WORST_CASE(Omega(n^1),O(n^2))
proof of input_KgMylBwUiW.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 235 ms]
    (10) BOUNDS(1, n^2)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 5 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 476 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 49 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 450 ms]
            (32) 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 TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where all 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) 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

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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

(9) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V2),0,[fun(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun3(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[x(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[activate(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, V2, Out),1,[activate(M1, Ret1),activate(N1, Ret2),fun1(0, Ret1, Ret2, Ret)],[Out = Ret,V = M1,V1 = 0,V2 = N1,M1 >= 0,N1 >= 0]).
eq(fun1(V1, V, V2, Out),1,[activate(N2, Ret10),activate(M2, Ret11),plus(Ret10, Ret11, Ret12)],[Out = 1 + Ret12,V = M2,V1 = 0,V2 = N2,M2 >= 0,N2 >= 0]).
eq(fun2(V1, V, V2, Out),1,[activate(M3, Ret13),activate(N3, Ret21),fun3(0, Ret13, Ret21, Ret3)],[Out = Ret3,V = M3,V1 = 0,V2 = N3,M3 >= 0,N3 >= 0]).
eq(fun3(V1, V, V2, Out),1,[activate(N4, Ret00),activate(M4, Ret01),x(Ret00, Ret01, Ret0),activate(N4, Ret14),plus(Ret0, Ret14, Ret4)],[Out = Ret4,V = M4,V1 = 0,V2 = N4,M4 >= 0,N4 >= 0]).
eq(plus(V1, V, Out),1,[],[Out = N5,V1 = N5,V = 0,N5 >= 0]).
eq(plus(V1, V, Out),1,[fun(0, M5, N6, Ret5)],[Out = Ret5,V = 1 + M5,V1 = N6,M5 >= 0,N6 >= 0]).
eq(x(V1, V, Out),1,[],[Out = 0,V1 = N7,V = 0,N7 >= 0]).
eq(x(V1, V, Out),1,[fun2(0, M6, N8, Ret6)],[Out = Ret6,V = 1 + M6,V1 = N8,M6 >= 0,N8 >= 0]).
eq(activate(V1, Out),1,[],[Out = X1,X1 >= 0,V1 = X1]).
input_output_vars(fun(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun1(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun2(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun3(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
input_output_vars(x(V1,V,Out),[V1,V],[Out]).
input_output_vars(activate(V1,Out),[V1],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. non_recursive  : [activate/2]
1. recursive  : [fun/4,fun1/4,plus/3]
2. recursive [non_tail] : [fun2/4,fun3/4,x/3]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into plus/3
2. SCC is partially evaluated into x/3
3. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations plus/3 
* CE 11 is refined into CE [12] 
* CE 10 is refined into CE [13] 


### Cost equations --> "Loop" of plus/3 
* CEs [13] --> Loop 6 
* CEs [12] --> Loop 7 

### Ranking functions of CR plus(V1,V,Out) 
* RF of phase [6]: [V]

#### Partial ranking functions of CR plus(V1,V,Out) 
* Partial RF of phase [6]:
  - RF of loop [6:1]:
    V


### Specialization of cost equations x/3 
* CE 9 is refined into CE [14] 
* CE 8 is refined into CE [15,16] 


### Cost equations --> "Loop" of x/3 
* CEs [16] --> Loop 8 
* CEs [15] --> Loop 9 
* CEs [14] --> Loop 10 

### Ranking functions of CR x(V1,V,Out) 
* RF of phase [8]: [V]
* RF of phase [9]: [V]

#### Partial ranking functions of CR x(V1,V,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [17,18,19,20] 
* CE 2 is refined into CE [21,22,23,24] 
* CE 3 is refined into CE [25,26] 
* CE 4 is refined into CE [27,28] 
* CE 5 is refined into CE [29,30] 
* CE 6 is refined into CE [31,32,33] 
* CE 7 is refined into CE [34] 


### Cost equations --> "Loop" of start/3 
* CEs [17,20,21,24,26,28,31] --> Loop 11 
* CEs [18,19,22,23,25,27,29,30,32,33,34] --> Loop 12 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


Computing Bounds
=====================================

#### Cost of chains of plus(V1,V,Out):
* Chain [[6],7]: 7*it(6)+1
  Such that:it(6) =< V

  with precondition: [V+V1=Out,V1>=0,V>=1] 

* Chain [7]: 1
  with precondition: [V=0,V1=Out,V1>=0] 


#### Cost of chains of x(V1,V,Out):
* Chain [[9],10]: 9*it(9)+1
  Such that:it(9) =< V

  with precondition: [V1=0,Out=0,V>=1] 

* Chain [[8],10]: 9*it(8)+7*s(3)+1
  Such that:aux(1) =< V1
it(8) =< V
s(3) =< it(8)*aux(1)

  with precondition: [V1>=1,V>=1,Out+1>=V+V1] 

* Chain [10]: 1
  with precondition: [V=0,Out=0,V1>=0] 


#### Cost of chains of start(V1,V,V2):
* Chain [12]: 14*s(4)+16*s(6)+7*s(9)+9
  Such that:s(7) =< V1
aux(2) =< V
aux(3) =< V2
s(6) =< aux(2)
s(4) =< aux(3)
s(9) =< s(6)*s(7)

  with precondition: [V1>=0] 

* Chain [11]: 59*s(10)+14*s(13)+14*s(14)+9
  Such that:aux(6) =< V
aux(7) =< V2
s(10) =< aux(6)
s(14) =< aux(7)
s(13) =< s(10)*aux(7)

  with precondition: [V1=0,V>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [12] with precondition: [V1>=0] 
    - Upper bound: 7*V1*nat(V)+9+nat(V)*16+nat(V2)*14 
    - Complexity: n^2 
* Chain [11] with precondition: [V1=0,V>=1] 
    - Upper bound: 59*V+9+14*V*nat(V2)+nat(V2)*14 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V2): nat(V)*16+9+nat(V2)*14+max([7*V1*nat(V),nat(V)*14*nat(V2)+nat(V)*43]) 
Asymptotic class: n^2 
* Total analysis performed in 203 ms.


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

(10)
BOUNDS(1, n^2)

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

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

(12)
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


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

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

(14)
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
----------------------------------------

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

(16)
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
----------------------------------------

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

(18)
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

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

(19) 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

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

(20)
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

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

(21) 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).
----------------------------------------

(22)
Complex Obligation (BEST)

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

(23)
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

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
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

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

(27) 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).
----------------------------------------

(28)
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

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

(29) 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).
----------------------------------------

(30)
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
----------------------------------------

(31) 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).
----------------------------------------

(32)
BOUNDS(1, INF)