WORST_CASE(?,O(n^1))
proof of input_Kzx8Ur0JaA.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 148 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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]

The TRS has the following type information:
admit :: carry -> nil:. -> nil:.
nil :: nil:.
. :: w -> nil:. -> nil:.
w :: w
cond :: =:true -> nil:. -> nil:.
= :: sum -> w -> =:true
sum :: carry -> w -> w -> sum
carry :: carry -> w -> w -> carry
true :: =:true

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:

   admit(v0, v1) -> null_admit [0]
   cond(v0, v1) -> null_cond [0]

And the following fresh constants:    null_admit, null_cond, 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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]
   admit(v0, v1) -> null_admit [0]
   cond(v0, v1) -> null_cond [0]

The TRS has the following type information:
admit :: carry -> nil:.:null_admit:null_cond -> nil:.:null_admit:null_cond
nil :: nil:.:null_admit:null_cond
. :: w -> nil:.:null_admit:null_cond -> nil:.:null_admit:null_cond
w :: w
cond :: =:true -> nil:.:null_admit:null_cond -> nil:.:null_admit:null_cond
= :: sum -> w -> =:true
sum :: carry -> w -> w -> sum
carry :: carry -> w -> w -> carry
true :: =:true
null_admit :: nil:.:null_admit:null_cond
null_cond :: nil:.:null_admit:null_cond
const :: carry
const1 :: sum

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:

   nil => 0
   w => 0
   true => 0
   null_admit => 0
   null_cond => 0
   const => 0
   const1 => 0

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

   admit(z', z'') -{ 1 }-> cond(1 + (1 + x + u + v) + 0, 1 + u + (1 + v + (1 + 0 + admit(1 + x + u + v, z)))) :|: v >= 0, z >= 0, z' = x, x >= 0, z'' = 1 + u + (1 + v + (1 + 0 + z)), u >= 0
   admit(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   admit(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   cond(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   cond(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,[admit(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[cond(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(admit(V, V1, Out),1,[],[Out = 0,V1 = 0,V = V2,V2 >= 0]).
eq(admit(V, V1, Out),1,[admit(1 + V5 + V3 + V4, V6, Ret1111),cond(1 + (1 + V5 + V3 + V4) + 0, 1 + V3 + (1 + V4 + (1 + 0 + Ret1111)), Ret)],[Out = Ret,V4 >= 0,V6 >= 0,V = V5,V5 >= 0,V1 = 3 + V3 + V4 + V6,V3 >= 0]).
eq(cond(V, V1, Out),1,[],[Out = V7,V1 = V7,V7 >= 0,V = 0]).
eq(admit(V, V1, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V8,V = V9]).
eq(cond(V, V1, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V10,V = V11]).
input_output_vars(admit(V,V1,Out),[V,V1],[Out]).
input_output_vars(cond(V,V1,Out),[V,V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [cond/3]
1. recursive [non_tail] : [admit/3]
2. non_recursive  : [start/2]

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

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

### Specialization of cost equations cond/3 
* CE 7 is refined into CE [8] 
* CE 6 is refined into CE [9] 


### Cost equations --> "Loop" of cond/3 
* CEs [8] --> Loop 6 
* CEs [9] --> Loop 7 

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

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


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


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

### Ranking functions of CR admit(V,V1,Out) 
* RF of phase [8]: [V/2+V1/2-1,V1/3-2/3]

#### Partial ranking functions of CR admit(V,V1,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V/2+V1/2-1
    V1/3-2/3


### Specialization of cost equations start/2 
* CE 1 is refined into CE [13] 
* CE 2 is refined into CE [14,15] 


### Cost equations --> "Loop" of start/2 
* CEs [13,14,15] --> Loop 10 

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

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


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

#### Cost of chains of cond(V,V1,Out):
* Chain [7]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

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


#### Cost of chains of admit(V,V1,Out):
* Chain [[8],9]: 1*it(8)+1
  Such that:it(8) =< V1/3

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

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


#### Cost of chains of start(V,V1):
* Chain [10]: 1*s(2)+1
  Such that:s(2) =< V1/3

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


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [10] with precondition: [V>=0,V1>=0] 
    - Upper bound: V1/3+1 
    - Complexity: n 

### Maximum cost of start(V,V1): V1/3+1 
Asymptotic class: n 
* Total analysis performed in 64 ms.


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(10)
BOUNDS(1, n^1)