WORST_CASE(?,O(n^1))
proof of input_YEkyqaiegG.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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, 257 ms]
(10) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
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(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:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) [1]

Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) [1]

The TRS has the following type information:
or :: true:false:= -> true:false:= -> true:false:=
true :: true:false:=
false :: true:false:=
mem :: a -> nil:set:union -> true:false:=
nil :: nil:set:union
set :: b -> nil:set:union
= :: a -> b -> true:false:=
union :: nil:set:union -> nil:set:union -> nil:set:union

Rewrite Strategy: INNERMOST
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(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:

   or(v0, v1) -> null_or [0]

And the following fresh constants:    null_or, const, const1

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(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:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) [1]
   or(v0, v1) -> null_or [0]

The TRS has the following type information:
or :: true:false:=:null_or -> true:false:=:null_or -> true:false:=:null_or
true :: true:false:=:null_or
false :: true:false:=:null_or
mem :: a -> nil:set:union -> true:false:=:null_or
nil :: nil:set:union
set :: b -> nil:set:union
= :: a -> b -> true:false:=:null_or
union :: nil:set:union -> nil:set:union -> nil:set:union
null_or :: true:false:=:null_or
const :: a
const1 :: b

Rewrite Strategy: INNERMOST
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(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 1
   false => 0
   nil => 0
   null_or => 0
   const => 0
   const1 => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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


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

#### Computed strongly connected components 
0. non_recursive  : [or/3]
1. recursive [non_tail,multiple] : [mem/3]
2. non_recursive  : [start/2]

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

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

### Specialization of cost equations or/3 
* CE 4 is refined into CE [10] 
* CE 3 is refined into CE [11] 
* CE 5 is refined into CE [12] 
* CE 6 is refined into CE [13] 


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

### Ranking functions of CR or(V,V1,Out) 

#### Partial ranking functions of CR or(V,V1,Out) 


### Specialization of cost equations mem/3 
* CE 9 is refined into CE [14,15,16] 
* CE 8 is refined into CE [17] 
* CE 7 is refined into CE [18] 


### Cost equations --> "Loop" of mem/3 
* CEs [17] --> Loop 11 
* CEs [18] --> Loop 12 
* CEs [15] --> Loop 13 
* CEs [14] --> Loop 14 
* CEs [16] --> Loop 15 

### Ranking functions of CR mem(V,V1,Out) 
* RF of phase [13,14,15]: [V1]

#### Partial ranking functions of CR mem(V,V1,Out) 
* Partial RF of phase [13,14,15]:
  - RF of loop [13:1,13:2,14:1,14:2,15:1,15:2]:
    V1


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


### Cost equations --> "Loop" of start/2 
* CEs [20] --> Loop 16 
* CEs [22] --> Loop 17 
* CEs [19,21,23,24] --> Loop 18 

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

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


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

#### Cost of chains of or(V,V1,Out):
* Chain [10]: 1
  with precondition: [Out=0,V>=0,V1>=0] 

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

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


#### Cost of chains of mem(V,V1,Out):
* Chain [12]: 1
  with precondition: [V1=0,Out=0,V>=0] 

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

* Chain [multiple([13,14,15],[[12],[11]])]: 6*it(13)+1*it([11])+1*it([12])+0
  Such that:it([12]) =< V1+1
it([11]) =< V1/2+1/2
aux(1) =< V1
it(13) =< aux(1)
it([11]) =< aux(1)

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


#### Cost of chains of start(V,V1):
* Chain [18]: 1*s(1)+1*s(2)+6*s(4)+1
  Such that:s(3) =< V1
s(1) =< V1+1
s(2) =< V1/2+1/2
s(4) =< s(3)
s(2) =< s(3)

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

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

* Chain [16]: 1
  with precondition: [V1=1,V>=0] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [18] with precondition: [V>=0,V1>=0] 
    - Upper bound: 15/2*V1+5/2 
    - Complexity: n 
* Chain [17] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [16] with precondition: [V1=1,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V,V1): 15/2*V1+5/2 
Asymptotic class: n 
* Total analysis performed in 171 ms.


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(10)
BOUNDS(1, n^1)