/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 462 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), 259 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) 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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr0(x), y, y)
   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   gr0(0) -> false
   gr0(s(x)) -> true
   p(0) -> 0
   p(s(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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
   cond2(true, x, y) -> cond1(gr0(x), y, y) [1]
   cond2(false, x, y) -> cond1(gr0(x), p(x), y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   gr0(0) -> false [1]
   gr0(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
   cond2(true, x, y) -> cond1(gr0(x), y, y) [1]
   cond2(false, x, y) -> cond1(gr0(x), p(x), y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   gr0(0) -> false [1]
   gr0(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2
gr :: 0:s -> 0:s -> true:false
gr0 :: 0:s -> true:false
false :: true:false
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s

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:

   cond1(v0, v1, v2) -> null_cond1 [0]

And the following fresh constants:    null_cond1

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

(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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
   cond2(true, x, y) -> cond1(gr0(x), y, y) [1]
   cond2(false, x, y) -> cond1(gr0(x), p(x), y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   gr0(0) -> false [1]
   gr0(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   cond1(v0, v1, v2) -> null_cond1 [0]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> null_cond1
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> null_cond1
gr :: 0:s -> 0:s -> true:false
gr0 :: 0:s -> true:false
false :: true:false
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
null_cond1 :: null_cond1

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
   false => 0
   0 => 0
   null_cond1 => 0

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

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

   cond1(z, z', z'') -{ 1 }-> cond2(gr(x, y), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   cond2(z, z', z'') -{ 1 }-> cond1(gr0(x), y, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond2(z, z', z'') -{ 1 }-> cond1(gr0(x), p(x), y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   gr0(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   gr0(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 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,[cond1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[cond2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[gr0(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
eq(cond1(V1, V, V2, Out),1,[gr(V4, V3, Ret0),cond2(Ret0, V4, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(cond2(V1, V, V2, Out),1,[gr0(V5, Ret01),cond1(Ret01, V6, V6, Ret1)],[Out = Ret1,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]).
eq(cond2(V1, V, V2, Out),1,[gr0(V8, Ret02),p(V8, Ret11),cond1(Ret02, Ret11, V7, Ret2)],[Out = Ret2,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 0,V = V9,V9 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = 1 + V10,V = 0]).
eq(gr(V1, V, Out),1,[gr(V12, V11, Ret3)],[Out = Ret3,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]).
eq(gr0(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(gr0(V1, Out),1,[],[Out = 1,V13 >= 0,V1 = 1 + V13]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V14,V14 >= 0,V1 = 1 + V14]).
eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V16 >= 0,V2 = V17,V15 >= 0,V1 = V16,V = V15,V17 >= 0]).
input_output_vars(cond1(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(cond2(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(gr0(V1,Out),[V1],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [gr0/2]
1. non_recursive  : [p/2]
2. recursive  : [gr/3]
3. recursive  : [cond1/4,cond2/4]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into gr0/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into gr/3
3. SCC is partially evaluated into cond2/4
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations gr0/2 
* CE 15 is refined into CE [18] 
* CE 14 is refined into CE [19] 


### Cost equations --> "Loop" of gr0/2 
* CEs [18] --> Loop 13 
* CEs [19] --> Loop 14 

### Ranking functions of CR gr0(V1,Out) 

#### Partial ranking functions of CR gr0(V1,Out) 


### Specialization of cost equations p/2 
* CE 17 is refined into CE [20] 
* CE 16 is refined into CE [21] 


### Cost equations --> "Loop" of p/2 
* CEs [20] --> Loop 15 
* CEs [21] --> Loop 16 

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations gr/3 
* CE 9 is refined into CE [22] 
* CE 8 is refined into CE [23] 
* CE 7 is refined into CE [24] 


### Cost equations --> "Loop" of gr/3 
* CEs [23] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [22] --> Loop 19 

### Ranking functions of CR gr(V1,V,Out) 
* RF of phase [19]: [V,V1]

#### Partial ranking functions of CR gr(V1,V,Out) 
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V
    V1


### Specialization of cost equations cond2/4 
* CE 13 is refined into CE [25,26] 
* CE 12 is refined into CE [27,28,29,30] 
* CE 11 is refined into CE [31,32] 
* CE 10 is refined into CE [33,34] 


### Cost equations --> "Loop" of cond2/4 
* CEs [32] --> Loop 20 
* CEs [31] --> Loop 21 
* CEs [34] --> Loop 22 
* CEs [33] --> Loop 23 
* CEs [26] --> Loop 24 
* CEs [25] --> Loop 25 
* CEs [30] --> Loop 26 
* CEs [29] --> Loop 27 
* CEs [28] --> Loop 28 
* CEs [27] --> Loop 29 

### Ranking functions of CR cond2(V1,V,V2,Out) 
* RF of phase [27]: [V-1]

#### Partial ranking functions of CR cond2(V1,V,V2,Out) 
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V-1


### Specialization of cost equations start/3 
* CE 1 is refined into CE [35] 
* CE 2 is refined into CE [36,37,38,39,40] 
* CE 3 is refined into CE [41,42,43,44,45,46,47] 
* CE 4 is refined into CE [48,49,50,51] 
* CE 5 is refined into CE [52,53] 
* CE 6 is refined into CE [54,55] 


### Cost equations --> "Loop" of start/3 
* CEs [40,47] --> Loop 30 
* CEs [35,37,38,39,46] --> Loop 31 
* CEs [36,45,49,50,51,53,55] --> Loop 32 
* CEs [41,42,43,44,48,52,54] --> Loop 33 

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

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


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

#### Cost of chains of gr0(V1,Out):
* Chain [14]: 1
  with precondition: [V1=0,Out=0] 

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


#### Cost of chains of p(V1,Out):
* Chain [16]: 1
  with precondition: [V1=0,Out=0] 

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


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

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

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

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

* Chain [18]: 1
  with precondition: [V1=0,Out=0,V>=0] 

* Chain [17]: 1
  with precondition: [V=0,Out=1,V1>=1] 


#### Cost of chains of cond2(V1,V,V2,Out):
* Chain [[27],29,23]: 5*it(27)+1*s(3)+8
  Such that:aux(3) =< V
it(27) =< aux(3)
s(3) =< it(27)*aux(3)

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

* Chain [[27],22]: 5*it(27)+1*s(3)+3
  Such that:aux(4) =< V
it(27) =< aux(4)
s(3) =< it(27)*aux(4)

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

* Chain [29,23]: 8
  with precondition: [V1=0,V=1,Out=0,V2>=0] 

* Chain [28,25,23]: 12
  with precondition: [V1=0,V2=0,Out=0,V>=2] 

* Chain [28,20]: 7
  with precondition: [V1=0,V2=0,Out=0,V>=2] 

* Chain [26,24,[27],29,23]: 7*it(27)+1*s(3)+17
  Such that:aux(6) =< V2
it(27) =< aux(6)
s(3) =< it(27)*aux(6)

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

* Chain [26,24,[27],22]: 7*it(27)+1*s(3)+12
  Such that:aux(8) =< V2
it(27) =< aux(8)
s(3) =< it(27)*aux(8)

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

* Chain [26,24,29,23]: 2*s(4)+17
  Such that:aux(9) =< 1
s(4) =< aux(9)

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

* Chain [26,24,22]: 2*s(4)+12
  Such that:aux(10) =< V2
s(4) =< aux(10)

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

* Chain [26,20]: 1*s(5)+7
  Such that:s(5) =< V2

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

* Chain [25,23]: 7
  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [24,[27],29,23]: 6*it(27)+1*s(3)+12
  Such that:aux(5) =< V2
it(27) =< aux(5)
s(3) =< it(27)*aux(5)

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

* Chain [24,[27],22]: 6*it(27)+1*s(3)+7
  Such that:aux(7) =< V2
it(27) =< aux(7)
s(3) =< it(27)*aux(7)

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

* Chain [24,29,23]: 1*s(4)+12
  Such that:s(4) =< 1

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

* Chain [24,22]: 1*s(4)+7
  Such that:s(4) =< V2

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

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

* Chain [22]: 3
  with precondition: [V1=0,Out=0,V>=1,V2>=0] 

* Chain [21]: 2
  with precondition: [V1=1,V=0,Out=0,V2>=0] 

* Chain [20]: 2
  with precondition: [V1=1,Out=0,V>=1,V2>=0] 


#### Cost of chains of start(V1,V,V2):
* Chain [33]: 17*s(30)+10*s(31)+2*s(32)+2*s(33)+2*s(35)+17
  Such that:s(34) =< 1
s(28) =< V
s(29) =< V2
s(35) =< s(34)
s(30) =< s(29)
s(31) =< s(28)
s(32) =< s(31)*s(28)
s(33) =< s(30)*s(29)

  with precondition: [V1=0] 

* Chain [32]: 1*s(36)+1*s(37)+5
  Such that:s(36) =< V1
s(37) =< V

  with precondition: [V1>=1] 

* Chain [31]: 11*s(41)+44*s(44)+2*s(46)+6*s(47)+19
  Such that:aux(14) =< V
aux(16) =< V2
s(44) =< aux(16)
s(47) =< s(44)*aux(16)
s(41) =< aux(14)
s(46) =< s(41)*aux(14)

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

* Chain [30]: 17
  with precondition: [V1=1,V2=1,V>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [33] with precondition: [V1=0] 
    - Upper bound: nat(V)*10+19+nat(V)*2*nat(V)+nat(V2)*17+nat(V2)*2*nat(V2) 
    - Complexity: n^2 
* Chain [32] with precondition: [V1>=1] 
    - Upper bound: V1+5+nat(V) 
    - Complexity: n 
* Chain [31] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 11*V+19+2*V*V+44*V2+6*V2*V2 
    - Complexity: n^2 
* Chain [30] with precondition: [V1=1,V2=1,V>=1] 
    - Upper bound: 17 
    - Complexity: constant 

### Maximum cost of start(V1,V,V2): max([12,nat(V)+max([V1,nat(V)*9+14+nat(V)*2*nat(V)+nat(V2)*17+nat(V2)*2*nat(V2)+(nat(V2)*27+nat(V)+nat(V2)*4*nat(V2))])])+5 
Asymptotic class: n^2 
* Total analysis performed in 373 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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr0(x), y, y)
   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   gr0(0') -> false
   gr0(s(x)) -> true
   p(0') -> 0'
   p(s(x)) -> x

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

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

(14)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1

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

(16)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s


Generator Equations:
gen_0':s4_3(0) <=> 0'
gen_0':s4_3(+(x, 1)) <=> s(gen_0':s4_3(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_3(n6_3), gen_0':s4_3(n6_3)) -> false, rt in Omega(1 + n6_3)

Induction Base:
gr(gen_0':s4_3(0), gen_0':s4_3(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_0':s4_3(+(n6_3, 1)), gen_0':s4_3(+(n6_3, 1))) ->_R^Omega(1)
gr(gen_0':s4_3(n6_3), gen_0':s4_3(n6_3)) ->_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:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s


Generator Equations:
gen_0':s4_3(0) <=> 0'
gen_0':s4_3(+(x, 1)) <=> s(gen_0':s4_3(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_3(n6_3), gen_0':s4_3(n6_3)) -> false, rt in Omega(1 + n6_3)


Generator Equations:
gen_0':s4_3(0) <=> 0'
gen_0':s4_3(+(x, 1)) <=> s(gen_0':s4_3(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2