/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [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 [FINISHED, 350 ms]
    (10) BOUNDS(1, n^1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative 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))
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))
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))
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 (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V12),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[fun(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]).
eq(le(V1, V, Out),1,[],[Out = 2,V5 >= 0,V1 = 0,V = V5]).
eq(le(V1, V, Out),1,[],[Out = 1,V6 >= 0,V1 = 1 + V6,V = 0]).
eq(le(V1, V, Out),1,[le(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
eq(quot(V1, V, Out),1,[le(1 + V10, V9, Ret0),fun(Ret0, V9, 1 + V10, Ret2)],[Out = Ret2,V = 1 + V10,V9 >= 0,V10 >= 0,V1 = V9]).
eq(fun(V1, V, V12, Out),1,[minus(V13, V11, Ret10),quot(Ret10, V11, Ret11)],[Out = 1 + Ret11,V1 = 2,V = V13,V12 = V11,V13 >= 0,V11 >= 0]).
eq(fun(V1, V, V12, Out),1,[],[Out = 0,V = V15,V12 = V14,V1 = 1,V15 >= 0,V14 >= 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V1 = V17,V = V16]).
eq(quot(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
eq(le(V1, V, Out),0,[],[Out = 0,V21 >= 0,V20 >= 0,V1 = V21,V = V20]).
eq(fun(V1, V, V12, Out),0,[],[Out = 0,V22 >= 0,V12 = V24,V23 >= 0,V1 = V22,V = V23,V24 >= 0]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(quot(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V12,Out),[V1,V,V12],[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 9 is refined into CE [18] 
* CE 7 is refined into CE [19] 
* CE 8 is refined into CE [20] 


### Cost equations --> "Loop" of minus/3 
* CEs [20] --> Loop 12 
* CEs [18] --> Loop 13 
* CEs [19] --> Loop 14 

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

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


### Specialization of cost equations le/3 
* CE 17 is refined into CE [21] 
* CE 15 is refined into CE [22] 
* CE 14 is refined into CE [23] 
* CE 16 is refined into CE [24] 


### Cost equations --> "Loop" of le/3 
* CEs [24] --> Loop 15 
* CEs [21] --> Loop 16 
* CEs [22] --> Loop 17 
* CEs [23] --> Loop 18 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR le(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
    V1


### Specialization of cost equations quot/3 
* CE 10 is refined into CE [25,26,27,28] 
* CE 11 is refined into CE [29,30] 
* CE 13 is refined into CE [31] 
* CE 12 is refined into CE [32,33] 


### Cost equations --> "Loop" of quot/3 
* CEs [33] --> Loop 19 
* CEs [32] --> Loop 20 
* CEs [25,26,27,28,29,30,31] --> Loop 21 

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

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


### Specialization of cost equations start/3 
* CE 3 is refined into CE [34,35,36,37] 
* CE 1 is refined into CE [38] 
* CE 2 is refined into CE [39] 
* CE 4 is refined into CE [40,41,42] 
* CE 5 is refined into CE [43,44,45,46,47] 
* CE 6 is refined into CE [48,49] 


### Cost equations --> "Loop" of start/3 
* CEs [40,44] --> Loop 22 
* CEs [34,35,36,37] --> Loop 23 
* CEs [39] --> Loop 24 
* CEs [38,41,42,43,45,46,47,48,49] --> Loop 25 

### Ranking functions of CR start(V1,V,V12) 

#### Partial ranking functions of CR start(V1,V,V12) 


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

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

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

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

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

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

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


#### Cost of chains of le(V1,V,Out):
* Chain [[15],18]: 1*it(15)+1
  Such that:it(15) =< V1

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

* Chain [[15],17]: 1*it(15)+1
  Such that:it(15) =< V

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

* Chain [[15],16]: 1*it(15)+0
  Such that:it(15) =< V

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

* Chain [18]: 1
  with precondition: [V1=0,Out=2,V>=0] 

* Chain [17]: 1
  with precondition: [V=0,Out=1,V1>=1] 

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


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

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

* Chain [[19],20,21]: 4*it(19)+3*s(5)+2*s(11)+6
  Such that:aux(3) =< V1
aux(7) =< V
aux(8) =< V1-V
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: [V>=1,Out>=2,V1+2>=2*V+Out] 

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

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

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

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


#### Cost of chains of start(V1,V,V12):
* Chain [25]: 12*s(27)+13*s(31)+4*s(39)+2*s(41)+6
  Such that:s(35) =< V1-V
aux(11) =< V1
aux(12) =< V
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: [V1>=0,V>=0] 

* Chain [24]: 1
  with precondition: [V1=1,V>=0,V12>=0] 

* Chain [23]: 3*s(45)+12*s(46)+12*s(53)+4*s(59)+2*s(61)+8
  Such that:s(44) =< V
s(55) =< V-2*V12
aux(16) =< V-V12
aux(17) =< V12
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: [V1=2,V>=0,V12>=0] 

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


Closed-form bounds of start(V1,V,V12): 
-------------------------------------
* Chain [25] with precondition: [V1>=0,V>=0] 
    - Upper bound: 15*V1+12*V+6+nat(V1-V)*4 
    - Complexity: n 
* Chain [24] with precondition: [V1=1,V>=0,V12>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [23] with precondition: [V1=2,V>=0,V12>=0] 
    - Upper bound: 3*V+12*V12+8+nat(V-V12)*14+nat(V-2*V12)*4 
    - Complexity: n 
* Chain [22] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V,V12): 3*V+5+max([15*V1+9*V+nat(V1-V)*4,nat(V12)*12+2+nat(V-V12)*14+nat(V-2*V12)*4])+1 
Asymptotic class: n 
* Total analysis performed in 269 ms.


----------------------------------------

(10)
BOUNDS(1, n^1)

----------------------------------------

(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

minus(s(x), s(y)) ->^+ minus(x, y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(x), y / s(y)].

The result substitution is [ ].




----------------------------------------

(14)
Complex Obligation (BEST)

----------------------------------------

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

----------------------------------------

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^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

S is empty.
Rewrite Strategy: INNERMOST