WORST_CASE(?,O(n^1))
proof of input_lcVQs3Sr20.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxRelTRS
(11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxWeightedTrs
(13) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxWeightedTrs
(15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTypedWeightedTrs
(17) CompletionProof [UPPER BOUND(ID), 0 ms]
(18) CpxTypedWeightedCompleteTrs
(19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) CompleteCoflocoProof [FINISHED, 376 ms]
(22) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   div(x, 0) -> 0
   div(0, y) -> 0
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))

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:
   -(z0, 0) -> z0
   -(0, s(z0)) -> 0
   -(s(z0), s(z1)) -> -(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   div(z0, 0) -> 0
   div(0, z0) -> 0
   div(s(z0), s(z1)) -> if(lt(z0, z1), 0, s(div(-(z0, z1), s(z1))))
Tuples:
   -'(z0, 0) -> c
   -'(0, s(z0)) -> c1
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(z0, 0) -> c3
   LT(0, s(z0)) -> c4
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   DIV(z0, 0) -> c8
   DIV(0, z0) -> c9
   DIV(s(z0), s(z1)) -> c10(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), DIV(-(z0, z1), s(z1)), -'(z0, z1))
S tuples:
   -'(z0, 0) -> c
   -'(0, s(z0)) -> c1
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(z0, 0) -> c3
   LT(0, s(z0)) -> c4
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   DIV(z0, 0) -> c8
   DIV(0, z0) -> c9
   DIV(s(z0), s(z1)) -> c10(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), DIV(-(z0, z1), s(z1)), -'(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, lt_2, if_3, div_2

Defined Pair Symbols:   -'_2, LT_2, IF_3, DIV_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7, c8, c9, c10_2, c11_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   DIV(z0, 0) -> c8
   -'(0, s(z0)) -> c1
   -'(z0, 0) -> c
   LT(z0, 0) -> c3
   IF(true, z0, z1) -> c6
   LT(0, s(z0)) -> c4
   DIV(0, z0) -> c9
   IF(false, z0, z1) -> c7

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(0, s(z0)) -> 0
   -(s(z0), s(z1)) -> -(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   div(z0, 0) -> 0
   div(0, z0) -> 0
   div(s(z0), s(z1)) -> if(lt(z0, z1), 0, s(div(-(z0, z1), s(z1))))
Tuples:
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), DIV(-(z0, z1), s(z1)), -'(z0, z1))
S tuples:
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(IF(lt(z0, z1), 0, s(div(-(z0, z1), s(z1)))), DIV(-(z0, z1), s(z1)), -'(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, lt_2, if_3, div_2

Defined Pair Symbols:   -'_2, LT_2, DIV_2

Compound Symbols:   c2_1, c5_1, c10_2, c11_3


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(0, s(z0)) -> 0
   -(s(z0), s(z1)) -> -(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   div(z0, 0) -> 0
   div(0, z0) -> 0
   div(s(z0), s(z1)) -> if(lt(z0, z1), 0, s(div(-(z0, z1), s(z1))))
Tuples:
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(DIV(-(z0, z1), s(z1)), -'(z0, z1))
S tuples:
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(DIV(-(z0, z1), s(z1)), -'(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, lt_2, if_3, div_2

Defined Pair Symbols:   -'_2, LT_2, DIV_2

Compound Symbols:   c2_1, c5_1, c10_1, c11_2


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

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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(0, s(z0)) -> 0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(DIV(-(z0, z1), s(z1)), -'(z0, z1))
S tuples:
   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(DIV(-(z0, z1), s(z1)), -'(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2

Defined Pair Symbols:   -'_2, LT_2, DIV_2

Compound Symbols:   c2_1, c5_1, c10_1, c11_2


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c2(-'(z0, z1))
   LT(s(z0), s(z1)) -> c5(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1))
   DIV(s(z0), s(z1)) -> c11(DIV(-(z0, z1), s(z1)), -'(z0, z1))

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

   -(z0, 0) -> z0
   -(0, s(z0)) -> 0
   -(s(z0), s(z1)) -> -(z0, z1)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c2(-'(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c5(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c11(DIV(-(z0, z1), s(z1)), -'(z0, z1)) [1]
   -(z0, 0) -> z0 [0]
   -(0, s(z0)) -> 0 [0]
   -(s(z0), s(z1)) -> -(z0, z1) [0]

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

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

- => minus

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c2(-'(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c5(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c11(DIV(minus(z0, z1), s(z1)), -'(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(0, s(z0)) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c2(-'(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c5(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c11(DIV(minus(z0, z1), s(z1)), -'(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(0, s(z0)) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
LT :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
DIV :: s:0 -> s:0 -> c10:c11
c10 :: c5 -> c10:c11
c11 :: c10:c11 -> c2 -> c10:c11
minus :: s:0 -> s:0 -> s:0
0 :: s:0

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

(17) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   minus(v0, v1) -> null_minus [0]
   -'(v0, v1) -> null_-' [0]
   LT(v0, v1) -> null_LT [0]
   DIV(v0, v1) -> null_DIV [0]

And the following fresh constants:    null_minus, null_-', null_LT, null_DIV

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

(18)
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:

   -'(s(z0), s(z1)) -> c2(-'(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c5(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c10(LT(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c11(DIV(minus(z0, z1), s(z1)), -'(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(0, s(z0)) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(v0, v1) -> null_minus [0]
   -'(v0, v1) -> null_-' [0]
   LT(v0, v1) -> null_LT [0]
   DIV(v0, v1) -> null_DIV [0]

The TRS has the following type information:
-' :: s:0:null_minus -> s:0:null_minus -> c2:null_-'
s :: s:0:null_minus -> s:0:null_minus
c2 :: c2:null_-' -> c2:null_-'
LT :: s:0:null_minus -> s:0:null_minus -> c5:null_LT
c5 :: c5:null_LT -> c5:null_LT
DIV :: s:0:null_minus -> s:0:null_minus -> c10:c11:null_DIV
c10 :: c5:null_LT -> c10:c11:null_DIV
c11 :: c10:c11:null_DIV -> c2:null_-' -> c10:c11:null_DIV
minus :: s:0:null_minus -> s:0:null_minus -> s:0:null_minus
0 :: s:0:null_minus
null_minus :: s:0:null_minus
null_-' :: c2:null_-'
null_LT :: c5:null_LT
null_DIV :: c10:c11:null_DIV

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

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

   0 => 0
   null_minus => 0
   null_-' => 0
   null_LT => 0
   null_DIV => 0

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   DIV(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   DIV(z, z') -{ 1 }-> 1 + LT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   DIV(z, z') -{ 1 }-> 1 + DIV(minus(z0, z1), 1 + z1) + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + 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
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z' = 1 + z0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[minus(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,[fun1(V5, V4, Ret11)],[Out = 1 + Ret11,V4 >= 0,V1 = 1 + V5,V5 >= 0,V = 1 + V4]).
eq(fun2(V1, V, Out),1,[fun1(V7, V6, Ret12)],[Out = 1 + Ret12,V6 >= 0,V1 = 1 + V7,V7 >= 0,V = 1 + V6]).
eq(fun2(V1, V, Out),1,[minus(V8, V9, Ret010),fun2(Ret010, 1 + V9, Ret01),fun(V8, V9, Ret13)],[Out = 1 + Ret01 + Ret13,V9 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V9]).
eq(minus(V1, V, Out),0,[],[Out = V10,V1 = V10,V10 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V11 >= 0,V = 1 + V11,V1 = 0]).
eq(minus(V1, V, Out),0,[minus(V13, V12, Ret)],[Out = Ret,V12 >= 0,V1 = 1 + V13,V13 >= 0,V = 1 + V12]).
eq(minus(V1, V, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V1 = V15,V = V14]).
eq(fun(V1, V, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V1 = V17,V = V16]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V20 >= 0,V21 >= 0,V1 = V20,V = V21]).
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,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into fun1/3
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into fun2/3
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 6 is refined into CE [15] 
* CE 5 is refined into CE [16] 


### Cost equations --> "Loop" of fun/3 
* CEs [16] --> Loop 12 
* CEs [15] --> Loop 13 

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

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


### Specialization of cost equations fun1/3 
* CE 8 is refined into CE [17] 
* CE 7 is refined into CE [18] 


### Cost equations --> "Loop" of fun1/3 
* CEs [18] --> Loop 14 
* CEs [17] --> Loop 15 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 13 is refined into CE [19] 
* CE 12 is refined into CE [20] 
* CE 14 is refined into CE [21] 


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

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [16]: [V,V1]

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


### Specialization of cost equations fun2/3 
* CE 9 is refined into CE [22,23] 
* CE 11 is refined into CE [24] 
* CE 10 is refined into CE [25,26,27,28,29] 


### Cost equations --> "Loop" of fun2/3 
* CEs [29] --> Loop 19 
* CEs [28] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [26] --> Loop 22 
* CEs [25] --> Loop 23 
* CEs [23] --> Loop 24 
* CEs [22] --> Loop 25 
* CEs [24] --> Loop 26 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [19,20]: [V1-1,V1-V+1]
* RF of phase [23]: [V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [19,20]:
  - RF of loop [19:1,20:1]:
    V1-1
    V1-V+1
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [30,31] 
* CE 2 is refined into CE [32,33] 
* CE 3 is refined into CE [34,35,36,37,38] 
* CE 4 is refined into CE [39,40,41] 


### Cost equations --> "Loop" of start/2 
* CEs [34] --> Loop 27 
* CEs [30,31,32,33,35,36,37,38,39,40,41] --> Loop 28 

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

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


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[12],13]: 1*it(12)+0
  Such that:it(12) =< Out

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

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< Out

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[16],18]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V] 

* Chain [[16],17]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

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


#### Cost of chains of fun2(V1,V,Out):
* Chain [[23],26]: 1*it(23)+0
  Such that:it(23) =< Out

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

* Chain [[23],25]: 1*it(23)+1
  Such that:it(23) =< Out

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

* Chain [[23],22,26]: 1*it(23)+1
  Such that:it(23) =< Out

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

* Chain [[19,20],26]: 2*it(19)+1*s(3)+0
  Such that:aux(2) =< V1-V+1
aux(5) =< V1
it(19) =< aux(5)
s(3) =< aux(5)
it(19) =< aux(2)

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

* Chain [[19,20],25]: 2*it(19)+1*s(3)+1
  Such that:aux(2) =< V1-V+1
aux(6) =< V1
it(19) =< aux(6)
s(3) =< aux(6)
it(19) =< aux(2)

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

* Chain [[19,20],24]: 2*it(19)+1*s(3)+1*s(4)+1
  Such that:aux(2) =< V1-V+1
s(4) =< V
aux(7) =< V1
it(19) =< aux(7)
s(3) =< aux(7)
it(19) =< aux(2)

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

* Chain [[19,20],22,26]: 2*it(19)+1*s(3)+1
  Such that:aux(2) =< V1-V+1
aux(8) =< V1
it(19) =< aux(8)
s(3) =< aux(8)
it(19) =< aux(2)

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

* Chain [[19,20],21,26]: 2*it(19)+2*s(3)+1
  Such that:aux(2) =< V1-V+1
aux(9) =< V1
s(3) =< aux(9)
it(19) =< aux(9)
it(19) =< aux(2)

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

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

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

* Chain [24]: 1*s(4)+1
  Such that:s(4) =< V

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

* Chain [22,26]: 1
  with precondition: [Out=1,V1>=1,V>=1] 

* Chain [21,26]: 1*s(5)+1
  Such that:s(5) =< V1

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


#### Cost of chains of start(V1,V):
* Chain [28]: 4*s(32)+7*s(37)+10*s(38)+1
  Such that:s(36) =< V1-V+1
aux(13) =< V1
aux(14) =< V
s(37) =< aux(13)
s(32) =< aux(14)
s(38) =< aux(13)
s(38) =< s(36)

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

* Chain [27]: 3*s(42)+1
  Such that:s(41) =< V1
s(42) =< s(41)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [28] with precondition: [V1>=0,V>=0] 
    - Upper bound: 17*V1+4*V+1 
    - Complexity: n 
* Chain [27] with precondition: [V=1,V1>=1] 
    - Upper bound: 3*V1+1 
    - Complexity: n 

### Maximum cost of start(V1,V): 17*V1+4*V+1 
Asymptotic class: n 
* Total analysis performed in 328 ms.


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

(22)
BOUNDS(1, n^1)