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:

f(true, x, y) → f(gt(x, y), trunc(x), s(y))
trunc(0) → 0
trunc(s(0)) → 0
trunc(s(s(x))) → s(s(trunc(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

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:

f(true, x, y) → f(gt(x, y), trunc(x), s(y)) [1]
trunc(0) → 0 [1]
trunc(s(0)) → 0 [1]
trunc(s(s(x))) → s(s(trunc(x))) [1]
gt(0, v) → false [1]
gt(s(u), 0) → true [1]
gt(s(u), s(v)) → gt(u, v) [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:

f(true, x, y) → f(gt(x, y), trunc(x), s(y)) [1]
trunc(0) → 0 [1]
trunc(s(0)) → 0 [1]
trunc(s(s(x))) → s(s(trunc(x))) [1]
gt(0, v) → false [1]
gt(s(u), 0) → true [1]
gt(s(u), s(v)) → gt(u, v) [1]

The TRS has the following type information:

f :: true:false → s:0 → s:0 → f true :: true:false gt :: s:0 → s:0 → true:false trunc :: s:0 → s:0 s :: s:0 → s:0 0 :: s:0 false :: true:false

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:

f(v0, v1, v2) → null_f [0]

And the following fresh constants:

null_f

(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: f(true, x, y) → f(gt(x, y), trunc(x), s(y)) [1] trunc(0) → 0 [1] trunc(s(0)) → 0 [1] trunc(s(s(x))) → s(s(trunc(x))) [1] gt(0, v) → false [1] gt(s(u), 0) → true [1] gt(s(u), s(v)) → gt(u, v) [1] f(v0, v1, v2) → null_f [0] The TRS has the following type information: f :: true:false → s:0 → s:0 → null_f true :: true:false gt :: s:0 → s:0 → true:false trunc :: s:0 → s:0 s :: s:0 → s:0 0 :: s:0 false :: true:false null_f :: null_f 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
0 => 0
false => 0
null_f => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z', z'') -{ 1 }→ f(gt(x, y), trunc(x), 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
f(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
gt(z, z') -{ 1 }→ gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
gt(z, z') -{ 1 }→ 1 :|: z = 1 + u, z' = 0, u >= 0
gt(z, z') -{ 1 }→ 0 :|: v >= 0, z' = v, z = 0
trunc(z) -{ 1 }→ 0 :|: z = 0
trunc(z) -{ 1 }→ 0 :|: z = 1 + 0
trunc(z) -{ 1 }→ 1 + (1 + trunc(x)) :|: x >= 0, z = 1 + (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, V1, V2),0,[f(V, V1, V2, Out)],[V >= 0,V1 >= 0,V2 >= 0]). eq(start(V, V1, V2),0,[trunc(V, Out)],[V >= 0]). eq(start(V, V1, V2),0,[gt(V, V1, Out)],[V >= 0,V1 >= 0]). eq(f(V, V1, V2, Out),1,[gt(V3, V4, Ret0),trunc(V3, Ret1),f(Ret0, Ret1, 1 + V4, Ret)],[Out = Ret,V1 = V3,V2 = V4,V = 1,V3 >= 0,V4 >= 0]). eq(trunc(V, Out),1,[],[Out = 0,V = 0]). eq(trunc(V, Out),1,[],[Out = 0,V = 1]). eq(trunc(V, Out),1,[trunc(V5, Ret11)],[Out = 2 + Ret11,V5 >= 0,V = 2 + V5]). eq(gt(V, V1, Out),1,[],[Out = 0,V6 >= 0,V1 = V6,V = 0]). eq(gt(V, V1, Out),1,[],[Out = 1,V = 1 + V7,V1 = 0,V7 >= 0]). eq(gt(V, V1, Out),1,[gt(V8, V9, Ret2)],[Out = Ret2,V9 >= 0,V1 = 1 + V9,V = 1 + V8,V8 >= 0]). eq(f(V, V1, V2, Out),0,[],[Out = 0,V10 >= 0,V2 = V11,V12 >= 0,V = V10,V1 = V12,V11 >= 0]). input_output_vars(f(V,V1,V2,Out),[V,V1,V2],[Out]). input_output_vars(trunc(V,Out),[V],[Out]). input_output_vars(gt(V,V1,Out),[V,V1],[Out]).

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

#### Computed strongly connected components
0. recursive : [gt/3]
1. recursive : [trunc/2]
2. recursive : [f/4]
3. non_recursive : [start/3]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into gt/3
1. SCC is partially evaluated into trunc/2
2. SCC is partially evaluated into f/4
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations gt/3
* CE 12 is refined into CE [13]
* CE 11 is refined into CE [14]
* CE 10 is refined into CE [15]


### Cost equations --> "Loop" of gt/3
* CEs [14] --> Loop 10
* CEs [15] --> Loop 11
* CEs [13] --> Loop 12

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

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


### Specialization of cost equations trunc/2
* CE 9 is refined into CE [16]
* CE 8 is refined into CE [17]
* CE 7 is refined into CE [18]


### Cost equations --> "Loop" of trunc/2
* CEs [17] --> Loop 13
* CEs [18] --> Loop 14
* CEs [16] --> Loop 15

### Ranking functions of CR trunc(V,Out)
* RF of phase [15]: [V-1]

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


### Specialization of cost equations f/4
* CE 6 is refined into CE [19]
* CE 5 is refined into CE [20,21,22,23,24,25,26,27,28]


### Cost equations --> "Loop" of f/4
* CEs [27] --> Loop 16
* CEs [28] --> Loop 17
* CEs [26] --> Loop 18
* CEs [25] --> Loop 19
* CEs [23] --> Loop 20
* CEs [22] --> Loop 21
* CEs [24] --> Loop 22
* CEs [21] --> Loop 23
* CEs [20] --> Loop 24
* CEs [19] --> Loop 25

### Ranking functions of CR f(V,V1,V2,Out)
* RF of phase [16,17]: [V1-V2]

#### Partial ranking functions of CR f(V,V1,V2,Out)
* Partial RF of phase [16,17]:
- RF of loop [16:1]:
V1-V2
- RF of loop [17:1]:
V1-2
V1/2-V2/2


### Specialization of cost equations start/3
* CE 2 is refined into CE [29,30,31,32,33,34]
* CE 3 is refined into CE [35,36,37,38]
* CE 4 is refined into CE [39,40,41,42]


### Cost equations --> "Loop" of start/3
* CEs [37,38,42] --> Loop 26
* CEs [41] --> Loop 27
* CEs [29] --> Loop 28
* CEs [40] --> Loop 29
* CEs [30,31,32,33,34,36] --> Loop 30
* CEs [35,39] --> Loop 31

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

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


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

#### Cost of chains of gt(V,V1,Out):
* Chain [[12],11]: 1*it(12)+1
Such that:it(12) =< V

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

* Chain [[12],10]: 1*it(12)+1
Such that:it(12) =< V1

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

* Chain [11]: 1
with precondition: [V=0,Out=0,V1>=0]

* Chain [10]: 1
with precondition: [V1=0,Out=1,V>=1]


#### Cost of chains of trunc(V,Out):
* Chain [[15],14]: 1*it(15)+1
Such that:it(15) =< Out

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

* Chain [[15],13]: 1*it(15)+1
Such that:it(15) =< Out

with precondition: [V=Out+1,V>=3]

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

* Chain [13]: 1
with precondition: [V=1,Out=0]


#### Cost of chains of f(V,V1,V2,Out):
* Chain [[16,17],25]: 3*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+0
Such that:aux(5) =< V1
aux(6) =< V1-V2
aux(7) =< V1-V2+1
it(17) =< V1/2-V2/2
it(17) =< aux(5)
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(7)
it(17) =< aux(7)
aux(4) =< aux(5)
aux(3) =< aux(5)-1
s(9) =< it(16)*aux(5)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

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

* Chain [[16,17],19,25]: 3*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+2*s(13)+3
Such that:aux(6) =< V1-V2
aux(9) =< V1
aux(10) =< 2*V1-V2
it(17) =< aux(9)
s(13) =< aux(10)
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(10)
it(17) =< aux(10)
aux(4) =< aux(9)
aux(3) =< aux(9)-1
s(9) =< it(16)*aux(9)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

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

* Chain [[16,17],18,25]: 3*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+2*s(15)+3
Such that:aux(6) =< V1-V2
aux(12) =< V1
aux(13) =< 2*V1-V2
it(17) =< aux(12)
s(15) =< aux(13)
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(13)
it(17) =< aux(13)
aux(4) =< aux(12)
aux(3) =< aux(12)-1
s(9) =< it(16)*aux(12)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

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

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

* Chain [24,25]: 3
with precondition: [V=1,V1=0,Out=0,V2>=0]

* Chain [23,25]: 3
with precondition: [V=1,V1=1,V2=0,Out=0]

* Chain [23,24,25]: 6
with precondition: [V=1,V1=1,V2=0,Out=0]

* Chain [22,25]: 1*s(17)+3
Such that:s(17) =< 1

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

* Chain [21,[16,17],25]: 4*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+3
Such that:it(17) =< V1/2
aux(14) =< V1
it(16) =< aux(14)
it(17) =< aux(14)
aux(4) =< aux(14)
aux(3) =< aux(14)-1
s(9) =< it(16)*aux(14)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

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

* Chain [21,[16,17],19,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(13)+1*s(18)+6
Such that:aux(10) =< 2*V1
aux(15) =< V1
s(18) =< aux(15)
it(16) =< aux(15)
s(13) =< aux(10)
it(16) =< aux(10)
aux(4) =< aux(15)
aux(3) =< aux(15)-1
s(9) =< it(16)*aux(15)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

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

* Chain [21,[16,17],18,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(15)+1*s(18)+6
Such that:aux(13) =< 2*V1
aux(16) =< V1
s(18) =< aux(16)
it(16) =< aux(16)
s(15) =< aux(13)
it(16) =< aux(13)
aux(4) =< aux(16)
aux(3) =< aux(16)-1
s(9) =< it(16)*aux(16)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

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

* Chain [21,25]: 1*s(18)+3
Such that:s(18) =< V1

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

* Chain [20,[16,17],25]: 4*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+3
Such that:it(17) =< V1/2
aux(17) =< V1
it(16) =< aux(17)
it(17) =< aux(17)
aux(4) =< aux(17)
aux(3) =< aux(17)-1
s(9) =< it(16)*aux(17)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

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

* Chain [20,[16,17],19,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(13)+1*s(19)+6
Such that:aux(10) =< 2*V1
aux(18) =< V1
s(19) =< aux(18)
it(16) =< aux(18)
s(13) =< aux(10)
it(16) =< aux(10)
aux(4) =< aux(18)
aux(3) =< aux(18)-1
s(9) =< it(16)*aux(18)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

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

* Chain [20,[16,17],18,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(15)+1*s(19)+6
Such that:aux(13) =< 2*V1
aux(19) =< V1
s(19) =< aux(19)
it(16) =< aux(19)
s(15) =< aux(13)
it(16) =< aux(13)
aux(4) =< aux(19)
aux(3) =< aux(19)-1
s(9) =< it(16)*aux(19)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

with precondition: [V=1,V2=0,Out=0,V1>=4]

* Chain [20,25]: 1*s(19)+3
Such that:s(19) =< V1

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

* Chain [19,25]: 2*s(13)+3
Such that:aux(8) =< V1
s(13) =< aux(8)

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

* Chain [18,25]: 2*s(15)+3
Such that:aux(11) =< V1
s(15) =< aux(11)

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


#### Cost of chains of start(V,V1,V2):
* Chain [31]: 1
with precondition: [V=0]

* Chain [30]: 1*s(114)+18*s(118)+6*s(119)+24*s(120)+8*s(121)+8*s(124)+4*s(125)+4*s(126)+4*s(127)+2*s(128)+2*s(129)+3*s(133)+3*s(137)+2*s(140)+1*s(141)+1*s(142)+6*s(143)+4*s(144)+6*s(145)+4*s(146)+2*s(147)+2*s(148)+6
Such that:s(114) =< 1
s(135) =< V1-V2
s(132) =< V1-V2+1
s(116) =< 2*V1
s(136) =< 2*V1-V2
s(117) =< V1/2
s(133) =< V1/2-V2/2
aux(27) =< V1
s(118) =< aux(27)
s(119) =< s(117)
s(120) =< aux(27)
s(121) =< s(116)
s(120) =< s(116)
s(122) =< aux(27)
s(123) =< aux(27)-1
s(124) =< s(120)*aux(27)
s(125) =< s(120)*s(122)
s(126) =< s(120)*s(123)
s(119) =< aux(27)
s(127) =< s(118)*aux(27)
s(128) =< s(119)*s(122)
s(129) =< s(119)*s(123)
s(133) =< aux(27)
s(137) =< s(135)
s(133) =< s(135)
s(137) =< s(132)
s(133) =< s(132)
s(140) =< s(137)*aux(27)
s(141) =< s(133)*s(122)
s(142) =< s(133)*s(123)
s(143) =< aux(27)
s(144) =< s(136)
s(145) =< s(135)
s(143) =< s(135)
s(145) =< s(136)
s(143) =< s(136)
s(146) =< s(145)*aux(27)
s(147) =< s(143)*s(122)
s(148) =< s(143)*s(123)

with precondition: [V=1]

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

* Chain [28]: 3
with precondition: [V>=0,V1>=0,V2>=0]

* Chain [27]: 1*s(149)+1
Such that:s(149) =< V

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

* Chain [26]: 2*s(150)+1*s(152)+1
Such that:s(152) =< V1
aux(28) =< V
s(150) =< aux(28)

with precondition: [V>=2]


Closed-form bounds of start(V,V1,V2):
-------------------------------------
* Chain [31] with precondition: [V=0]
- Upper bound: 1
- Complexity: constant
* Chain [30] with precondition: [V=1]
- Upper bound: nat(V1)*48+7+nat(V1)*18*nat(V1)+nat(V1)*6*nat(V1-V2)+nat(V1)*2*nat(V1/2)+nat(nat(V1)+ -1)*6*nat(V1)+nat(V1/2-V2/2)*nat(nat(V1)+ -1)+nat(nat(V1)+ -1)*2*nat(V1/2)+nat(2*V1)*8+nat(V1-V2)*9+nat(2*V1-V2)*4+nat(V1/2-V2/2)*3+nat(V1/2-V2/2)*nat(V1)+nat(V1/2)*6
- Complexity: n^2
* Chain [29] with precondition: [V1=0,V>=1]
- Upper bound: 1
- Complexity: constant
* Chain [28] with precondition: [V>=0,V1>=0,V2>=0]
- Upper bound: 3
- Complexity: constant
* Chain [27] with precondition: [V>=1,V1>=V]
- Upper bound: V+1
- Complexity: n
* Chain [26] with precondition: [V>=2]
- Upper bound: 2*V+1+nat(V1)
- Complexity: n

### Maximum cost of start(V,V1,V2): max([max([2,nat(V1)*48+6+nat(V1)*18*nat(V1)+nat(V1)*6*nat(V1-V2)+nat(V1)*2*nat(V1/2)+nat(nat(V1)+ -1)*6*nat(V1)+nat(V1/2-V2/2)*nat(nat(V1)+ -1)+nat(nat(V1)+ -1)*2*nat(V1/2)+nat(2*V1)*8+nat(V1-V2)*9+nat(2*V1-V2)*4+nat(V1/2-V2/2)*3+nat(V1/2-V2/2)*nat(V1)+nat(V1/2)*6]),nat(V1)+V+V])+1
Asymptotic class: n^2
* Total analysis performed in 528 ms.

(10) BOUNDS(1, n^2)