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

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS 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)
cond2(true, x, y) → cond1(gr(add(x, y), 0), p(x), y)
cond2(false, x, y) → cond3(eq(x, y), x, y)
cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y)
cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
add(0, x) → x
add(s(x), y) → s(add(x, y))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed 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(gr(add(x, y), 0), p(x), y) [1]
cond2(false, x, y) → cond3(eq(x, y), x, y) [1]
cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1]
cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y)) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
add(0, x) → x [1]
add(s(x), y) → s(add(x, y)) [1]
eq(0, 0) → true [1]
eq(0, s(x)) → false [1]
eq(s(x), 0) → false [1]
eq(s(x), s(y)) → eq(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

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(gr(add(x, y), 0), p(x), y) [1]
cond2(false, x, y) → cond3(eq(x, y), x, y) [1]
cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1]
cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y)) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
add(0, x) → x [1]
add(s(x), y) → s(add(x, y)) [1]
eq(0, 0) → true [1]
eq(0, s(x)) → false [1]
eq(s(x), 0) → false [1]
eq(s(x), s(y)) → eq(x, y) [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:cond3 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2:cond3 gr :: 0:s → 0:s → true:false add :: 0:s → 0:s → 0:s 0 :: 0:s p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 eq :: 0:s → 0:s → true:false s :: 0:s → 0:s

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

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(gr(add(x, y), 0), p(x), y) [1] cond2(false, x, y) → cond3(eq(x, y), x, y) [1] cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1] cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y)) [1] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] add(0, x) → x [1] add(s(x), y) → s(add(x, y)) [1] eq(0, 0) → true [1] eq(0, s(x)) → false [1] eq(s(x), 0) → false [1] eq(s(x), s(y)) → eq(x, y) [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 add :: 0:s → 0:s → 0:s 0 :: 0:s p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → null_cond1 eq :: 0:s → 0:s → true:false s :: 0:s → 0:s null_cond1 :: null_cond1 Rewrite Strategy: INNERMOST

(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

true => 1
0 => 0
false => 0
null_cond1 => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ x :|: z' = x, x >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
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 }→ cond3(eq(x, y), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
cond2(z, z', z'') -{ 1 }→ cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
cond3(z, z', z'') -{ 1 }→ cond1(gr(add(x, y), 0), x, p(y)) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
cond3(z, z', z'') -{ 1 }→ cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
eq(z, z') -{ 1 }→ eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' = 1 + x, x >= 0, z = 0
eq(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, 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
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 (EQUIVALENT transformation)

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

eq(start(V, V1, V2),0,[cond1(V, V1, V2, Out)],[V >= 0,V1 >= 0,V2 >= 0]). eq(start(V, V1, V2),0,[cond2(V, V1, V2, Out)],[V >= 0,V1 >= 0,V2 >= 0]). eq(start(V, V1, V2),0,[cond3(V, V1, V2, Out)],[V >= 0,V1 >= 0,V2 >= 0]). eq(start(V, V1, V2),0,[gr(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V2),0,[add(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V2),0,[eq(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V2),0,[p(V, Out)],[V >= 0]). eq(cond1(V, V1, V2, Out),1,[gr(V3, V4, Ret0),cond2(Ret0, V3, V4, Ret)],[Out = Ret,V1 = V3,V2 = V4,V = 1,V3 >= 0,V4 >= 0]). eq(cond2(V, V1, V2, Out),1,[add(V5, V6, Ret00),gr(Ret00, 0, Ret01),p(V5, Ret1),cond1(Ret01, Ret1, V6, Ret2)],[Out = Ret2,V1 = V5,V2 = V6,V = 1,V5 >= 0,V6 >= 0]). eq(cond2(V, V1, V2, Out),1,[eq(V7, V8, Ret02),cond3(Ret02, V7, V8, Ret3)],[Out = Ret3,V1 = V7,V2 = V8,V7 >= 0,V8 >= 0,V = 0]). eq(cond3(V, V1, V2, Out),1,[add(V9, V10, Ret001),gr(Ret001, 0, Ret03),p(V9, Ret11),cond1(Ret03, Ret11, V10, Ret4)],[Out = Ret4,V1 = V9,V2 = V10,V = 1,V9 >= 0,V10 >= 0]). eq(cond3(V, V1, V2, Out),1,[add(V11, V12, Ret002),gr(Ret002, 0, Ret04),p(V12, Ret21),cond1(Ret04, V11, Ret21, Ret5)],[Out = Ret5,V1 = V11,V2 = V12,V11 >= 0,V12 >= 0,V = 0]). eq(gr(V, V1, Out),1,[],[Out = 0,V1 = V13,V13 >= 0,V = 0]). eq(gr(V, V1, Out),1,[],[Out = 1,V14 >= 0,V = 1 + V14,V1 = 0]). eq(gr(V, V1, Out),1,[gr(V15, V16, Ret6)],[Out = Ret6,V1 = 1 + V16,V15 >= 0,V16 >= 0,V = 1 + V15]). eq(add(V, V1, Out),1,[],[Out = V17,V1 = V17,V17 >= 0,V = 0]). eq(add(V, V1, Out),1,[add(V18, V19, Ret12)],[Out = 1 + Ret12,V18 >= 0,V19 >= 0,V = 1 + V18,V1 = V19]). eq(eq(V, V1, Out),1,[],[Out = 1,V = 0,V1 = 0]). eq(eq(V, V1, Out),1,[],[Out = 0,V1 = 1 + V20,V20 >= 0,V = 0]). eq(eq(V, V1, Out),1,[],[Out = 0,V21 >= 0,V = 1 + V21,V1 = 0]). eq(eq(V, V1, Out),1,[eq(V22, V23, Ret7)],[Out = Ret7,V1 = 1 + V23,V22 >= 0,V23 >= 0,V = 1 + V22]). eq(p(V, Out),1,[],[Out = 0,V = 0]). eq(p(V, Out),1,[],[Out = V24,V24 >= 0,V = 1 + V24]). eq(cond1(V, V1, V2, Out),0,[],[Out = 0,V25 >= 0,V2 = V26,V27 >= 0,V = V25,V1 = V27,V26 >= 0]). input_output_vars(cond1(V,V1,V2,Out),[V,V1,V2],[Out]). input_output_vars(cond2(V,V1,V2,Out),[V,V1,V2],[Out]). input_output_vars(cond3(V,V1,V2,Out),[V,V1,V2],[Out]). input_output_vars(gr(V,V1,Out),[V,V1],[Out]). input_output_vars(add(V,V1,Out),[V,V1],[Out]). input_output_vars(eq(V,V1,Out),[V,V1],[Out]). input_output_vars(p(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. recursive : [add/3]
1. recursive : [gr/3]
2. non_recursive : [p/2]
3. recursive : [eq/3]
4. recursive : [cond1/4,cond2/4,cond3/4]
5. non_recursive : [start/3]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into add/3
1. SCC is partially evaluated into gr/3
2. SCC is partially evaluated into p/2
3. SCC is partially evaluated into eq/3
4. SCC is partially evaluated into cond1/4
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations add/3
* CE 12 is refined into CE [26]
* CE 11 is refined into CE [27]


### Cost equations --> "Loop" of add/3
* CEs [27] --> Loop 17
* CEs [26] --> Loop 18

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

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


### Specialization of cost equations gr/3
* CE 15 is refined into CE [28]
* CE 14 is refined into CE [29]
* CE 13 is refined into CE [30]


### Cost equations --> "Loop" of gr/3
* CEs [29] --> Loop 19
* CEs [30] --> Loop 20
* CEs [28] --> Loop 21

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

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


### Specialization of cost equations p/2
* CE 17 is refined into CE [31]
* CE 16 is refined into CE [32]


### Cost equations --> "Loop" of p/2
* CEs [31] --> Loop 22
* CEs [32] --> Loop 23

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

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


### Specialization of cost equations eq/3
* CE 25 is refined into CE [33]
* CE 24 is refined into CE [34]
* CE 23 is refined into CE [35]
* CE 22 is refined into CE [36]


### Cost equations --> "Loop" of eq/3
* CEs [34] --> Loop 24
* CEs [35] --> Loop 25
* CEs [36] --> Loop 26
* CEs [33] --> Loop 27

### Ranking functions of CR eq(V,V1,Out)
* RF of phase [27]: [V,V1]

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


### Specialization of cost equations cond1/4
* CE 21 is refined into CE [37]
* CE 18 is refined into CE [38,39]
* CE 19 is refined into CE [40,41]
* CE 20 is refined into CE [42,43]


### Cost equations --> "Loop" of cond1/4
* CEs [43] --> Loop 28
* CEs [39] --> Loop 29
* CEs [41] --> Loop 30
* CEs [38] --> Loop 31
* CEs [42] --> Loop 32
* CEs [40] --> Loop 33
* CEs [37] --> Loop 34

### Ranking functions of CR cond1(V,V1,V2,Out)
* RF of phase [28,30]: [V1+V2-1]
* RF of phase [29]: [V1-1,V1-V2]
* RF of phase [31]: [V1]
* RF of phase [32]: [V2]

#### Partial ranking functions of CR cond1(V,V1,V2,Out)
* Partial RF of phase [28,30]:
- RF of loop [28:1]:
-V1+V2 depends on loops [30:1]
V2-1
- RF of loop [30:1]:
V1
V1-V2+1 depends on loops [28:1]
* Partial RF of phase [29]:
- RF of loop [29:1]:
V1-1
V1-V2
* Partial RF of phase [31]:
- RF of loop [31:1]:
V1
* Partial RF of phase [32]:
- RF of loop [32:1]:
V2


### Specialization of cost equations start/3
* CE 2 is refined into CE [44,45,46,47,48,49,50,51]
* CE 3 is refined into CE [52,53,54,55]
* CE 4 is refined into CE [56,57,58,59,60,61,62,63,64]
* CE 5 is refined into CE [65,66,67,68,69,70,71,72,73]
* CE 6 is refined into CE [74,75,76,77,78]
* CE 7 is refined into CE [79,80,81,82]
* CE 8 is refined into CE [83,84]
* CE 9 is refined into CE [85,86,87,88,89,90]
* CE 10 is refined into CE [91,92]


### Cost equations --> "Loop" of start/3
* CEs [90] --> Loop 35
* CEs [49,76] --> Loop 36
* CEs [47,48,50,51,77,78] --> Loop 37
* CEs [45,46,74,75] --> Loop 38
* CEs [44,80,81,82,84,87,88,89,92] --> Loop 39
* CEs [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,79,83,85,86,91] --> Loop 40

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

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


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

#### Cost of chains of add(V,V1,Out):
* Chain [[18],17]: 1*it(18)+1
Such that:it(18) =< -V1+Out

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

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


#### Cost of chains of gr(V,V1,Out):
* Chain [[21],20]: 1*it(21)+1
Such that:it(21) =< V

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

* Chain [[21],19]: 1*it(21)+1
Such that:it(21) =< V1

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

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

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


#### Cost of chains of p(V,Out):
* Chain [23]: 1
with precondition: [V=0,Out=0]

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


#### Cost of chains of eq(V,V1,Out):
* Chain [[27],26]: 1*it(27)+1
Such that:it(27) =< V

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

* Chain [[27],25]: 1*it(27)+1
Such that:it(27) =< V

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

* Chain [[27],24]: 1*it(27)+1
Such that:it(27) =< V1

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

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

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

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


#### Cost of chains of cond1(V,V1,V2,Out):
* Chain [[32],34]: 8*it(32)+0
Such that:it(32) =< V2

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

* Chain [[32],33,34]: 8*it(32)+8
Such that:it(32) =< V2

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

* Chain [[31],34]: 6*it(31)+1*s(3)+0
Such that:aux(3) =< V1
it(31) =< aux(3)
s(3) =< it(31)*aux(3)

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

* Chain [[31],33,34]: 6*it(31)+1*s(3)+8
Such that:aux(4) =< V1
it(31) =< aux(4)
s(3) =< it(31)*aux(4)

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

* Chain [[29],[28,30],[32],34]: 16*it(28)+6*it(29)+8*it(32)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+0
Such that:aux(20) =< V1
it(29) =< V1-V2
aux(17) =< 2*V2
aux(21) =< V2
it(28) =< aux(21)
it(32) =< aux(17)
it(28) =< aux(17)
s(17) =< it(28)*aux(17)
s(15) =< it(28)*aux(21)
s(16) =< s(17)
s(14) =< s(15)
it(29) =< aux(20)
s(22) =< it(29)*aux(21)
s(23) =< it(29)*aux(20)

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

* Chain [[29],[28,30],[32],33,34]: 16*it(28)+6*it(29)+8*it(32)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+8
Such that:aux(20) =< V1
it(29) =< V1-V2
aux(23) =< 2*V2
aux(24) =< V2
it(28) =< aux(24)
it(32) =< aux(23)
it(28) =< aux(23)
s(17) =< it(28)*aux(23)
s(15) =< it(28)*aux(24)
s(16) =< s(17)
s(14) =< s(15)
it(29) =< aux(20)
s(22) =< it(29)*aux(24)
s(23) =< it(29)*aux(20)

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

* Chain [[29],[28,30],34]: 16*it(28)+6*it(29)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+0
Such that:aux(20) =< V1
it(29) =< V1-V2
aux(26) =< 2*V2
aux(27) =< V2
it(28) =< aux(27)
it(28) =< aux(26)
s(17) =< it(28)*aux(26)
s(15) =< it(28)*aux(27)
s(16) =< s(17)
s(14) =< s(15)
it(29) =< aux(20)
s(22) =< it(29)*aux(27)
s(23) =< it(29)*aux(20)

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

* Chain [[29],34]: 6*it(29)+1*s(22)+1*s(23)+0
Such that:aux(20) =< V1
it(29) =< V1-V2
aux(19) =< V2
it(29) =< aux(20)
s(22) =< it(29)*aux(19)
s(23) =< it(29)*aux(20)

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

* Chain [[28,30],[32],34]: 8*it(28)+8*it(30)+8*it(32)+3*s(14)+3*s(16)+0
Such that:it(28) =< V2
aux(16) =< V1
aux(17) =< V1+V2
it(30) =< aux(16)
it(32) =< aux(17)
it(28) =< aux(17)
it(30) =< aux(17)
s(17) =< it(30)*aux(17)
s(15) =< it(28)*aux(16)
s(16) =< s(17)
s(14) =< s(15)

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

* Chain [[28,30],[32],33,34]: 8*it(28)+8*it(30)+8*it(32)+3*s(14)+3*s(16)+8
Such that:it(28) =< V2
aux(22) =< V1
aux(23) =< V1+V2
it(30) =< aux(22)
it(32) =< aux(23)
it(28) =< aux(23)
it(30) =< aux(23)
s(17) =< it(30)*aux(23)
s(15) =< it(28)*aux(22)
s(16) =< s(17)
s(14) =< s(15)

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

* Chain [[28,30],34]: 8*it(28)+8*it(30)+3*s(14)+3*s(16)+0
Such that:it(28) =< V2
aux(25) =< V1
aux(26) =< V1+V2
it(30) =< aux(25)
it(28) =< aux(26)
it(30) =< aux(26)
s(17) =< it(30)*aux(26)
s(15) =< it(28)*aux(25)
s(16) =< s(17)
s(14) =< s(15)

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

* Chain [34]: 0
with precondition: [Out=0,V>=0,V1>=0,V2>=0]

* Chain [33,34]: 8
with precondition: [V=1,V1=0,V2=0,Out=0]


#### Cost of chains of start(V,V1,V2):
* Chain [40]: 38*s(98)+19*s(100)+144*s(109)+27*s(113)+27*s(114)+48*s(115)+63*s(118)+8*s(122)+48*s(130)+48*s(131)+18*s(134)+18*s(135)+32*s(136)+48*s(150)+8*s(151)+8*s(152)+14
Such that:aux(50) =< 1
aux(51) =< V1
aux(52) =< V1-V2+1
aux(53) =< V1+V2
aux(54) =< V2
aux(55) =< 2*V2
s(100) =< aux(50)
s(118) =< aux(51)
s(98) =< aux(54)
s(122) =< s(118)*aux(51)
s(150) =< aux(52)
s(150) =< aux(51)
s(151) =< s(150)*aux(54)
s(152) =< s(150)*aux(51)
s(109) =< aux(54)
s(109) =< aux(55)
s(111) =< s(109)*aux(55)
s(112) =< s(109)*aux(54)
s(113) =< s(111)
s(114) =< s(112)
s(115) =< aux(55)
s(130) =< aux(54)
s(131) =< aux(51)
s(130) =< aux(53)
s(131) =< aux(53)
s(132) =< s(131)*aux(53)
s(133) =< s(130)*aux(51)
s(134) =< s(132)
s(135) =< s(133)
s(136) =< aux(53)

with precondition: [V=0]

* Chain [39]: 3*s(196)+2*s(197)+12
Such that:aux(56) =< V
aux(57) =< V1
s(196) =< aux(56)
s(197) =< aux(57)

with precondition: [V>=1]

* Chain [38]: 32*s(202)+12
Such that:aux(58) =< V2
s(202) =< aux(58)

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

* Chain [37]: 3*s(205)+1*s(206)+16*s(208)+48*s(213)+48*s(214)+18*s(217)+18*s(218)+32*s(219)+48*s(225)+8*s(226)+8*s(227)+96*s(228)+18*s(231)+18*s(232)+32*s(233)+12
Such that:s(206) =< 1
aux(61) =< V1
aux(62) =< V1-V2
aux(63) =< V1+V2
aux(64) =< V2
aux(65) =< 2*V2
s(205) =< aux(61)
s(208) =< aux(64)
s(213) =< aux(64)
s(214) =< aux(61)
s(213) =< aux(63)
s(214) =< aux(63)
s(215) =< s(214)*aux(63)
s(216) =< s(213)*aux(61)
s(217) =< s(215)
s(218) =< s(216)
s(219) =< aux(63)
s(225) =< aux(62)
s(225) =< aux(61)
s(226) =< s(225)*aux(64)
s(227) =< s(225)*aux(61)
s(228) =< aux(64)
s(228) =< aux(65)
s(229) =< s(228)*aux(65)
s(230) =< s(228)*aux(64)
s(231) =< s(229)
s(232) =< s(230)
s(233) =< aux(65)

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

* Chain [36]: 25*s(257)+4*s(260)+12
Such that:aux(67) =< V1
s(257) =< aux(67)
s(260) =< s(257)*aux(67)

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

* Chain [35]: 1*s(264)+1
Such that:s(264) =< V1

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


Closed-form bounds of start(V,V1,V2):
-------------------------------------
* Chain [40] with precondition: [V=0]
- Upper bound: nat(V1)*111+33+nat(V1)*8*nat(V1)+nat(V1)*18*nat(V2)+nat(V1)*8*nat(V1-V2+1)+nat(V2)*230+nat(V2)*27*nat(V2)+nat(V2)*8*nat(V1-V2+1)+nat(2*V2)*48+nat(2*V2)*27*nat(V2)+nat(V1+V2)*32+nat(V1+V2)*18*nat(V1)+nat(V1-V2+1)*48
- Complexity: n^2
* Chain [39] with precondition: [V>=1]
- Upper bound: 3*V+12+nat(V1)*2
- Complexity: n
* Chain [38] with precondition: [V>=0,V1>=0,V2>=0]
- Upper bound: 32*V2+12
- Complexity: n
* Chain [37] with precondition: [V=1,V1>=1,V2>=0]
- Upper bound: 51*V1+13+18*V1*V2+8*V1*nat(V1-V2)+160*V2+18*V2*V2+8*V2*nat(V1-V2)+64*V2+36*V2*V2+ (32*V1+32*V2)+ (18*V1+18*V2)*V1+nat(V1-V2)*48
- Complexity: n^2
* Chain [36] with precondition: [V=1,V2=0,V1>=1]
- Upper bound: 25*V1+12+4*V1*V1
- Complexity: n^2
* Chain [35] with precondition: [V=V1,V>=1]
- Upper bound: V1+1
- Complexity: n

### Maximum cost of start(V,V1,V2): max([nat(V2)*32+11,nat(V1)+11+max([3*V,nat(V1)*23+max([nat(V1)*4*nat(V1),nat(V1)*26+1+nat(V1)*18*nat(V2)+nat(V2)*160+nat(V2)*18*nat(V2)+nat(2*V2)*32+nat(2*V2)*18*nat(V2)+nat(V1+V2)*32+nat(V1+V2)*18*nat(V1)+max([nat(V2)*8*nat(V1-V2)+nat(V1)*8*nat(V1-V2)+nat(V1-V2)*48,nat(V1)*60+20+nat(V1)*8*nat(V1)+nat(V1)*8*nat(V1-V2+1)+nat(V2)*70+nat(V2)*9*nat(V2)+nat(V2)*8*nat(V1-V2+1)+nat(2*V2)*16+nat(2*V2)*9*nat(V2)+nat(V1-V2+1)*48])])])+nat(V1)])+1
Asymptotic class: n^2
* Total analysis performed in 1079 ms.

(10) BOUNDS(1, n^2)