0 CpxTRS
↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTypedWeightedTrs
↳5 CompletionProof (UPPER BOUND(ID), 0 ms)
↳6 CpxTypedWeightedCompleteTrs
↳7 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳8 CpxRNTS
↳9 CompleteCoflocoProof (⇔, 523 ms)
↳10 BOUNDS(1, n^1)
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)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
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]
log(s(0)) → 0 [1]
log(s(s(x))) → s(log(s(quot(x, s(s(0)))))) [1]
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]
log(s(0)) → 0 [1]
log(s(s(x))) → s(log(s(quot(x, s(s(0)))))) [1]
| 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 log :: s:0 → s:0 |
pred(v0) → null_pred [0]
quot(v0, v1) → null_quot [0]
log(v0) → null_log [0]
minus(v0, v1) → null_minus [0]
null_pred, null_quot, null_log, null_minus
| Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules:
The TRS has the following type information:
Rewrite Strategy: INNERMOST |
0 => 0
null_pred => 0
null_quot => 0
null_log => 0
null_minus => 0
log(z) -{ 1 }→ 0 :|: z = 1 + 0
log(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
log(z) -{ 1 }→ 1 + log(1 + quot(x, 1 + (1 + 0))) :|: x >= 0, z = 1 + (1 + x)
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
| 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(start(V, V2),0,[log(V, Out)],[V >= 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(log(V, Out),1,[],[Out = 0,V = 1]). eq(log(V, Out),1,[quot(V9, 1 + (1 + 0), Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V9 >= 0,V = 2 + V9]). eq(pred(V, Out),0,[],[Out = 0,V10 >= 0,V = V10]). eq(quot(V, V2, Out),0,[],[Out = 0,V11 >= 0,V12 >= 0,V = V11,V2 = V12]). eq(log(V, Out),0,[],[Out = 0,V13 >= 0,V = V13]). eq(minus(V, V2, Out),0,[],[Out = 0,V14 >= 0,V15 >= 0,V = V14,V2 = V15]). 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]). input_output_vars(log(V,Out),[V],[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. recursive : [log/2]
4. 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 log/2
4. SCC is partially evaluated into start/2
Control-Flow Refinement of Cost Relations
=====================================
### Specialization of cost equations pred/2
* CE 6 is refined into CE [17]
* CE 7 is refined into CE [18]
### Cost equations --> "Loop" of pred/2
* CEs [17] --> Loop 11
* CEs [18] --> Loop 12
### Ranking functions of CR pred(V,Out)
#### Partial ranking functions of CR pred(V,Out)
### 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 [21,22]
### Cost equations --> "Loop" of minus/3
* CEs [22] --> Loop 13
* CEs [21] --> Loop 14
* CEs [19] --> Loop 15
* CEs [20] --> Loop 16
### Ranking functions of CR minus(V,V2,Out)
* RF of phase [13]: [V2]
* RF of phase [14]: [V2]
#### Partial ranking functions of CR minus(V,V2,Out)
* Partial RF of phase [13]:
- RF of loop [13:1]:
V2
* Partial RF of phase [14]:
- RF of loop [14:1]:
V2
### Specialization of cost equations quot/3
* CE 11 is refined into CE [23]
* CE 13 is refined into CE [24]
* CE 12 is refined into CE [25,26,27]
### Cost equations --> "Loop" of quot/3
* CEs [27] --> Loop 17
* CEs [26] --> Loop 18
* CEs [25] --> Loop 19
* CEs [23,24] --> Loop 20
### Ranking functions of CR quot(V,V2,Out)
* RF of phase [17]: [V-1,V-V2+1]
* RF of phase [19]: [V]
#### Partial ranking functions of CR quot(V,V2,Out)
* Partial RF of phase [17]:
- RF of loop [17:1]:
V-1
V-V2+1
* Partial RF of phase [19]:
- RF of loop [19:1]:
V
### Specialization of cost equations log/2
* CE 14 is refined into CE [28]
* CE 16 is refined into CE [29]
* CE 15 is refined into CE [30,31,32,33]
### Cost equations --> "Loop" of log/2
* CEs [33] --> Loop 21
* CEs [32] --> Loop 22
* CEs [31] --> Loop 23
* CEs [30] --> Loop 24
* CEs [28,29] --> Loop 25
### Ranking functions of CR log(V,Out)
* RF of phase [21,22]: [V-3,V/2-3/2]
#### Partial ranking functions of CR log(V,Out)
* Partial RF of phase [21,22]:
- RF of loop [21:1]:
V/2-2
- RF of loop [22:1]:
V-3
### Specialization of cost equations start/2
* CE 2 is refined into CE [34,35]
* CE 3 is refined into CE [36,37,38]
* CE 4 is refined into CE [39,40,41,42,43]
* CE 5 is refined into CE [44,45,46,47,48,49]
### Cost equations --> "Loop" of start/2
* CEs [39] --> Loop 26
* CEs [34,35,36,37,38,40,41,42,43,44,45,46,47,48,49] --> Loop 27
### 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 [12]: 0
with precondition: [Out=0,V>=0]
* Chain [11]: 1
with precondition: [V=Out+1,V>=1]
#### Cost of chains of minus(V,V2,Out):
* Chain [[14],[13],16]: 3*it(13)+1
Such that:aux(1) =< V2
it(13) =< aux(1)
with precondition: [Out=0,V>=1,V2>=2]
* Chain [[14],16]: 1*it(14)+1
Such that:it(14) =< V2
with precondition: [Out=0,V>=0,V2>=1]
* Chain [[14],15]: 1*it(14)+0
Such that:it(14) =< V2
with precondition: [Out=0,V>=0,V2>=1]
* Chain [[13],16]: 2*it(13)+1
Such that:it(13) =< V2
with precondition: [V=Out+V2,V2>=1,V>=V2]
* Chain [16]: 1
with precondition: [V2=0,V=Out,V>=0]
* Chain [15]: 0
with precondition: [Out=0,V>=0,V2>=0]
#### Cost of chains of quot(V,V2,Out):
* Chain [[19],20]: 2*it(19)+1
Such that:it(19) =< Out
with precondition: [V2=1,Out>=1,V>=Out]
* Chain [[19],18,20]: 2*it(19)+5*s(6)+3
Such that:s(5) =< 1
it(19) =< Out
s(6) =< s(5)
with precondition: [V2=1,Out>=2,V>=Out]
* Chain [[17],20]: 2*it(17)+2*s(9)+1
Such that:it(17) =< V-V2+1
aux(5) =< V
it(17) =< aux(5)
s(9) =< aux(5)
with precondition: [V2>=2,Out>=1,V+2>=2*Out+V2]
* Chain [[17],18,20]: 2*it(17)+5*s(6)+2*s(9)+3
Such that:it(17) =< V-V2+1
s(5) =< V2
aux(6) =< V
s(6) =< s(5)
it(17) =< aux(6)
s(9) =< aux(6)
with precondition: [V2>=2,Out>=2,V+3>=2*Out+V2]
* Chain [20]: 1
with precondition: [Out=0,V>=0,V2>=0]
* Chain [18,20]: 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 log(V,Out):
* Chain [[21,22],25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+1
Such that:s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(17) =< V
aux(18) =< V/2
aux(10) =< aux(17)
it(21) =< aux(17)
it(22) =< aux(17)
aux(10) =< aux(18)
it(21) =< aux(18)
it(22) =< aux(18)
it(22) =< aux(15)
s(31) =< aux(15)
it(22) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)
with precondition: [Out>=1,V>=3*Out+1]
* Chain [[21,22],24,25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+3
Such that:s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(19) =< V
aux(20) =< V/2
aux(10) =< aux(19)
it(21) =< aux(19)
it(22) =< aux(19)
aux(10) =< aux(20)
it(21) =< aux(20)
it(22) =< aux(20)
it(22) =< aux(15)
s(31) =< aux(15)
it(22) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)
with precondition: [Out>=2,V+2>=3*Out]
* Chain [[21,22],23,25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+5*s(35)+5
Such that:s(34) =< 2
s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(21) =< V
aux(22) =< V/2
s(35) =< s(34)
aux(10) =< aux(21)
it(21) =< aux(21)
it(22) =< aux(21)
aux(10) =< aux(22)
it(21) =< aux(22)
it(22) =< aux(22)
it(22) =< aux(15)
s(31) =< aux(15)
it(22) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)
with precondition: [Out>=2,V+3>=4*Out]
* Chain [[21,22],23,24,25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+5*s(35)+7
Such that:s(34) =< 2
s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(23) =< V
aux(24) =< V/2
s(35) =< s(34)
aux(10) =< aux(23)
it(21) =< aux(23)
it(22) =< aux(23)
aux(10) =< aux(24)
it(21) =< aux(24)
it(22) =< aux(24)
it(22) =< aux(15)
s(31) =< aux(15)
it(22) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)
with precondition: [Out>=3,V+7>=4*Out]
* Chain [25]: 1
with precondition: [Out=0,V>=0]
* Chain [24,25]: 3
with precondition: [Out=1,V>=2]
* Chain [23,25]: 5*s(35)+5
Such that:s(34) =< 2
s(35) =< s(34)
with precondition: [Out=1,V>=3]
* Chain [23,24,25]: 5*s(35)+7
Such that:s(34) =< 2
s(35) =< s(34)
with precondition: [Out=2,V>=3]
#### Cost of chains of start(V,V2):
* Chain [27]: 17*s(67)+4*s(71)+4*s(73)+15*s(80)+16*s(87)+8*s(88)+16*s(91)+16*s(92)+20*s(93)+15
Such that:aux(30) =< 2
aux(31) =< V
aux(32) =< V-V2+1
aux(33) =< 2*V
aux(34) =< V/2
aux(35) =< 5/2*V
aux(36) =< 5/2*V+27/2
aux(37) =< V2
s(71) =< aux(32)
s(67) =< aux(37)
s(80) =< aux(30)
s(86) =< aux(31)
s(87) =< aux(31)
s(88) =< aux(31)
s(86) =< aux(34)
s(87) =< aux(34)
s(88) =< aux(34)
s(88) =< aux(36)
s(89) =< aux(36)
s(88) =< aux(35)
s(89) =< aux(35)
s(90) =< s(86)*2
s(91) =< aux(33)
s(92) =< s(89)
s(93) =< s(90)
s(71) =< aux(31)
s(73) =< aux(31)
with precondition: [V>=0]
* Chain [26]: 4*s(126)+5*s(127)+3
Such that:s(124) =< 1
s(125) =< V
s(126) =< s(125)
s(127) =< s(124)
with precondition: [V2=1,V>=1]
Closed-form bounds of start(V,V2):
-------------------------------------
* Chain [27] with precondition: [V>=0]
- Upper bound: 68*V+45+nat(V2)*17+32*V+ (40*V+216)+nat(V-V2+1)*4
- Complexity: n
* Chain [26] with precondition: [V2=1,V>=1]
- Upper bound: 4*V+8
- Complexity: n
### Maximum cost of start(V,V2): 64*V+37+nat(V2)*17+32*V+ (40*V+216)+nat(V-V2+1)*4+ (4*V+8)
Asymptotic class: n
* Total analysis performed in 411 ms.