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

(0) Obligation:

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


The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(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^1).


The TRS R consists of the following rules:

le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]
gcd(0, y) → y [1]
gcd(s(x), 0) → s(x) [1]
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) [1]
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) [1]
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(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:

le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]
gcd(0, y) → y [1]
gcd(s(x), 0) → s(x) [1]
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) [1]
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) [1]
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x)) [1]

The TRS has the following type information:

le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s gcd :: 0:s → 0:s → 0:s if_gcd :: true:false → 0: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:

minus(v0, v1) → null_minus [0]
if_gcd(v0, v1, v2) → null_if_gcd [0]
le(v0, v1) → null_le [0]
gcd(v0, v1) → null_gcd [0]

And the following fresh constants:

null_minus, null_if_gcd, null_le, null_gcd

(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: le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] gcd(0, y) → y [1] gcd(s(x), 0) → s(x) [1] gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) [1] if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) [1] if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x)) [1] minus(v0, v1) → null_minus [0] if_gcd(v0, v1, v2) → null_if_gcd [0] le(v0, v1) → null_le [0] gcd(v0, v1) → null_gcd [0] The TRS has the following type information: le :: 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd → true:false:null_le 0 :: 0:s:null_minus:null_if_gcd:null_gcd true :: true:false:null_le s :: 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd false :: true:false:null_le minus :: 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd gcd :: 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd if_gcd :: true:false:null_le → 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd → 0:s:null_minus:null_if_gcd:null_gcd null_minus :: 0:s:null_minus:null_if_gcd:null_gcd null_if_gcd :: 0:s:null_minus:null_if_gcd:null_gcd null_le :: true:false:null_le null_gcd :: 0:s:null_minus:null_if_gcd:null_gcd 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:

0 => 0
true => 2
false => 1
null_minus => 0
null_if_gcd => 0
null_le => 0
null_gcd => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

gcd(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
gcd(z, z') -{ 1 }→ if_gcd(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
gcd(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
gcd(z, z') -{ 1 }→ 1 + x :|: x >= 0, z = 1 + x, z' = 0
if_gcd(z, z', z'') -{ 1 }→ gcd(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
if_gcd(z, z', z'') -{ 1 }→ gcd(minus(y, x), 1 + x) :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y
if_gcd(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
le(z, z') -{ 1 }→ le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
le(z, z') -{ 1 }→ 2 :|: y >= 0, z = 0, z' = y
le(z, z') -{ 1 }→ 1 :|: x >= 0, z = 1 + x, z' = 0
le(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
minus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
minus(z, z') -{ 1 }→ minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
minus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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, V13),0,[le(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V13),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V13),0,[gcd(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V13),0,[fun(V, V1, V13, Out)],[V >= 0,V1 >= 0,V13 >= 0]). eq(le(V, V1, Out),1,[],[Out = 2,V2 >= 0,V = 0,V1 = V2]). eq(le(V, V1, Out),1,[],[Out = 1,V3 >= 0,V = 1 + V3,V1 = 0]). eq(le(V, V1, Out),1,[le(V4, V5, Ret)],[Out = Ret,V1 = 1 + V5,V4 >= 0,V5 >= 0,V = 1 + V4]). eq(minus(V, V1, Out),1,[],[Out = V6,V6 >= 0,V = V6,V1 = 0]). eq(minus(V, V1, Out),1,[minus(V7, V8, Ret1)],[Out = Ret1,V1 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]). eq(gcd(V, V1, Out),1,[],[Out = V9,V9 >= 0,V = 0,V1 = V9]). eq(gcd(V, V1, Out),1,[],[Out = 1 + V10,V10 >= 0,V = 1 + V10,V1 = 0]). eq(gcd(V, V1, Out),1,[le(V11, V12, Ret0),fun(Ret0, 1 + V12, 1 + V11, Ret2)],[Out = Ret2,V1 = 1 + V11,V12 >= 0,V11 >= 0,V = 1 + V12]). eq(fun(V, V1, V13, Out),1,[minus(V14, V15, Ret01),gcd(Ret01, 1 + V15, Ret3)],[Out = Ret3,V = 2,V1 = 1 + V14,V14 >= 0,V15 >= 0,V13 = 1 + V15]). eq(fun(V, V1, V13, Out),1,[minus(V16, V17, Ret02),gcd(Ret02, 1 + V17, Ret4)],[Out = Ret4,V1 = 1 + V17,V = 1,V17 >= 0,V16 >= 0,V13 = 1 + V16]). eq(minus(V, V1, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V = V18,V1 = V19]). eq(fun(V, V1, V13, Out),0,[],[Out = 0,V20 >= 0,V13 = V21,V22 >= 0,V = V20,V1 = V22,V21 >= 0]). eq(le(V, V1, Out),0,[],[Out = 0,V23 >= 0,V24 >= 0,V = V23,V1 = V24]). eq(gcd(V, V1, Out),0,[],[Out = 0,V25 >= 0,V26 >= 0,V = V25,V1 = V26]). input_output_vars(le(V,V1,Out),[V,V1],[Out]). input_output_vars(minus(V,V1,Out),[V,V1],[Out]). input_output_vars(gcd(V,V1,Out),[V,V1],[Out]). input_output_vars(fun(V,V1,V13,Out),[V,V1,V13],[Out]).

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

#### Computed strongly connected components
0. recursive : [le/3]
1. recursive : [minus/3]
2. recursive : [fun/4,gcd/3]
3. non_recursive : [start/3]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into le/3
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into gcd/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations le/3
* CE 20 is refined into CE [21]
* CE 18 is refined into CE [22]
* CE 17 is refined into CE [23]
* CE 19 is refined into CE [24]


### Cost equations --> "Loop" of le/3
* CEs [24] --> Loop 15
* CEs [21] --> Loop 16
* CEs [22] --> Loop 17
* CEs [23] --> Loop 18

### Ranking functions of CR le(V,V1,Out)
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR le(V,V1,Out)
* Partial RF of phase [15]:
- RF of loop [15:1]:
V
V1


### Specialization of cost equations minus/3
* CE 10 is refined into CE [25]
* CE 8 is refined into CE [26]
* CE 9 is refined into CE [27]


### Cost equations --> "Loop" of minus/3
* CEs [27] --> Loop 19
* CEs [25] --> Loop 20
* CEs [26] --> Loop 21

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

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


### Specialization of cost equations gcd/3
* CE 11 is refined into CE [28,29,30,31,32]
* CE 16 is refined into CE [33]
* CE 15 is refined into CE [34]
* CE 14 is refined into CE [35]
* CE 13 is refined into CE [36,37,38,39]
* CE 12 is refined into CE [40,41,42,43]


### Cost equations --> "Loop" of gcd/3
* CEs [43] --> Loop 22
* CEs [39] --> Loop 23
* CEs [42] --> Loop 24
* CEs [38] --> Loop 25
* CEs [36] --> Loop 26
* CEs [37] --> Loop 27
* CEs [40] --> Loop 28
* CEs [41] --> Loop 29
* CEs [28] --> Loop 30
* CEs [34] --> Loop 31
* CEs [29,30,31,32,33] --> Loop 32
* CEs [35] --> Loop 33

### Ranking functions of CR gcd(V,V1,Out)
* RF of phase [22,23]: [V+V1-3]
* RF of phase [26]: [V]

#### Partial ranking functions of CR gcd(V,V1,Out)
* Partial RF of phase [22,23]:
- RF of loop [22:1]:
V/2+V1/2-2
V1-2
- RF of loop [23:1]:
V-1 depends on loops [22:1]
V-V1+1 depends on loops [22:1]
* Partial RF of phase [26]:
- RF of loop [26:1]:
V


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


### Cost equations --> "Loop" of start/3
* CEs [81,82] --> Loop 34
* CEs [70,74,80] --> Loop 35
* CEs [52] --> Loop 36
* CEs [50] --> Loop 37
* CEs [44,45,46,47,48,49,51,53,54,55] --> Loop 38
* CEs [63] --> Loop 39
* CEs [65,79] --> Loop 40
* CEs [57,58,59,60,61,62,64,66,67,68] --> Loop 41
* CEs [56,69,71,72,73,75,76,77,78,83,84,85] --> Loop 42

### Ranking functions of CR start(V,V1,V13)

#### Partial ranking functions of CR start(V,V1,V13)


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

#### Cost of chains of le(V,V1,Out):
* Chain [[15],18]: 1*it(15)+1
Such that:it(15) =< V

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

* Chain [[15],17]: 1*it(15)+1
Such that:it(15) =< V1

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

* Chain [[15],16]: 1*it(15)+0
Such that:it(15) =< V1

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

* Chain [18]: 1
with precondition: [V=0,Out=2,V1>=0]

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

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


#### Cost of chains of minus(V,V1,Out):
* Chain [[19],21]: 1*it(19)+1
Such that:it(19) =< V1

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

* Chain [[19],20]: 1*it(19)+0
Such that:it(19) =< V1

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

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

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


#### Cost of chains of gcd(V,V1,Out):
* Chain [[26],33]: 4*it(26)+1
Such that:it(26) =< V

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

* Chain [[26],32]: 6*it(26)+1*s(5)+2
Such that:s(5) =< 1
aux(2) =< V
it(26) =< aux(2)

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

* Chain [[26],30]: 4*it(26)+2
Such that:it(26) =< V

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

* Chain [[26],27,33]: 4*it(26)+4
Such that:it(26) =< V

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

* Chain [[26],27,32]: 4*it(26)+1*s(5)+5
Such that:s(5) =< 1
it(26) =< V

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

* Chain [[22,23],33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+1
Such that:aux(8) =< V-V1+1
aux(20) =< V+V1
aux(21) =< V+V1-Out
it(22) =< V/2+V1/2
aux(23) =< V/2+V1/2-Out/2
aux(24) =< V1
aux(25) =< V1-Out
aux(7) =< 2*V1-2*Out
aux(26) =< V
it(22) =< aux(20)
it(23) =< aux(20)
s(16) =< aux(20)
it(22) =< aux(21)
it(23) =< aux(21)
s(16) =< aux(21)
it(22) =< aux(23)
it(23) =< aux(23)
aux(5) =< aux(24)
it(22) =< aux(24)
aux(5) =< aux(25)
it(22) =< aux(25)
it(23) =< aux(7)+aux(8)
it(23) =< aux(5)+aux(26)
s(18) =< aux(5)+aux(26)
s(18) =< it(23)*aux(24)
s(17) =< s(18)
s(15) =< s(16)

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

* Chain [[22,23],32]: 4*it(22)+4*it(23)+5*s(3)+2*s(17)+2
Such that:aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(7) =< 2*V1
aux(27) =< V
aux(28) =< V+V1
aux(29) =< V1
s(3) =< aux(28)
it(22) =< aux(28)
it(23) =< aux(28)
it(22) =< aux(29)
it(23) =< aux(7)+aux(8)
it(23) =< aux(29)+aux(27)
s(18) =< aux(29)+aux(27)
s(18) =< it(23)*aux(29)
s(17) =< s(18)

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

* Chain [[22,23],29,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+4
Such that:aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(30) =< V
aux(31) =< V+V1
aux(32) =< V/2+V1/2
aux(33) =< V1
it(22) =< aux(32)
it(22) =< aux(31)
it(23) =< aux(31)
it(23) =< aux(32)
it(22) =< aux(33)
it(23) =< aux(7)+aux(8)
it(23) =< aux(33)+aux(30)
s(18) =< aux(33)+aux(30)
s(18) =< it(23)*aux(33)
s(17) =< s(18)
s(15) =< aux(31)

with precondition: [Out=1,V>=2,V1>=2,V+V1>=5]

* Chain [[22,23],29,32]: 4*it(22)+4*it(23)+1*s(5)+2*s(15)+2*s(17)+5
Such that:s(5) =< 1
aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(34) =< V
aux(35) =< V+V1
aux(36) =< V/2+V1/2
aux(37) =< V1
it(22) =< aux(36)
it(22) =< aux(35)
it(23) =< aux(35)
it(23) =< aux(36)
it(22) =< aux(37)
it(23) =< aux(7)+aux(8)
it(23) =< aux(37)+aux(34)
s(18) =< aux(37)+aux(34)
s(18) =< it(23)*aux(37)
s(17) =< s(18)
s(15) =< aux(35)

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

* Chain [[22,23],28,[26],33]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+5
Such that:aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(7) =< 2*V1
aux(38) =< V
aux(39) =< V+V1
aux(40) =< V1
it(26) =< aux(40)
it(22) =< aux(39)
it(23) =< aux(39)
it(22) =< aux(40)
it(23) =< aux(7)+aux(8)
it(23) =< aux(40)+aux(38)
s(18) =< aux(40)+aux(38)
s(18) =< it(23)*aux(40)
s(17) =< s(18)
s(15) =< aux(39)

with precondition: [Out=1,V>=2,V1>=2,V+V1>=5]

* Chain [[22,23],28,[26],32]: 4*it(22)+4*it(23)+6*it(26)+1*s(5)+2*s(15)+2*s(17)+6
Such that:s(5) =< 1
aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(41) =< V
aux(42) =< V+V1
aux(43) =< V1
aux(44) =< 2*V1
it(26) =< aux(44)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(43)
it(23) =< aux(44)+aux(8)
it(23) =< aux(43)+aux(41)
s(18) =< aux(43)+aux(41)
s(18) =< it(23)*aux(43)
s(17) =< s(18)
s(15) =< aux(42)

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

* Chain [[22,23],28,[26],30]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+6
Such that:aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(7) =< 2*V1
aux(45) =< V
aux(46) =< V+V1
aux(47) =< V1
it(26) =< aux(47)
it(22) =< aux(46)
it(23) =< aux(46)
it(22) =< aux(47)
it(23) =< aux(7)+aux(8)
it(23) =< aux(47)+aux(45)
s(18) =< aux(47)+aux(45)
s(18) =< it(23)*aux(47)
s(17) =< s(18)
s(15) =< aux(46)

with precondition: [Out=0,V>=3,V1>=3,V+V1>=7]

* Chain [[22,23],28,[26],27,33]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+8
Such that:aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(7) =< 2*V1
aux(48) =< V
aux(49) =< V+V1
aux(50) =< V1
it(26) =< aux(50)
it(22) =< aux(49)
it(23) =< aux(49)
it(22) =< aux(50)
it(23) =< aux(7)+aux(8)
it(23) =< aux(50)+aux(48)
s(18) =< aux(50)+aux(48)
s(18) =< it(23)*aux(50)
s(17) =< s(18)
s(15) =< aux(49)

with precondition: [Out=1,V>=3,V1>=3,V+V1>=7]

* Chain [[22,23],28,[26],27,32]: 4*it(22)+4*it(23)+4*it(26)+1*s(5)+2*s(15)+2*s(17)+9
Such that:s(5) =< 1
aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(7) =< 2*V1
aux(51) =< V
aux(52) =< V+V1
aux(53) =< V1
it(26) =< aux(53)
it(22) =< aux(52)
it(23) =< aux(52)
it(22) =< aux(53)
it(23) =< aux(7)+aux(8)
it(23) =< aux(53)+aux(51)
s(18) =< aux(53)+aux(51)
s(18) =< it(23)*aux(53)
s(17) =< s(18)
s(15) =< aux(52)

with precondition: [Out=0,V>=3,V1>=3,V+V1>=7]

* Chain [[22,23],28,32]: 4*it(22)+4*it(23)+2*s(3)+1*s(5)+2*s(15)+2*s(17)+6
Such that:s(5) =< 1
aux(8) =< V-V1+1
it(22) =< V/2+V1/2
aux(54) =< V
aux(55) =< V+V1
aux(56) =< V1
aux(57) =< 2*V1
s(3) =< aux(57)
it(22) =< aux(55)
it(23) =< aux(55)
it(22) =< aux(56)
it(23) =< aux(57)+aux(8)
it(23) =< aux(56)+aux(54)
s(18) =< aux(56)+aux(54)
s(18) =< it(23)*aux(56)
s(17) =< s(18)
s(15) =< aux(55)

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

* Chain [[22,23],28,30]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+6
Such that:aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(58) =< V
aux(59) =< V+V1
aux(60) =< V/2+V1/2
aux(61) =< V1
it(22) =< aux(60)
it(22) =< aux(59)
it(23) =< aux(59)
it(23) =< aux(60)
it(22) =< aux(61)
it(23) =< aux(7)+aux(8)
it(23) =< aux(61)+aux(58)
s(18) =< aux(61)+aux(58)
s(18) =< it(23)*aux(61)
s(17) =< s(18)
s(15) =< aux(59)

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

* Chain [[22,23],28,27,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+8
Such that:aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(62) =< V
aux(63) =< V+V1
aux(64) =< V/2+V1/2
aux(65) =< V1
it(22) =< aux(64)
it(22) =< aux(63)
it(23) =< aux(63)
it(23) =< aux(64)
it(22) =< aux(65)
it(23) =< aux(7)+aux(8)
it(23) =< aux(65)+aux(62)
s(18) =< aux(65)+aux(62)
s(18) =< it(23)*aux(65)
s(17) =< s(18)
s(15) =< aux(63)

with precondition: [Out=1,V>=2,V1>=2,V+V1>=5]

* Chain [[22,23],28,27,32]: 4*it(22)+4*it(23)+1*s(5)+2*s(15)+2*s(17)+9
Such that:s(5) =< 1
aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(66) =< V
aux(67) =< V+V1
aux(68) =< V/2+V1/2
aux(69) =< V1
it(22) =< aux(68)
it(22) =< aux(67)
it(23) =< aux(67)
it(23) =< aux(68)
it(22) =< aux(69)
it(23) =< aux(7)+aux(8)
it(23) =< aux(69)+aux(66)
s(18) =< aux(69)+aux(66)
s(18) =< it(23)*aux(69)
s(17) =< s(18)
s(15) =< aux(67)

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

* Chain [[22,23],25,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+2*s(20)+4
Such that:aux(19) =< V
aux(8) =< V-V1+1
aux(20) =< V+V1
aux(21) =< V+V1-2*Out
aux(22) =< V-Out
it(22) =< V/2+V1/2
aux(23) =< V/2+V1/2-Out
aux(24) =< V1
aux(25) =< V1-Out
aux(7) =< 2*V1-2*Out
aux(70) =< Out
s(20) =< aux(70)
it(22) =< aux(20)
it(23) =< aux(20)
s(16) =< aux(20)
it(22) =< aux(21)
it(23) =< aux(21)
s(16) =< aux(21)
it(22) =< aux(23)
it(23) =< aux(23)
aux(5) =< aux(24)
it(22) =< aux(24)
aux(5) =< aux(25)
it(22) =< aux(25)
it(23) =< aux(7)+aux(8)
it(23) =< aux(5)+aux(19)
s(18) =< aux(5)+aux(22)
s(18) =< aux(5)+aux(19)
it(23) =< aux(5)+aux(22)
s(18) =< it(23)*aux(24)
s(17) =< s(18)
s(15) =< s(16)

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

* Chain [[22,23],25,32]: 4*it(22)+4*it(23)+3*s(5)+2*s(15)+2*s(17)+5
Such that:aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(72) =< V
aux(73) =< V+V1
aux(74) =< V/2+V1/2
aux(75) =< V1
it(22) =< aux(74)
s(5) =< aux(75)
it(22) =< aux(73)
it(23) =< aux(73)
it(23) =< aux(74)
it(22) =< aux(75)
it(23) =< aux(7)+aux(8)
it(23) =< aux(75)+aux(72)
s(18) =< aux(75)+aux(72)
s(18) =< it(23)*aux(75)
s(17) =< s(18)
s(15) =< aux(73)

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

* Chain [[22,23],24,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+2*s(22)+4
Such that:aux(19) =< V
aux(8) =< V-V1+1
aux(20) =< V+V1
aux(21) =< V+V1-2*Out
aux(22) =< V-Out
it(22) =< V/2+V1/2
aux(23) =< V/2+V1/2-Out
aux(24) =< V1
aux(25) =< V1-Out
aux(7) =< 2*V1-2*Out
aux(76) =< Out
s(22) =< aux(76)
it(22) =< aux(20)
it(23) =< aux(20)
s(16) =< aux(20)
it(22) =< aux(21)
it(23) =< aux(21)
s(16) =< aux(21)
it(22) =< aux(23)
it(23) =< aux(23)
aux(5) =< aux(24)
it(22) =< aux(24)
aux(5) =< aux(25)
it(22) =< aux(25)
it(23) =< aux(7)+aux(8)
it(23) =< aux(5)+aux(19)
s(18) =< aux(5)+aux(22)
s(18) =< aux(5)+aux(19)
it(23) =< aux(5)+aux(22)
s(18) =< it(23)*aux(24)
s(17) =< s(18)
s(15) =< s(16)

with precondition: [Out>=2,V>=Out+1,V1>=Out+1,V+V1>=3*Out+2]

* Chain [[22,23],24,32]: 4*it(22)+4*it(23)+3*s(5)+2*s(15)+2*s(17)+5
Such that:aux(8) =< V-V1+1
aux(7) =< 2*V1
aux(78) =< V
aux(79) =< V+V1
aux(80) =< V/2+V1/2
aux(81) =< V1
it(22) =< aux(80)
s(5) =< aux(78)
it(22) =< aux(79)
it(23) =< aux(79)
it(23) =< aux(80)
it(22) =< aux(81)
it(23) =< aux(7)+aux(8)
it(23) =< aux(81)+aux(78)
s(18) =< aux(81)+aux(78)
s(18) =< it(23)*aux(81)
s(17) =< s(18)
s(15) =< aux(79)

with precondition: [Out=0,V>=3,V1>=3,V+V1>=8]

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

* Chain [32]: 2*s(3)+1*s(5)+2
Such that:s(5) =< V1
aux(1) =< V
s(3) =< aux(1)

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

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

* Chain [30]: 2
with precondition: [V1=1,Out=0,V>=1]

* Chain [29,33]: 4
with precondition: [V=1,Out=1,V1>=2]

* Chain [29,32]: 1*s(5)+5
Such that:s(5) =< 1

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

* Chain [28,[26],33]: 4*it(26)+5
Such that:it(26) =< V1

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

* Chain [28,[26],32]: 6*it(26)+1*s(5)+6
Such that:s(5) =< 1
aux(2) =< V1
it(26) =< aux(2)

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

* Chain [28,[26],30]: 4*it(26)+6
Such that:it(26) =< V1

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

* Chain [28,[26],27,33]: 4*it(26)+8
Such that:it(26) =< V1

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

* Chain [28,[26],27,32]: 4*it(26)+1*s(5)+9
Such that:s(5) =< 1
it(26) =< V1

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

* Chain [28,32]: 2*s(3)+1*s(5)+6
Such that:s(5) =< 1
aux(1) =< V1
s(3) =< aux(1)

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

* Chain [28,30]: 6
with precondition: [V=1,Out=0,V1>=2]

* Chain [28,27,33]: 8
with precondition: [V=1,Out=1,V1>=2]

* Chain [28,27,32]: 1*s(5)+9
Such that:s(5) =< 1

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

* Chain [27,33]: 4
with precondition: [V1=1,Out=1,V>=1]

* Chain [27,32]: 1*s(5)+5
Such that:s(5) =< 1

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

* Chain [25,33]: 2*s(20)+4
Such that:aux(70) =< Out
s(20) =< aux(70)

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

* Chain [25,32]: 3*s(5)+5
Such that:aux(71) =< V1
s(5) =< aux(71)

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

* Chain [24,33]: 2*s(22)+4
Such that:aux(76) =< Out
s(22) =< aux(76)

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

* Chain [24,32]: 3*s(5)+5
Such that:aux(77) =< V
s(5) =< aux(77)

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


#### Cost of chains of start(V,V1,V13):
* Chain [42]: 45*s(266)+9*s(268)+10*s(278)+68*s(279)+36*s(281)+18*s(283)+43*s(284)+8*s(285)+32*s(286)+16*s(288)+9
Such that:s(271) =< 1
aux(114) =< V
aux(115) =< V-V1+1
aux(116) =< V+V1
aux(117) =< V/2+V1/2
aux(118) =< V1
aux(119) =< 2*V1
s(268) =< aux(114)
s(266) =< aux(118)
s(278) =< s(271)
s(279) =< aux(117)
s(279) =< aux(116)
s(281) =< aux(116)
s(281) =< aux(117)
s(279) =< aux(118)
s(281) =< aux(119)+aux(115)
s(281) =< aux(118)+aux(114)
s(282) =< aux(118)+aux(114)
s(282) =< s(281)*aux(118)
s(283) =< s(282)
s(284) =< aux(116)
s(285) =< aux(119)
s(286) =< aux(116)
s(286) =< aux(119)+aux(115)
s(286) =< aux(118)+aux(114)
s(287) =< aux(118)+aux(114)
s(287) =< s(286)*aux(118)
s(288) =< s(287)

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

* Chain [41]: 64*s(341)+40*s(342)+20*s(344)+10*s(346)+96*s(347)+8*s(348)+20*s(349)+10*s(351)+101*s(359)+40*s(369)+20*s(371)+10*s(373)+16*s(375)+20*s(376)+10*s(378)+68*s(389)+36*s(391)+18*s(393)+32*s(396)+16*s(398)+8*s(399)+11
Such that:s(340) =< 2
s(363) =< -V1+1
s(365) =< V1/2
aux(130) =< 1
aux(131) =< -2*V1+V13+1
aux(132) =< -V1+V13
aux(133) =< V1
aux(134) =< 2*V1
aux(135) =< V13
aux(136) =< V13/2
s(359) =< aux(133)
s(341) =< aux(130)
s(347) =< aux(135)
s(369) =< s(365)
s(369) =< aux(133)
s(371) =< aux(133)
s(371) =< s(365)
s(371) =< aux(134)+s(363)
s(372) =< aux(133)
s(372) =< s(371)*aux(133)
s(373) =< s(372)
s(375) =< aux(134)
s(376) =< aux(133)
s(376) =< aux(134)+s(363)
s(377) =< aux(133)
s(377) =< s(376)*aux(133)
s(378) =< s(377)
s(389) =< aux(136)
s(389) =< aux(135)
s(391) =< aux(135)
s(391) =< aux(136)
s(389) =< aux(133)
s(391) =< aux(134)+aux(131)
s(391) =< aux(133)+aux(132)
s(392) =< aux(133)+aux(132)
s(392) =< s(391)*aux(133)
s(393) =< s(392)
s(396) =< aux(135)
s(396) =< aux(134)+aux(131)
s(396) =< aux(133)+aux(132)
s(397) =< aux(133)+aux(132)
s(397) =< s(396)*aux(133)
s(398) =< s(397)
s(399) =< aux(132)
s(342) =< aux(136)
s(342) =< aux(135)
s(344) =< aux(135)
s(344) =< aux(136)
s(342) =< aux(130)
s(344) =< s(340)+aux(135)
s(344) =< aux(130)+aux(135)
s(345) =< aux(130)+aux(135)
s(345) =< s(344)*aux(130)
s(346) =< s(345)
s(348) =< s(340)
s(349) =< aux(135)
s(349) =< s(340)+aux(135)
s(349) =< aux(130)+aux(135)
s(350) =< aux(130)+aux(135)
s(350) =< s(349)*aux(130)
s(351) =< s(350)

with precondition: [V=1,V1>=1,V13>=1]

* Chain [40]: 9*s(447)+8*s(451)+10
Such that:s(450) =< V1
aux(137) =< V13
s(451) =< s(450)
s(447) =< aux(137)

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

* Chain [39]: 1*s(452)+3
Such that:s(452) =< V13

with precondition: [V=1,V1=V13,V1>=2]

* Chain [38]: 64*s(460)+40*s(461)+20*s(463)+10*s(465)+96*s(466)+8*s(467)+20*s(468)+10*s(470)+101*s(478)+40*s(488)+20*s(490)+10*s(492)+16*s(494)+20*s(495)+10*s(497)+68*s(508)+36*s(510)+18*s(512)+32*s(515)+16*s(517)+8*s(518)+11
Such that:s(459) =< 2
s(482) =< -V13+1
s(484) =< V13/2
aux(148) =< 1
aux(149) =< V1
aux(150) =< V1-2*V13+1
aux(151) =< V1-V13
aux(152) =< V1/2
aux(153) =< V13
aux(154) =< 2*V13
s(478) =< aux(153)
s(460) =< aux(148)
s(466) =< aux(149)
s(488) =< s(484)
s(488) =< aux(153)
s(490) =< aux(153)
s(490) =< s(484)
s(490) =< aux(154)+s(482)
s(491) =< aux(153)
s(491) =< s(490)*aux(153)
s(492) =< s(491)
s(494) =< aux(154)
s(495) =< aux(153)
s(495) =< aux(154)+s(482)
s(496) =< aux(153)
s(496) =< s(495)*aux(153)
s(497) =< s(496)
s(508) =< aux(152)
s(508) =< aux(149)
s(510) =< aux(149)
s(510) =< aux(152)
s(508) =< aux(153)
s(510) =< aux(154)+aux(150)
s(510) =< aux(153)+aux(151)
s(511) =< aux(153)+aux(151)
s(511) =< s(510)*aux(153)
s(512) =< s(511)
s(515) =< aux(149)
s(515) =< aux(154)+aux(150)
s(515) =< aux(153)+aux(151)
s(516) =< aux(153)+aux(151)
s(516) =< s(515)*aux(153)
s(517) =< s(516)
s(518) =< aux(151)
s(461) =< aux(152)
s(461) =< aux(149)
s(463) =< aux(149)
s(463) =< aux(152)
s(461) =< aux(148)
s(463) =< s(459)+aux(149)
s(463) =< aux(148)+aux(149)
s(464) =< aux(148)+aux(149)
s(464) =< s(463)*aux(148)
s(465) =< s(464)
s(467) =< s(459)
s(468) =< aux(149)
s(468) =< s(459)+aux(149)
s(468) =< aux(148)+aux(149)
s(469) =< aux(148)+aux(149)
s(469) =< s(468)*aux(148)
s(470) =< s(469)

with precondition: [V=2,V1>=1,V13>=1]

* Chain [37]: 1*s(566)+3
Such that:s(566) =< V13

with precondition: [V=2,V1=V13,V1>=2]

* Chain [36]: 9*s(567)+10
Such that:aux(155) =< V13
s(567) =< aux(155)

with precondition: [V=2,V1=V13+1,V1>=3]

* Chain [35]: 1
with precondition: [V1=0,V>=0]

* Chain [34]: 3*s(572)+22*s(573)+5
Such that:s(570) =< 1
aux(156) =< V
s(572) =< s(570)
s(573) =< aux(156)

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


Closed-form bounds of start(V,V1,V13):
-------------------------------------
* Chain [42] with precondition: [V>=0,V1>=0]
- Upper bound: 188*V+240*V1+19
- Complexity: n
* Chain [41] with precondition: [V=1,V1>=1,V13>=1]
- Upper bound: 227*V1+224*V13+151+nat(-V1+V13)*42+20*V1+34*V13
- Complexity: n
* Chain [40] with precondition: [V=1,V1>=2]
- Upper bound: 8*V1+10+nat(V13)*9
- Complexity: n
* Chain [39] with precondition: [V=1,V1=V13,V1>=2]
- Upper bound: V13+3
- Complexity: n
* Chain [38] with precondition: [V=2,V1>=1,V13>=1]
- Upper bound: 224*V1+227*V13+151+nat(V1-V13)*42+34*V1+20*V13
- Complexity: n
* Chain [37] with precondition: [V=2,V1=V13,V1>=2]
- Upper bound: V13+3
- Complexity: n
* Chain [36] with precondition: [V=2,V1=V13+1,V1>=3]
- Upper bound: 9*V13+10
- Complexity: n
* Chain [35] with precondition: [V1=0,V>=0]
- Upper bound: 1
- Complexity: constant
* Chain [34] with precondition: [V1=1,V>=1]
- Upper bound: 22*V+8
- Complexity: n

### Maximum cost of start(V,V1,V13): max([188*V+240*V1+18,187*V1+141+nat(V13)*186+20*V1+nat(V13/2)*40+max([nat(V13)*29+32*V1+nat(-V1+V13)*42+nat(V13/2)*28,nat(2*V13)*16+29*V1+nat(V1-V13)*42+14*V1])+8*V1+ (nat(V13)*8+7)+ (nat(V13)+2)])+1
Asymptotic class: n
* Total analysis performed in 1389 ms.

(10) BOUNDS(1, n^1)