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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 221 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), 232 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), 59 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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(s(x), s(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
le :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
quot :: 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:

   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   le(v0, v1) -> null_le [0]

And the following fresh constants:    null_minus, null_quot, null_le

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

(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) [1]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   le(v0, v1) -> null_le [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
0 :: 0:s:null_minus:null_quot
s :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
le :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> true:false:null_le
true :: true:false:null_le
false :: true:false:null_le
quot :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
null_minus :: 0:s:null_minus:null_quot
null_quot :: 0:s:null_minus:null_quot
null_le :: true:false:null_le

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:

   0 => 0
   true => 2
   false => 1
   null_minus => 0
   null_quot => 0
   null_le => 0

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

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

   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 1 + quot(minus(1 + x, 1 + y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

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,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]).
eq(le(V1, V, Out),1,[],[Out = 2,V5 >= 0,V1 = 0,V = V5]).
eq(le(V1, V, Out),1,[],[Out = 1,V6 >= 0,V1 = 1 + V6,V = 0]).
eq(le(V1, V, Out),1,[le(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V9,V9 >= 0,V1 = 0]).
eq(quot(V1, V, Out),1,[minus(1 + V11, 1 + V10, Ret10),quot(Ret10, 1 + V10, Ret11)],[Out = 1 + Ret11,V = 1 + V10,V11 >= 0,V10 >= 0,V1 = 1 + V11]).
eq(minus(V1, V, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V1 = V13,V = V12]).
eq(quot(V1, V, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V1 = V15,V = V14]).
eq(le(V1, V, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V1 = V17,V = V16]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(quot(V1,V,Out),[V1,V],[Out]).


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

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

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

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

### Specialization of cost equations le/3 
* CE 10 is refined into CE [14] 
* CE 8 is refined into CE [15] 
* CE 7 is refined into CE [16] 
* CE 9 is refined into CE [17] 


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

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

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


### Specialization of cost equations minus/3 
* CE 6 is refined into CE [18] 
* CE 4 is refined into CE [19] 
* CE 5 is refined into CE [20] 


### Cost equations --> "Loop" of minus/3 
* CEs [20] --> Loop 15 
* CEs [18] --> Loop 16 
* CEs [19] --> Loop 17 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
    V1


### Specialization of cost equations quot/3 
* CE 11 is refined into CE [21] 
* CE 13 is refined into CE [22] 
* CE 12 is refined into CE [23,24] 


### Cost equations --> "Loop" of quot/3 
* CEs [24] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [21,22] --> Loop 20 

### Ranking functions of CR quot(V1,V,Out) 
* RF of phase [18]: [V1,V1-V+1]

#### Partial ranking functions of CR quot(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1
    V1-V+1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [25,26,27] 
* CE 2 is refined into CE [28,29,30,31,32] 
* CE 3 is refined into CE [33,34,35] 


### Cost equations --> "Loop" of start/2 
* CEs [25,29] --> Loop 21 
* CEs [26,27,28,30,31,32,33,34,35] --> Loop 22 

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

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


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

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

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

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

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

* Chain [[11],12]: 1*it(11)+0
  Such that:it(11) =< V

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

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

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[15],17]: 1*it(15)+1
  Such that:it(15) =< V

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

* Chain [[15],16]: 1*it(15)+0
  Such that:it(15) =< V

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

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

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


#### Cost of chains of quot(V1,V,Out):
* Chain [[18],20]: 2*it(18)+1*s(5)+1
  Such that:it(18) =< V1-V+1
aux(3) =< V1
it(18) =< aux(3)
s(5) =< aux(3)

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

* Chain [[18],19,20]: 3*it(18)+1*s(6)+2
  Such that:s(6) =< V
aux(4) =< V1
it(18) =< aux(4)

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

* Chain [20]: 1
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [19,20]: 1*s(6)+2
  Such that:s(6) =< V

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


#### Cost of chains of start(V1,V):
* Chain [22]: 6*s(13)+5*s(17)+2*s(19)+2
  Such that:s(19) =< V1-V+1
aux(6) =< V1
aux(7) =< V
s(17) =< aux(6)
s(13) =< aux(7)
s(19) =< aux(6)

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

* Chain [21]: 1
  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [22] with precondition: [V1>=0,V>=0] 
    - Upper bound: 5*V1+6*V+2+nat(V1-V+1)*2 
    - Complexity: n 
* Chain [21] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V): 5*V1+6*V+1+nat(V1-V+1)*2+1 
Asymptotic class: n 
* Total analysis performed in 200 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:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y)))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
quot :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, le, quot

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

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

(16)
Obligation:
Innermost TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
quot :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, le, quot

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)

Induction Base:
minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
minus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
gen_0':s3_0(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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
quot :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, le, quot

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

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
quot :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, quot
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n257_0), gen_0':s3_0(n257_0)) -> true, rt in Omega(1 + n257_0)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s3_0(+(n257_0, 1)), gen_0':s3_0(+(n257_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n257_0), gen_0':s3_0(n257_0)) ->_IH
true

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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
quot :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
le(gen_0':s3_0(n257_0), gen_0':s3_0(n257_0)) -> true, rt in Omega(1 + n257_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
quot