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(gr0(x), y, y)
cond2(false, x, y) → cond1(gr0(x), p(x), y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
gr0(0) → false
gr0(s(x)) → true
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(gr0(x), y, y) [1]
cond2(false, x, y) → cond1(gr0(x), p(x), y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
gr0(0) → false [1]
gr0(s(x)) → true [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(gr0(x), y, y) [1]
cond2(false, x, y) → cond1(gr0(x), p(x), y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
gr0(0) → false [1]
gr0(s(x)) → true [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 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2 gr :: 0:s → 0:s → true:false gr0 :: 0:s → true:false false :: true:false p :: 0:s → 0:s 0 :: 0:s 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(gr0(x), y, y) [1] cond2(false, x, y) → cond1(gr0(x), p(x), y) [1] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] gr0(0) → false [1] gr0(s(x)) → true [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 gr0 :: 0:s → true:false false :: true:false p :: 0:s → 0:s 0 :: 0:s 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
false => 0
0 => 0
null_cond1 => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

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 }→ cond1(gr0(x), y, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
cond2(z, z', z'') -{ 1 }→ cond1(gr0(x), p(x), y) :|: z' = x, z'' = y, x >= 0, y >= 0, 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
gr0(z) -{ 1 }→ 1 :|: x >= 0, z = 1 + x
gr0(z) -{ 1 }→ 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,[gr(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V2),0,[gr0(V, Out)],[V >= 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,[gr0(V5, Ret01),cond1(Ret01, V6, V6, Ret1)],[Out = Ret1,V1 = V5,V2 = V6,V = 1,V5 >= 0,V6 >= 0]). eq(cond2(V, V1, V2, Out),1,[gr0(V7, Ret02),p(V7, Ret11),cond1(Ret02, Ret11, V8, Ret2)],[Out = Ret2,V1 = V7,V2 = V8,V7 >= 0,V8 >= 0,V = 0]). eq(gr(V, V1, Out),1,[],[Out = 0,V1 = V9,V9 >= 0,V = 0]). eq(gr(V, V1, Out),1,[],[Out = 1,V10 >= 0,V = 1 + V10,V1 = 0]). eq(gr(V, V1, Out),1,[gr(V11, V12, Ret3)],[Out = Ret3,V1 = 1 + V12,V11 >= 0,V12 >= 0,V = 1 + V11]). eq(gr0(V, Out),1,[],[Out = 0,V = 0]). eq(gr0(V, Out),1,[],[Out = 1,V13 >= 0,V = 1 + V13]). eq(p(V, Out),1,[],[Out = 0,V = 0]). eq(p(V, Out),1,[],[Out = V14,V14 >= 0,V = 1 + V14]). eq(cond1(V, V1, V2, Out),0,[],[Out = 0,V15 >= 0,V2 = V16,V17 >= 0,V = V15,V1 = V17,V16 >= 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(gr(V,V1,Out),[V,V1],[Out]). input_output_vars(gr0(V,Out),[V],[Out]). input_output_vars(p(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. non_recursive : [gr0/2]
1. non_recursive : [p/2]
2. recursive : [gr/3]
3. recursive : [cond1/4,cond2/4]
4. non_recursive : [start/3]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into gr0/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into gr/3
3. SCC is partially evaluated into cond2/4
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations gr0/2
* CE 16 is refined into CE [19]
* CE 15 is refined into CE [20]


### Cost equations --> "Loop" of gr0/2
* CEs [19] --> Loop 13
* CEs [20] --> Loop 14

### Ranking functions of CR gr0(V,Out)

#### Partial ranking functions of CR gr0(V,Out)


### Specialization of cost equations p/2
* CE 18 is refined into CE [21]
* CE 17 is refined into CE [22]


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

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

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


### Specialization of cost equations gr/3
* CE 10 is refined into CE [23]
* CE 9 is refined into CE [24]
* CE 8 is refined into CE [25]


### Cost equations --> "Loop" of gr/3
* CEs [24] --> Loop 17
* CEs [25] --> Loop 18
* CEs [23] --> Loop 19

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

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


### Specialization of cost equations cond2/4
* CE 14 is refined into CE [26,27]
* CE 13 is refined into CE [28,29,30,31]
* CE 12 is refined into CE [32,33]
* CE 11 is refined into CE [34,35]


### Cost equations --> "Loop" of cond2/4
* CEs [33] --> Loop 20
* CEs [32] --> Loop 21
* CEs [35] --> Loop 22
* CEs [34] --> Loop 23
* CEs [27] --> Loop 24
* CEs [26] --> Loop 25
* CEs [31] --> Loop 26
* CEs [30] --> Loop 27
* CEs [29] --> Loop 28
* CEs [28] --> Loop 29

### Ranking functions of CR cond2(V,V1,V2,Out)
* RF of phase [27]: [V1-1]

#### Partial ranking functions of CR cond2(V,V1,V2,Out)
* Partial RF of phase [27]:
- RF of loop [27:1]:
V1-1


### Specialization of cost equations start/3
* CE 2 is refined into CE [36]
* CE 3 is refined into CE [37,38,39,40,41]
* CE 4 is refined into CE [42,43,44,45,46,47,48]
* CE 5 is refined into CE [49,50,51,52]
* CE 6 is refined into CE [53,54]
* CE 7 is refined into CE [55,56]


### Cost equations --> "Loop" of start/3
* CEs [41,48] --> Loop 30
* CEs [36,38,39,40,47] --> Loop 31
* CEs [37,46,50,51,52,54,56] --> Loop 32
* CEs [42,43,44,45,49,53,55] --> Loop 33

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

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


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

#### Cost of chains of gr0(V,Out):
* Chain [14]: 1
with precondition: [V=0,Out=0]

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


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

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


#### Cost of chains of gr(V,V1,Out):
* Chain [[19],18]: 1*it(19)+1
Such that:it(19) =< V

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

* Chain [[19],17]: 1*it(19)+1
Such that:it(19) =< V1

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

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

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


#### Cost of chains of cond2(V,V1,V2,Out):
* Chain [[27],29,23]: 5*it(27)+1*s(3)+8
Such that:aux(3) =< V1
it(27) =< aux(3)
s(3) =< it(27)*aux(3)

with precondition: [V=0,Out=0,V1>=2,V2+1>=V1]

* Chain [[27],22]: 5*it(27)+1*s(3)+3
Such that:aux(4) =< V1
it(27) =< aux(4)
s(3) =< it(27)*aux(4)

with precondition: [V=0,Out=0,V1>=2,V2+1>=V1]

* Chain [29,23]: 8
with precondition: [V=0,V1=1,Out=0,V2>=0]

* Chain [28,25,23]: 12
with precondition: [V=0,V2=0,Out=0,V1>=2]

* Chain [28,20]: 7
with precondition: [V=0,V2=0,Out=0,V1>=2]

* Chain [26,24,[27],29,23]: 7*it(27)+1*s(3)+17
Such that:aux(6) =< V2
it(27) =< aux(6)
s(3) =< it(27)*aux(6)

with precondition: [V=0,Out=0,V2>=2,V1>=V2+2]

* Chain [26,24,[27],22]: 7*it(27)+1*s(3)+12
Such that:aux(8) =< V2
it(27) =< aux(8)
s(3) =< it(27)*aux(8)

with precondition: [V=0,Out=0,V2>=2,V1>=V2+2]

* Chain [26,24,29,23]: 2*s(4)+17
Such that:aux(9) =< 1
s(4) =< aux(9)

with precondition: [V=0,V2=1,Out=0,V1>=3]

* Chain [26,24,22]: 2*s(4)+12
Such that:aux(10) =< V2
s(4) =< aux(10)

with precondition: [V=0,Out=0,V2>=1,V1>=V2+2]

* Chain [26,20]: 1*s(5)+7
Such that:s(5) =< V2

with precondition: [V=0,Out=0,V2>=1,V1>=V2+2]

* Chain [25,23]: 7
with precondition: [V=1,V2=0,Out=0,V1>=1]

* Chain [24,[27],29,23]: 6*it(27)+1*s(3)+12
Such that:aux(5) =< V2
it(27) =< aux(5)
s(3) =< it(27)*aux(5)

with precondition: [V=1,Out=0,V1>=1,V2>=2]

* Chain [24,[27],22]: 6*it(27)+1*s(3)+7
Such that:aux(7) =< V2
it(27) =< aux(7)
s(3) =< it(27)*aux(7)

with precondition: [V=1,Out=0,V1>=1,V2>=2]

* Chain [24,29,23]: 1*s(4)+12
Such that:s(4) =< 1

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

* Chain [24,22]: 1*s(4)+7
Such that:s(4) =< V2

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

* Chain [23]: 3
with precondition: [V=0,V1=0,Out=0,V2>=0]

* Chain [22]: 3
with precondition: [V=0,Out=0,V1>=1,V2>=0]

* Chain [21]: 2
with precondition: [V=1,V1=0,Out=0,V2>=0]

* Chain [20]: 2
with precondition: [V=1,Out=0,V1>=1,V2>=0]


#### Cost of chains of start(V,V1,V2):
* Chain [33]: 17*s(30)+10*s(31)+2*s(32)+2*s(33)+2*s(35)+17
Such that:s(34) =< 1
s(28) =< V1
s(29) =< V2
s(35) =< s(34)
s(30) =< s(29)
s(31) =< s(28)
s(32) =< s(31)*s(28)
s(33) =< s(30)*s(29)

with precondition: [V=0]

* Chain [32]: 1*s(36)+1*s(37)+5
Such that:s(36) =< V
s(37) =< V1

with precondition: [V>=1]

* Chain [31]: 11*s(41)+44*s(44)+2*s(46)+6*s(47)+19
Such that:aux(14) =< V1
aux(16) =< V2
s(44) =< aux(16)
s(47) =< s(44)*aux(16)
s(41) =< aux(14)
s(46) =< s(41)*aux(14)

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

* Chain [30]: 17
with precondition: [V=1,V2=1,V1>=1]


Closed-form bounds of start(V,V1,V2):
-------------------------------------
* Chain [33] with precondition: [V=0]
- Upper bound: nat(V1)*10+19+nat(V1)*2*nat(V1)+nat(V2)*17+nat(V2)*2*nat(V2)
- Complexity: n^2
* Chain [32] with precondition: [V>=1]
- Upper bound: V+5+nat(V1)
- Complexity: n
* Chain [31] with precondition: [V>=0,V1>=0,V2>=0]
- Upper bound: 11*V1+19+2*V1*V1+44*V2+6*V2*V2
- Complexity: n^2
* Chain [30] with precondition: [V=1,V2=1,V1>=1]
- Upper bound: 17
- Complexity: constant

### Maximum cost of start(V,V1,V2): max([12,nat(V1)+max([V,nat(V1)*9+14+nat(V1)*2*nat(V1)+nat(V2)*17+nat(V2)*2*nat(V2)+ (nat(V2)*27+nat(V1)+nat(V2)*4*nat(V2))])])+5
Asymptotic class: n^2
* Total analysis performed in 426 ms.

(10) BOUNDS(1, n^2)