WORST_CASE(Omega(n^1),O(n^2))
proof of input_NW5VRId6rE.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 150 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 661 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 298 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 540 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 227 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
    (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxRelTRS
    (39) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) typed CpxTrs
    (43) OrderProof [LOWER BOUND(ID), 0 ms]
    (44) typed CpxTrs
    (45) RewriteLemmaProof [LOWER BOUND(ID), 487 ms]
    (46) BEST
        (47) proven lower bound
            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
            (49) BOUNDS(n^1, INF)
        (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 72 ms]
            (52) typed CpxTrs
            (53) RewriteLemmaProof [LOWER BOUND(ID), 108 ms]
            (54) typed CpxTrs
            (55) RewriteLemmaProof [LOWER BOUND(ID), 523 ms]
            (56) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N)) [1]
   U12(tt, M, N) -> s(plus(activate(N), activate(M))) [1]
   U21(tt, M, N) -> U22(tt, activate(M), activate(N)) [1]
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> U21(tt, M, N) [1]
   activate(X) -> X [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N)) [1]
   U12(tt, M, N) -> s(plus(activate(N), activate(M))) [1]
   U21(tt, M, N) -> U22(tt, activate(M), activate(N)) [1]
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> U21(tt, M, N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
U11 :: tt -> s:0 -> s:0 -> s:0
tt :: tt
U12 :: tt -> s:0 -> s:0 -> s:0
activate :: s:0 -> s:0
s :: s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
U21 :: tt -> s:0 -> s:0 -> s:0
U22 :: tt -> s:0 -> s:0 -> s:0
x :: s:0 -> s:0 -> s:0
0 :: s:0

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

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   activate_1
   x_2
   U21_3
   U22_3
   plus_2
   U11_3
   U12_3

Due to the following rules being added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N)) [1]
   U12(tt, M, N) -> s(plus(activate(N), activate(M))) [1]
   U21(tt, M, N) -> U22(tt, activate(M), activate(N)) [1]
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> U21(tt, M, N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
U11 :: tt -> s:0 -> s:0 -> s:0
tt :: tt
U12 :: tt -> s:0 -> s:0 -> s:0
activate :: s:0 -> s:0
s :: s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
U21 :: tt -> s:0 -> s:0 -> s:0
U22 :: tt -> s:0 -> s:0 -> s:0
x :: s:0 -> s:0 -> s:0
0 :: s:0

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

(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, M, N) [3]
   U12(tt, M, N) -> s(plus(N, M)) [3]
   U21(tt, M, N) -> U22(tt, M, N) [3]
   U22(tt, M, N) -> plus(x(N, M), N) [4]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> U21(tt, M, N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
U11 :: tt -> s:0 -> s:0 -> s:0
tt :: tt
U12 :: tt -> s:0 -> s:0 -> s:0
activate :: s:0 -> s:0
s :: s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
U21 :: tt -> s:0 -> s:0 -> s:0
U22 :: tt -> s:0 -> s:0 -> s:0
x :: s:0 -> s:0 -> s:0
0 :: s:0

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

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   tt => 0
   0 => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, M, N) :|: z' = M, z = 0, z'' = N, M >= 0, N >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(N, M) :|: z' = M, z = 0, z'' = N, M >= 0, N >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, M, N) :|: z' = M, z = 0, z'' = N, M >= 0, N >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(N, M), N) :|: z' = M, z = 0, z'' = N, M >= 0, N >= 0
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> U11(0, M, N) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 1 }-> U21(0, M, N) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0


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

(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0


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

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { activate }
   { plus, U12, U11 }
   { U22, x, U21 }

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {plus,U12,U11}, {U22,x,U21}

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

(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {plus,U12,U11}, {U22,x,U21}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {plus,U12,U11}, {U22,x,U21}
Previous analysis results are:
  activate: runtime: ?, size: O(n^1) [z]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus,U12,U11}, {U22,x,U21}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]

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

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus,U12,U11}, {U22,x,U21}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

Computed SIZE bound using CoFloCo for: U12
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

Computed SIZE bound using CoFloCo for: U11
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus,U12,U11}, {U22,x,U21}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: ?, size: O(n^1) [z + z']
  U12: runtime: ?, size: O(n^1) [1 + z' + z'']
  U11: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 11 + 7*z'

Computed RUNTIME bound using CoFloCo for: U12
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 14 + 7*z'

Computed RUNTIME bound using CoFloCo for: U11
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 17 + 7*z'

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 3 }-> U12(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {U22,x,U21}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']

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

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {U22,x,U21}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: U22
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z'*z'' + z''

Computed SIZE bound using KoAT for: x
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z + z*z'

Computed SIZE bound using KoAT for: U21
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z'*z'' + z''

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {U22,x,U21}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  U22: runtime: ?, size: O(n^2) [z'*z'' + z'']
  x: runtime: ?, size: O(n^2) [z + z*z']
  U21: runtime: ?, size: O(n^2) [z'*z'' + z'']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: U22
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 35 + 19*z' + 7*z'*z'' + 14*z''

Computed RUNTIME bound using KoAT for: x
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 40 + 14*z + 7*z*z' + 19*z'

Computed RUNTIME bound using KoAT for: U21
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 38 + 19*z' + 7*z'*z'' + 14*z''

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 0, z' >= 0, z'' >= 0
   U21(z, z', z'') -{ 3 }-> U22(0, z', z'') :|: z = 0, z' >= 0, z'' >= 0
   U22(z, z', z'') -{ 4 }-> plus(x(z'', z'), z'') :|: z = 0, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   x(z, z') -{ 1 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  U22: runtime: O(n^2) [35 + 19*z' + 7*z'*z'' + 14*z''], size: O(n^2) [z'*z'' + z'']
  x: runtime: O(n^2) [40 + 14*z + 7*z*z' + 19*z'], size: O(n^2) [z + z*z']
  U21: runtime: O(n^2) [38 + 19*z' + 7*z'*z'' + 14*z''], size: O(n^2) [z'*z'' + z'']

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

(33) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(34)
BOUNDS(1, n^2)

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

(35) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0
Tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13
S tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13
K tuples:none
Defined Rule Symbols:   U11_3, U12_3, U21_3, U22_3, plus_2, x_2, activate_1

Defined Pair Symbols:   U11'_3, U12'_3, U21'_3, U22'_3, PLUS_2, X_2, ACTIVATE_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5_2, c6_3, c7_3, c8_2, c9, c10_1, c11, c12_1, c13


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

(37) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(38)
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:

   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13

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

   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0

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

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

(40)
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:

   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0') -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0') -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13

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

   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0') -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0') -> 0'
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0

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

(41) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(42)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s

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

(43) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PLUS, x, X, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

(44)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
PLUS, x, X, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)

Induction Base:
PLUS(gen_0':s10_14(a), gen_0':s10_14(0))

Induction Step:
PLUS(gen_0':s10_14(a), gen_0':s10_14(+(n12_14, 1))) ->_R^Omega(1)
c10(U11'(tt, gen_0':s10_14(n12_14), gen_0':s10_14(a))) ->_R^Omega(1)
c10(c(U12'(tt, activate(gen_0':s10_14(n12_14)), activate(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(U12'(tt, gen_0':s10_14(n12_14), activate(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(U12'(tt, gen_0':s10_14(n12_14), gen_0':s10_14(a)), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(1)
c10(c(c2(PLUS(activate(gen_0':s10_14(a)), activate(gen_0':s10_14(n12_14))), ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(c2(PLUS(gen_0':s10_14(a), activate(gen_0':s10_14(n12_14))), ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(c2(PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)), ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_IH
c10(c(c2(*11_14, ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(1)
c10(c(c2(*11_14, c13), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(1)
c10(c(c2(*11_14, c13), c13))

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

(46)
Complex Obligation (BEST)

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

(47)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
PLUS, x, X, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

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

(49)
BOUNDS(n^1, INF)

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

(50)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Lemmas:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
plus, x, X

They will be analysed ascendingly in the following order:
x < X
plus < x

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14)) -> gen_0':s10_14(+(n2625_14, a)), rt in Omega(0)

Induction Base:
plus(gen_0':s10_14(a), gen_0':s10_14(0)) ->_R^Omega(0)
gen_0':s10_14(a)

Induction Step:
plus(gen_0':s10_14(a), gen_0':s10_14(+(n2625_14, 1))) ->_R^Omega(0)
U11(tt, gen_0':s10_14(n2625_14), gen_0':s10_14(a)) ->_R^Omega(0)
U12(tt, activate(gen_0':s10_14(n2625_14)), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U12(tt, gen_0':s10_14(n2625_14), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U12(tt, gen_0':s10_14(n2625_14), gen_0':s10_14(a)) ->_R^Omega(0)
s(plus(activate(gen_0':s10_14(a)), activate(gen_0':s10_14(n2625_14)))) ->_R^Omega(0)
s(plus(gen_0':s10_14(a), activate(gen_0':s10_14(n2625_14)))) ->_R^Omega(0)
s(plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14))) ->_IH
s(gen_0':s10_14(+(a, c2626_14)))

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

(52)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Lemmas:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)
plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14)) -> gen_0':s10_14(+(n2625_14, a)), rt in Omega(0)


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
x, X

They will be analysed ascendingly in the following order:
x < X

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_0':s10_14(a), gen_0':s10_14(n4085_14)) -> gen_0':s10_14(*(n4085_14, a)), rt in Omega(0)

Induction Base:
x(gen_0':s10_14(a), gen_0':s10_14(0)) ->_R^Omega(0)
0'

Induction Step:
x(gen_0':s10_14(a), gen_0':s10_14(+(n4085_14, 1))) ->_R^Omega(0)
U21(tt, gen_0':s10_14(n4085_14), gen_0':s10_14(a)) ->_R^Omega(0)
U22(tt, activate(gen_0':s10_14(n4085_14)), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U22(tt, gen_0':s10_14(n4085_14), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U22(tt, gen_0':s10_14(n4085_14), gen_0':s10_14(a)) ->_R^Omega(0)
plus(x(activate(gen_0':s10_14(a)), activate(gen_0':s10_14(n4085_14))), activate(gen_0':s10_14(a))) ->_R^Omega(0)
plus(x(gen_0':s10_14(a), activate(gen_0':s10_14(n4085_14))), activate(gen_0':s10_14(a))) ->_R^Omega(0)
plus(x(gen_0':s10_14(a), gen_0':s10_14(n4085_14)), activate(gen_0':s10_14(a))) ->_IH
plus(gen_0':s10_14(*(c4086_14, a)), activate(gen_0':s10_14(a))) ->_R^Omega(0)
plus(gen_0':s10_14(*(n4085_14, a)), gen_0':s10_14(a)) ->_L^Omega(0)
gen_0':s10_14(+(a, *(n4085_14, a)))

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

(54)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Lemmas:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)
plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14)) -> gen_0':s10_14(+(n2625_14, a)), rt in Omega(0)
x(gen_0':s10_14(a), gen_0':s10_14(n4085_14)) -> gen_0':s10_14(*(n4085_14, a)), rt in Omega(0)


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
X(gen_0':s10_14(0), gen_0':s10_14(n6197_14)) -> *11_14, rt in Omega(n6197_14)

Induction Base:
X(gen_0':s10_14(0), gen_0':s10_14(0))

Induction Step:
X(gen_0':s10_14(0), gen_0':s10_14(+(n6197_14, 1))) ->_R^Omega(1)
c12(U21'(tt, gen_0':s10_14(n6197_14), gen_0':s10_14(0))) ->_R^Omega(1)
c12(c4(U22'(tt, activate(gen_0':s10_14(n6197_14)), activate(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(U22'(tt, gen_0':s10_14(n6197_14), activate(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(U22'(tt, gen_0':s10_14(n6197_14), gen_0':s10_14(0)), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(PLUS(x(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(PLUS(x(gen_0':s10_14(0), activate(gen_0':s10_14(n6197_14))), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(PLUS(x(gen_0':s10_14(0), gen_0':s10_14(n6197_14)), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_L^Omega(0)
c12(c4(c6(PLUS(gen_0':s10_14(*(n6197_14, 0)), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(PLUS(gen_0':s10_14(0), gen_0':s10_14(0)), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(c9, X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(c9, X(gen_0':s10_14(0), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(c9, X(gen_0':s10_14(0), gen_0':s10_14(n6197_14)), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_IH
c12(c4(c6(c9, *11_14, ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(c9, *11_14, c13), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(c9, *11_14, c13), c13))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(56)
BOUNDS(1, INF)