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)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y))
if_mod(true, s(x), s(y)) → mod(minus(x, y), s(y))
if_mod(false, s(x), s(y)) → 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]
mod(0, y) → 0 [1]
mod(s(x), 0) → 0 [1]
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y)) [1]
if_mod(true, s(x), s(y)) → mod(minus(x, y), s(y)) [1]
if_mod(false, s(x), s(y)) → 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]
mod(0, y) → 0 [1]
mod(s(x), 0) → 0 [1]
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y)) [1]
if_mod(true, s(x), s(y)) → mod(minus(x, y), s(y)) [1]
if_mod(false, s(x), s(y)) → 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 mod :: 0:s → 0:s → 0:s if_mod :: 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_mod(v0, v1, v2) → null_if_mod [0]
le(v0, v1) → null_le [0]
mod(v0, v1) → null_mod [0]

And the following fresh constants:

null_minus, null_if_mod, null_le, null_mod

(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] mod(0, y) → 0 [1] mod(s(x), 0) → 0 [1] mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y)) [1] if_mod(true, s(x), s(y)) → mod(minus(x, y), s(y)) [1] if_mod(false, s(x), s(y)) → s(x) [1] minus(v0, v1) → null_minus [0] if_mod(v0, v1, v2) → null_if_mod [0] le(v0, v1) → null_le [0] mod(v0, v1) → null_mod [0] The TRS has the following type information: le :: 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod → true:false:null_le 0 :: 0:s:null_minus:null_if_mod:null_mod true :: true:false:null_le s :: 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod false :: true:false:null_le minus :: 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod mod :: 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod if_mod :: true:false:null_le → 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod → 0:s:null_minus:null_if_mod:null_mod null_minus :: 0:s:null_minus:null_if_mod:null_mod null_if_mod :: 0:s:null_minus:null_if_mod:null_mod null_le :: true:false:null_le null_mod :: 0:s:null_minus:null_if_mod:null_mod 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_mod => 0
null_le => 0
null_mod => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

if_mod(z, z', z'') -{ 1 }→ mod(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
if_mod(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
if_mod(z, z', z'') -{ 1 }→ 1 + x :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y
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
mod(z, z') -{ 1 }→ if_mod(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
mod(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y
mod(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
mod(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,[mod(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(mod(V, V1, Out),1,[],[Out = 0,V9 >= 0,V = 0,V1 = V9]). eq(mod(V, V1, Out),1,[],[Out = 0,V10 >= 0,V = 1 + V10,V1 = 0]). eq(mod(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),mod(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,[],[Out = 1 + V16,V1 = 1 + V16,V = 1,V16 >= 0,V17 >= 0,V13 = 1 + V17]). 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(mod(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(mod(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 : [minus/3]
1. recursive : [le/3]
2. recursive : [fun/4, (mod)/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 (mod)/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 [21]
* CE 8 is refined into CE [22]
* CE 9 is refined into CE [23]


### Cost equations --> "Loop" of minus/3
* CEs [23] --> Loop 14
* CEs [21] --> Loop 15
* CEs [22] --> Loop 16

### Ranking functions of CR minus(V,V1,Out)
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR minus(V,V1,Out)
* Partial RF of phase [14]:
- RF of loop [14:1]:
V
V1


### Specialization of cost equations le/3
* CE 20 is refined into CE [24]
* CE 18 is refined into CE [25]
* CE 17 is refined into CE [26]
* CE 19 is refined into CE [27]


### Cost equations --> "Loop" of le/3
* CEs [27] --> Loop 17
* CEs [24] --> Loop 18
* CEs [25] --> Loop 19
* CEs [26] --> Loop 20

### Ranking functions of CR le(V,V1,Out)
* RF of phase [17]: [V,V1]

#### Partial ranking functions of CR le(V,V1,Out)
* Partial RF of phase [17]:
- RF of loop [17:1]:
V
V1


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


### Cost equations --> "Loop" of (mod)/3
* CEs [41] --> Loop 21
* CEs [40] --> Loop 22
* CEs [38] --> Loop 23
* CEs [39] --> Loop 24
* CEs [29] --> Loop 25
* CEs [31] --> Loop 26
* CEs [30] --> Loop 27
* CEs [28] --> Loop 28
* CEs [32] --> Loop 29
* CEs [33,34,35,36,37] --> Loop 30

### Ranking functions of CR mod(V,V1,Out)
* RF of phase [21]: [V-1,V-V1+1]
* RF of phase [23]: [V]

#### Partial ranking functions of CR mod(V,V1,Out)
* Partial RF of phase [21]:
- RF of loop [21:1]:
V-1
V-V1+1
* Partial RF of phase [23]:
- RF of loop [23:1]:
V


### Specialization of cost equations start/3
* CE 4 is refined into CE [42,43,44,45,46,47,48,49]
* CE 2 is refined into CE [50]
* CE 3 is refined into CE [51]
* CE 5 is refined into CE [52,53,54,55,56]
* CE 6 is refined into CE [57,58,59]
* CE 7 is refined into CE [60,61,62,63,64,65,66]


### Cost equations --> "Loop" of start/3
* CEs [63] --> Loop 31
* CEs [53,57,62] --> Loop 32
* CEs [46] --> Loop 33
* CEs [42,43,44,45,47,48,49] --> Loop 34
* CEs [61] --> Loop 35
* CEs [51] --> Loop 36
* CEs [50,52,54,55,56,58,59,60,64,65,66] --> Loop 37

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

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


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

#### Cost of chains of minus(V,V1,Out):
* Chain [[14],16]: 1*it(14)+1
Such that:it(14) =< V1

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

* Chain [[14],15]: 1*it(14)+0
Such that:it(14) =< V1

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

* Chain [16]: 1
with precondition: [V1=0,V=Out,V>=0]

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


#### Cost of chains of le(V,V1,Out):
* Chain [[17],20]: 1*it(17)+1
Such that:it(17) =< V

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

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

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

* Chain [[17],18]: 1*it(17)+0
Such that:it(17) =< V1

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

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

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

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


#### Cost of chains of mod(V,V1,Out):
* Chain [[23],30]: 6*it(23)+1*s(5)+2
Such that:s(5) =< 1
aux(2) =< V
it(23) =< aux(2)

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

* Chain [[23],26]: 4*it(23)+2
Such that:it(23) =< V

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

* Chain [[23],24,30]: 4*it(23)+1*s(5)+5
Such that:s(5) =< 1
it(23) =< V

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

* Chain [[21],30]: 8*it(21)+1*s(5)+2
Such that:s(5) =< V1
aux(6) =< V
it(21) =< aux(6)

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

* Chain [[21],29]: 4*it(21)+2*s(11)+2
Such that:it(21) =< V-V1+1
aux(7) =< V
it(21) =< aux(7)
s(11) =< aux(7)

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

* Chain [[21],28]: 4*it(21)+2*s(11)+3
Such that:it(21) =< V-V1+1
aux(8) =< V
it(21) =< aux(8)
s(11) =< aux(8)

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

* Chain [[21],25]: 4*it(21)+2*s(11)+1*s(13)+3
Such that:aux(4) =< V
it(21) =< V-V1+1
aux(5) =< V-Out
s(13) =< Out
it(21) =< aux(4)
s(12) =< aux(4)
it(21) =< aux(5)
s(12) =< aux(5)
s(11) =< s(12)

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

* Chain [[21],22,30]: 4*it(21)+3*s(5)+2*s(11)+5
Such that:aux(4) =< V
aux(10) =< V1
aux(11) =< V-V1
it(21) =< aux(11)
s(5) =< aux(10)
it(21) =< aux(4)
s(12) =< aux(4)
s(12) =< aux(11)
s(11) =< s(12)

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

* Chain [30]: 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 [29]: 2
with precondition: [V=1,Out=0,V1>=2]

* Chain [28]: 3
with precondition: [V=1,Out=1,V1>=2]

* Chain [27]: 1
with precondition: [V1=0,Out=0,V>=1]

* Chain [26]: 2
with precondition: [V1=1,Out=0,V>=1]

* Chain [25]: 1*s(13)+3
Such that:s(13) =< V

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

* Chain [24,30]: 1*s(5)+5
Such that:s(5) =< 1

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

* Chain [22,30]: 3*s(5)+5
Such that:aux(10) =< V1
s(5) =< aux(10)

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


#### Cost of chains of start(V,V1,V13):
* Chain [37]: 12*s(41)+19*s(43)+12*s(47)+4*s(51)+2*s(53)+5
Such that:s(46) =< V-V1
aux(17) =< V
aux(18) =< V-V1+1
aux(19) =< V1
s(43) =< aux(17)
s(47) =< aux(18)
s(41) =< aux(19)
s(51) =< s(46)
s(51) =< aux(17)
s(52) =< aux(17)
s(52) =< s(46)
s(53) =< s(52)
s(47) =< aux(17)

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

* Chain [36]: 1
with precondition: [V=1,V1>=1,V13>=1]

* Chain [35]: 3
with precondition: [V=1,V1>=2]

* Chain [34]: 36*s(66)+11*s(69)+21*s(78)+12*s(90)+4*s(94)+2*s(96)+18*s(97)+7
Such that:s(89) =< V1-2*V13
aux(24) =< 1
aux(25) =< V1
aux(26) =< V1-2*V13+1
aux(27) =< V1-V13
aux(28) =< V13
s(90) =< aux(26)
s(97) =< aux(27)
s(78) =< aux(28)
s(66) =< aux(25)
s(69) =< aux(24)
s(94) =< s(89)
s(94) =< aux(27)
s(95) =< aux(27)
s(95) =< s(89)
s(96) =< s(95)
s(90) =< aux(27)

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

* Chain [33]: 1*s(111)+5
Such that:s(111) =< V13

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

* Chain [32]: 1
with precondition: [V1=0,V>=0]

* Chain [31]: 3*s(114)+14*s(115)+5
Such that:s(112) =< 1
s(113) =< V
s(114) =< s(112)
s(115) =< s(113)

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


Closed-form bounds of start(V,V1,V13):
-------------------------------------
* Chain [37] with precondition: [V>=0,V1>=0]
- Upper bound: 21*V+12*V1+5+nat(V-V1+1)*12+nat(V-V1)*4
- Complexity: n
* Chain [36] with precondition: [V=1,V1>=1,V13>=1]
- Upper bound: 1
- Complexity: constant
* Chain [35] with precondition: [V=1,V1>=2]
- Upper bound: 3
- Complexity: constant
* Chain [34] with precondition: [V=2,V1>=1,V13>=1]
- Upper bound: 36*V1+21*V13+18+nat(V1-2*V13+1)*12+nat(V1-V13)*20+nat(V1-2*V13)*4
- Complexity: n
* Chain [33] with precondition: [V=2,V1=V13+1,V1>=3]
- Upper bound: V13+5
- Complexity: n
* Chain [32] with precondition: [V1=0,V>=0]
- Upper bound: 1
- Complexity: constant
* Chain [31] with precondition: [V1=1,V>=1]
- Upper bound: 14*V+8
- Complexity: n

### Maximum cost of start(V,V1,V13): max([14*V+2+max([3,7*V+12*V1+nat(V-V1+1)*12+nat(V-V1)*4]),36*V1+13+nat(V13)*20+nat(V1-2*V13+1)*12+nat(V1-V13)*20+nat(V1-2*V13)*4+ (nat(V13)+2)])+3
Asymptotic class: n
* Total analysis performed in 446 ms.

(10) BOUNDS(1, n^1)