/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^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 288 ms]
    (14) BOUNDS(1, n^1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 278 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), 52 ms]
            (28) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0)))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of quot: minus
The following defined symbols can occur below the 0th argument of minus: minus

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0)))

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

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))

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

(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
quot(s(x), s([]))


The defined contexts are:
quot([], s(x1))
minus([], x1)


[] just represents basic- or constructor-terms in the following defined contexts:
quot([], s(x1))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))

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

(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

(9) 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:

   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_minus, null_quot, null_plus

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

(10)
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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus
0 :: 0:s:null_minus:null_quot:null_plus
s :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus
quot :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus
plus :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus
null_minus :: 0:s:null_minus:null_quot:null_plus
null_quot :: 0:s:null_minus:null_quot:null_plus
null_plus :: 0:s:null_minus:null_quot:null_plus

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

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

   0 => 0
   null_minus => 0
   null_quot => 0
   null_plus => 0

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

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

   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]).
eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V5,V5 >= 0,V1 = 0]).
eq(quot(V1, V, Out),1,[minus(V7, V6, Ret10),quot(Ret10, 1 + V6, Ret1)],[Out = 1 + Ret1,V = 1 + V6,V7 >= 0,V6 >= 0,V1 = 1 + V7]).
eq(plus(V1, V, Out),1,[],[Out = V8,V8 >= 0,V1 = 0,V = V8]).
eq(plus(V1, V, Out),1,[plus(V9, V10, Ret11)],[Out = 1 + Ret11,V9 >= 0,V10 >= 0,V1 = 1 + V9,V = V10]).
eq(minus(V1, V, Out),0,[],[Out = 0,V12 >= 0,V11 >= 0,V1 = V12,V = V11]).
eq(quot(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(plus(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(quot(V1,V,Out),[V1,V],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [minus/3]
1. recursive  : [plus/3]
2. recursive  : [quot/3]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into minus/3
1. SCC is partially evaluated into plus/3
2. SCC is partially evaluated into quot/3
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations minus/3 
* CE 6 is refined into CE [13] 
* CE 4 is refined into CE [14] 
* CE 5 is refined into CE [15] 


### Cost equations --> "Loop" of minus/3 
* CEs [15] --> Loop 10 
* CEs [13] --> Loop 11 
* CEs [14] --> Loop 12 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V
    V1


### Specialization of cost equations plus/3 
* CE 12 is refined into CE [16] 
* CE 10 is refined into CE [17] 
* CE 11 is refined into CE [18] 


### Cost equations --> "Loop" of plus/3 
* CEs [18] --> Loop 13 
* CEs [16] --> Loop 14 
* CEs [17] --> Loop 15 

### Ranking functions of CR plus(V1,V,Out) 
* RF of phase [13]: [V1]

#### Partial ranking functions of CR plus(V1,V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    V1


### Specialization of cost equations quot/3 
* CE 7 is refined into CE [19] 
* CE 9 is refined into CE [20] 
* CE 8 is refined into CE [21,22,23] 


### Cost equations --> "Loop" of quot/3 
* CEs [23] --> Loop 16 
* CEs [22] --> Loop 17 
* CEs [21] --> Loop 18 
* CEs [19,20] --> Loop 19 

### Ranking functions of CR quot(V1,V,Out) 
* RF of phase [16]: [V1-1,V1-V+1]
* RF of phase [18]: [V1]

#### Partial ranking functions of CR quot(V1,V,Out) 
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V1-1
    V1-V+1
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [24,25,26] 
* CE 2 is refined into CE [27,28,29,30,31] 
* CE 3 is refined into CE [32,33,34,35] 


### Cost equations --> "Loop" of start/2 
* CEs [27] --> Loop 20 
* CEs [24] --> Loop 21 
* CEs [25,26,28,29,30,31,32,33,34,35] --> Loop 22 

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

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


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

#### Cost of chains of minus(V1,V,Out):
* Chain [[10],12]: 1*it(10)+1
  Such that:it(10) =< V

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

* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< V

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

* Chain [12]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

* Chain [11]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of plus(V1,V,Out):
* Chain [[13],15]: 1*it(13)+1
  Such that:it(13) =< -V+Out

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

* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< Out

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

* Chain [15]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [14]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of quot(V1,V,Out):
* Chain [[18],19]: 2*it(18)+1
  Such that:it(18) =< Out

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

* Chain [[18],17,19]: 2*it(18)+1*s(2)+2
  Such that:s(2) =< 1
it(18) =< Out

  with precondition: [V=1,Out>=2,V1>=Out] 

* Chain [[16],19]: 2*it(16)+1*s(5)+1
  Such that:it(16) =< V1-V+1
aux(3) =< V1
it(16) =< aux(3)
s(5) =< aux(3)

  with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 

* Chain [[16],17,19]: 2*it(16)+1*s(2)+1*s(5)+2
  Such that:it(16) =< V1-V+1
s(2) =< V
aux(4) =< V1
it(16) =< aux(4)
s(5) =< aux(4)

  with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] 

* Chain [19]: 1
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [17,19]: 1*s(2)+2
  Such that:s(2) =< V

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


#### Cost of chains of start(V1,V):
* Chain [22]: 4*s(9)+4*s(12)+4*s(14)+2
  Such that:aux(6) =< V1
aux(7) =< V1-V+1
aux(8) =< V
s(14) =< aux(6)
s(12) =< aux(7)
s(9) =< aux(8)
s(12) =< aux(6)

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

* Chain [21]: 1
  with precondition: [V=0,V1>=0] 

* Chain [20]: 1*s(21)+4*s(23)+2
  Such that:s(21) =< 1
s(22) =< V1
s(23) =< s(22)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [22] with precondition: [V1>=0,V>=0] 
    - Upper bound: 4*V1+4*V+2+nat(V1-V+1)*4 
    - Complexity: n 
* Chain [21] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [20] with precondition: [V=1,V1>=1] 
    - Upper bound: 4*V1+3 
    - Complexity: n 

### Maximum cost of start(V1,V): 4*V1+1+max([1,nat(V1-V+1)*4+4*V])+1 
Asymptotic class: n 
* Total analysis performed in 217 ms.


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

(14)
BOUNDS(1, n^1)

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

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

(16)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0')))

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

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

(18)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0')))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, quot, plus

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

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

(20)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0')))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, quot, plus

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
gen_0':s2_0(0)

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:

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0')))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, quot, plus

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

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0')))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
quot, plus
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s2_0(n296_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n296_0, b)), rt in Omega(1 + n296_0)

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n296_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
s(plus(gen_0':s2_0(n296_0), gen_0':s2_0(b))) ->_IH
s(gen_0':s2_0(+(b, c297_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(28)
BOUNDS(1, INF)