/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^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 1 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 288 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 272 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) 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:

   f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

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

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

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


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


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

   f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

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:

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

   f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [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
s :: s:0 -> s:0
0 :: s:0
false :: true:false

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:

   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_f

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

(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:

   f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [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
s :: s:0 -> s:0
0 :: s:0
false :: true:false
null_f :: null_f

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:

   true => 1
   0 => 0
   false => 0
   null_f => 0

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

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

   f(z, z', z'') -{ 1 }-> f(gt(x, y), 1 + x, 1 + (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

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


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

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

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

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

### Specialization of cost equations gt/3 
* CE 7 is refined into CE [8] 
* CE 6 is refined into CE [9] 
* CE 5 is refined into CE [10] 


### Cost equations --> "Loop" of gt/3 
* CEs [9] --> Loop 7 
* CEs [10] --> Loop 8 
* CEs [8] --> Loop 9 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [9]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V
    V1


### Specialization of cost equations f/4 
* CE 4 is refined into CE [11] 
* CE 3 is refined into CE [12,13,14,15] 


### Cost equations --> "Loop" of f/4 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [13] --> Loop 12 
* CEs [12] --> Loop 13 
* CEs [11] --> Loop 14 

### Ranking functions of CR f(V1,V,V2,Out) 
* RF of phase [10]: [V-V2]

#### Partial ranking functions of CR f(V1,V,V2,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V-V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [16,17,18,19] 
* CE 2 is refined into CE [20,21,22,23] 


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

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

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


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

#### Cost of chains of gt(V1,V,Out):
* Chain [[9],8]: 1*it(9)+1
  Such that:it(9) =< V1

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

* Chain [[9],7]: 1*it(9)+1
  Such that:it(9) =< V

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

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

* Chain [7]: 1
  with precondition: [V=0,Out=1,V1>=1] 


#### Cost of chains of f(V1,V,V2,Out):
* Chain [[10],14]: 2*it(10)+1*s(3)+0
  Such that:it(10) =< V-V2
aux(1) =< 2*V-V2
s(3) =< it(10)*aux(1)

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

* Chain [[10],11,14]: 2*it(10)+1*s(3)+1*s(4)+2
  Such that:it(10) =< V-V2
aux(1) =< 2*V-V2
s(4) =< 2*V-V2+1
s(3) =< it(10)*aux(1)

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

* Chain [14]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [13,14]: 2
  with precondition: [V1=1,V=0,Out=0,V2>=0] 

* Chain [12,[10],14]: 2*it(10)+1*s(3)+2
  Such that:it(10) =< V
aux(1) =< 2*V
s(3) =< it(10)*aux(1)

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

* Chain [12,[10],11,14]: 2*it(10)+1*s(3)+1*s(4)+4
  Such that:it(10) =< V
aux(1) =< 2*V
s(4) =< 2*V+1
s(3) =< it(10)*aux(1)

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

* Chain [12,14]: 2
  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [12,11,14]: 1*s(4)+4
  Such that:s(4) =< 3

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

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

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


#### Cost of chains of start(V1,V,V2):
* Chain [20]: 1
  with precondition: [V1=0,V>=0] 

* Chain [19]: 1*s(20)+1*s(21)+4*s(24)+2*s(25)+4
  Such that:s(20) =< 3
s(22) =< V
s(23) =< 2*V
s(21) =< 2*V+1
s(24) =< s(22)
s(25) =< s(24)*s(23)

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

* Chain [18]: 1*s(26)+1*s(27)+2
  Such that:s(27) =< V1
s(26) =< V+1

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

* Chain [17]: 1*s(28)+4*s(31)+2*s(32)+2
  Such that:s(29) =< V-V2
s(30) =< 2*V-V2
s(28) =< 2*V-V2+1
s(31) =< s(29)
s(32) =< s(31)*s(30)

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

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

* Chain [15]: 1*s(33)+1
  Such that:s(33) =< V

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


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [20] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [19] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 4*V+7+2*V*(2*V)+(2*V+1) 
    - Complexity: n^2 
* Chain [18] with precondition: [V1>=1,V>=V1] 
    - Upper bound: V1+V+3 
    - Complexity: n 
* Chain [17] with precondition: [V1=1,V2>=1,V>=V2+1] 
    - Upper bound: 6*V-5*V2+3+(2*V-2*V2)*(2*V-V2) 
    - Complexity: n^2 
* Chain [16] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [15] with precondition: [V>=1,V1>=V+1] 
    - Upper bound: V+1 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([max([V1+V+2,nat(2*V-V2+1)+1+nat(V-V2)*4+nat(V-V2)*2*nat(2*V-V2)]),3*V+6+2*V*(2*V)+(2*V+1)+V])+1 
Asymptotic class: n^2 
* Total analysis performed in 206 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:

   f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(u, v)

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

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

(16)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, gt

They will be analysed ascendingly in the following order:
gt < f

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

(18)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f

They will be analysed ascendingly in the following order:
gt < f

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
false

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f

They will be analysed ascendingly in the following order:
gt < f

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
f