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


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


WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, 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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 490 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), 247 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), 129 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
   round(0) -> 0
   round(s(0)) -> s(s(0))
   round(s(s(x))) -> s(s(round(x)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

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:

   f(true, x, y) -> f(gt(x, y), x, round(s(y))) [1]
   round(0) -> 0 [1]
   round(s(0)) -> s(s(0)) [1]
   round(s(s(x))) -> s(s(round(x))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [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:

   f(true, x, y) -> f(gt(x, y), x, round(s(y))) [1]
   round(0) -> 0 [1]
   round(s(0)) -> s(s(0)) [1]
   round(s(s(x))) -> s(s(round(x))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]

The TRS has the following type information:
f :: true:false -> s:0 -> s:0 -> f
true :: true:false
gt :: s:0 -> s:0 -> true:false
round :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
false :: true:false

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:

   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_f

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

(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:

   f(true, x, y) -> f(gt(x, y), x, round(s(y))) [1]
   round(0) -> 0 [1]
   round(s(0)) -> s(s(0)) [1]
   round(s(s(x))) -> s(s(round(x))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
f :: true:false -> s:0 -> s:0 -> null_f
true :: true:false
gt :: s:0 -> s:0 -> true:false
round :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
false :: true:false
null_f :: null_f

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:

   true => 1
   0 => 0
   false => 0
   null_f => 0

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

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

   f(z, z', z'') -{ 1 }-> f(gt(x, y), x, round(1 + y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
   gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0
   gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0
   round(z) -{ 1 }-> 0 :|: z = 0
   round(z) -{ 1 }-> 1 + (1 + round(x)) :|: x >= 0, z = 1 + (1 + x)
   round(z) -{ 1 }-> 1 + (1 + 0) :|: z = 1 + 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,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[round(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(f(V1, V, V2, Out),1,[gt(V4, V3, Ret0),round(1 + V3, Ret2),f(Ret0, V4, Ret2, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(round(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(round(V1, Out),1,[],[Out = 2,V1 = 1]).
eq(round(V1, Out),1,[round(V5, Ret11)],[Out = 2 + Ret11,V5 >= 0,V1 = 2 + V5]).
eq(gt(V1, V, Out),1,[],[Out = 0,V6 >= 0,V = V6,V1 = 0]).
eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V7,V = 0,V7 >= 0]).
eq(gt(V1, V, Out),1,[gt(V8, V9, Ret1)],[Out = Ret1,V9 >= 0,V = 1 + V9,V1 = 1 + V8,V8 >= 0]).
eq(f(V1, V, V2, Out),0,[],[Out = 0,V11 >= 0,V2 = V12,V10 >= 0,V1 = V11,V = V10,V12 >= 0]).
input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(round(V1,Out),[V1],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [gt/3]
1. recursive  : [round/2]
2. recursive  : [f/4]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into gt/3
1. SCC is partially evaluated into round/2
2. SCC is partially evaluated into f/4
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations gt/3 
* CE 11 is refined into CE [12] 
* CE 10 is refined into CE [13] 
* CE 9 is refined into CE [14] 


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

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations round/2 
* CE 8 is refined into CE [15] 
* CE 7 is refined into CE [16] 
* CE 6 is refined into CE [17] 


### Cost equations --> "Loop" of round/2 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [15] --> Loop 15 

### Ranking functions of CR round(V1,Out) 
* RF of phase [15]: [V1-1]

#### Partial ranking functions of CR round(V1,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V1-1


### Specialization of cost equations f/4 
* CE 5 is refined into CE [18] 
* CE 4 is refined into CE [19,20,21,22,23,24,25,26] 


### Cost equations --> "Loop" of f/4 
* CEs [25] --> Loop 16 
* CEs [26] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [22] --> Loop 20 
* CEs [21] --> Loop 21 
* CEs [20] --> Loop 22 
* CEs [19] --> Loop 23 
* CEs [18] --> Loop 24 

### Ranking functions of CR f(V1,V,V2,Out) 
* RF of phase [16,17]: [V-V2]

#### Partial ranking functions of CR f(V1,V,V2,Out) 
* Partial RF of phase [16,17]:
  - RF of loop [16:1]:
    V/2-V2/2
  - RF of loop [17:1]:
    V-V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [27,28,29,30,31] 
* CE 2 is refined into CE [32,33,34,35] 
* CE 3 is refined into CE [36,37,38,39] 


### Cost equations --> "Loop" of start/3 
* CEs [34,35,39] --> Loop 25 
* CEs [38] --> Loop 26 
* CEs [27] --> Loop 27 
* CEs [37] --> Loop 28 
* CEs [28,29,30,31,33] --> Loop 29 
* CEs [32,36] --> Loop 30 

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

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


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

#### Cost of chains of gt(V1,V,Out):
* Chain [[12],11]: 1*it(12)+1
  Such that:it(12) =< V1

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

* Chain [[12],10]: 1*it(12)+1
  Such that:it(12) =< V

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

* Chain [11]: 1
  with precondition: [V1=0,Out=0,V>=0] 

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


#### Cost of chains of round(V1,Out):
* Chain [[15],14]: 1*it(15)+1
  Such that:it(15) =< Out

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

* Chain [[15],13]: 1*it(15)+1
  Such that:it(15) =< Out

  with precondition: [V1+1=Out,V1>=3] 

* Chain [14]: 1
  with precondition: [V1=0,Out=0] 

* Chain [13]: 1
  with precondition: [V1=1,Out=2] 


#### Cost of chains of f(V1,V,V2,Out):
* Chain [[16,17],24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+0
  Such that:aux(3) =< V+1
aux(5) =< V-V2
aux(6) =< V-V2+1
it(16) =< V/2-V2/2
it(16) =< aux(5)
it(17) =< aux(5)
it(16) =< aux(6)
it(17) =< aux(6)
aux(4) =< aux(3)-1
s(10) =< it(16)*aux(3)
s(12) =< it(17)*aux(4)
s(11) =< s(12)
s(9) =< s(10)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [[16,17],19,24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+1*s(13)+1*s(14)+3
  Such that:s(13) =< V
aux(3) =< V+1
s(14) =< V+3
aux(5) =< V-V2
aux(6) =< V-V2+1
it(16) =< V/2-V2/2
it(16) =< aux(5)
it(17) =< aux(5)
it(16) =< aux(6)
it(17) =< aux(6)
aux(4) =< aux(3)-1
s(10) =< it(16)*aux(3)
s(12) =< it(17)*aux(4)
s(11) =< s(12)
s(9) =< s(10)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [[16,17],18,24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+1*s(15)+1*s(16)+3
  Such that:s(15) =< V
aux(3) =< V+1
s(16) =< V+2
aux(5) =< V-V2
aux(6) =< V-V2+1
it(16) =< V/2-V2/2
it(16) =< aux(5)
it(17) =< aux(5)
it(16) =< aux(6)
it(17) =< aux(6)
aux(4) =< aux(3)-1
s(10) =< it(16)*aux(3)
s(12) =< it(17)*aux(4)
s(11) =< s(12)
s(9) =< s(10)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [24]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [23,24]: 3
  with precondition: [V1=1,V=0,V2=0,Out=0] 

* Chain [22,24]: 1*s(17)+3
  Such that:s(17) =< V2+2

  with precondition: [V1=1,V=0,Out=0,V2>=2] 

* Chain [21,24]: 1*s(18)+3
  Such that:s(18) =< V2+1

  with precondition: [V1=1,V=0,Out=0,V2>=1] 

* Chain [20,[16,17],24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+3
  Such that:aux(3) =< V+1
it(16) =< V/2
aux(7) =< V
it(16) =< aux(7)
it(17) =< aux(7)
aux(4) =< aux(3)-1
s(10) =< it(16)*aux(3)
s(12) =< it(17)*aux(4)
s(11) =< s(12)
s(9) =< s(10)

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [20,[16,17],19,24]: 3*it(16)+4*it(17)+2*s(9)+2*s(11)+1*s(14)+6
  Such that:aux(3) =< V+1
s(14) =< V+3
it(16) =< V/2
aux(8) =< V
it(17) =< aux(8)
it(16) =< aux(8)
aux(4) =< aux(3)-1
s(10) =< it(16)*aux(3)
s(12) =< it(17)*aux(4)
s(11) =< s(12)
s(9) =< s(10)

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [20,[16,17],18,24]: 3*it(16)+4*it(17)+2*s(9)+2*s(11)+1*s(16)+6
  Such that:aux(3) =< V+1
s(16) =< V+2
it(16) =< V/2
aux(9) =< V
it(17) =< aux(9)
it(16) =< aux(9)
aux(4) =< aux(3)-1
s(10) =< it(16)*aux(3)
s(12) =< it(17)*aux(4)
s(11) =< s(12)
s(9) =< s(10)

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [20,24]: 3
  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [20,19,24]: 1*s(13)+1*s(14)+6
  Such that:s(14) =< 4
s(13) =< V

  with precondition: [V1=1,V2=0,Out=0,2>=V,V>=1] 

* Chain [20,18,24]: 1*s(15)+1*s(16)+6
  Such that:s(16) =< 3
s(15) =< V

  with precondition: [V1=1,V2=0,Out=0,2>=V,V>=1] 

* Chain [19,24]: 1*s(13)+1*s(14)+3
  Such that:s(13) =< V
s(14) =< V2+2

  with precondition: [V1=1,Out=0,V>=1,V2>=2,V2>=V] 

* Chain [18,24]: 1*s(15)+1*s(16)+3
  Such that:s(15) =< V
s(16) =< V2+1

  with precondition: [V1=1,Out=0,V>=1,V2>=V] 


#### Cost of chains of start(V1,V,V2):
* Chain [30]: 1
  with precondition: [V1=0] 

* Chain [29]: 2*s(92)+2*s(93)+1*s(94)+1*s(95)+2*s(96)+2*s(97)+17*s(101)+9*s(102)+6*s(106)+6*s(107)+9*s(120)+9*s(121)+6*s(125)+6*s(126)+6
  Such that:s(94) =< 3
s(95) =< 4
s(116) =< V-V2
s(117) =< V-V2+1
s(100) =< V/2
s(118) =< V/2-V2/2
aux(19) =< V
aux(20) =< V+1
aux(21) =< V+2
aux(22) =< V+3
aux(23) =< V2+1
aux(24) =< V2+2
s(96) =< aux(21)
s(97) =< aux(22)
s(92) =< aux(23)
s(93) =< aux(24)
s(101) =< aux(19)
s(102) =< s(100)
s(102) =< aux(19)
s(103) =< aux(20)-1
s(104) =< s(102)*aux(20)
s(105) =< s(101)*s(103)
s(106) =< s(105)
s(107) =< s(104)
s(120) =< s(118)
s(120) =< s(116)
s(121) =< s(116)
s(120) =< s(117)
s(121) =< s(117)
s(123) =< s(120)*aux(20)
s(124) =< s(121)*s(103)
s(125) =< s(124)
s(126) =< s(123)

  with precondition: [V1=1] 

* Chain [28]: 1
  with precondition: [V=0,V1>=1] 

* Chain [27]: 3
  with precondition: [V1>=0,V>=0,V2>=0] 

* Chain [26]: 1*s(127)+1
  Such that:s(127) =< V1

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

* Chain [25]: 1*s(128)+1*s(129)+1*s(130)+1
  Such that:s(129) =< V1
s(128) =< V1+1
s(130) =< V

  with precondition: [V1>=2] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [30] with precondition: [V1=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [29] with precondition: [V1=1] 
    - Upper bound: nat(V)*17+13+nat(V)*6*nat(nat(V+1)+ -1)+nat(nat(V+1)+ -1)*6*nat(V-V2)+nat(V+1)*6*nat(V/2-V2/2)+nat(V+1)*6*nat(V/2)+nat(V+2)*2+nat(V+3)*2+nat(V2+1)*2+nat(V2+2)*2+nat(V-V2)*9+nat(V/2-V2/2)*9+nat(V/2)*9 
    - Complexity: n^2 
* Chain [28] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [27] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [26] with precondition: [V1>=1,V>=V1] 
    - Upper bound: V1+1 
    - Complexity: n 
* Chain [25] with precondition: [V1>=2] 
    - Upper bound: V1+1+nat(V)+(V1+1) 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([max([2,nat(V)*17+12+nat(V)*6*nat(nat(V+1)+ -1)+nat(nat(V+1)+ -1)*6*nat(V-V2)+nat(V+1)*6*nat(V/2-V2/2)+nat(V+1)*6*nat(V/2)+nat(V+2)*2+nat(V+3)*2+nat(V2+1)*2+nat(V2+2)*2+nat(V-V2)*9+nat(V/2-V2/2)*9+nat(V/2)*9]),V1+1+nat(V)+V1])+1 
Asymptotic class: n^2 
* Total analysis performed in 404 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^1, INF).


The TRS R consists of the following rules:

   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
   round(0') -> 0'
   round(s(0')) -> s(s(0'))
   round(s(s(x))) -> s(s(round(x)))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(u, v)

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

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

(14)
Obligation:
Innermost TRS:
Rules:
f(true, x, y) -> f(gt(x, y), x, round(s(y)))
round(0') -> 0'
round(s(0')) -> s(s(0'))
round(s(s(x))) -> s(s(round(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
round :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, gt, round

They will be analysed ascendingly in the following order:
gt < f
round < f

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

(16)
Obligation:
Innermost TRS:
Rules:
f(true, x, y) -> f(gt(x, y), x, round(s(y)))
round(0') -> 0'
round(s(0')) -> s(s(0'))
round(s(s(x))) -> s(s(round(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
round :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f, round

They will be analysed ascendingly in the following order:
gt < f
round < f

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
false

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:
f(true, x, y) -> f(gt(x, y), x, round(s(y)))
round(0') -> 0'
round(s(0')) -> s(s(0'))
round(s(s(x))) -> s(s(round(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
round :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f, round

They will be analysed ascendingly in the following order:
gt < f
round < f

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
f(true, x, y) -> f(gt(x, y), x, round(s(y)))
round(0') -> 0'
round(s(0')) -> s(s(0'))
round(s(s(x))) -> s(s(round(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
round :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
round, f

They will be analysed ascendingly in the following order:
round < f

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
round(gen_s:0'4_0(*(2, n253_0))) -> gen_s:0'4_0(*(2, n253_0)), rt in Omega(1 + n253_0)

Induction Base:
round(gen_s:0'4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
round(gen_s:0'4_0(*(2, +(n253_0, 1)))) ->_R^Omega(1)
s(s(round(gen_s:0'4_0(*(2, n253_0))))) ->_IH
s(s(gen_s:0'4_0(*(2, c254_0))))

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

(24)
Obligation:
Innermost TRS:
Rules:
f(true, x, y) -> f(gt(x, y), x, round(s(y)))
round(0') -> 0'
round(s(0')) -> s(s(0'))
round(s(s(x))) -> s(s(round(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
round :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
round(gen_s:0'4_0(*(2, n253_0))) -> gen_s:0'4_0(*(2, n253_0)), rt in Omega(1 + n253_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
f