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:

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

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:

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

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

The TRS has the following type information:

pred :: s:0 → s:0 s :: s:0 → s:0 minus :: s:0 → s:0 → s:0 0 :: s:0 quot :: s:0 → s:0 → s:0

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:

pred(v0) → null_pred [0]
quot(v0, v1) → null_quot [0]
minus(v0, v1) → null_minus [0]

And the following fresh constants:

null_pred, null_quot, null_minus

(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: pred(s(x)) → x [1] minus(x, 0) → x [1] minus(x, s(y)) → pred(minus(x, y)) [1] quot(0, s(y)) → 0 [1] quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) [1] pred(v0) → null_pred [0] quot(v0, v1) → null_quot [0] minus(v0, v1) → null_minus [0] The TRS has the following type information: pred :: s:0:null_pred:null_quot:null_minus → s:0:null_pred:null_quot:null_minus s :: s:0:null_pred:null_quot:null_minus → s:0:null_pred:null_quot:null_minus minus :: s:0:null_pred:null_quot:null_minus → s:0:null_pred:null_quot:null_minus → s:0:null_pred:null_quot:null_minus 0 :: s:0:null_pred:null_quot:null_minus quot :: s:0:null_pred:null_quot:null_minus → s:0:null_pred:null_quot:null_minus → s:0:null_pred:null_quot:null_minus null_pred :: s:0:null_pred:null_quot:null_minus null_quot :: s:0:null_pred:null_quot:null_minus null_minus :: s:0:null_pred:null_quot:null_minus 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
null_pred => 0
null_quot => 0
null_minus => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

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

(9) CompleteCoflocoProof (EQUIVALENT transformation)

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

eq(start(V, V2),0,[pred(V, Out)],[V >= 0]). eq(start(V, V2),0,[minus(V, V2, Out)],[V >= 0,V2 >= 0]). eq(start(V, V2),0,[quot(V, V2, Out)],[V >= 0,V2 >= 0]). eq(pred(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]). eq(minus(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = V3,V2 = 0]). eq(minus(V, V2, Out),1,[minus(V4, V5, Ret0),pred(Ret0, Ret)],[Out = Ret,V2 = 1 + V5,V4 >= 0,V5 >= 0,V = V4]). eq(quot(V, V2, Out),1,[],[Out = 0,V2 = 1 + V6,V6 >= 0,V = 0]). eq(quot(V, V2, Out),1,[minus(V7, V8, Ret10),quot(Ret10, 1 + V8, Ret1)],[Out = 1 + Ret1,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]). eq(pred(V, Out),0,[],[Out = 0,V9 >= 0,V = V9]). eq(quot(V, V2, Out),0,[],[Out = 0,V10 >= 0,V11 >= 0,V = V10,V2 = V11]). eq(minus(V, V2, Out),0,[],[Out = 0,V12 >= 0,V13 >= 0,V = V12,V2 = V13]). input_output_vars(pred(V,Out),[V],[Out]). input_output_vars(minus(V,V2,Out),[V,V2],[Out]). input_output_vars(quot(V,V2,Out),[V,V2],[Out]).

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

#### Computed strongly connected components
0. non_recursive : [pred/2]
1. recursive [non_tail] : [minus/3]
2. recursive : [quot/3]
3. non_recursive : [start/2]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into pred/2
1. SCC is partially evaluated into minus/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 pred/2
* CE 5 is refined into CE [13]
* CE 6 is refined into CE [14]


### Cost equations --> "Loop" of pred/2
* CEs [13] --> Loop 9
* CEs [14] --> Loop 10

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

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


### Specialization of cost equations minus/3
* CE 9 is refined into CE [15]
* CE 7 is refined into CE [16]
* CE 8 is refined into CE [17,18]


### Cost equations --> "Loop" of minus/3
* CEs [18] --> Loop 11
* CEs [17] --> Loop 12
* CEs [15] --> Loop 13
* CEs [16] --> Loop 14

### Ranking functions of CR minus(V,V2,Out)
* RF of phase [11]: [V2]
* RF of phase [12]: [V2]

#### Partial ranking functions of CR minus(V,V2,Out)
* Partial RF of phase [11]:
- RF of loop [11:1]:
V2
* Partial RF of phase [12]:
- RF of loop [12:1]:
V2


### Specialization of cost equations quot/3
* CE 10 is refined into CE [19]
* CE 12 is refined into CE [20]
* CE 11 is refined into CE [21,22,23]


### Cost equations --> "Loop" of quot/3
* CEs [23] --> Loop 15
* CEs [22] --> Loop 16
* CEs [21] --> Loop 17
* CEs [19,20] --> Loop 18

### Ranking functions of CR quot(V,V2,Out)
* RF of phase [15]: [V-1,V-V2+1]
* RF of phase [17]: [V]

#### Partial ranking functions of CR quot(V,V2,Out)
* Partial RF of phase [15]:
- RF of loop [15:1]:
V-1
V-V2+1
* Partial RF of phase [17]:
- RF of loop [17:1]:
V


### Specialization of cost equations start/2
* CE 2 is refined into CE [24,25]
* CE 3 is refined into CE [26,27,28]
* CE 4 is refined into CE [29,30,31,32,33]


### Cost equations --> "Loop" of start/2
* CEs [29] --> Loop 19
* CEs [24,25,26,27,28,30,31,32,33] --> Loop 20

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

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


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

#### Cost of chains of pred(V,Out):
* Chain [10]: 0
with precondition: [Out=0,V>=0]

* Chain [9]: 1
with precondition: [V=Out+1,V>=1]


#### Cost of chains of minus(V,V2,Out):
* Chain [[12],[11],14]: 3*it(11)+1
Such that:aux(1) =< V2
it(11) =< aux(1)

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

* Chain [[12],14]: 1*it(12)+1
Such that:it(12) =< V2

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

* Chain [[12],13]: 1*it(12)+0
Such that:it(12) =< V2

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

* Chain [[11],14]: 2*it(11)+1
Such that:it(11) =< V2

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

* Chain [14]: 1
with precondition: [V2=0,V=Out,V>=0]

* Chain [13]: 0
with precondition: [Out=0,V>=0,V2>=0]


#### Cost of chains of quot(V,V2,Out):
* Chain [[17],18]: 2*it(17)+1
Such that:it(17) =< Out

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

* Chain [[17],16,18]: 2*it(17)+5*s(6)+3
Such that:s(5) =< 1
it(17) =< Out
s(6) =< s(5)

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

* Chain [[15],18]: 2*it(15)+2*s(9)+1
Such that:it(15) =< V-V2+1
aux(5) =< V
it(15) =< aux(5)
s(9) =< aux(5)

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

* Chain [[15],16,18]: 2*it(15)+5*s(6)+2*s(9)+3
Such that:it(15) =< V-V2+1
s(5) =< V2
aux(6) =< V
s(6) =< s(5)
it(15) =< aux(6)
s(9) =< aux(6)

with precondition: [V2>=2,Out>=2,V+3>=2*Out+V2]

* Chain [18]: 1
with precondition: [Out=0,V>=0,V2>=0]

* Chain [16,18]: 5*s(6)+3
Such that:s(5) =< V2
s(6) =< s(5)

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


#### Cost of chains of start(V,V2):
* Chain [20]: 17*s(15)+4*s(19)+4*s(21)+3
Such that:aux(8) =< V
aux(9) =< V-V2+1
aux(10) =< V2
s(19) =< aux(9)
s(15) =< aux(10)
s(19) =< aux(8)
s(21) =< aux(8)

with precondition: [V>=0]

* Chain [19]: 4*s(29)+5*s(30)+3
Such that:s(27) =< 1
s(28) =< V
s(29) =< s(28)
s(30) =< s(27)

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


Closed-form bounds of start(V,V2):
-------------------------------------
* Chain [20] with precondition: [V>=0]
- Upper bound: 4*V+3+nat(V2)*17+nat(V-V2+1)*4
- Complexity: n
* Chain [19] with precondition: [V2=1,V>=1]
- Upper bound: 4*V+8
- Complexity: n

### Maximum cost of start(V,V2): 4*V+3+max([5,nat(V-V2+1)*4+nat(V2)*17])
Asymptotic class: n
* Total analysis performed in 234 ms.

(10) BOUNDS(1, n^1)