/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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, 293 ms]
    (12) BOUNDS(1, n^1)
(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), 1070 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (26) BOUNDS(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:

   minus(X, 0) -> X
   minus(s(X), s(Y)) -> p(minus(X, Y))
   p(s(X)) -> X
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

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

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

The duplicating contexts are:
div(s(X), s([]))


The defined contexts are:
p([])
div([], s(x1))
minus([], x1)


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:

   minus(X, 0) -> X
   minus(s(X), s(Y)) -> p(minus(X, Y))
   p(s(X)) -> X
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

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:

   minus(X, 0) -> X [1]
   minus(s(X), s(Y)) -> p(minus(X, Y)) [1]
   p(s(X)) -> X [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) [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:

   minus(X, 0) -> X [1]
   minus(s(X), s(Y)) -> p(minus(X, Y)) [1]
   p(s(X)) -> X [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) [1]

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

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:

   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   div(v0, v1) -> null_div [0]

And the following fresh constants:    null_minus, null_p, null_div

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

(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:

   minus(X, 0) -> X [1]
   minus(s(X), s(Y)) -> p(minus(X, Y)) [1]
   p(s(X)) -> X [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) [1]
   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   div(v0, v1) -> null_div [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div
0 :: 0:s:null_minus:null_p:null_div
s :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div
p :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div
div :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div
null_minus :: 0:s:null_minus:null_p:null_div
null_p :: 0:s:null_minus:null_p:null_div
null_div :: 0:s:null_minus:null_p:null_div

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
   null_minus => 0
   null_p => 0
   null_div => 0

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

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

   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   div(z, z') -{ 1 }-> 1 + div(minus(X, Y), 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   minus(z, z') -{ 1 }-> p(minus(X, Y)) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]).
eq(minus(V1, V, Out),1,[minus(X2, Y1, Ret0),p(Ret0, Ret)],[Out = Ret,V1 = 1 + X2,Y1 >= 0,V = 1 + Y1,X2 >= 0]).
eq(p(V1, Out),1,[],[Out = X3,V1 = 1 + X3,X3 >= 0]).
eq(div(V1, V, Out),1,[],[Out = 0,Y2 >= 0,V = 1 + Y2,V1 = 0]).
eq(div(V1, V, Out),1,[minus(X4, Y3, Ret10),div(Ret10, 1 + Y3, Ret1)],[Out = 1 + Ret1,V1 = 1 + X4,Y3 >= 0,V = 1 + Y3,X4 >= 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(p(V1, Out),0,[],[Out = 0,V4 >= 0,V1 = V4]).
eq(div(V1, V, Out),0,[],[Out = 0,V6 >= 0,V5 >= 0,V1 = V6,V = V5]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [p/2]
1. recursive [non_tail] : [minus/3]
2. recursive  : [(div)/3]
3. 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 (div)/3
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations p/2 
* CE 7 is refined into CE [12] 
* CE 8 is refined into CE [13] 


### Cost equations --> "Loop" of p/2 
* CEs [12] --> Loop 9 
* CEs [13] --> Loop 10 

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

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


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


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

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [11]: [V,V1]
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V
    V1
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations (div)/3 
* CE 9 is refined into CE [18] 
* CE 11 is refined into CE [19] 
* CE 10 is refined into CE [20,21,22] 


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

### Ranking functions of CR div(V1,V,Out) 
* RF of phase [15]: [V1/3-2/3,V1/3-2/3*V+2/3]
* RF of phase [17]: [V1]

#### Partial ranking functions of CR div(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V1/3-2/3
    V1/3-2/3*V+2/3
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [23,24,25] 
* CE 2 is refined into CE [26,27] 
* CE 3 is refined into CE [28,29,30,31,32] 


### Cost equations --> "Loop" of start/2 
* CEs [28] --> Loop 19 
* CEs [23,24,25,26,27,29,30,31,32] --> Loop 20 

### 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 [10]: 0
  with precondition: [Out=0,V1>=0] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[12],[11],14]: 3*it(11)+1
  Such that:aux(1) =< V
it(11) =< aux(1)

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

* Chain [[12],14]: 1*it(12)+1
  Such that:it(12) =< V

  with precondition: [Out=0,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 [[11],14]: 2*it(11)+1
  Such that:it(11) =< V

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

* 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 div(V1,V,Out):
* Chain [[17],18]: 2*it(17)+1
  Such that:it(17) =< Out

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

* Chain [[17],16,18]: 2*it(17)+5*s(6)+3
  Such that:s(5) =< 1
it(17) =< Out
s(6) =< s(5)

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

* Chain [[15],18]: 2*it(15)+2*s(9)+1
  Such that:it(15) =< V1/3-2/3*V+2/3
s(9) =< 2/3*V1-V/3+2/3

  with precondition: [V>=2,Out>=1,V1+4>=3*Out+2*V] 

* Chain [[15],16,18]: 2*it(15)+5*s(6)+2*s(9)+3
  Such that:it(15) =< V1/3-2/3*V+2/3
s(9) =< 2/3*V1
s(5) =< V
s(6) =< s(5)

  with precondition: [V>=2,Out>=2,V1+6>=3*Out+2*V] 

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

* Chain [16,18]: 5*s(6)+3
  Such that:s(5) =< V
s(6) =< s(5)

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


#### Cost of chains of start(V1,V):
* Chain [20]: 17*s(15)+4*s(19)+2*s(20)+2*s(22)+3
  Such that:s(22) =< 2/3*V1
s(20) =< 2/3*V1-V/3+2/3
aux(4) =< V1/3-2/3*V+2/3
aux(5) =< V
s(19) =< aux(4)
s(15) =< aux(5)

  with precondition: [V1>=0] 

* Chain [19]: 4*s(27)+5*s(28)+3
  Such that:s(25) =< 1
s(26) =< V1
s(27) =< s(26)
s(28) =< s(25)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [20] with precondition: [V1>=0] 
    - Upper bound: nat(V)*17+3+4/3*V1+nat(2/3*V1-V/3+2/3)*2+nat(V1/3-2/3*V+2/3)*4 
    - Complexity: n 
* Chain [19] with precondition: [V=1,V1>=1] 
    - Upper bound: 4*V1+8 
    - Complexity: n 

### Maximum cost of start(V1,V): max([4*V1+5,4/3*V1+nat(V)*17+nat(2/3*V1-V/3+2/3)*2+nat(V1/3-2/3*V+2/3)*4])+3 
Asymptotic class: n 
* Total analysis performed in 217 ms.


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

(12)
BOUNDS(1, n^1)

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

(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:

   minus(X, 0') -> X
   minus(s(X), s(Y)) -> p(minus(X, Y))
   p(s(X)) -> X
   div(0', s(Y)) -> 0'
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

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

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

(16)
Obligation:
TRS:
Rules:
minus(X, 0') -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

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

They will be analysed ascendingly in the following order:
minus < div

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

(18)
Obligation:
TRS:
Rules:
minus(X, 0') -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, div

They will be analysed ascendingly in the following order:
minus < div

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
minus(gen_0':s2_0(+(1, 0)), gen_0':s2_0(+(1, 0)))

Induction Step:
minus(gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
p(minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0)))) ->_IH
p(*3_0)

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:
minus(X, 0') -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, div

They will be analysed ascendingly in the following order:
minus < div

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
minus(X, 0') -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0', s(Y)) -> 0'
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div(gen_0':s2_0(n1747_0), gen_0':s2_0(1)) -> gen_0':s2_0(n1747_0), rt in Omega(1 + n1747_0)

Induction Base:
div(gen_0':s2_0(0), gen_0':s2_0(1)) ->_R^Omega(1)
0'

Induction Step:
div(gen_0':s2_0(+(n1747_0, 1)), gen_0':s2_0(1)) ->_R^Omega(1)
s(div(minus(gen_0':s2_0(n1747_0), gen_0':s2_0(0)), s(gen_0':s2_0(0)))) ->_R^Omega(1)
s(div(gen_0':s2_0(n1747_0), s(gen_0':s2_0(0)))) ->_IH
s(gen_0':s2_0(c1748_0))

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

(26)
BOUNDS(1, INF)