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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 199 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 260 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


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)
   int(x, y) -> if(le(x, y), x, y)
   if(true, x, y) -> cons(x, int(s(x), y))
   if(false, x, y) -> nil

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
int([], y)
int(x, [])
if(true, [], y)


The defined contexts are:
if([], x1, x2)


[] just represents basic- or constructor-terms in the following defined contexts:
if([], x1, x2)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   int(x, y) -> if(le(x, y), x, y)
   if(true, x, y) -> cons(x, int(s(x), y))
   if(false, x, y) -> nil

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

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   int(x, y) -> if(le(x, y), x, y) [1]
   if(true, x, y) -> cons(x, int(s(x), y)) [1]
   if(false, x, y) -> nil [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
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]
   int(x, y) -> if(le(x, y), x, y) [1]
   if(true, x, y) -> cons(x, int(s(x), y)) [1]
   if(false, x, y) -> nil [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
int :: 0:s -> 0:s -> cons:nil
if :: true:false -> 0:s -> 0:s -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil

Rewrite Strategy: INNERMOST
----------------------------------------

(7) 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:
none

And the following fresh constants: none

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

(8)
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]
   int(x, y) -> if(le(x, y), x, y) [1]
   if(true, x, y) -> cons(x, int(s(x), y)) [1]
   if(false, x, y) -> nil [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
int :: 0:s -> 0:s -> cons:nil
if :: true:false -> 0:s -> 0:s -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil

Rewrite Strategy: INNERMOST
----------------------------------------

(9) 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 => 1
   false => 0
   nil => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + x + int(1 + x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   int(z, z') -{ 1 }-> if(le(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0

Only complete derivations are relevant for the runtime complexity.

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

(11) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V10),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V10),0,[int(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V10),0,[if(V1, V, V10, Out)],[V1 >= 0,V >= 0,V10 >= 0]).
eq(le(V1, V, Out),1,[],[Out = 1,V2 >= 0,V1 = 0,V = V2]).
eq(le(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 1 + V3,V = 0]).
eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(int(V1, V, Out),1,[le(V7, V6, Ret0),if(Ret0, V7, V6, Ret1)],[Out = Ret1,V7 >= 0,V6 >= 0,V1 = V7,V = V6]).
eq(if(V1, V, V10, Out),1,[int(1 + V8, V9, Ret11)],[Out = 1 + Ret11 + V8,V = V8,V10 = V9,V1 = 1,V8 >= 0,V9 >= 0]).
eq(if(V1, V, V10, Out),1,[],[Out = 0,V = V12,V10 = V11,V12 >= 0,V11 >= 0,V1 = 0]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(int(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V10,Out),[V1,V,V10],[Out]).


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

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

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

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

### Specialization of cost equations le/3 
* CE 9 is refined into CE [10] 
* CE 8 is refined into CE [11] 
* CE 7 is refined into CE [12] 


### Cost equations --> "Loop" of le/3 
* CEs [11] --> Loop 8 
* CEs [12] --> Loop 9 
* CEs [10] --> Loop 10 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR le(V1,V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V
    V1


### Specialization of cost equations int/3 
* CE 6 is refined into CE [13,14] 
* CE 5 is refined into CE [15,16] 


### Cost equations --> "Loop" of int/3 
* CEs [16] --> Loop 11 
* CEs [15] --> Loop 12 
* CEs [14] --> Loop 13 
* CEs [13] --> Loop 14 

### Ranking functions of CR int(V1,V,Out) 
* RF of phase [13]: [-V1+V+1]

#### Partial ranking functions of CR int(V1,V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    -V1+V+1


### Specialization of cost equations start/3 
* CE 2 is refined into CE [17,18,19] 
* CE 1 is refined into CE [20] 
* CE 3 is refined into CE [21,22,23,24] 
* CE 4 is refined into CE [25,26,27,28,29] 


### Cost equations --> "Loop" of start/3 
* CEs [23,28] --> Loop 15 
* CEs [22,27] --> Loop 16 
* CEs [18,24,29] --> Loop 17 
* CEs [19] --> Loop 18 
* CEs [17] --> Loop 19 
* CEs [20,21,25,26] --> Loop 20 

### Ranking functions of CR start(V1,V,V10) 

#### Partial ranking functions of CR start(V1,V,V10) 


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

#### Cost of chains of le(V1,V,Out):
* Chain [[10],9]: 1*it(10)+1
  Such that:it(10) =< V1

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

* Chain [[10],8]: 1*it(10)+1
  Such that:it(10) =< V

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

* Chain [9]: 1
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [8]: 1
  with precondition: [V=0,Out=0,V1>=1] 


#### Cost of chains of int(V1,V,Out):
* Chain [[13],11]: 3*it(13)+1*s(1)+1*s(4)+3
  Such that:it(13) =< -V1+V+1
s(1) =< V
aux(1) =< V+1
s(4) =< it(13)*aux(1)

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

* Chain [14,[13],11]: 4*it(13)+1*s(4)+6
  Such that:aux(1) =< V+1
aux(2) =< V
it(13) =< aux(2)
s(4) =< it(13)*aux(1)

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

* Chain [14,12]: 6
  with precondition: [V1=0,V=0,Out=1] 

* Chain [12]: 3
  with precondition: [V=0,Out=0,V1>=1] 

* Chain [11]: 1*s(1)+3
  Such that:s(1) =< V

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


#### Cost of chains of start(V1,V,V10):
* Chain [20]: 4*s(7)+1*s(8)+6
  Such that:s(6) =< V
s(5) =< V+1
s(7) =< s(6)
s(8) =< s(7)*s(5)

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

* Chain [19]: 4
  with precondition: [V1=1,V10=0,V>=0] 

* Chain [18]: 3*s(9)+1*s(10)+1*s(12)+4
  Such that:s(9) =< -V+V10
s(10) =< V10
s(11) =< V10+1
s(12) =< s(9)*s(11)

  with precondition: [V1=1,V>=0,V10>=V+1] 

* Chain [17]: 1*s(13)+1*s(14)+3*s(15)+1*s(16)+1*s(18)+4
  Such that:s(15) =< -V1+V+1
s(14) =< V1
s(16) =< V
s(17) =< V+1
s(13) =< V10
s(18) =< s(15)*s(17)

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

* Chain [16]: 3
  with precondition: [V=0,V1>=1] 

* Chain [15]: 2*s(19)+3
  Such that:aux(3) =< V
s(19) =< aux(3)

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


Closed-form bounds of start(V1,V,V10): 
-------------------------------------
* Chain [20] with precondition: [V1=0,V>=0] 
    - Upper bound: 4*V+6+(V+1)*V 
    - Complexity: n^2 
* Chain [19] with precondition: [V1=1,V10=0,V>=0] 
    - Upper bound: 4 
    - Complexity: constant 
* Chain [18] with precondition: [V1=1,V>=0,V10>=V+1] 
    - Upper bound: V10+4+(-V+V10)*(V10+1)+(-3*V+3*V10) 
    - Complexity: n^2 
* Chain [17] with precondition: [V1>=1,V>=V1] 
    - Upper bound: V1+V+4+nat(V10)+(-V1+V+1)*(V+1)+(-3*V1+3*V+3) 
    - Complexity: n^2 
* Chain [16] with precondition: [V=0,V1>=1] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [15] with precondition: [V>=1,V1>=V+1] 
    - Upper bound: 2*V+3 
    - Complexity: n 

### Maximum cost of start(V1,V,V10): max([max([1,nat(V10)+1+nat(-V+V10)*nat(V10+1)+nat(-V+V10)*3]),max([2*V+3+(V+1)*V+V,V1+1+nat(V10)+(V+1)*nat(-V1+V+1)+nat(-V1+V+1)*3])+V])+3 
Asymptotic class: n^2 
* Total analysis performed in 170 ms.


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

(12)
BOUNDS(1, n^2)

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

(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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)
   int(x, y) -> if(le(x, y), x, y)
   if(true, x, y) -> cons(x, int(s(x), y))
   if(false, x, y) -> nil

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

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/0

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

(16)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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)
   int(x, y) -> if(le(x, y), x, y)
   if(true, x, y) -> cons(int(s(x), y))
   if(false, x, y) -> nil

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

(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(18)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
int(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> cons(int(s(x), y))
if(false, x, y) -> nil

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: cons:nil -> cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, int

They will be analysed ascendingly in the following order:
le < int

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

(20)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
int(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> cons(int(s(x), y))
if(false, x, y) -> nil

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: cons:nil -> cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
le, int

They will be analysed ascendingly in the following order:
le < int

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
int(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> cons(int(s(x), y))
if(false, x, y) -> nil

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: cons:nil -> cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
le, int

They will be analysed ascendingly in the following order:
le < int

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
int(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> cons(int(s(x), y))
if(false, x, y) -> nil

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: cons:nil -> cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
int