/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^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(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), 1 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 239 ms]
    (12) BOUNDS(1, n^2)
(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), 290 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   minus(x, y) -> cond(gt(x, y), x, y)
   cond(false, x, y) -> 0
   cond(true, x, y) -> s(minus(x, s(y)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

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

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

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


The defined contexts are:
cond([], x1, x2)


[] just represents basic- or constructor-terms in the following defined contexts:
cond([], x1, x2)


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^2).


The TRS R consists of the following rules:

   minus(x, y) -> cond(gt(x, y), x, y)
   cond(false, x, y) -> 0
   cond(true, x, y) -> s(minus(x, s(y)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

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^2).


The TRS R consists of the following rules:

   minus(x, y) -> cond(gt(x, y), x, y) [1]
   cond(false, x, y) -> 0 [1]
   cond(true, x, y) -> s(minus(x, s(y))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [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, y) -> cond(gt(x, y), x, y) [1]
   cond(false, x, y) -> 0 [1]
   cond(true, x, y) -> s(minus(x, s(y))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
cond :: false:true -> 0:s -> 0:s -> 0:s
gt :: 0:s -> 0:s -> false:true
false :: false:true
0 :: 0:s
true :: false:true
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:
none

And the following fresh constants: none

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

(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, y) -> cond(gt(x, y), x, y) [1]
   cond(false, x, y) -> 0 [1]
   cond(true, x, y) -> s(minus(x, s(y))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
cond :: false:true -> 0:s -> 0:s -> 0:s
gt :: 0:s -> 0:s -> false:true
false :: false:true
0 :: 0:s
true :: false:true
s :: 0:s -> 0:s

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:

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

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

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

   cond(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   cond(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
   gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0
   gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0
   minus(z, z') -{ 1 }-> cond(gt(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y

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


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

#### Computed strongly connected components 
0. recursive  : [gt/3]
1. recursive  : [cond/4,minus/3]
2. non_recursive  : [start/3]

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

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

### Specialization of cost equations gt/3 
* CE 9 is refined into CE [10] 
* CE 8 is refined into CE [11] 
* CE 7 is refined into CE [12] 


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

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

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


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


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

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

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [17,18,19] 
* CE 2 is refined into CE [20] 
* CE 3 is refined into CE [21,22,23,24,25] 
* CE 4 is refined into CE [26,27,28,29] 


### Cost equations --> "Loop" of start/3 
* CEs [25,29] --> Loop 15 
* CEs [19] --> Loop 16 
* CEs [18,24,28] --> Loop 17 
* CEs [17,22,23,27] --> Loop 18 
* CEs [20,21,26] --> Loop 19 

### Ranking functions of CR start(V1,V,V5) 

#### Partial ranking functions of CR start(V1,V,V5) 


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

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

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

* Chain [[10],8]: 1*it(10)+1
  Such that:it(10) =< V

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

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[11],13]: 3*it(11)+1*s(1)+1*s(4)+3
  Such that:it(11) =< Out
aux(2) =< V+Out
s(1) =< aux(2)
s(4) =< it(11)*aux(2)

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

* Chain [14]: 3
  with precondition: [V1=0,Out=0,V>=0] 

* Chain [13]: 1*s(1)+3
  Such that:s(1) =< V1

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

* Chain [12,[11],13]: 4*it(11)+1*s(4)+6
  Such that:aux(3) =< Out
it(11) =< aux(3)
s(4) =< it(11)*aux(3)

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

* Chain [12,13]: 1*s(1)+6
  Such that:s(1) =< 1

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


#### Cost of chains of start(V1,V,V5):
* Chain [19]: 3
  with precondition: [V1=0,V>=0] 

* Chain [18]: 1*s(5)+4*s(7)+1*s(8)+6
  Such that:s(5) =< 1
s(6) =< V1
s(7) =< s(6)
s(8) =< s(7)*s(6)

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

* Chain [17]: 1*s(9)+2*s(10)+4
  Such that:s(9) =< V
aux(4) =< V1
s(10) =< aux(4)

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

* Chain [16]: 3*s(12)+1*s(14)+1*s(15)+4
  Such that:s(13) =< V
s(12) =< V-V5
s(14) =< s(13)
s(15) =< s(12)*s(13)

  with precondition: [V1=1,V5>=0,V>=V5+2] 

* Chain [15]: 3*s(16)+1*s(18)+1*s(19)+1*s(20)+3
  Such that:s(17) =< V1
s(16) =< V1-V
s(20) =< V
s(18) =< s(17)
s(19) =< s(16)*s(17)

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


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [19] with precondition: [V1=0,V>=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [18] with precondition: [V=0,V1>=1] 
    - Upper bound: 4*V1+7+V1*V1 
    - Complexity: n^2 
* Chain [17] with precondition: [V1>=1,V>=V1] 
    - Upper bound: 2*V1+V+4 
    - Complexity: n 
* Chain [16] with precondition: [V1=1,V5>=0,V>=V5+2] 
    - Upper bound: 3*V-3*V5+(V+4+(V-V5)*V) 
    - Complexity: n^2 
* Chain [15] with precondition: [V>=1,V1>=V+1] 
    - Upper bound: 3*V1-3*V+(V1+3+(V1-V)*V1+V) 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V5): max([max([V1+1+max([V,2*V1+3+V1*V1]),nat(V1-V)*V1+V+nat(V1-V)*3])+V1,V+1+nat(V-V5)*V+nat(V-V5)*3])+3 
Asymptotic class: n^2 
* Total analysis performed in 167 ms.


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

(12)
BOUNDS(1, n^2)

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

(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, y) -> cond(gt(x, y), x, y)
   cond(false, x, y) -> 0'
   cond(true, x, y) -> s(minus(x, s(y)))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(u, v)

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

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

(16)
Obligation:
TRS:
Rules:
minus(x, y) -> cond(gt(x, y), x, y)
cond(false, x, y) -> 0'
cond(true, x, y) -> s(minus(x, s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
minus :: 0':s -> 0':s -> 0':s
cond :: false:true -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s

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

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

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

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

(18)
Obligation:
TRS:
Rules:
minus(x, y) -> cond(gt(x, y), x, y)
cond(false, x, y) -> 0'
cond(true, x, y) -> s(minus(x, s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
minus :: 0':s -> 0':s -> 0':s
cond :: false:true -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
gt, minus

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

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

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

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

Induction Step:
gt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
false

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, y) -> cond(gt(x, y), x, y)
cond(false, x, y) -> 0'
cond(true, x, y) -> s(minus(x, s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
minus :: 0':s -> 0':s -> 0':s
cond :: false:true -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
gt, minus

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

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
minus(x, y) -> cond(gt(x, y), x, y)
cond(false, x, y) -> 0'
cond(true, x, y) -> s(minus(x, s(y)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
minus :: 0':s -> 0':s -> 0':s
cond :: false:true -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, 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:
minus