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:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of quot: minus
The following defined symbols can occur below the 0th argument of minus: minus

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

(2) 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:

quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(s(x), y) → s(plus(x, y))
plus(0, y) → y
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
quot(0, s(y)) → 0

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(4) 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:

quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) [1]
plus(s(x), y) → s(plus(x, y)) [1]
plus(0, y) → y [1]
minus(s(x), s(y)) → minus(x, y) [1]
minus(x, 0) → x [1]
quot(0, s(y)) → 0 [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) [1]
plus(s(x), y) → s(plus(x, y)) [1]
plus(0, y) → y [1]
minus(s(x), s(y)) → minus(x, y) [1]
minus(x, 0) → x [1]
quot(0, s(y)) → 0 [1]

The TRS has the following type information:

quot :: s:0 → s:0 → s:0 s :: s:0 → s:0 minus :: s:0 → s:0 → s:0 plus :: s:0 → s:0 → s:0 0 :: s:0

Rewrite Strategy: INNERMOST

(7) 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:

quot(v0, v1) → null_quot [0]
minus(v0, v1) → null_minus [0]
plus(v0, v1) → null_plus [0]

And the following fresh constants:

null_quot, null_minus, null_plus

(8) 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: quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) [1] plus(s(x), y) → s(plus(x, y)) [1] plus(0, y) → y [1] minus(s(x), s(y)) → minus(x, y) [1] minus(x, 0) → x [1] quot(0, s(y)) → 0 [1] quot(v0, v1) → null_quot [0] minus(v0, v1) → null_minus [0] plus(v0, v1) → null_plus [0] The TRS has the following type information: quot :: s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus s :: s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus minus :: s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus plus :: s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus → s:0:null_quot:null_minus:null_plus 0 :: s:0:null_quot:null_minus:null_plus null_quot :: s:0:null_quot:null_minus:null_plus null_minus :: s:0:null_quot:null_minus:null_plus null_plus :: s:0:null_quot:null_minus:null_plus Rewrite Strategy: INNERMOST

(9) 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
null_quot => 0
null_minus => 0
null_plus => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

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
plus(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
quot(z, z') -{ 1 }→ 0 :|: z' = 1 + y, y >= 0, z = 0
quot(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
quot(z, z') -{ 1 }→ 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

(11) CompleteCoflocoProof (EQUIVALENT transformation)

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

eq(start(V, V1),0,[quot(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]). eq(quot(V, V1, Out),1,[minus(V2, V3, Ret10),quot(Ret10, 1 + V3, Ret1)],[Out = 1 + Ret1,V1 = 1 + V3,V2 >= 0,V3 >= 0,V = 1 + V2]). eq(plus(V, V1, Out),1,[plus(V4, V5, Ret11)],[Out = 1 + Ret11,V4 >= 0,V5 >= 0,V = 1 + V4,V1 = V5]). eq(plus(V, V1, Out),1,[],[Out = V6,V6 >= 0,V = 0,V1 = V6]). eq(minus(V, V1, Out),1,[minus(V7, V8, Ret)],[Out = Ret,V1 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]). eq(minus(V, V1, Out),1,[],[Out = V9,V9 >= 0,V = V9,V1 = 0]). eq(quot(V, V1, Out),1,[],[Out = 0,V1 = 1 + V10,V10 >= 0,V = 0]). eq(quot(V, V1, Out),0,[],[Out = 0,V11 >= 0,V12 >= 0,V = V11,V1 = V12]). eq(minus(V, V1, Out),0,[],[Out = 0,V13 >= 0,V14 >= 0,V = V13,V1 = V14]). eq(plus(V, V1, Out),0,[],[Out = 0,V15 >= 0,V16 >= 0,V = V15,V1 = V16]). input_output_vars(quot(V,V1,Out),[V,V1],[Out]). input_output_vars(plus(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. recursive : [minus/3]
1. recursive : [plus/3]
2. recursive : [quot/3]
3. non_recursive : [start/2]

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

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

### Specialization of cost equations minus/3
* CE 13 is refined into CE [14]
* CE 12 is refined into CE [15]
* CE 11 is refined into CE [16]


### Cost equations --> "Loop" of minus/3
* CEs [16] --> Loop 10
* CEs [14] --> Loop 11
* CEs [15] --> Loop 12

### Ranking functions of CR minus(V,V1,Out)
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR minus(V,V1,Out)
* Partial RF of phase [10]:
- RF of loop [10:1]:
V
V1


### Specialization of cost equations plus/3
* CE 10 is refined into CE [17]
* CE 9 is refined into CE [18]
* CE 8 is refined into CE [19]


### Cost equations --> "Loop" of plus/3
* CEs [19] --> Loop 13
* CEs [17] --> Loop 14
* CEs [18] --> Loop 15

### Ranking functions of CR plus(V,V1,Out)
* RF of phase [13]: [V]

#### Partial ranking functions of CR plus(V,V1,Out)
* Partial RF of phase [13]:
- RF of loop [13:1]:
V


### Specialization of cost equations quot/3
* CE 6 is refined into CE [20]
* CE 7 is refined into CE [21]
* CE 5 is refined into CE [22,23,24]


### Cost equations --> "Loop" of quot/3
* CEs [24] --> Loop 16
* CEs [23] --> Loop 17
* CEs [22] --> Loop 18
* CEs [20,21] --> Loop 19

### Ranking functions of CR quot(V,V1,Out)
* RF of phase [16]: [V-1,V-V1+1]
* RF of phase [18]: [V]

#### Partial ranking functions of CR quot(V,V1,Out)
* Partial RF of phase [16]:
- RF of loop [16:1]:
V-1
V-V1+1
* Partial RF of phase [18]:
- RF of loop [18:1]:
V


### Specialization of cost equations start/2
* CE 2 is refined into CE [25,26,27,28,29]
* CE 3 is refined into CE [30,31,32,33]
* CE 4 is refined into CE [34,35,36]


### Cost equations --> "Loop" of start/2
* CEs [25] --> Loop 20
* CEs [34] --> Loop 21
* CEs [26,27,28,29,30,31,32,33,35,36] --> Loop 22

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

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


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

#### Cost of chains of minus(V,V1,Out):
* Chain [[10],12]: 1*it(10)+1
Such that:it(10) =< V1

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

* Chain [[10],11]: 1*it(10)+0
Such that:it(10) =< V1

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

* Chain [12]: 1
with precondition: [V1=0,V=Out,V>=0]

* Chain [11]: 0
with precondition: [Out=0,V>=0,V1>=0]


#### Cost of chains of plus(V,V1,Out):
* Chain [[13],15]: 1*it(13)+1
Such that:it(13) =< -V1+Out

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

* Chain [[13],14]: 1*it(13)+0
Such that:it(13) =< Out

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

* Chain [15]: 1
with precondition: [V=0,V1=Out,V1>=0]

* Chain [14]: 0
with precondition: [Out=0,V>=0,V1>=0]


#### Cost of chains of quot(V,V1,Out):
* Chain [[18],19]: 2*it(18)+1
Such that:it(18) =< Out

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

* Chain [[18],17,19]: 2*it(18)+1*s(2)+2
Such that:s(2) =< 1
it(18) =< Out

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

* Chain [[16],19]: 2*it(16)+1*s(5)+1
Such that:it(16) =< V-V1+1
aux(3) =< V
it(16) =< aux(3)
s(5) =< aux(3)

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

* Chain [[16],17,19]: 2*it(16)+1*s(2)+1*s(5)+2
Such that:it(16) =< V-V1+1
s(2) =< V1
aux(4) =< V
it(16) =< aux(4)
s(5) =< aux(4)

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

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

* Chain [17,19]: 1*s(2)+2
Such that:s(2) =< V1

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


#### Cost of chains of start(V,V1):
* Chain [22]: 4*s(9)+4*s(10)+4*s(12)+2
Such that:aux(6) =< V
aux(7) =< V-V1+1
aux(8) =< V1
s(12) =< aux(6)
s(10) =< aux(7)
s(9) =< aux(8)
s(10) =< aux(6)

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

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

* Chain [20]: 1*s(21)+4*s(23)+2
Such that:s(21) =< 1
s(22) =< V
s(23) =< s(22)

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


Closed-form bounds of start(V,V1):
-------------------------------------
* Chain [22] with precondition: [V>=0,V1>=0]
- Upper bound: 4*V+4*V1+2+nat(V-V1+1)*4
- Complexity: n
* Chain [21] with precondition: [V1=0,V>=0]
- Upper bound: 1
- Complexity: constant
* Chain [20] with precondition: [V1=1,V>=1]
- Upper bound: 4*V+3
- Complexity: n

### Maximum cost of start(V,V1): 4*V+1+max([1,nat(V-V1+1)*4+4*V1])+1
Asymptotic class: n
* Total analysis performed in 242 ms.

(12) BOUNDS(1, n^1)