
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32 :: Nat -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
A__U11, a__isNat, A__ISNAT, A__U31, MARK, A__U41, A__U42, A__PLUS, mark, a__U11, a__U31, a__U41, a__U42, a__plus

They will be analysed ascendingly in the following order:
a__isNat < A__U11
A__U11 = A__ISNAT
A__U11 < MARK
a__isNat < A__ISNAT
a__isNat < A__U41
a__isNat < A__PLUS
a__isNat < mark
a__isNat = a__U11
a__isNat < a__U41
a__isNat < a__plus
A__ISNAT < MARK
A__ISNAT < A__U41
A__ISNAT < A__PLUS
A__U31 = MARK
A__U31 = A__U41
A__U31 = A__U42
A__U31 = A__PLUS
MARK = A__U41
MARK = A__U42
MARK = A__PLUS
mark < MARK
A__U41 = A__U42
A__U41 = A__PLUS
A__U42 = A__PLUS
mark < A__U42
a__U11 < mark
mark = a__U31
mark = a__U41
mark = a__U42
mark = a__plus
a__U31 = a__U41
a__U31 = a__U42
a__U31 = a__plus
a__U41 = a__U42
a__U41 = a__plus
a__U42 = a__plus

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

(8)
Obligation:
Innermost TRS:
Rules:
A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
A__U11(z0, z1) -> c1
A__U12(tt) -> c2
A__U12(z0) -> c3
A__U21(tt) -> c4
A__U21(z0) -> c5
A__U31(tt, z0) -> c6(MARK(z0))
A__U31(z0, z1) -> c7
A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
A__U41(z0, z1, z2) -> c9
A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
A__U42(z0, z1, z2) -> c12
A__ISNAT(0') -> c13
A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
A__ISNAT(z0) -> c16
A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
A__PLUS(z0, z1) -> c19
MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
MARK(isNat(z0)) -> c22(A__ISNAT(z0))
MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
MARK(tt) -> c29
MARK(s(z0)) -> c30(MARK(z0))
MARK(0') -> c31
a__U11(tt, z0) -> a__U12(a__isNat(z0))
a__U11(z0, z1) -> U11(z0, z1)
a__U12(tt) -> tt
a__U12(z0) -> U12(z0)
a__U21(tt) -> tt
a__U21(z0) -> U21(z0)
a__U31(tt, z0) -> mark(z0)
a__U31(z0, z1) -> U31(z0, z1)
a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
a__U41(z0, z1, z2) -> U41(z0, z1, z2)
a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
a__U42(z0, z1, z2) -> U42(z0, z1, z2)
a__isNat(0') -> tt
a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
a__isNat(s(z0)) -> a__U21(a__isNat(z0))
a__isNat(z0) -> isNat(z0)
a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
a__plus(z0, z1) -> plus(z0, z1)
mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
mark(U12(z0)) -> a__U12(mark(z0))
mark(isNat(z0)) -> a__isNat(z0)
mark(U21(z0)) -> a__U21(mark(z0))
mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
mark(tt) -> tt
mark(s(z0)) -> s(mark(z0))
mark(0') -> 0'

Types:
A__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c:c1
tt :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c :: c2:c3 -> c13:c14:c15:c16 -> c:c1
A__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c2:c3
a__isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
A__ISNAT :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c13:c14:c15:c16
c1 :: c:c1
c2 :: c2:c3
c3 :: c2:c3
A__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
A__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c6:c7
c6 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c6:c7
MARK :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c7 :: c6:c7
A__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c8:c9
c8 :: c10:c11:c12 -> c13:c14:c15:c16 -> c8:c9
A__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c10:c11:c12
c9 :: c8:c9
c10 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
A__PLUS :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c17:c18:c19
mark :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c11 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
c12 :: c10:c11:c12
0' :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c13 :: c13:c14:c15:c16
plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c14 :: c:c1 -> c13:c14:c15:c16 -> c13:c14:c15:c16
s :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c15 :: c4:c5 -> c13:c14:c15:c16 -> c13:c14:c15:c16
c16 :: c13:c14:c15:c16
c17 :: c6:c7 -> c13:c14:c15:c16 -> c17:c18:c19
c18 :: c8:c9 -> c13:c14:c15:c16 -> c17:c18:c19
c19 :: c17:c18:c19
U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c20 :: c:c1 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c21 :: c2:c3 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c22 :: c13:c14:c15:c16 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c23 :: c4:c5 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c24 :: c6:c7 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c25 :: c8:c9 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c26 :: c10:c11:c12 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c27 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c28 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c29 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c30 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c31 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
a__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c:c11_32 :: c:c1
hole_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U422_32 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c2:c33_32 :: c2:c3
hole_c13:c14:c15:c164_32 :: c13:c14:c15:c16
hole_c4:c55_32 :: c4:c5
hole_c6:c76_32 :: c6:c7
hole_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c317_32 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
hole_c8:c98_32 :: c8:c9
hole_c10:c11:c129_32 :: c10:c11:c12
hole_c17:c18:c1910_32 :: c17:c18:c19
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32 :: Nat -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
gen_c13:c14:c15:c1612_32 :: Nat -> c13:c14:c15:c16
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32 :: Nat -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31


Generator Equations:
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0) <=> tt
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(x, 1)) <=> plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(x), tt)
gen_c13:c14:c15:c1612_32(0) <=> c13
gen_c13:c14:c15:c1612_32(+(x, 1)) <=> c14(c(c2, c13), gen_c13:c14:c15:c1612_32(x))
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(0) <=> c22(c13)
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(+(x, 1)) <=> c20(c(c2, c13), gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(x))


The following defined symbols remain to be analysed:
a__U11, A__U11, a__isNat, A__ISNAT, A__U31, MARK, A__U41, A__U42, A__PLUS, mark, a__U31, a__U41, a__U42, a__plus

They will be analysed ascendingly in the following order:
a__isNat < A__U11
A__U11 = A__ISNAT
A__U11 < MARK
a__isNat < A__ISNAT
a__isNat < A__U41
a__isNat < A__PLUS
a__isNat < mark
a__isNat = a__U11
a__isNat < a__U41
a__isNat < a__plus
A__ISNAT < MARK
A__ISNAT < A__U41
A__ISNAT < A__PLUS
A__U31 = MARK
A__U31 = A__U41
A__U31 = A__U42
A__U31 = A__PLUS
MARK = A__U41
MARK = A__U42
MARK = A__PLUS
mark < MARK
A__U41 = A__U42
A__U41 = A__PLUS
A__U42 = A__PLUS
mark < A__U42
a__U11 < mark
mark = a__U31
mark = a__U41
mark = a__U42
mark = a__plus
a__U31 = a__U41
a__U31 = a__U42
a__U31 = a__plus
a__U41 = a__U42
a__U41 = a__plus
a__U42 = a__plus

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n963_32))) -> *14_32, rt in Omega(0)

Induction Base:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, 0)))

Induction Step:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, +(n963_32, 1)))) ->_R^Omega(0)
a__U11(a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n963_32))), tt) ->_IH
a__U11(*14_32, tt)

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

(10)
Obligation:
Innermost TRS:
Rules:
A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
A__U11(z0, z1) -> c1
A__U12(tt) -> c2
A__U12(z0) -> c3
A__U21(tt) -> c4
A__U21(z0) -> c5
A__U31(tt, z0) -> c6(MARK(z0))
A__U31(z0, z1) -> c7
A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
A__U41(z0, z1, z2) -> c9
A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
A__U42(z0, z1, z2) -> c12
A__ISNAT(0') -> c13
A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
A__ISNAT(z0) -> c16
A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
A__PLUS(z0, z1) -> c19
MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
MARK(isNat(z0)) -> c22(A__ISNAT(z0))
MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
MARK(tt) -> c29
MARK(s(z0)) -> c30(MARK(z0))
MARK(0') -> c31
a__U11(tt, z0) -> a__U12(a__isNat(z0))
a__U11(z0, z1) -> U11(z0, z1)
a__U12(tt) -> tt
a__U12(z0) -> U12(z0)
a__U21(tt) -> tt
a__U21(z0) -> U21(z0)
a__U31(tt, z0) -> mark(z0)
a__U31(z0, z1) -> U31(z0, z1)
a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
a__U41(z0, z1, z2) -> U41(z0, z1, z2)
a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
a__U42(z0, z1, z2) -> U42(z0, z1, z2)
a__isNat(0') -> tt
a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
a__isNat(s(z0)) -> a__U21(a__isNat(z0))
a__isNat(z0) -> isNat(z0)
a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
a__plus(z0, z1) -> plus(z0, z1)
mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
mark(U12(z0)) -> a__U12(mark(z0))
mark(isNat(z0)) -> a__isNat(z0)
mark(U21(z0)) -> a__U21(mark(z0))
mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
mark(tt) -> tt
mark(s(z0)) -> s(mark(z0))
mark(0') -> 0'

Types:
A__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c:c1
tt :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c :: c2:c3 -> c13:c14:c15:c16 -> c:c1
A__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c2:c3
a__isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
A__ISNAT :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c13:c14:c15:c16
c1 :: c:c1
c2 :: c2:c3
c3 :: c2:c3
A__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
A__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c6:c7
c6 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c6:c7
MARK :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c7 :: c6:c7
A__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c8:c9
c8 :: c10:c11:c12 -> c13:c14:c15:c16 -> c8:c9
A__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c10:c11:c12
c9 :: c8:c9
c10 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
A__PLUS :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c17:c18:c19
mark :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c11 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
c12 :: c10:c11:c12
0' :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c13 :: c13:c14:c15:c16
plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c14 :: c:c1 -> c13:c14:c15:c16 -> c13:c14:c15:c16
s :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c15 :: c4:c5 -> c13:c14:c15:c16 -> c13:c14:c15:c16
c16 :: c13:c14:c15:c16
c17 :: c6:c7 -> c13:c14:c15:c16 -> c17:c18:c19
c18 :: c8:c9 -> c13:c14:c15:c16 -> c17:c18:c19
c19 :: c17:c18:c19
U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c20 :: c:c1 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c21 :: c2:c3 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c22 :: c13:c14:c15:c16 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c23 :: c4:c5 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c24 :: c6:c7 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c25 :: c8:c9 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c26 :: c10:c11:c12 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c27 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c28 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c29 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c30 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c31 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
a__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c:c11_32 :: c:c1
hole_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U422_32 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c2:c33_32 :: c2:c3
hole_c13:c14:c15:c164_32 :: c13:c14:c15:c16
hole_c4:c55_32 :: c4:c5
hole_c6:c76_32 :: c6:c7
hole_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c317_32 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
hole_c8:c98_32 :: c8:c9
hole_c10:c11:c129_32 :: c10:c11:c12
hole_c17:c18:c1910_32 :: c17:c18:c19
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32 :: Nat -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
gen_c13:c14:c15:c1612_32 :: Nat -> c13:c14:c15:c16
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32 :: Nat -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31


Lemmas:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n963_32))) -> *14_32, rt in Omega(0)


Generator Equations:
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0) <=> tt
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(x, 1)) <=> plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(x), tt)
gen_c13:c14:c15:c1612_32(0) <=> c13
gen_c13:c14:c15:c1612_32(+(x, 1)) <=> c14(c(c2, c13), gen_c13:c14:c15:c1612_32(x))
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(0) <=> c22(c13)
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(+(x, 1)) <=> c20(c(c2, c13), gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(x))


The following defined symbols remain to be analysed:
a__U11, A__U11, A__ISNAT, A__U31, MARK, A__U41, A__U42, A__PLUS, mark, a__U31, a__U41, a__U42, a__plus

They will be analysed ascendingly in the following order:
a__isNat < A__U11
A__U11 = A__ISNAT
A__U11 < MARK
a__isNat < A__ISNAT
a__isNat < A__U41
a__isNat < A__PLUS
a__isNat < mark
a__isNat = a__U11
a__isNat < a__U41
a__isNat < a__plus
A__ISNAT < MARK
A__ISNAT < A__U41
A__ISNAT < A__PLUS
A__U31 = MARK
A__U31 = A__U41
A__U31 = A__U42
A__U31 = A__PLUS
MARK = A__U41
MARK = A__U42
MARK = A__PLUS
mark < MARK
A__U41 = A__U42
A__U41 = A__PLUS
A__U42 = A__PLUS
mark < A__U42
a__U11 < mark
mark = a__U31
mark = a__U41
mark = a__U42
mark = a__plus
a__U31 = a__U41
a__U31 = a__U42
a__U31 = a__plus
a__U41 = a__U42
a__U41 = a__plus
a__U42 = a__plus

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32)) -> gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32), rt in Omega(0)

Induction Base:
mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0)) ->_R^Omega(0)
tt

Induction Step:
mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(n558289_32, 1))) ->_R^Omega(0)
a__plus(mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32)), mark(tt)) ->_IH
a__plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(c558290_32), mark(tt)) ->_R^Omega(0)
a__plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32), tt) ->_R^Omega(0)
plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32), tt)

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

(12)
Obligation:
Innermost TRS:
Rules:
A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
A__U11(z0, z1) -> c1
A__U12(tt) -> c2
A__U12(z0) -> c3
A__U21(tt) -> c4
A__U21(z0) -> c5
A__U31(tt, z0) -> c6(MARK(z0))
A__U31(z0, z1) -> c7
A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
A__U41(z0, z1, z2) -> c9
A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
A__U42(z0, z1, z2) -> c12
A__ISNAT(0') -> c13
A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
A__ISNAT(z0) -> c16
A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
A__PLUS(z0, z1) -> c19
MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
MARK(isNat(z0)) -> c22(A__ISNAT(z0))
MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
MARK(tt) -> c29
MARK(s(z0)) -> c30(MARK(z0))
MARK(0') -> c31
a__U11(tt, z0) -> a__U12(a__isNat(z0))
a__U11(z0, z1) -> U11(z0, z1)
a__U12(tt) -> tt
a__U12(z0) -> U12(z0)
a__U21(tt) -> tt
a__U21(z0) -> U21(z0)
a__U31(tt, z0) -> mark(z0)
a__U31(z0, z1) -> U31(z0, z1)
a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
a__U41(z0, z1, z2) -> U41(z0, z1, z2)
a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
a__U42(z0, z1, z2) -> U42(z0, z1, z2)
a__isNat(0') -> tt
a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
a__isNat(s(z0)) -> a__U21(a__isNat(z0))
a__isNat(z0) -> isNat(z0)
a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
a__plus(z0, z1) -> plus(z0, z1)
mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
mark(U12(z0)) -> a__U12(mark(z0))
mark(isNat(z0)) -> a__isNat(z0)
mark(U21(z0)) -> a__U21(mark(z0))
mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
mark(tt) -> tt
mark(s(z0)) -> s(mark(z0))
mark(0') -> 0'

Types:
A__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c:c1
tt :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c :: c2:c3 -> c13:c14:c15:c16 -> c:c1
A__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c2:c3
a__isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
A__ISNAT :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c13:c14:c15:c16
c1 :: c:c1
c2 :: c2:c3
c3 :: c2:c3
A__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
A__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c6:c7
c6 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c6:c7
MARK :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c7 :: c6:c7
A__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c8:c9
c8 :: c10:c11:c12 -> c13:c14:c15:c16 -> c8:c9
A__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c10:c11:c12
c9 :: c8:c9
c10 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
A__PLUS :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c17:c18:c19
mark :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c11 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
c12 :: c10:c11:c12
0' :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c13 :: c13:c14:c15:c16
plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c14 :: c:c1 -> c13:c14:c15:c16 -> c13:c14:c15:c16
s :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c15 :: c4:c5 -> c13:c14:c15:c16 -> c13:c14:c15:c16
c16 :: c13:c14:c15:c16
c17 :: c6:c7 -> c13:c14:c15:c16 -> c17:c18:c19
c18 :: c8:c9 -> c13:c14:c15:c16 -> c17:c18:c19
c19 :: c17:c18:c19
U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c20 :: c:c1 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c21 :: c2:c3 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c22 :: c13:c14:c15:c16 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c23 :: c4:c5 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c24 :: c6:c7 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c25 :: c8:c9 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c26 :: c10:c11:c12 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c27 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c28 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c29 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c30 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c31 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
a__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c:c11_32 :: c:c1
hole_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U422_32 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c2:c33_32 :: c2:c3
hole_c13:c14:c15:c164_32 :: c13:c14:c15:c16
hole_c4:c55_32 :: c4:c5
hole_c6:c76_32 :: c6:c7
hole_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c317_32 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
hole_c8:c98_32 :: c8:c9
hole_c10:c11:c129_32 :: c10:c11:c12
hole_c17:c18:c1910_32 :: c17:c18:c19
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32 :: Nat -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
gen_c13:c14:c15:c1612_32 :: Nat -> c13:c14:c15:c16
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32 :: Nat -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31


Lemmas:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n963_32))) -> *14_32, rt in Omega(0)
mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32)) -> gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32), rt in Omega(0)


Generator Equations:
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0) <=> tt
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(x, 1)) <=> plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(x), tt)
gen_c13:c14:c15:c1612_32(0) <=> c13
gen_c13:c14:c15:c1612_32(+(x, 1)) <=> c14(c(c2, c13), gen_c13:c14:c15:c1612_32(x))
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(0) <=> c22(c13)
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(+(x, 1)) <=> c20(c(c2, c13), gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(x))


The following defined symbols remain to be analysed:
a__U41, A__U11, A__ISNAT, A__U31, MARK, A__U41, A__U42, A__PLUS, a__U31, a__U42, a__plus

They will be analysed ascendingly in the following order:
A__U11 = A__ISNAT
A__U11 < MARK
A__ISNAT < MARK
A__ISNAT < A__U41
A__ISNAT < A__PLUS
A__U31 = MARK
A__U31 = A__U41
A__U31 = A__U42
A__U31 = A__PLUS
MARK = A__U41
MARK = A__U42
MARK = A__PLUS
mark < MARK
A__U41 = A__U42
A__U41 = A__PLUS
A__U42 = A__PLUS
mark < A__U42
mark = a__U31
mark = a__U41
mark = a__U42
mark = a__plus
a__U31 = a__U41
a__U31 = a__U42
a__U31 = a__plus
a__U41 = a__U42
a__U41 = a__plus
a__U42 = a__plus

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32))) -> *14_32, rt in Omega(n744339_32)

Induction Base:
MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, 0)))

Induction Step:
MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, +(n744339_32, 1)))) ->_R^Omega(1)
c27(A__PLUS(mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32))), mark(tt)), MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32)))) ->_L^Omega(0)
c27(A__PLUS(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32)), mark(tt)), MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32)))) ->_L^Omega(0)
c27(A__PLUS(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32)), gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0)), MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32)))) ->_R^Omega(1)
c27(c19, MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32)))) ->_IH
c27(c19, *14_32)

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
A__U11(z0, z1) -> c1
A__U12(tt) -> c2
A__U12(z0) -> c3
A__U21(tt) -> c4
A__U21(z0) -> c5
A__U31(tt, z0) -> c6(MARK(z0))
A__U31(z0, z1) -> c7
A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
A__U41(z0, z1, z2) -> c9
A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
A__U42(z0, z1, z2) -> c12
A__ISNAT(0') -> c13
A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
A__ISNAT(z0) -> c16
A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
A__PLUS(z0, z1) -> c19
MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
MARK(isNat(z0)) -> c22(A__ISNAT(z0))
MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
MARK(tt) -> c29
MARK(s(z0)) -> c30(MARK(z0))
MARK(0') -> c31
a__U11(tt, z0) -> a__U12(a__isNat(z0))
a__U11(z0, z1) -> U11(z0, z1)
a__U12(tt) -> tt
a__U12(z0) -> U12(z0)
a__U21(tt) -> tt
a__U21(z0) -> U21(z0)
a__U31(tt, z0) -> mark(z0)
a__U31(z0, z1) -> U31(z0, z1)
a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
a__U41(z0, z1, z2) -> U41(z0, z1, z2)
a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
a__U42(z0, z1, z2) -> U42(z0, z1, z2)
a__isNat(0') -> tt
a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
a__isNat(s(z0)) -> a__U21(a__isNat(z0))
a__isNat(z0) -> isNat(z0)
a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
a__plus(z0, z1) -> plus(z0, z1)
mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
mark(U12(z0)) -> a__U12(mark(z0))
mark(isNat(z0)) -> a__isNat(z0)
mark(U21(z0)) -> a__U21(mark(z0))
mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
mark(tt) -> tt
mark(s(z0)) -> s(mark(z0))
mark(0') -> 0'

Types:
A__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c:c1
tt :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c :: c2:c3 -> c13:c14:c15:c16 -> c:c1
A__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c2:c3
a__isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
A__ISNAT :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c13:c14:c15:c16
c1 :: c:c1
c2 :: c2:c3
c3 :: c2:c3
A__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
A__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c6:c7
c6 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c6:c7
MARK :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c7 :: c6:c7
A__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c8:c9
c8 :: c10:c11:c12 -> c13:c14:c15:c16 -> c8:c9
A__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c10:c11:c12
c9 :: c8:c9
c10 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
A__PLUS :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c17:c18:c19
mark :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c11 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
c12 :: c10:c11:c12
0' :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c13 :: c13:c14:c15:c16
plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c14 :: c:c1 -> c13:c14:c15:c16 -> c13:c14:c15:c16
s :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c15 :: c4:c5 -> c13:c14:c15:c16 -> c13:c14:c15:c16
c16 :: c13:c14:c15:c16
c17 :: c6:c7 -> c13:c14:c15:c16 -> c17:c18:c19
c18 :: c8:c9 -> c13:c14:c15:c16 -> c17:c18:c19
c19 :: c17:c18:c19
U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c20 :: c:c1 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c21 :: c2:c3 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c22 :: c13:c14:c15:c16 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c23 :: c4:c5 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c24 :: c6:c7 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c25 :: c8:c9 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c26 :: c10:c11:c12 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c27 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c28 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c29 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c30 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c31 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
a__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c:c11_32 :: c:c1
hole_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U422_32 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c2:c33_32 :: c2:c3
hole_c13:c14:c15:c164_32 :: c13:c14:c15:c16
hole_c4:c55_32 :: c4:c5
hole_c6:c76_32 :: c6:c7
hole_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c317_32 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
hole_c8:c98_32 :: c8:c9
hole_c10:c11:c129_32 :: c10:c11:c12
hole_c17:c18:c1910_32 :: c17:c18:c19
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32 :: Nat -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
gen_c13:c14:c15:c1612_32 :: Nat -> c13:c14:c15:c16
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32 :: Nat -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31


Lemmas:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n963_32))) -> *14_32, rt in Omega(0)
mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32)) -> gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32), rt in Omega(0)


Generator Equations:
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0) <=> tt
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(x, 1)) <=> plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(x), tt)
gen_c13:c14:c15:c1612_32(0) <=> c13
gen_c13:c14:c15:c1612_32(+(x, 1)) <=> c14(c(c2, c13), gen_c13:c14:c15:c1612_32(x))
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(0) <=> c22(c13)
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(+(x, 1)) <=> c20(c(c2, c13), gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(x))


The following defined symbols remain to be analysed:
MARK, A__U31, A__U41, A__U42, A__PLUS

They will be analysed ascendingly in the following order:
A__U31 = MARK
A__U31 = A__U41
A__U31 = A__U42
A__U31 = A__PLUS
MARK = A__U41
MARK = A__U42
MARK = A__PLUS
A__U41 = A__U42
A__U41 = A__PLUS
A__U42 = A__PLUS

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
Innermost TRS:
Rules:
A__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0))
A__U11(z0, z1) -> c1
A__U12(tt) -> c2
A__U12(z0) -> c3
A__U21(tt) -> c4
A__U21(z0) -> c5
A__U31(tt, z0) -> c6(MARK(z0))
A__U31(z0, z1) -> c7
A__U41(tt, z0, z1) -> c8(A__U42(a__isNat(z1), z0, z1), A__ISNAT(z1))
A__U41(z0, z1, z2) -> c9
A__U42(tt, z0, z1) -> c10(A__PLUS(mark(z1), mark(z0)), MARK(z1))
A__U42(tt, z0, z1) -> c11(A__PLUS(mark(z1), mark(z0)), MARK(z0))
A__U42(z0, z1, z2) -> c12
A__ISNAT(0') -> c13
A__ISNAT(plus(z0, z1)) -> c14(A__U11(a__isNat(z0), z1), A__ISNAT(z0))
A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)), A__ISNAT(z0))
A__ISNAT(z0) -> c16
A__PLUS(z0, 0') -> c17(A__U31(a__isNat(z0), z0), A__ISNAT(z0))
A__PLUS(z0, s(z1)) -> c18(A__U41(a__isNat(z1), z1, z0), A__ISNAT(z1))
A__PLUS(z0, z1) -> c19
MARK(U11(z0, z1)) -> c20(A__U11(mark(z0), z1), MARK(z0))
MARK(U12(z0)) -> c21(A__U12(mark(z0)), MARK(z0))
MARK(isNat(z0)) -> c22(A__ISNAT(z0))
MARK(U21(z0)) -> c23(A__U21(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) -> c24(A__U31(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) -> c25(A__U41(mark(z0), z1, z2), MARK(z0))
MARK(U42(z0, z1, z2)) -> c26(A__U42(mark(z0), z1, z2), MARK(z0))
MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z0))
MARK(plus(z0, z1)) -> c28(A__PLUS(mark(z0), mark(z1)), MARK(z1))
MARK(tt) -> c29
MARK(s(z0)) -> c30(MARK(z0))
MARK(0') -> c31
a__U11(tt, z0) -> a__U12(a__isNat(z0))
a__U11(z0, z1) -> U11(z0, z1)
a__U12(tt) -> tt
a__U12(z0) -> U12(z0)
a__U21(tt) -> tt
a__U21(z0) -> U21(z0)
a__U31(tt, z0) -> mark(z0)
a__U31(z0, z1) -> U31(z0, z1)
a__U41(tt, z0, z1) -> a__U42(a__isNat(z1), z0, z1)
a__U41(z0, z1, z2) -> U41(z0, z1, z2)
a__U42(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
a__U42(z0, z1, z2) -> U42(z0, z1, z2)
a__isNat(0') -> tt
a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1)
a__isNat(s(z0)) -> a__U21(a__isNat(z0))
a__isNat(z0) -> isNat(z0)
a__plus(z0, 0') -> a__U31(a__isNat(z0), z0)
a__plus(z0, s(z1)) -> a__U41(a__isNat(z1), z1, z0)
a__plus(z0, z1) -> plus(z0, z1)
mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
mark(U12(z0)) -> a__U12(mark(z0))
mark(isNat(z0)) -> a__isNat(z0)
mark(U21(z0)) -> a__U21(mark(z0))
mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
mark(tt) -> tt
mark(s(z0)) -> s(mark(z0))
mark(0') -> 0'

Types:
A__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c:c1
tt :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c :: c2:c3 -> c13:c14:c15:c16 -> c:c1
A__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c2:c3
a__isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
A__ISNAT :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c13:c14:c15:c16
c1 :: c:c1
c2 :: c2:c3
c3 :: c2:c3
A__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
A__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c6:c7
c6 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c6:c7
MARK :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c7 :: c6:c7
A__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c8:c9
c8 :: c10:c11:c12 -> c13:c14:c15:c16 -> c8:c9
A__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c10:c11:c12
c9 :: c8:c9
c10 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
A__PLUS :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> c17:c18:c19
mark :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c11 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c10:c11:c12
c12 :: c10:c11:c12
0' :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c13 :: c13:c14:c15:c16
plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c14 :: c:c1 -> c13:c14:c15:c16 -> c13:c14:c15:c16
s :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c15 :: c4:c5 -> c13:c14:c15:c16 -> c13:c14:c15:c16
c16 :: c13:c14:c15:c16
c17 :: c6:c7 -> c13:c14:c15:c16 -> c17:c18:c19
c18 :: c8:c9 -> c13:c14:c15:c16 -> c17:c18:c19
c19 :: c17:c18:c19
U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c20 :: c:c1 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c21 :: c2:c3 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
isNat :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c22 :: c13:c14:c15:c16 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c23 :: c4:c5 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c24 :: c6:c7 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c25 :: c8:c9 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
c26 :: c10:c11:c12 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c27 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c28 :: c17:c18:c19 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c29 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c30 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31 -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
c31 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
a__U11 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U12 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U21 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U31 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U41 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__U42 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
a__plus :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42 -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c:c11_32 :: c:c1
hole_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U422_32 :: tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
hole_c2:c33_32 :: c2:c3
hole_c13:c14:c15:c164_32 :: c13:c14:c15:c16
hole_c4:c55_32 :: c4:c5
hole_c6:c76_32 :: c6:c7
hole_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c317_32 :: c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31
hole_c8:c98_32 :: c8:c9
hole_c10:c11:c129_32 :: c10:c11:c12
hole_c17:c18:c1910_32 :: c17:c18:c19
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32 :: Nat -> tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U42
gen_c13:c14:c15:c1612_32 :: Nat -> c13:c14:c15:c16
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32 :: Nat -> c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c31


Lemmas:
a__isNat(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n963_32))) -> *14_32, rt in Omega(0)
mark(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32)) -> gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(n558289_32), rt in Omega(0)
MARK(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(1, n744339_32))) -> *14_32, rt in Omega(n744339_32)


Generator Equations:
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(0) <=> tt
gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(+(x, 1)) <=> plus(gen_tt:0':plus:s:U11:U12:isNat:U21:U31:U41:U4211_32(x), tt)
gen_c13:c14:c15:c1612_32(0) <=> c13
gen_c13:c14:c15:c1612_32(+(x, 1)) <=> c14(c(c2, c13), gen_c13:c14:c15:c1612_32(x))
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(0) <=> c22(c13)
gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(+(x, 1)) <=> c20(c(c2, c13), gen_c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30:c3113_32(x))


The following defined symbols remain to be analysed:
A__U31, A__U41, A__U42, A__PLUS

They will be analysed ascendingly in the following order:
A__U31 = MARK
A__U31 = A__U41
A__U31 = A__U42
A__U31 = A__PLUS
MARK = A__U41
MARK = A__U42
MARK = A__PLUS
A__U41 = A__U42
A__U41 = A__PLUS
A__U42 = A__PLUS