/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^2), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, n^2).

(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), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 175 ms]
    (10) BOUNDS(1, n^2)
(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), 292 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), 84 ms]
            (24) proven lower bound
            (25) LowerBoundPropagationProof [FINISHED, 0 ms]
            (26) BOUNDS(n^2, INF)


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

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


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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^2).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0: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:
none

And the following fresh constants:    const

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

(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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0:s -> 0:s -> 0:s
const :: and:activate

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:

   tt => 0
   0 => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 1 }-> plus(x(N, M), N) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 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),0,[and(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[x(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[activate(V1, Out)],[V1 >= 0]).
eq(and(V1, V, Out),1,[activate(X1, Ret)],[Out = Ret,V = X1,X1 >= 0,V1 = 0]).
eq(plus(V1, V, Out),1,[],[Out = N1,V1 = N1,V = 0,N1 >= 0]).
eq(plus(V1, V, Out),1,[plus(N2, M1, Ret1)],[Out = 1 + Ret1,V = 1 + M1,V1 = N2,M1 >= 0,N2 >= 0]).
eq(x(V1, V, Out),1,[],[Out = 0,V1 = N3,V = 0,N3 >= 0]).
eq(x(V1, V, Out),1,[x(N4, M2, Ret0),plus(Ret0, N4, Ret2)],[Out = Ret2,V = 1 + M2,V1 = N4,M2 >= 0,N4 >= 0]).
eq(activate(V1, Out),1,[],[Out = X2,X2 >= 0,V1 = X2]).
input_output_vars(and(V1,V,Out),[V1,V],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
input_output_vars(x(V1,V,Out),[V1,V],[Out]).
input_output_vars(activate(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [activate/2]
1. non_recursive  : [and/3]
2. recursive  : [plus/3]
3. recursive [non_tail] : [x/3]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is completely evaluated into other SCCs
2. SCC is partially evaluated into plus/3
3. SCC is partially evaluated into x/3
4. SCC is partially evaluated into start/2

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

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


### Cost equations --> "Loop" of plus/3 
* CEs [10] --> Loop 6 
* CEs [9] --> Loop 7 

### Ranking functions of CR plus(V1,V,Out) 
* RF of phase [7]: [V]

#### Partial ranking functions of CR plus(V1,V,Out) 
* Partial RF of phase [7]:
  - RF of loop [7:1]:
    V


### Specialization of cost equations x/3 
* CE 8 is refined into CE [11,12] 
* CE 7 is refined into CE [13] 


### Cost equations --> "Loop" of x/3 
* CEs [13] --> Loop 8 
* CEs [12] --> Loop 9 
* CEs [11] --> Loop 10 

### Ranking functions of CR x(V1,V,Out) 
* RF of phase [9]: [V]
* RF of phase [10]: [V]

#### Partial ranking functions of CR x(V1,V,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [14] 
* CE 2 is refined into CE [15,16] 
* CE 3 is refined into CE [17,18,19] 
* CE 4 is refined into CE [20] 


### Cost equations --> "Loop" of start/2 
* CEs [15,18] --> Loop 11 
* CEs [14,16,17,19,20] --> Loop 12 

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

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


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

#### Cost of chains of plus(V1,V,Out):
* Chain [[7],6]: 1*it(7)+1
  Such that:it(7) =< V

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

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


#### Cost of chains of x(V1,V,Out):
* Chain [[10],8]: 2*it(10)+1
  Such that:it(10) =< V

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

* Chain [[9],8]: 2*it(9)+1*s(3)+1
  Such that:aux(1) =< V1
it(9) =< V
s(3) =< it(9)*aux(1)

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

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


#### Cost of chains of start(V1,V):
* Chain [12]: 5*s(4)+1*s(8)+2
  Such that:s(6) =< V1
aux(2) =< V
s(4) =< aux(2)
s(8) =< s(4)*s(6)

  with precondition: [V1>=0] 

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [12] with precondition: [V1>=0] 
    - Upper bound: nat(V)*V1+2+nat(V)*5 
    - Complexity: n^2 
* Chain [11] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V): nat(V)*V1+1+nat(V)*5+1 
Asymptotic class: n^2 
* Total analysis performed in 111 ms.


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

(10)
BOUNDS(1, n^2)

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

(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^2, INF).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0') -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0') -> 0'
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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

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

(14)
Obligation:
Innermost TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
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:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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

(16)
Obligation:
Innermost TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
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:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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

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

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

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) ->_IH
s(gen_0':s4_0(+(a, c7_0)))

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:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
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:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), 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:
x
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_0':s4_0(a), gen_0':s4_0(n473_0)) -> gen_0':s4_0(*(n473_0, a)), rt in Omega(1 + a*n473_0 + n473_0)

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

Induction Step:
x(gen_0':s4_0(a), gen_0':s4_0(+(n473_0, 1))) ->_R^Omega(1)
plus(x(gen_0':s4_0(a), gen_0':s4_0(n473_0)), gen_0':s4_0(a)) ->_IH
plus(gen_0':s4_0(*(c474_0, a)), gen_0':s4_0(a)) ->_L^Omega(1 + a)
gen_0':s4_0(+(a, *(n473_0, a)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(24)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), 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:
x
----------------------------------------

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

(26)
BOUNDS(n^2, INF)