WORST_CASE(?,O(n^1))
proof of input_qTCkWAAziP.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 156 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:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
   u21(ackout(X), Y) -> u22(ackin(Y, X))

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:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
   u21(ackout(X), Y) -> u22(ackin(Y, X)) [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:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]

The TRS has the following type information:
ackin :: s -> s -> ackout:u22
s :: s -> s
u21 :: ackout:u22 -> s -> ackout:u22
ackout :: s -> ackout:u22
u22 :: ackout:u22 -> ackout:u22

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:

   ackin(v0, v1) -> null_ackin [0]
   u21(v0, v1) -> null_u21 [0]

And the following fresh constants:    null_ackin, null_u21, const

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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:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
   ackin(v0, v1) -> null_ackin [0]
   u21(v0, v1) -> null_u21 [0]

The TRS has the following type information:
ackin :: s -> s -> ackout:u22:null_ackin:null_u21
s :: s -> s
u21 :: ackout:u22:null_ackin:null_u21 -> s -> ackout:u22:null_ackin:null_u21
ackout :: s -> ackout:u22:null_ackin:null_u21
u22 :: ackout:u22:null_ackin:null_u21 -> ackout:u22:null_ackin:null_u21
null_ackin :: ackout:u22:null_ackin:null_u21
null_u21 :: ackout:u22:null_ackin:null_u21
const :: s

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:

   null_ackin => 0
   null_u21 => 0
   const => 0

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

   ackin(z, z') -{ 1 }-> u21(ackin(1 + X, Y), X) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   ackin(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   u21(z, z') -{ 1 }-> 1 + ackin(Y, X) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0

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(V1, V),0,[ackin(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[u21(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(ackin(V1, V, Out),1,[ackin(1 + X1, Y1, Ret0),u21(Ret0, X1, Ret)],[Out = Ret,V1 = 1 + X1,Y1 >= 0,V = 1 + Y1,X1 >= 0]).
eq(u21(V1, V, Out),1,[ackin(Y2, X2, Ret1)],[Out = 1 + Ret1,V1 = 1 + X2,V = Y2,Y2 >= 0,X2 >= 0]).
eq(ackin(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(u21(V1, V, Out),0,[],[Out = 0,V5 >= 0,V4 >= 0,V1 = V5,V = V4]).
input_output_vars(ackin(V1,V,Out),[V1,V],[Out]).
input_output_vars(u21(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [ackin/3,u21/3]
1. non_recursive  : [start/2]

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

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

### Specialization of cost equations ackin/3 
* CE 6 is refined into CE [7] 
* CE 5 is refined into CE [8] 
* CE 4 is refined into CE [9] 


### Cost equations --> "Loop" of ackin/3 
* CEs [9] --> Loop 5 
* CEs [8] --> Loop 6 
* CEs [7] --> Loop 7 

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

#### Partial ranking functions of CR ackin(V1,V,Out) 
* Partial RF of phase [5,6]:
  - RF of loop [5:1,6:1]:
    V depends on loops [6:2] 
  - RF of loop [6:2]:
    V1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [10] 
* CE 2 is refined into CE [11] 
* CE 3 is refined into CE [12] 


### Cost equations --> "Loop" of start/2 
* CEs [10,11,12] --> Loop 8 

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

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


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

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

* Chain [multiple([5,6],[[7]])]: 1*it(5)+0
  Such that:it(5) =< V

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


#### Cost of chains of start(V1,V):
* Chain [8]: 1*s(2)+1*s(3)+1
  Such that:s(2) =< V1
s(3) =< V

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [8] with precondition: [V1>=0,V>=0] 
    - Upper bound: V1+V+1 
    - Complexity: n 

### Maximum cost of start(V1,V): V1+V+1 
Asymptotic class: n 
* Total analysis performed in 75 ms.


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(10)
BOUNDS(1, n^1)