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


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 3 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 280 ms]
    (10) BOUNDS(1, n^2)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 241 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 79 ms]
            (24) proven lower bound
            (25) LowerBoundPropagationProof [FINISHED, 0 ms]
            (26) BOUNDS(n^2, INF)


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

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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 199 ms.


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

(10)
BOUNDS(1, n^2)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


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: INNERMOST
----------------------------------------

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

(14)
Obligation:
Innermost TRS:
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

Types:
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'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'

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

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

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

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

(16)
Obligation:
Innermost TRS:
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

Types:
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'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
U11(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) ->_R^Omega(1)
U12(tt, activate(gen_s:0'3_0(n5_0)), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U12(tt, gen_s:0'3_0(n5_0), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U12(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) ->_R^Omega(1)
s(plus(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n5_0)))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), activate(gen_s:0'3_0(n5_0)))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) ->_IH
s(gen_s:0'3_0(+(a, c6_0)))

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
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

Types:
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'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

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

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
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

Types:
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'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n742_0)) -> gen_s:0'3_0(*(n742_0, a)), rt in Omega(1 + a*n742_0 + n742_0)

Induction Base:
x(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
0'

Induction Step:
x(gen_s:0'3_0(a), gen_s:0'3_0(+(n742_0, 1))) ->_R^Omega(1)
U21(tt, gen_s:0'3_0(n742_0), gen_s:0'3_0(a)) ->_R^Omega(1)
U22(tt, activate(gen_s:0'3_0(n742_0)), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U22(tt, gen_s:0'3_0(n742_0), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U22(tt, gen_s:0'3_0(n742_0), gen_s:0'3_0(a)) ->_R^Omega(1)
plus(x(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n742_0))), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
plus(x(gen_s:0'3_0(a), activate(gen_s:0'3_0(n742_0))), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
plus(x(gen_s:0'3_0(a), gen_s:0'3_0(n742_0)), activate(gen_s:0'3_0(a))) ->_IH
plus(gen_s:0'3_0(*(c743_0, a)), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
plus(gen_s:0'3_0(*(n742_0, a)), gen_s:0'3_0(a)) ->_L^Omega(1 + a)
gen_s:0'3_0(+(a, *(n742_0, a)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(24)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
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

Types:
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'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
x
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(25) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(26)
BOUNDS(n^2, INF)