WORST_CASE(Omega(n^1),O(n^2))
proof of input_spvTLqQwvB.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 286 ms]
    (18) BOUNDS(1, n^2)
(19) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
    (21) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRelTRS
    (23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 12 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 433 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
            (36) typed CpxTrs


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

(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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fac(x) -> help(x, 0)
   help(x, c) -> if(lt(c, x), x, c)
   if(true, x, c) -> times(s(c), help(x, s(c)))
   if(false, x, c) -> s(0)

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fac(z0) -> help(z0, 0)
   help(z0, z1) -> if(lt(z1, z0), z0, z1)
   if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
   if(false, z0, z1) -> s(0)
Tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FAC(z0) -> c3(HELP(z0, 0))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
   IF(false, z0, z1) -> c6
S tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FAC(z0) -> c3(HELP(z0, 0))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
   IF(false, z0, z1) -> c6
K tuples:none
Defined Rule Symbols:   lt_2, fac_1, help_2, if_3

Defined Pair Symbols:   LT_2, FAC_1, HELP_2, IF_3

Compound Symbols:   c, c1, c2_1, c3_1, c4_2, c5_1, c6


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FAC(z0) -> c3(HELP(z0, 0))
Removed 3 trailing nodes:
   IF(false, z0, z1) -> c6
   LT(z0, 0) -> c1
   LT(0, s(z0)) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fac(z0) -> help(z0, 0)
   help(z0, z1) -> if(lt(z1, z0), z0, z1)
   if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
   if(false, z0, z1) -> s(0)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, fac_1, help_2, if_3

Defined Pair Symbols:   LT_2, HELP_2, IF_3

Compound Symbols:   c2_1, c4_2, c5_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fac(z0) -> help(z0, 0)
   help(z0, z1) -> if(lt(z1, z0), z0, z1)
   if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
   if(false, z0, z1) -> s(0)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2

Defined Pair Symbols:   LT_2, HELP_2, IF_3

Compound Symbols:   c2_1, c4_2, c5_1


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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))

The (relative) TRS S consists of the following rules:

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)

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

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

(10)
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:

   LT(s(z0), s(z1)) -> c2(LT(z0, z1)) [1]
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0)) [1]
   IF(true, z0, z1) -> c5(HELP(z0, s(z1))) [1]
   lt(0, s(z0)) -> true [0]
   lt(z0, 0) -> false [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LT(s(z0), s(z1)) -> c2(LT(z0, z1)) [1]
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0)) [1]
   IF(true, z0, z1) -> c5(HELP(z0, s(z1))) [1]
   lt(0, s(z0)) -> true [0]
   lt(z0, 0) -> false [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]

The TRS has the following type information:
LT :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
HELP :: s:0 -> s:0 -> c4
c4 :: c5 -> c2 -> c4
IF :: true:false -> s:0 -> s:0 -> c5
lt :: s:0 -> s:0 -> true:false
true :: true:false
c5 :: c4 -> c5
0 :: s:0
false :: true:false

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

(13) 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:

   lt(v0, v1) -> null_lt [0]
   LT(v0, v1) -> null_LT [0]
   IF(v0, v1, v2) -> null_IF [0]

And the following fresh constants:    null_lt, null_LT, null_IF, const

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

(14)
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:

   LT(s(z0), s(z1)) -> c2(LT(z0, z1)) [1]
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0)) [1]
   IF(true, z0, z1) -> c5(HELP(z0, s(z1))) [1]
   lt(0, s(z0)) -> true [0]
   lt(z0, 0) -> false [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]
   lt(v0, v1) -> null_lt [0]
   LT(v0, v1) -> null_LT [0]
   IF(v0, v1, v2) -> null_IF [0]

The TRS has the following type information:
LT :: s:0 -> s:0 -> c2:null_LT
s :: s:0 -> s:0
c2 :: c2:null_LT -> c2:null_LT
HELP :: s:0 -> s:0 -> c4
c4 :: c5:null_IF -> c2:null_LT -> c4
IF :: true:false:null_lt -> s:0 -> s:0 -> c5:null_IF
lt :: s:0 -> s:0 -> true:false:null_lt
true :: true:false:null_lt
c5 :: c4 -> c5:null_IF
0 :: s:0
false :: true:false:null_lt
null_lt :: true:false:null_lt
null_LT :: c2:null_LT
null_IF :: c5:null_IF
const :: c4

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

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

   true => 2
   0 => 0
   false => 1
   null_lt => 0
   null_LT => 0
   null_IF => 0
   const => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   HELP(z, z') -{ 1 }-> 1 + IF(lt(z1, z0), z0, z1) + LT(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   IF(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + HELP(z0, 1 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   LT(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   LT(z, z') -{ 1 }-> 1 + LT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   lt(z, z') -{ 0 }-> lt(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   lt(z, z') -{ 0 }-> 2 :|: z0 >= 0, z' = 1 + z0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z = z0, z0 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V6),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun2(V1, V, V6, Out)],[V1 >= 0,V >= 0,V6 >= 0]).
eq(start(V1, V, V6),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[lt(V4, V5, Ret010),fun2(Ret010, V5, V4, Ret01),fun(V4, V5, Ret11)],[Out = 1 + Ret01 + Ret11,V1 = V5,V4 >= 0,V = V4,V5 >= 0]).
eq(fun2(V1, V, V6, Out),1,[fun1(V8, 1 + V7, Ret12)],[Out = 1 + Ret12,V1 = 2,V7 >= 0,V8 >= 0,V = V8,V6 = V7]).
eq(lt(V1, V, Out),0,[],[Out = 2,V9 >= 0,V = 1 + V9,V1 = 0]).
eq(lt(V1, V, Out),0,[],[Out = 1,V1 = V10,V10 >= 0,V = 0]).
eq(lt(V1, V, Out),0,[lt(V11, V12, Ret)],[Out = Ret,V12 >= 0,V1 = 1 + V11,V11 >= 0,V = 1 + V12]).
eq(lt(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun2(V1, V, V6, Out),0,[],[Out = 0,V18 >= 0,V6 = V19,V17 >= 0,V1 = V18,V = V17,V19 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,V6,Out),[V1,V,V6],[Out]).
input_output_vars(lt(V1,V,Out),[V1,V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into lt/3
2. SCC is partially evaluated into fun1/3
3. SCC is partially evaluated into start/3

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

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


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

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

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


### Specialization of cost equations lt/3 
* CE 13 is refined into CE [16] 
* CE 11 is refined into CE [17] 
* CE 10 is refined into CE [18] 
* CE 12 is refined into CE [19] 


### Cost equations --> "Loop" of lt/3 
* CEs [19] --> Loop 12 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [18] --> Loop 15 

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

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


### Specialization of cost equations fun1/3 
* CE 7 is refined into CE [20,21,22] 
* CE 6 is refined into CE [23,24,25,26,27,28,29,30] 


### Cost equations --> "Loop" of fun1/3 
* CEs [30] --> Loop 16 
* CEs [26,28] --> Loop 17 
* CEs [23] --> Loop 18 
* CEs [24,25,27,29] --> Loop 19 
* CEs [21] --> Loop 20 
* CEs [22] --> Loop 21 
* CEs [20] --> Loop 22 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [20,21]: [V1-V]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [20,21]:
  - RF of loop [20:1,21:1]:
    V1-V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [31] 
* CE 2 is refined into CE [32,33,34] 
* CE 3 is refined into CE [35,36] 
* CE 4 is refined into CE [37,38,39,40,41,42] 
* CE 5 is refined into CE [43,44,45,46,47] 


### Cost equations --> "Loop" of start/3 
* CEs [38,39,40,44] --> Loop 23 
* CEs [32,33,34] --> Loop 24 
* CEs [31,35,36,37,41,42,43,45,46,47] --> Loop 25 

### Ranking functions of CR start(V1,V,V6) 

#### Partial ranking functions of CR start(V1,V,V6) 


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< Out

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

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


#### Cost of chains of lt(V1,V,Out):
* Chain [[12],15]: 0
  with precondition: [Out=2,V1>=1,V>=V1+1] 

* Chain [[12],14]: 0
  with precondition: [Out=1,V>=1,V1>=V] 

* Chain [[12],13]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [15]: 0
  with precondition: [V1=0,Out=2,V>=1] 

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

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,V,Out):
* Chain [[20,21],19]: 4*it(20)+1*s(3)+1
  Such that:aux(1) =< V1
aux(4) =< V1-V
it(20) =< aux(4)
s(3) =< it(20)*aux(1)

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

* Chain [[20,21],17]: 4*it(20)+1*s(3)+2*s(4)+1
  Such that:aux(5) =< V1
aux(6) =< V1-V
s(4) =< aux(5)
it(20) =< aux(6)
s(3) =< it(20)*aux(5)

  with precondition: [V>=1,Out>=4,V1>=V+1] 

* Chain [[20,21],16]: 4*it(20)+1*s(3)+1*s(6)+1
  Such that:aux(7) =< V1
aux(8) =< V1-V
s(6) =< aux(7)
it(20) =< aux(8)
s(3) =< it(20)*aux(7)

  with precondition: [V>=1,Out>=4,V1>=V+2] 

* Chain [22,[20,21],19]: 4*it(20)+1*s(3)+3
  Such that:aux(9) =< V1
it(20) =< aux(9)
s(3) =< it(20)*aux(9)

  with precondition: [V=0,V1>=2,Out>=5] 

* Chain [22,[20,21],17]: 6*it(20)+1*s(3)+3
  Such that:aux(10) =< V1
it(20) =< aux(10)
s(3) =< it(20)*aux(10)

  with precondition: [V=0,V1>=2,Out>=6] 

* Chain [22,[20,21],16]: 5*it(20)+1*s(3)+3
  Such that:aux(11) =< V1
it(20) =< aux(11)
s(3) =< it(20)*aux(11)

  with precondition: [V=0,V1>=3,Out>=6] 

* Chain [22,19]: 3
  with precondition: [V=0,Out=3,V1>=1] 

* Chain [22,17]: 1*s(4)+1*s(5)+3
  Such that:s(4) =< 1
s(5) =< V1

  with precondition: [V=0,Out=4,V1>=1] 

* Chain [22,16]: 1*s(6)+3
  Such that:s(6) =< 1

  with precondition: [V=0,Out=4,V1>=2] 

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

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

* Chain [17]: 1*s(4)+1*s(5)+1
  Such that:s(5) =< V1
s(4) =< V

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

* Chain [16]: 1*s(6)+1
  Such that:s(6) =< V

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


#### Cost of chains of start(V1,V,V6):
* Chain [25]: 3*s(36)+4*s(39)+12*s(40)+3*s(41)+1
  Such that:s(38) =< V1-V
aux(17) =< V1
aux(18) =< V
s(39) =< aux(17)
s(36) =< aux(18)
s(40) =< s(38)
s(41) =< s(40)*aux(17)

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

* Chain [24]: 4*s(47)+12*s(48)+3*s(49)+2*s(52)+2
  Such that:s(46) =< V-V6
s(51) =< V6+1
aux(19) =< V
s(47) =< aux(19)
s(48) =< s(46)
s(49) =< s(48)*aux(19)
s(52) =< s(51)

  with precondition: [V1=2,V>=0,V6>=0] 

* Chain [23]: 16*s(53)+2*s(55)+3*s(58)+3
  Such that:s(54) =< 1
aux(20) =< V1
s(53) =< aux(20)
s(55) =< s(54)
s(58) =< s(53)*aux(20)

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


Closed-form bounds of start(V1,V,V6): 
-------------------------------------
* Chain [25] with precondition: [V1>=0,V>=0] 
    - Upper bound: 4*V1+1+3*V1*nat(V1-V)+3*V+nat(V1-V)*12 
    - Complexity: n^2 
* Chain [24] with precondition: [V1=2,V>=0,V6>=0] 
    - Upper bound: 4*V+2+3*V*nat(V-V6)+(2*V6+2)+nat(V-V6)*12 
    - Complexity: n^2 
* Chain [23] with precondition: [V=0,V1>=0] 
    - Upper bound: 16*V1+5+3*V1*V1 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V6): max([4*V1+max([12*V1+4+3*V1*V1,3*V1*nat(V1-V)+3*V+nat(V1-V)*12]),4*V+1+3*V*nat(V-V6)+nat(V6+1)*2+nat(V-V6)*12])+1 
Asymptotic class: n^2 
* Total analysis performed in 306 ms.


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

(18)
BOUNDS(1, n^2)

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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fac(z0) -> help(z0, 0)
   help(z0, z1) -> if(lt(z1, z0), z0, z1)
   if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
   if(false, z0, z1) -> s(0)
Tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FAC(z0) -> c3(HELP(z0, 0))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
   IF(false, z0, z1) -> c6
S tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FAC(z0) -> c3(HELP(z0, 0))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
   IF(false, z0, z1) -> c6
K tuples:none
Defined Rule Symbols:   lt_2, fac_1, help_2, if_3

Defined Pair Symbols:   LT_2, FAC_1, HELP_2, IF_3

Compound Symbols:   c, c1, c2_1, c3_1, c4_2, c5_1, c6


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

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

(22)
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:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FAC(z0) -> c3(HELP(z0, 0))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
   IF(false, z0, z1) -> c6

The (relative) TRS S consists of the following rules:

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fac(z0) -> help(z0, 0)
   help(z0, z1) -> if(lt(z1, z0), z0, z1)
   if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
   if(false, z0, z1) -> s(0)

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

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

(24)
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:

   LT(0', s(z0)) -> c
   LT(z0, 0') -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FAC(z0) -> c3(HELP(z0, 0'))
   HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
   IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
   IF(false, z0, z1) -> c6

The (relative) TRS S consists of the following rules:

   lt(0', s(z0)) -> true
   lt(z0, 0') -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fac(z0) -> help(z0, 0')
   help(z0, z1) -> if(lt(z1, z0), z0, z1)
   if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
   if(false, z0, z1) -> s(0')

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

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

(26)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
FAC(z0) -> c3(HELP(z0, 0'))
HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
IF(false, z0, z1) -> c6
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fac(z0) -> help(z0, 0')
help(z0, z1) -> if(lt(z1, z0), z0, z1)
if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
if(false, z0, z1) -> s(0')

Types:
LT :: 0':s:times -> 0':s:times -> c:c1:c2
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
FAC :: 0':s:times -> c3
c3 :: c4 -> c3
HELP :: 0':s:times -> 0':s:times -> c4
c4 :: c5:c6 -> c:c1:c2 -> c4
IF :: true:false -> 0':s:times -> 0':s:times -> c5:c6
lt :: 0':s:times -> 0':s:times -> true:false
true :: true:false
c5 :: c4 -> c5:c6
false :: true:false
c6 :: c5:c6
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times -> 0':s:times
hole_c:c1:c21_7 :: c:c1:c2
hole_0':s:times2_7 :: 0':s:times
hole_c33_7 :: c3
hole_c44_7 :: c4
hole_c5:c65_7 :: c5:c6
hole_true:false6_7 :: true:false
gen_c:c1:c27_7 :: Nat -> c:c1:c2
gen_0':s:times8_7 :: Nat -> 0':s:times

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LT, HELP, lt, help

They will be analysed ascendingly in the following order:
LT < HELP
lt < HELP
lt < help

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

(28)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
FAC(z0) -> c3(HELP(z0, 0'))
HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
IF(false, z0, z1) -> c6
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fac(z0) -> help(z0, 0')
help(z0, z1) -> if(lt(z1, z0), z0, z1)
if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
if(false, z0, z1) -> s(0')

Types:
LT :: 0':s:times -> 0':s:times -> c:c1:c2
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
FAC :: 0':s:times -> c3
c3 :: c4 -> c3
HELP :: 0':s:times -> 0':s:times -> c4
c4 :: c5:c6 -> c:c1:c2 -> c4
IF :: true:false -> 0':s:times -> 0':s:times -> c5:c6
lt :: 0':s:times -> 0':s:times -> true:false
true :: true:false
c5 :: c4 -> c5:c6
false :: true:false
c6 :: c5:c6
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times -> 0':s:times
hole_c:c1:c21_7 :: c:c1:c2
hole_0':s:times2_7 :: 0':s:times
hole_c33_7 :: c3
hole_c44_7 :: c4
hole_c5:c65_7 :: c5:c6
hole_true:false6_7 :: true:false
gen_c:c1:c27_7 :: Nat -> c:c1:c2
gen_0':s:times8_7 :: Nat -> 0':s:times


Generator Equations:
gen_c:c1:c27_7(0) <=> c
gen_c:c1:c27_7(+(x, 1)) <=> c2(gen_c:c1:c27_7(x))
gen_0':s:times8_7(0) <=> 0'
gen_0':s:times8_7(+(x, 1)) <=> s(gen_0':s:times8_7(x))


The following defined symbols remain to be analysed:
LT, HELP, lt, help

They will be analysed ascendingly in the following order:
LT < HELP
lt < HELP
lt < help

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT(gen_0':s:times8_7(n10_7), gen_0':s:times8_7(+(1, n10_7))) -> gen_c:c1:c27_7(n10_7), rt in Omega(1 + n10_7)

Induction Base:
LT(gen_0':s:times8_7(0), gen_0':s:times8_7(+(1, 0))) ->_R^Omega(1)
c

Induction Step:
LT(gen_0':s:times8_7(+(n10_7, 1)), gen_0':s:times8_7(+(1, +(n10_7, 1)))) ->_R^Omega(1)
c2(LT(gen_0':s:times8_7(n10_7), gen_0':s:times8_7(+(1, n10_7)))) ->_IH
c2(gen_c:c1:c27_7(c11_7))

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

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
FAC(z0) -> c3(HELP(z0, 0'))
HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
IF(false, z0, z1) -> c6
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fac(z0) -> help(z0, 0')
help(z0, z1) -> if(lt(z1, z0), z0, z1)
if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
if(false, z0, z1) -> s(0')

Types:
LT :: 0':s:times -> 0':s:times -> c:c1:c2
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
FAC :: 0':s:times -> c3
c3 :: c4 -> c3
HELP :: 0':s:times -> 0':s:times -> c4
c4 :: c5:c6 -> c:c1:c2 -> c4
IF :: true:false -> 0':s:times -> 0':s:times -> c5:c6
lt :: 0':s:times -> 0':s:times -> true:false
true :: true:false
c5 :: c4 -> c5:c6
false :: true:false
c6 :: c5:c6
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times -> 0':s:times
hole_c:c1:c21_7 :: c:c1:c2
hole_0':s:times2_7 :: 0':s:times
hole_c33_7 :: c3
hole_c44_7 :: c4
hole_c5:c65_7 :: c5:c6
hole_true:false6_7 :: true:false
gen_c:c1:c27_7 :: Nat -> c:c1:c2
gen_0':s:times8_7 :: Nat -> 0':s:times


Generator Equations:
gen_c:c1:c27_7(0) <=> c
gen_c:c1:c27_7(+(x, 1)) <=> c2(gen_c:c1:c27_7(x))
gen_0':s:times8_7(0) <=> 0'
gen_0':s:times8_7(+(x, 1)) <=> s(gen_0':s:times8_7(x))


The following defined symbols remain to be analysed:
LT, HELP, lt, help

They will be analysed ascendingly in the following order:
LT < HELP
lt < HELP
lt < help

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

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

(33)
BOUNDS(n^1, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
FAC(z0) -> c3(HELP(z0, 0'))
HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
IF(false, z0, z1) -> c6
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fac(z0) -> help(z0, 0')
help(z0, z1) -> if(lt(z1, z0), z0, z1)
if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
if(false, z0, z1) -> s(0')

Types:
LT :: 0':s:times -> 0':s:times -> c:c1:c2
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
FAC :: 0':s:times -> c3
c3 :: c4 -> c3
HELP :: 0':s:times -> 0':s:times -> c4
c4 :: c5:c6 -> c:c1:c2 -> c4
IF :: true:false -> 0':s:times -> 0':s:times -> c5:c6
lt :: 0':s:times -> 0':s:times -> true:false
true :: true:false
c5 :: c4 -> c5:c6
false :: true:false
c6 :: c5:c6
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times -> 0':s:times
hole_c:c1:c21_7 :: c:c1:c2
hole_0':s:times2_7 :: 0':s:times
hole_c33_7 :: c3
hole_c44_7 :: c4
hole_c5:c65_7 :: c5:c6
hole_true:false6_7 :: true:false
gen_c:c1:c27_7 :: Nat -> c:c1:c2
gen_0':s:times8_7 :: Nat -> 0':s:times


Lemmas:
LT(gen_0':s:times8_7(n10_7), gen_0':s:times8_7(+(1, n10_7))) -> gen_c:c1:c27_7(n10_7), rt in Omega(1 + n10_7)


Generator Equations:
gen_c:c1:c27_7(0) <=> c
gen_c:c1:c27_7(+(x, 1)) <=> c2(gen_c:c1:c27_7(x))
gen_0':s:times8_7(0) <=> 0'
gen_0':s:times8_7(+(x, 1)) <=> s(gen_0':s:times8_7(x))


The following defined symbols remain to be analysed:
lt, HELP, help

They will be analysed ascendingly in the following order:
lt < HELP
lt < help

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s:times8_7(n517_7), gen_0':s:times8_7(+(1, n517_7))) -> true, rt in Omega(0)

Induction Base:
lt(gen_0':s:times8_7(0), gen_0':s:times8_7(+(1, 0))) ->_R^Omega(0)
true

Induction Step:
lt(gen_0':s:times8_7(+(n517_7, 1)), gen_0':s:times8_7(+(1, +(n517_7, 1)))) ->_R^Omega(0)
lt(gen_0':s:times8_7(n517_7), gen_0':s:times8_7(+(1, n517_7))) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(36)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
FAC(z0) -> c3(HELP(z0, 0'))
HELP(z0, z1) -> c4(IF(lt(z1, z0), z0, z1), LT(z1, z0))
IF(true, z0, z1) -> c5(HELP(z0, s(z1)))
IF(false, z0, z1) -> c6
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fac(z0) -> help(z0, 0')
help(z0, z1) -> if(lt(z1, z0), z0, z1)
if(true, z0, z1) -> times(s(z1), help(z0, s(z1)))
if(false, z0, z1) -> s(0')

Types:
LT :: 0':s:times -> 0':s:times -> c:c1:c2
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
FAC :: 0':s:times -> c3
c3 :: c4 -> c3
HELP :: 0':s:times -> 0':s:times -> c4
c4 :: c5:c6 -> c:c1:c2 -> c4
IF :: true:false -> 0':s:times -> 0':s:times -> c5:c6
lt :: 0':s:times -> 0':s:times -> true:false
true :: true:false
c5 :: c4 -> c5:c6
false :: true:false
c6 :: c5:c6
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times -> 0':s:times
hole_c:c1:c21_7 :: c:c1:c2
hole_0':s:times2_7 :: 0':s:times
hole_c33_7 :: c3
hole_c44_7 :: c4
hole_c5:c65_7 :: c5:c6
hole_true:false6_7 :: true:false
gen_c:c1:c27_7 :: Nat -> c:c1:c2
gen_0':s:times8_7 :: Nat -> 0':s:times


Lemmas:
LT(gen_0':s:times8_7(n10_7), gen_0':s:times8_7(+(1, n10_7))) -> gen_c:c1:c27_7(n10_7), rt in Omega(1 + n10_7)
lt(gen_0':s:times8_7(n517_7), gen_0':s:times8_7(+(1, n517_7))) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c27_7(0) <=> c
gen_c:c1:c27_7(+(x, 1)) <=> c2(gen_c:c1:c27_7(x))
gen_0':s:times8_7(0) <=> 0'
gen_0':s:times8_7(+(x, 1)) <=> s(gen_0':s:times8_7(x))


The following defined symbols remain to be analysed:
HELP, help