/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 180 ms]
    (14) BOUNDS(1, n^1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) SlicingProof [LOWER BOUND(ID), 0 ms]
    (18) CpxTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 1024 ms]
    (24) proven lower bound
    (25) LowerBoundPropagationProof [FINISHED, 0 ms]
    (26) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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

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

The duplicating contexts are:
-([], s(y))
-(x, s([]))


The defined contexts are:
-(x0, [])
p(s([]))


[] just represents basic- or constructor-terms in the following defined contexts:
-(x0, [])


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^1).


The TRS R consists of the following rules:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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^1).


The TRS R consists of the following rules:

   -(0, y) -> 0 [1]
   -(x, 0) -> x [1]
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

(6)
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(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
minus :: 0:s:if -> 0:s:if -> 0:s:if
0 :: 0:s:if
s :: 0:s:if -> 0:s:if
if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
greater :: 0:s:if -> 0:s:if -> greater
p :: 0:s:if -> 0:s:if

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

(9) 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]
   p(v0) -> null_p [0]

And the following fresh constants:    null_minus, null_p, const

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

(10)
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(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]

The TRS has the following type information:
minus :: 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p
0 :: 0:s:if:null_minus:null_p
s :: 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p
if :: greater -> 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p
greater :: 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p -> greater
p :: 0:s:if:null_minus:null_p -> 0:s:if:null_minus:null_p
null_minus :: 0:s:if:null_minus:null_p
null_p :: 0:s:if:null_minus:null_p
const :: greater

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

(11) 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
   null_minus => 0
   null_p => 0
   const => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 1 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, p(1 + y))) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(minus(V1, V, Out),1,[],[Out = 0,V2 >= 0,V1 = 0,V = V2]).
eq(minus(V1, V, Out),1,[],[Out = V3,V3 >= 0,V1 = V3,V = 0]).
eq(minus(V1, V, Out),1,[p(1 + V5, Ret0111),minus(V4, Ret0111, Ret011)],[Out = 4 + Ret011 + V4 + V5,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = V4]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V6,V6 >= 0,V1 = 1 + V6]).
eq(minus(V1, V, Out),0,[],[Out = 0,V8 >= 0,V7 >= 0,V1 = V8,V = V7]).
eq(p(V1, Out),0,[],[Out = 0,V9 >= 0,V1 = V9]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [p/2]
1. recursive  : [minus/3]
2. non_recursive  : [start/2]

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

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

### Specialization of cost equations p/2 
* CE 8 is refined into CE [10] 
* CE 7 is refined into CE [11] 
* CE 9 is refined into CE [12] 


### Cost equations --> "Loop" of p/2 
* CEs [10] --> Loop 7 
* CEs [11,12] --> Loop 8 

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations minus/3 
* CE 4 is refined into CE [13] 
* CE 3 is refined into CE [14] 
* CE 6 is refined into CE [15] 
* CE 5 is refined into CE [16,17] 


### Cost equations --> "Loop" of minus/3 
* CEs [17] --> Loop 9 
* CEs [16] --> Loop 10 
* CEs [13] --> Loop 11 
* CEs [14,15] --> Loop 12 

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [18,19,20,21,22] 
* CE 2 is refined into CE [23,24] 


### Cost equations --> "Loop" of start/2 
* CEs [18,19,20,21,22,23,24] --> Loop 13 

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

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


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

#### Cost of chains of p(V1,Out):
* Chain [8]: 1
  with precondition: [Out=0,V1>=0] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[9],12]: 2*it(9)+1
  Such that:it(9) =< V

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

* Chain [[9],11]: 2*it(9)+1
  Such that:it(9) =< V

  with precondition: [V1>=0,V>=1,Out+1>=5*V+2*V1] 

* Chain [[9],10,12]: 2*it(9)+3
  Such that:it(9) =< V

  with precondition: [V1>=0,V>=2,Out>=2*V+2*V1+5] 

* Chain [[9],10,11]: 2*it(9)+3
  Such that:it(9) =< V

  with precondition: [V1>=0,V>=2,Out>=3*V1+2*V+5] 

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

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

* Chain [10,12]: 3
  with precondition: [V+V1+3=Out,V>=1,Out>=V+3] 

* Chain [10,11]: 3
  with precondition: [V+2*V1+3=Out,V1>=0,V>=1] 


#### Cost of chains of start(V1,V):
* Chain [13]: 8*s(5)+3
  Such that:aux(2) =< V
s(5) =< aux(2)

  with precondition: [V1>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [13] with precondition: [V1>=0] 
    - Upper bound: nat(V)*8+3 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V)*8+3 
Asymptotic class: n 
* Total analysis performed in 107 ms.


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

(14)
BOUNDS(1, n^1)

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

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

(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:

   -(0', y) -> 0'
   -(x, 0') -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
   p(0') -> 0'
   p(s(x)) -> x

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

(17) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
if/0
if/2
greater/0
greater/1

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

(18)
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:

   -(0', y) -> 0'
   -(x, 0') -> x
   -(x, s(y)) -> if(s(-(x, p(s(y)))))
   p(0') -> 0'
   p(s(x)) -> x

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

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

(20)
Obligation:
TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(s(-(x, p(s(y)))))
p(0') -> 0'
p(s(x)) -> x

Types:
- :: 0':s:if -> 0':s:if -> 0':s:if
0' :: 0':s:if
s :: 0':s:if -> 0':s:if
if :: 0':s:if -> 0':s:if
p :: 0':s:if -> 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat -> 0':s:if

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-
----------------------------------------

(22)
Obligation:
TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(s(-(x, p(s(y)))))
p(0') -> 0'
p(s(x)) -> x

Types:
- :: 0':s:if -> 0':s:if -> 0':s:if
0' :: 0':s:if
s :: 0':s:if -> 0':s:if
if :: 0':s:if -> 0':s:if
p :: 0':s:if -> 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat -> 0':s:if


Generator Equations:
gen_0':s:if2_0(0) <=> 0'
gen_0':s:if2_0(+(x, 1)) <=> s(gen_0':s:if2_0(x))


The following defined symbols remain to be analysed:
-
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(0))

Induction Step:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(+(n4_0, 1))) ->_R^Omega(1)
if(s(-(gen_0':s:if2_0(a), p(s(gen_0':s:if2_0(n4_0)))))) ->_R^Omega(1)
if(s(-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)))) ->_IH
if(s(*3_0))

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

(24)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(s(-(x, p(s(y)))))
p(0') -> 0'
p(s(x)) -> x

Types:
- :: 0':s:if -> 0':s:if -> 0':s:if
0' :: 0':s:if
s :: 0':s:if -> 0':s:if
if :: 0':s:if -> 0':s:if
p :: 0':s:if -> 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat -> 0':s:if


Generator Equations:
gen_0':s:if2_0(0) <=> 0'
gen_0':s:if2_0(+(x, 1)) <=> s(gen_0':s:if2_0(x))


The following defined symbols remain to be analysed:
-
----------------------------------------

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

(26)
BOUNDS(n^1, INF)