/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^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 11 ms]
(2) CpxTRS
    (3) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
    (4) CpxRelTRS
    (5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) CpxWeightedTrsRenamingProof [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), 2 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 399 ms]
    (18) BOUNDS(1, n^2)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 1119 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (34) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(O(x), O(y)) -> O(+(x, y))
   +(O(x), I(y)) -> I(+(x, y))
   +(I(x), O(y)) -> I(+(x, y))
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(O(x), y) -> O(*(x, y))
   *(I(x), y) -> +(O(*(x, y)), y)
   -(x, 0) -> x
   -(0, x) -> 0
   -(O(x), O(y)) -> O(-(x, y))
   -(O(x), I(y)) -> I(-(-(x, y), I(1)))
   -(I(x), O(y)) -> I(-(x, y))
   -(I(x), I(y)) -> O(-(x, y))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of O: O, +, -, *
The following defined symbols can occur below the 0th argument of +: O, +, *

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
+(O(x), O(y)) -> O(+(x, y))
+(I(x), O(y)) -> I(+(x, y))
*(O(x), y) -> O(*(x, y))
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1)))
-(I(x), O(y)) -> I(-(x, y))

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

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


The TRS R consists of the following rules:

   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(O(x), I(y)) -> I(+(x, y))
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(I(x), y) -> +(O(*(x, y)), y)
   -(x, 0) -> x
   -(0, x) -> 0
   -(I(x), I(y)) -> O(-(x, y))

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

(3) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(I(x), y) -> +(O(*(x, y)), y)
   -(x, 0) -> x
   -(0, x) -> 0
   -(I(x), I(y)) -> O(-(x, y))
   +(c_O(x), I(y)) -> I(+(x, y))

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

   O(x0) -> c_O(x0)

Rewrite Strategy: FULL
----------------------------------------

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

The duplicating contexts are:
*(I(x), [])


The defined contexts are:
O([])
+([], x1)
+([], I(0))


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

(6)
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:

   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(I(x), y) -> +(O(*(x, y)), y)
   -(x, 0) -> x
   -(0, x) -> 0
   -(I(x), I(y)) -> O(-(x, y))
   +(c_O(x), I(y)) -> I(+(x, y))

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

   O(x0) -> c_O(x0)

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

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

(8)
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:

   O(0) -> 0 [1]
   +(0, x) -> x [1]
   +(x, 0) -> x [1]
   +(I(x), I(y)) -> O(+(+(x, y), I(0))) [1]
   *(0, x) -> 0 [1]
   *(x, 0) -> 0 [1]
   *(I(x), y) -> +(O(*(x, y)), y) [1]
   -(x, 0) -> x [1]
   -(0, x) -> 0 [1]
   -(I(x), I(y)) -> O(-(x, y)) [1]
   +(c_O(x), I(y)) -> I(+(x, y)) [1]
   O(x0) -> c_O(x0) [0]

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

(9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus
* => times
- => minus

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

(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:

   O(0) -> 0 [1]
   plus(0, x) -> x [1]
   plus(x, 0) -> x [1]
   plus(I(x), I(y)) -> O(plus(plus(x, y), I(0))) [1]
   times(0, x) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(I(x), y) -> plus(O(times(x, y)), y) [1]
   minus(x, 0) -> x [1]
   minus(0, x) -> 0 [1]
   minus(I(x), I(y)) -> O(minus(x, y)) [1]
   plus(c_O(x), I(y)) -> I(plus(x, y)) [1]
   O(x0) -> c_O(x0) [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:

   O(0) -> 0 [1]
   plus(0, x) -> x [1]
   plus(x, 0) -> x [1]
   plus(I(x), I(y)) -> O(plus(plus(x, y), I(0))) [1]
   times(0, x) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(I(x), y) -> plus(O(times(x, y)), y) [1]
   minus(x, 0) -> x [1]
   minus(0, x) -> 0 [1]
   minus(I(x), I(y)) -> O(minus(x, y)) [1]
   plus(c_O(x), I(y)) -> I(plus(x, y)) [1]
   O(x0) -> c_O(x0) [0]

The TRS has the following type information:
O :: 0:I:c_O -> 0:I:c_O
0 :: 0:I:c_O
plus :: 0:I:c_O -> 0:I:c_O -> 0:I:c_O
I :: 0:I:c_O -> 0:I:c_O
times :: 0:I:c_O -> 0:I:c_O -> 0:I:c_O
minus :: 0:I:c_O -> 0:I:c_O -> 0:I:c_O
c_O :: 0:I:c_O -> 0:I:c_O

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:

   O(v0) -> null_O [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   minus(v0, v1) -> null_minus [0]

And the following fresh constants:    null_O, null_plus, null_times, null_minus

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

(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:

   O(0) -> 0 [1]
   plus(0, x) -> x [1]
   plus(x, 0) -> x [1]
   plus(I(x), I(y)) -> O(plus(plus(x, y), I(0))) [1]
   times(0, x) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(I(x), y) -> plus(O(times(x, y)), y) [1]
   minus(x, 0) -> x [1]
   minus(0, x) -> 0 [1]
   minus(I(x), I(y)) -> O(minus(x, y)) [1]
   plus(c_O(x), I(y)) -> I(plus(x, y)) [1]
   O(x0) -> c_O(x0) [0]
   O(v0) -> null_O [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   minus(v0, v1) -> null_minus [0]

The TRS has the following type information:
O :: 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus
0 :: 0:I:c_O:null_O:null_plus:null_times:null_minus
plus :: 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus
I :: 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus
times :: 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus
minus :: 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus
c_O :: 0:I:c_O:null_O:null_plus:null_times:null_minus -> 0:I:c_O:null_O:null_plus:null_times:null_minus
null_O :: 0:I:c_O:null_O:null_plus:null_times:null_minus
null_plus :: 0:I:c_O:null_O:null_plus:null_times:null_minus
null_times :: 0:I:c_O:null_O:null_plus:null_times:null_minus
null_minus :: 0:I:c_O:null_O:null_plus:null_times:null_minus

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:

   0 => 0
   null_O => 0
   null_plus => 0
   null_times => 0
   null_minus => 0

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

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

   O(z) -{ 1 }-> 0 :|: z = 0
   O(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   O(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> O(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> O(plus(plus(x, y), 1 + 0)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   times(z, z') -{ 1 }-> plus(O(times(x, y)), y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   times(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(V, V1),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V1),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[times(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(fun(V, Out),1,[],[Out = 0,V = 0]).
eq(plus(V, V1, Out),1,[],[Out = V2,V1 = V2,V2 >= 0,V = 0]).
eq(plus(V, V1, Out),1,[],[Out = V3,V3 >= 0,V = V3,V1 = 0]).
eq(plus(V, V1, Out),1,[plus(V4, V5, Ret00),plus(Ret00, 1 + 0, Ret0),fun(Ret0, Ret)],[Out = Ret,V1 = 1 + V5,V4 >= 0,V5 >= 0,V = 1 + V4]).
eq(times(V, V1, Out),1,[],[Out = 0,V1 = V6,V6 >= 0,V = 0]).
eq(times(V, V1, Out),1,[],[Out = 0,V7 >= 0,V = V7,V1 = 0]).
eq(times(V, V1, Out),1,[times(V8, V9, Ret001),fun(Ret001, Ret01),plus(Ret01, V9, Ret1)],[Out = Ret1,V8 >= 0,V9 >= 0,V = 1 + V8,V1 = V9]).
eq(minus(V, V1, Out),1,[],[Out = V10,V10 >= 0,V = V10,V1 = 0]).
eq(minus(V, V1, Out),1,[],[Out = 0,V1 = V11,V11 >= 0,V = 0]).
eq(minus(V, V1, Out),1,[minus(V13, V12, Ret02),fun(Ret02, Ret2)],[Out = Ret2,V1 = 1 + V12,V13 >= 0,V12 >= 0,V = 1 + V13]).
eq(plus(V, V1, Out),1,[plus(V15, V14, Ret11)],[Out = 1 + Ret11,V1 = 1 + V14,V15 >= 0,V14 >= 0,V = 1 + V15]).
eq(fun(V, Out),0,[],[Out = 1 + V16,V = V16,V16 >= 0]).
eq(fun(V, Out),0,[],[Out = 0,V17 >= 0,V = V17]).
eq(plus(V, V1, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V = V19,V1 = V18]).
eq(times(V, V1, Out),0,[],[Out = 0,V21 >= 0,V20 >= 0,V = V21,V1 = V20]).
eq(minus(V, V1, Out),0,[],[Out = 0,V22 >= 0,V23 >= 0,V = V22,V1 = V23]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(plus(V,V1,Out),[V,V1],[Out]).
input_output_vars(times(V,V1,Out),[V,V1],[Out]).
input_output_vars(minus(V,V1,Out),[V,V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [fun/2]
1. recursive [non_tail] : [minus/3]
2. recursive [non_tail,multiple] : [plus/3]
3. recursive [non_tail] : [times/3]
4. non_recursive  : [start/2]

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

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

### Specialization of cost equations fun/2 
* CE 6 is refined into CE [21] 
* CE 5 is refined into CE [22] 
* CE 7 is refined into CE [23] 


### Cost equations --> "Loop" of fun/2 
* CEs [21] --> Loop 15 
* CEs [22,23] --> Loop 16 

### Ranking functions of CR fun(V,Out) 

#### Partial ranking functions of CR fun(V,Out) 


### Specialization of cost equations minus/3 
* CE 17 is refined into CE [24] 
* CE 18 is refined into CE [25] 
* CE 20 is refined into CE [26] 
* CE 19 is refined into CE [27,28] 


### Cost equations --> "Loop" of minus/3 
* CEs [28] --> Loop 17 
* CEs [27] --> Loop 18 
* CEs [24] --> Loop 19 
* CEs [25,26] --> Loop 20 

### Ranking functions of CR minus(V,V1,Out) 
* RF of phase [17,18]: [V,V1]

#### Partial ranking functions of CR minus(V,V1,Out) 
* Partial RF of phase [17,18]:
  - RF of loop [17:1,18:1]:
    V
    V1


### Specialization of cost equations plus/3 
* CE 12 is refined into CE [29] 
* CE 9 is refined into CE [30] 
* CE 8 is refined into CE [31] 
* CE 11 is refined into CE [32] 
* CE 10 is refined into CE [33,34] 


### Cost equations --> "Loop" of plus/3 
* CEs [34] --> Loop 21 
* CEs [33] --> Loop 22 
* CEs [32] --> Loop 23 
* CEs [29] --> Loop 24 
* CEs [30] --> Loop 25 
* CEs [31] --> Loop 26 

### Ranking functions of CR plus(V,V1,Out) 

#### Partial ranking functions of CR plus(V,V1,Out) 
* Partial RF of phase [21,22,23]:
  - RF of loop [21:1,22:1,23:1]:
    V depends on loops [21:2,22:2] 
    V1


### Specialization of cost equations times/3 
* CE 14 is refined into CE [35] 
* CE 13 is refined into CE [36] 
* CE 16 is refined into CE [37] 
* CE 15 is refined into CE [38,39,40,41,42,43] 


### Cost equations --> "Loop" of times/3 
* CEs [43] --> Loop 27 
* CEs [38] --> Loop 28 
* CEs [41] --> Loop 29 
* CEs [39,40,42] --> Loop 30 
* CEs [35] --> Loop 31 
* CEs [36,37] --> Loop 32 

### Ranking functions of CR times(V,V1,Out) 
* RF of phase [27,28,29,30]: [V]

#### Partial ranking functions of CR times(V,V1,Out) 
* Partial RF of phase [27,28,29,30]:
  - RF of loop [27:1,28:1,29:1,30:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [44,45] 
* CE 2 is refined into CE [46,47,48,49] 
* CE 3 is refined into CE [50,51] 
* CE 4 is refined into CE [52,53,54,55] 


### Cost equations --> "Loop" of start/2 
* CEs [47,52] --> Loop 33 
* CEs [44,45,46,48,50,51,53,54,55] --> Loop 34 
* CEs [49] --> Loop 35 

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

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


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

#### Cost of chains of fun(V,Out):
* Chain [16]: 1
  with precondition: [Out=0,V>=0] 

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


#### Cost of chains of minus(V,V1,Out):
* Chain [[17,18],20]: 3*it(17)+1
  Such that:aux(3) =< V1
aux(5) =< V
it(17) =< aux(5)
it(17) =< aux(3)

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

* Chain [[17,18],19]: 3*it(17)+1
  Such that:aux(1) =< V
aux(6) =< V1
it(17) =< aux(1)
it(17) =< aux(6)

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

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

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


#### Cost of chains of plus(V,V1,Out):
* Chain [multiple([21,22,23],[[],[26],[25],[24]])]...: 4*it(21)+2*it([25])+0
  Such that:aux(7) =< 1
aux(8) =< 2*V1
it(21) =< aux(8)
it([25]) =< it(21)+it(21)+aux(7)

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

* Chain [26]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

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

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


#### Cost of chains of times(V,V1,Out):
* Chain [[27,28,29,30],32]: 9*it(27)+4*s(9)+2*s(10)+1
  Such that:aux(10) =< 2*V1
aux(13) =< V
it(27) =< aux(13)
s(12) =< it(27)*aux(10)
s(9) =< s(12)
s(10) =< s(9)+s(9)+aux(13)

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

* Chain [[27,28,29,30],31]: 9*it(27)+2*s(10)+1
  Such that:aux(14) =< V
it(27) =< aux(14)
s(10) =< aux(14)

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

* Chain [32]: 1
  with precondition: [Out=0,V>=0,V1>=0] 

* Chain [31]: 1
  with precondition: [V1=0,Out=0,V>=0] 


#### Cost of chains of start(V,V1):
* Chain [35]...: 4*s(24)+2*s(25)+0
  Such that:s(22) =< 1
s(23) =< 2*V1
s(24) =< s(23)
s(25) =< s(24)+s(24)+s(22)

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

* Chain [34]: 20*s(28)+4*s(30)+2*s(31)+6*s(34)+1
  Such that:s(26) =< 2*V1
aux(16) =< V
aux(17) =< V1
s(28) =< aux(16)
s(29) =< s(28)*s(26)
s(30) =< s(29)
s(31) =< s(30)+s(30)+aux(16)
s(34) =< aux(16)
s(34) =< aux(17)

  with precondition: [V>=0] 

* Chain [33]: 1
  with precondition: [V1=0,V>=0] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [35]... with precondition: [V>=1,V1>=1] 
    - Upper bound: 16*V1+2 
    - Complexity: n 
* Chain [34] with precondition: [V>=0] 
    - Upper bound: 28*V+1+8*V*nat(2*V1) 
    - Complexity: n^2 
* Chain [33] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V,V1): max([nat(2*V1)*8+1,8*V*nat(2*V1)+28*V])+1 
Asymptotic class: n^2 
* Total analysis performed in 303 ms.


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

(18)
BOUNDS(1, n^2)

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

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

(20)
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:

   O(0') -> 0'
   +'(0', x) -> x
   +'(x, 0') -> x
   +'(O(x), O(y)) -> O(+'(x, y))
   +'(O(x), I(y)) -> I(+'(x, y))
   +'(I(x), O(y)) -> I(+'(x, y))
   +'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
   *'(0', x) -> 0'
   *'(x, 0') -> 0'
   *'(O(x), y) -> O(*'(x, y))
   *'(I(x), y) -> +'(O(*'(x, y)), y)
   -(x, 0') -> x
   -(0', x) -> 0'
   -(O(x), O(y)) -> O(-(x, y))
   -(O(x), I(y)) -> I(-(-(x, y), I(1')))
   -(I(x), O(y)) -> I(-(x, y))
   -(I(x), I(y)) -> O(-(x, y))

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

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

(22)
Obligation:
TRS:
Rules:
O(0') -> 0'
+'(0', x) -> x
+'(x, 0') -> x
+'(O(x), O(y)) -> O(+'(x, y))
+'(O(x), I(y)) -> I(+'(x, y))
+'(I(x), O(y)) -> I(+'(x, y))
+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
*'(0', x) -> 0'
*'(x, 0') -> 0'
*'(O(x), y) -> O(*'(x, y))
*'(I(x), y) -> +'(O(*'(x, y)), y)
-(x, 0') -> x
-(0', x) -> 0'
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1')))
-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))

Types:
O :: 0':I:1' -> 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' -> 0':I:1' -> 0':I:1'
I :: 0':I:1' -> 0':I:1'
*' :: 0':I:1' -> 0':I:1' -> 0':I:1'
- :: 0':I:1' -> 0':I:1' -> 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat -> 0':I:1'

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', *', -

They will be analysed ascendingly in the following order:
+' < *'

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

(24)
Obligation:
TRS:
Rules:
O(0') -> 0'
+'(0', x) -> x
+'(x, 0') -> x
+'(O(x), O(y)) -> O(+'(x, y))
+'(O(x), I(y)) -> I(+'(x, y))
+'(I(x), O(y)) -> I(+'(x, y))
+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
*'(0', x) -> 0'
*'(x, 0') -> 0'
*'(O(x), y) -> O(*'(x, y))
*'(I(x), y) -> +'(O(*'(x, y)), y)
-(x, 0') -> x
-(0', x) -> 0'
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1')))
-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))

Types:
O :: 0':I:1' -> 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' -> 0':I:1' -> 0':I:1'
I :: 0':I:1' -> 0':I:1'
*' :: 0':I:1' -> 0':I:1' -> 0':I:1'
- :: 0':I:1' -> 0':I:1' -> 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat -> 0':I:1'


Generator Equations:
gen_0':I:1'2_0(0) <=> 0'
gen_0':I:1'2_0(+(x, 1)) <=> I(gen_0':I:1'2_0(x))


The following defined symbols remain to be analysed:
+', *', -

They will be analysed ascendingly in the following order:
+' < *'

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':I:1'2_0(n4_0), gen_0':I:1'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
+'(gen_0':I:1'2_0(0), gen_0':I:1'2_0(0))

Induction Step:
+'(gen_0':I:1'2_0(+(n4_0, 1)), gen_0':I:1'2_0(+(n4_0, 1))) ->_R^Omega(1)
O(+'(+'(gen_0':I:1'2_0(n4_0), gen_0':I:1'2_0(n4_0)), I(0'))) ->_IH
O(+'(*3_0, I(0')))

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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
O(0') -> 0'
+'(0', x) -> x
+'(x, 0') -> x
+'(O(x), O(y)) -> O(+'(x, y))
+'(O(x), I(y)) -> I(+'(x, y))
+'(I(x), O(y)) -> I(+'(x, y))
+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
*'(0', x) -> 0'
*'(x, 0') -> 0'
*'(O(x), y) -> O(*'(x, y))
*'(I(x), y) -> +'(O(*'(x, y)), y)
-(x, 0') -> x
-(0', x) -> 0'
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1')))
-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))

Types:
O :: 0':I:1' -> 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' -> 0':I:1' -> 0':I:1'
I :: 0':I:1' -> 0':I:1'
*' :: 0':I:1' -> 0':I:1' -> 0':I:1'
- :: 0':I:1' -> 0':I:1' -> 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat -> 0':I:1'


Generator Equations:
gen_0':I:1'2_0(0) <=> 0'
gen_0':I:1'2_0(+(x, 1)) <=> I(gen_0':I:1'2_0(x))


The following defined symbols remain to be analysed:
+', *', -

They will be analysed ascendingly in the following order:
+' < *'

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

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
TRS:
Rules:
O(0') -> 0'
+'(0', x) -> x
+'(x, 0') -> x
+'(O(x), O(y)) -> O(+'(x, y))
+'(O(x), I(y)) -> I(+'(x, y))
+'(I(x), O(y)) -> I(+'(x, y))
+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
*'(0', x) -> 0'
*'(x, 0') -> 0'
*'(O(x), y) -> O(*'(x, y))
*'(I(x), y) -> +'(O(*'(x, y)), y)
-(x, 0') -> x
-(0', x) -> 0'
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1')))
-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))

Types:
O :: 0':I:1' -> 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' -> 0':I:1' -> 0':I:1'
I :: 0':I:1' -> 0':I:1'
*' :: 0':I:1' -> 0':I:1' -> 0':I:1'
- :: 0':I:1' -> 0':I:1' -> 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat -> 0':I:1'


Lemmas:
+'(gen_0':I:1'2_0(n4_0), gen_0':I:1'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_0':I:1'2_0(0) <=> 0'
gen_0':I:1'2_0(+(x, 1)) <=> I(gen_0':I:1'2_0(x))


The following defined symbols remain to be analysed:
*', -
----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':I:1'2_0(n40693_0), gen_0':I:1'2_0(0)) -> gen_0':I:1'2_0(0), rt in Omega(1 + n40693_0)

Induction Base:
*'(gen_0':I:1'2_0(0), gen_0':I:1'2_0(0)) ->_R^Omega(1)
0'

Induction Step:
*'(gen_0':I:1'2_0(+(n40693_0, 1)), gen_0':I:1'2_0(0)) ->_R^Omega(1)
+'(O(*'(gen_0':I:1'2_0(n40693_0), gen_0':I:1'2_0(0))), gen_0':I:1'2_0(0)) ->_IH
+'(O(gen_0':I:1'2_0(0)), gen_0':I:1'2_0(0)) ->_R^Omega(1)
+'(0', gen_0':I:1'2_0(0)) ->_R^Omega(1)
gen_0':I:1'2_0(0)

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

(32)
Obligation:
TRS:
Rules:
O(0') -> 0'
+'(0', x) -> x
+'(x, 0') -> x
+'(O(x), O(y)) -> O(+'(x, y))
+'(O(x), I(y)) -> I(+'(x, y))
+'(I(x), O(y)) -> I(+'(x, y))
+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
*'(0', x) -> 0'
*'(x, 0') -> 0'
*'(O(x), y) -> O(*'(x, y))
*'(I(x), y) -> +'(O(*'(x, y)), y)
-(x, 0') -> x
-(0', x) -> 0'
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1')))
-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))

Types:
O :: 0':I:1' -> 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' -> 0':I:1' -> 0':I:1'
I :: 0':I:1' -> 0':I:1'
*' :: 0':I:1' -> 0':I:1' -> 0':I:1'
- :: 0':I:1' -> 0':I:1' -> 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat -> 0':I:1'


Lemmas:
+'(gen_0':I:1'2_0(n4_0), gen_0':I:1'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
*'(gen_0':I:1'2_0(n40693_0), gen_0':I:1'2_0(0)) -> gen_0':I:1'2_0(0), rt in Omega(1 + n40693_0)


Generator Equations:
gen_0':I:1'2_0(0) <=> 0'
gen_0':I:1'2_0(+(x, 1)) <=> I(gen_0':I:1'2_0(x))


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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':I:1'2_0(n46179_0), gen_0':I:1'2_0(n46179_0)) -> gen_0':I:1'2_0(0), rt in Omega(1 + n46179_0)

Induction Base:
-(gen_0':I:1'2_0(0), gen_0':I:1'2_0(0)) ->_R^Omega(1)
gen_0':I:1'2_0(0)

Induction Step:
-(gen_0':I:1'2_0(+(n46179_0, 1)), gen_0':I:1'2_0(+(n46179_0, 1))) ->_R^Omega(1)
O(-(gen_0':I:1'2_0(n46179_0), gen_0':I:1'2_0(n46179_0))) ->_IH
O(gen_0':I:1'2_0(0)) ->_R^Omega(1)
0'

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

(34)
BOUNDS(1, INF)