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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 5 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 273 ms]
    (10) BOUNDS(1, n^1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (24) typed CpxTrs


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

(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:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))

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:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y)) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   add(0, x) -> x [1]
   add(s(x), y) -> s(add(x, y)) [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:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y)) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   add(0, x) -> x [1]
   add(s(x), y) -> s(add(x, y)) [1]

The TRS has the following type information:
cond :: true:false -> 0:s -> 0:s -> cond
true :: true:false
gr :: 0:s -> 0:s -> true:false
add :: 0:s -> 0:s -> 0:s
0 :: 0:s
false :: true:false
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:

   cond(v0, v1, v2) -> null_cond [0]

And the following fresh constants:    null_cond

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

(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:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y)) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   add(0, x) -> x [1]
   add(s(x), y) -> s(add(x, y)) [1]
   cond(v0, v1, v2) -> null_cond [0]

The TRS has the following type information:
cond :: true:false -> 0:s -> 0:s -> null_cond
true :: true:false
gr :: 0:s -> 0:s -> true:false
add :: 0:s -> 0:s -> 0:s
0 :: 0:s
false :: true:false
s :: 0:s -> 0:s
null_cond :: null_cond

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:

   true => 1
   0 => 0
   false => 0
   null_cond => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   cond(z, z', z'') -{ 1 }-> cond(gr(x, y), x, add(x, y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0

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, V2),0,[cond(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[add(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(cond(V1, V, V2, Out),1,[gr(V4, V3, Ret0),add(V4, V3, Ret2),cond(Ret0, V4, Ret2, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(gr(V1, V, Out),1,[],[Out = 0,V = V5,V5 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 1,V6 >= 0,V1 = 1 + V6,V = 0]).
eq(gr(V1, V, Out),1,[gr(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
eq(add(V1, V, Out),1,[],[Out = V9,V = V9,V9 >= 0,V1 = 0]).
eq(add(V1, V, Out),1,[add(V11, V10, Ret11)],[Out = 1 + Ret11,V11 >= 0,V10 >= 0,V1 = 1 + V11,V = V10]).
eq(cond(V1, V, V2, Out),0,[],[Out = 0,V13 >= 0,V2 = V14,V12 >= 0,V1 = V13,V = V12,V14 >= 0]).
input_output_vars(cond(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(add(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [add/3]
1. recursive  : [gr/3]
2. recursive  : [cond/4]
3. non_recursive  : [start/3]

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

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

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


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

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

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


### Specialization of cost equations gr/3 
* CE 8 is refined into CE [13] 
* CE 7 is refined into CE [14] 
* CE 6 is refined into CE [15] 


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

### Ranking functions of CR gr(V1,V,Out) 
* RF of phase [13]: [V,V1]

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


### Specialization of cost equations cond/4 
* CE 5 is refined into CE [16] 
* CE 4 is refined into CE [17,18,19,20] 


### Cost equations --> "Loop" of cond/4 
* CEs [20] --> Loop 14 
* CEs [19] --> Loop 15 
* CEs [18] --> Loop 16 
* CEs [17] --> Loop 17 
* CEs [16] --> Loop 18 

### Ranking functions of CR cond(V1,V,V2,Out) 

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [21,22,23,24] 
* CE 2 is refined into CE [25,26,27,28] 
* CE 3 is refined into CE [29,30] 


### Cost equations --> "Loop" of start/3 
* CEs [26] --> Loop 19 
* CEs [24] --> Loop 20 
* CEs [23,27,28,30] --> Loop 21 
* CEs [21,22] --> Loop 22 
* CEs [25,29] --> Loop 23 

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

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


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

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

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

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


#### Cost of chains of gr(V1,V,Out):
* Chain [[13],12]: 1*it(13)+1
  Such that:it(13) =< V1

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

* Chain [[13],11]: 1*it(13)+1
  Such that:it(13) =< V

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

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

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


#### Cost of chains of cond(V1,V,V2,Out):
* Chain [18]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [17,18]: 3
  with precondition: [V1=1,V=0,Out=0,V2>=0] 

* Chain [16,18]: 1*s(1)+3
  Such that:s(1) =< V

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

* Chain [16,15,18]: 3*s(1)+6
  Such that:aux(2) =< V
s(1) =< aux(2)

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

* Chain [15,18]: 2*s(2)+3
  Such that:aux(1) =< V
s(2) =< aux(1)

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

* Chain [14,18]: 1*s(4)+1*s(5)+3
  Such that:s(5) =< V
s(4) =< V2

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

* Chain [14,15,18]: 3*s(2)+1*s(4)+6
  Such that:s(4) =< V2
aux(3) =< V
s(2) =< aux(3)

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


#### Cost of chains of start(V1,V,V2):
* Chain [23]: 1
  with precondition: [V1=0,V>=0] 

* Chain [22]: 4*s(15)+6
  Such that:s(14) =< V
s(15) =< s(14)

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

* Chain [21]: 3*s(17)+2*s(18)+3
  Such that:aux(7) =< V1
aux(8) =< V
s(18) =< aux(7)
s(17) =< aux(8)

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

* Chain [20]: 4*s(23)+2*s(24)+6
  Such that:s(21) =< V
s(22) =< V2
s(23) =< s(21)
s(24) =< s(22)

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

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


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [23] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [22] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 4*V+6 
    - Complexity: n 
* Chain [21] with precondition: [V1>=1,V>=0] 
    - Upper bound: 2*V1+3*V+3 
    - Complexity: n 
* Chain [20] with precondition: [V1=1,V2>=1,V>=V2+1] 
    - Upper bound: 4*V+2*V2+6 
    - Complexity: n 
* Chain [19] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V,V2): 3*V+2+max([2*V1,V+3+nat(V2)*2])+1 
Asymptotic class: n 
* Total analysis performed in 205 ms.


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

(10)
BOUNDS(1, n^1)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0', x) -> x
   add(s(x), y) -> s(add(x, y))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond, gr, add

They will be analysed ascendingly in the following order:
gr < cond
add < cond

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

(16)
Obligation:
Innermost TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


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


The following defined symbols remain to be analysed:
gr, cond, add

They will be analysed ascendingly in the following order:
gr < cond
add < cond

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

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

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_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).
----------------------------------------

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


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


The following defined symbols remain to be analysed:
gr, cond, add

They will be analysed ascendingly in the following order:
gr < cond
add < cond

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


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


The following defined symbols remain to be analysed:
add, cond

They will be analysed ascendingly in the following order:
add < cond

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s4_0(n241_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n241_0, b)), rt in Omega(1 + n241_0)

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

Induction Step:
add(gen_0':s4_0(+(n241_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
s(add(gen_0':s4_0(n241_0), gen_0':s4_0(b))) ->_IH
s(gen_0':s4_0(+(b, c242_0)))

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

(24)
Obligation:
Innermost TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
add(gen_0':s4_0(n241_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n241_0, b)), rt in Omega(1 + n241_0)


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


The following defined symbols remain to be analysed:
cond