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)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, s(y)) → if_quot(le(s(y), x), x, s(y))
if_quot(true, x, y) → s(quot(minus(x, y), y))
if_quot(false, x, y) → 0

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:

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

minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]
le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
quot(x, s(y)) → if_quot(le(s(y), x), x, s(y)) [1]
if_quot(true, x, y) → s(quot(minus(x, y), y)) [1]
if_quot(false, x, y) → 0 [1]

The TRS has the following type information:

minus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false quot :: 0:s → 0:s → 0:s if_quot :: 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]
quot(v0, v1) → null_quot [0]
le(v0, v1) → null_le [0]
if_quot(v0, v1, v2) → null_if_quot [0]

And the following fresh constants:

null_minus, null_quot, null_le, null_if_quot

(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: minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] quot(x, s(y)) → if_quot(le(s(y), x), x, s(y)) [1] if_quot(true, x, y) → s(quot(minus(x, y), y)) [1] if_quot(false, x, y) → 0 [1] minus(v0, v1) → null_minus [0] quot(v0, v1) → null_quot [0] le(v0, v1) → null_le [0] if_quot(v0, v1, v2) → null_if_quot [0] The TRS has the following type information: minus :: 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot 0 :: 0:s:null_minus:null_quot:null_if_quot s :: 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot le :: 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot → true:false:null_le true :: true:false:null_le false :: true:false:null_le quot :: 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot if_quot :: true:false:null_le → 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot → 0:s:null_minus:null_quot:null_if_quot null_minus :: 0:s:null_minus:null_quot:null_if_quot null_quot :: 0:s:null_minus:null_quot:null_if_quot null_le :: true:false:null_le null_if_quot :: 0:s:null_minus:null_quot:null_if_quot 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_quot => 0
null_le => 0
null_if_quot => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

if_quot(z, z', z'') -{ 1 }→ 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
if_quot(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
if_quot(z, z', z'') -{ 1 }→ 1 + quot(minus(x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 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
quot(z, z') -{ 1 }→ if_quot(le(1 + y, x), x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
quot(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, V11),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V11),0,[le(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V11),0,[quot(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V11),0,[fun(V, V1, V11, Out)],[V >= 0,V1 >= 0,V11 >= 0]). eq(minus(V, V1, Out),1,[],[Out = V2,V2 >= 0,V = V2,V1 = 0]). eq(minus(V, V1, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V1 = 1 + V4,V3 >= 0,V4 >= 0,V = 1 + V3]). eq(le(V, V1, Out),1,[],[Out = 2,V5 >= 0,V = 0,V1 = V5]). eq(le(V, V1, Out),1,[],[Out = 1,V6 >= 0,V = 1 + V6,V1 = 0]). eq(le(V, V1, Out),1,[le(V7, V8, Ret1)],[Out = Ret1,V1 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]). eq(quot(V, V1, Out),1,[le(1 + V9, V10, Ret0),fun(Ret0, V10, 1 + V9, Ret2)],[Out = Ret2,V1 = 1 + V9,V10 >= 0,V9 >= 0,V = V10]). eq(fun(V, V1, V11, Out),1,[minus(V12, V13, Ret10),quot(Ret10, V13, Ret11)],[Out = 1 + Ret11,V = 2,V1 = V12,V11 = V13,V12 >= 0,V13 >= 0]). eq(fun(V, V1, V11, Out),1,[],[Out = 0,V1 = V14,V11 = V15,V = 1,V14 >= 0,V15 >= 0]). eq(minus(V, V1, Out),0,[],[Out = 0,V16 >= 0,V17 >= 0,V = V16,V1 = V17]). eq(quot(V, V1, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V = V18,V1 = V19]). eq(le(V, V1, Out),0,[],[Out = 0,V20 >= 0,V21 >= 0,V = V20,V1 = V21]). eq(fun(V, V1, V11, Out),0,[],[Out = 0,V22 >= 0,V11 = V23,V24 >= 0,V = V22,V1 = V24,V23 >= 0]). input_output_vars(minus(V,V1,Out),[V,V1],[Out]). input_output_vars(le(V,V1,Out),[V,V1],[Out]). input_output_vars(quot(V,V1,Out),[V,V1],[Out]). input_output_vars(fun(V,V1,V11,Out),[V,V1,V11],[Out]).

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

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

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

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

### Specialization of cost equations minus/3
* CE 10 is refined into CE [19]
* CE 8 is refined into CE [20]
* CE 9 is refined into CE [21]


### Cost equations --> "Loop" of minus/3
* CEs [21] --> Loop 12
* CEs [19] --> Loop 13
* CEs [20] --> Loop 14

### Ranking functions of CR minus(V,V1,Out)
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR minus(V,V1,Out)
* Partial RF of phase [12]:
- RF of loop [12:1]:
V
V1


### Specialization of cost equations le/3
* CE 18 is refined into CE [22]
* CE 16 is refined into CE [23]
* CE 15 is refined into CE [24]
* CE 17 is refined into CE [25]


### Cost equations --> "Loop" of le/3
* CEs [25] --> Loop 15
* CEs [22] --> Loop 16
* CEs [23] --> Loop 17
* CEs [24] --> 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 quot/3
* CE 11 is refined into CE [26,27,28,29]
* CE 12 is refined into CE [30,31]
* CE 14 is refined into CE [32]
* CE 13 is refined into CE [33,34]


### Cost equations --> "Loop" of quot/3
* CEs [34] --> Loop 19
* CEs [33] --> Loop 20
* CEs [26,27,28,29,30,31,32] --> Loop 21

### Ranking functions of CR quot(V,V1,Out)
* RF of phase [19]: [V,V-V1+1]

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


### Specialization of cost equations start/3
* CE 4 is refined into CE [35,36,37,38]
* CE 2 is refined into CE [39]
* CE 3 is refined into CE [40]
* CE 5 is refined into CE [41,42,43]
* CE 6 is refined into CE [44,45,46,47,48]
* CE 7 is refined into CE [49,50]


### Cost equations --> "Loop" of start/3
* CEs [41,45] --> Loop 22
* CEs [35,36,37,38] --> Loop 23
* CEs [40] --> Loop 24
* CEs [39,42,43,44,46,47,48,49,50] --> Loop 25

### Ranking functions of CR start(V,V1,V11)

#### Partial ranking functions of CR start(V,V1,V11)


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

#### Cost of chains of minus(V,V1,Out):
* Chain [[12],14]: 1*it(12)+1
Such that:it(12) =< V1

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

* Chain [[12],13]: 1*it(12)+0
Such that:it(12) =< V1

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

* Chain [14]: 1
with precondition: [V1=0,V=Out,V>=0]

* Chain [13]: 0
with precondition: [Out=0,V>=0,V1>=0]


#### 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 quot(V,V1,Out):
* Chain [[19],21]: 9*it(19)+1*s(5)+3
Such that:s(5) =< V1
aux(5) =< V
it(19) =< aux(5)

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

* Chain [[19],20,21]: 4*it(19)+3*s(5)+2*s(11)+6
Such that:aux(3) =< V
aux(7) =< V1
aux(8) =< V-V1
it(19) =< aux(8)
s(5) =< aux(7)
it(19) =< aux(3)
s(12) =< aux(3)
s(12) =< aux(8)
s(11) =< s(12)

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

* Chain [21]: 3*s(3)+1*s(5)+3
Such that:s(5) =< V1
aux(1) =< V
s(3) =< aux(1)

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

* Chain [20,21]: 3*s(5)+6
Such that:aux(7) =< V1
s(5) =< aux(7)

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


#### Cost of chains of start(V,V1,V11):
* Chain [25]: 12*s(27)+13*s(31)+4*s(39)+2*s(41)+6
Such that:s(35) =< V-V1
aux(11) =< V
aux(12) =< V1
s(31) =< aux(11)
s(27) =< aux(12)
s(39) =< s(35)
s(39) =< aux(11)
s(40) =< aux(11)
s(40) =< s(35)
s(41) =< s(40)

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

* Chain [24]: 1
with precondition: [V=1,V1>=0,V11>=0]

* Chain [23]: 3*s(45)+12*s(46)+12*s(53)+4*s(59)+2*s(61)+8
Such that:s(44) =< V1
s(55) =< V1-2*V11
aux(16) =< V1-V11
aux(17) =< V11
s(45) =< s(44)
s(46) =< aux(17)
s(59) =< s(55)
s(59) =< aux(16)
s(60) =< aux(16)
s(60) =< s(55)
s(61) =< s(60)
s(53) =< aux(16)

with precondition: [V=2,V1>=0,V11>=0]

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


Closed-form bounds of start(V,V1,V11):
-------------------------------------
* Chain [25] with precondition: [V>=0,V1>=0]
- Upper bound: 15*V+12*V1+6+nat(V-V1)*4
- Complexity: n
* Chain [24] with precondition: [V=1,V1>=0,V11>=0]
- Upper bound: 1
- Complexity: constant
* Chain [23] with precondition: [V=2,V1>=0,V11>=0]
- Upper bound: 3*V1+12*V11+8+nat(V1-V11)*14+nat(V1-2*V11)*4
- Complexity: n
* Chain [22] with precondition: [V1=0,V>=0]
- Upper bound: 1
- Complexity: constant

### Maximum cost of start(V,V1,V11): 3*V1+5+max([15*V+9*V1+nat(V-V1)*4,nat(V11)*12+2+nat(V1-V11)*14+nat(V1-2*V11)*4])+1
Asymptotic class: n
* Total analysis performed in 302 ms.

(10) BOUNDS(1, n^1)