/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 177 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 391 ms]
    (14) BOUNDS(1, n^2)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
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:

   f(x, a) -> x [1]
   f(x, g(y)) -> f(g(x), y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, a) -> x [1]
   f(x, g(y)) -> f(g(x), y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]

The TRS has the following type information:
f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
a :: a:g:cons_f
g :: a:g:cons_f -> a:g:cons_f
encArg :: a:g:cons_f -> a:g:cons_f
cons_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_a :: a:g:cons_f
encode_g :: a:g:cons_f -> a:g:cons_f

Rewrite Strategy: INNERMOST
----------------------------------------

(9) 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:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_g(v0) -> null_encode_g [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_g, null_f

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

(10)
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:

   f(x, a) -> x [1]
   f(x, g(y)) -> f(g(x), y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_g(v0) -> null_encode_g [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
a :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
g :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
encArg :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
cons_f :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
encode_f :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
encode_a :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
encode_g :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f -> a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
null_encArg :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
null_encode_f :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
null_encode_a :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
null_encode_g :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f
null_f :: a:g:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_g:null_f

Rewrite Strategy: INNERMOST
----------------------------------------

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

   a => 0
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_g => 0
   null_f => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_g(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   f(z, z') -{ 1 }-> f(1 + x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(Out)],[]).
eq(start(V1, V),0,[fun2(V1, Out)],[V1 >= 0]).
eq(f(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(f(V1, V, Out),1,[f(1 + V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = V3]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret1)],[Out = 1 + Ret1,V1 = 1 + V5,V5 >= 0]).
eq(encArg(V1, Out),0,[encArg(V6, Ret0),encArg(V7, Ret11),f(Ret0, Ret11, Ret2)],[Out = Ret2,V6 >= 0,V1 = 1 + V6 + V7,V7 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V8, Ret01),encArg(V9, Ret12),f(Ret01, Ret12, Ret3)],[Out = Ret3,V8 >= 0,V9 >= 0,V1 = V8,V = V9]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(V1, Out),0,[encArg(V10, Ret13)],[Out = 1 + Ret13,V10 >= 0,V1 = V10]).
eq(encArg(V1, Out),0,[],[Out = 0,V11 >= 0,V1 = V11]).
eq(fun(V1, V, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V1 = V13,V = V12]).
eq(fun2(V1, Out),0,[],[Out = 0,V14 >= 0,V1 = V14]).
eq(f(V1, V, Out),0,[],[Out = 0,V15 >= 0,V16 >= 0,V1 = V15,V = V16]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(Out),[],[Out]).
input_output_vars(fun2(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [f/3]
1. recursive [non_tail,multiple] : [encArg/2]
2. non_recursive  : [fun/3]
3. non_recursive  : [fun1/1]
4. non_recursive  : [fun2/2]
5. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into f/3
1. SCC is partially evaluated into encArg/2
2. SCC is partially evaluated into fun/3
3. SCC is completely evaluated into other SCCs
4. SCC is partially evaluated into fun2/2
5. SCC is partially evaluated into start/2

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

### Specialization of cost equations f/3 
* CE 8 is refined into CE [16] 
* CE 6 is refined into CE [17] 
* CE 7 is refined into CE [18] 


### Cost equations --> "Loop" of f/3 
* CEs [18] --> Loop 10 
* CEs [16] --> Loop 11 
* CEs [17] --> Loop 12 

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

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


### Specialization of cost equations encArg/2 
* CE 9 is refined into CE [19] 
* CE 11 is refined into CE [20,21,22] 
* CE 10 is refined into CE [23] 


### Cost equations --> "Loop" of encArg/2 
* CEs [23] --> Loop 13 
* CEs [22] --> Loop 14 
* CEs [20] --> Loop 15 
* CEs [21] --> Loop 16 
* CEs [19] --> Loop 17 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [13,14,15,16]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [13,14,15,16]:
  - RF of loop [13:1,14:1,14:2,15:1,15:2,16:1,16:2]:
    V1


### Specialization of cost equations fun/3 
* CE 12 is refined into CE [24,25,26,27,28,29,30,31,32,33] 
* CE 13 is refined into CE [34] 


### Cost equations --> "Loop" of fun/3 
* CEs [33] --> Loop 18 
* CEs [29,31] --> Loop 19 
* CEs [28] --> Loop 20 
* CEs [24,25,26,27,30,32,34] --> Loop 21 

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

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


### Specialization of cost equations fun2/2 
* CE 14 is refined into CE [35,36] 
* CE 15 is refined into CE [37] 


### Cost equations --> "Loop" of fun2/2 
* CEs [36] --> Loop 22 
* CEs [35] --> Loop 23 
* CEs [37] --> Loop 24 

### Ranking functions of CR fun2(V1,Out) 

#### Partial ranking functions of CR fun2(V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [38,39,40] 
* CE 2 is refined into CE [41,42] 
* CE 3 is refined into CE [43,44,45,46] 
* CE 4 is refined into CE [47] 
* CE 5 is refined into CE [48,49,50] 


### Cost equations --> "Loop" of start/2 
* CEs [38,39,40,41,42,43,44,45,46,47,48,49,50] --> Loop 25 

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

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


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

#### Cost of chains of f(V1,V,Out):
* Chain [[10],12]: 1*it(10)+1
  Such that:it(10) =< V

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

* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< V

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

* Chain [12]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

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


#### Cost of chains of encArg(V1,Out):
* Chain [17]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([13,14,15,16],[[17]])]: 1*it(14)+1*it(15)+1*s(6)+1*s(7)+0
  Such that:aux(7) =< V1
aux(8) =< 2/3*V1
it(13) =< aux(7)
it(14) =< aux(7)
it(15) =< aux(7)
it(14) =< aux(8)
aux(3) =< aux(7)
s(7) =< it(13)*aux(3)
s(6) =< it(14)*aux(7)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [21]: 3*s(12)+5*s(13)+3*s(15)+3*s(16)+2*s(29)+2*s(30)+2*s(32)+2*s(33)+1
  Such that:aux(11) =< V1
aux(12) =< 2/3*V1
aux(13) =< V
aux(14) =< 2/3*V
s(29) =< aux(11)
s(30) =< aux(11)
s(29) =< aux(12)
s(31) =< aux(11)
s(32) =< aux(11)*s(31)
s(33) =< s(29)*aux(11)
s(13) =< aux(13)
s(12) =< aux(13)
s(12) =< aux(14)
s(14) =< aux(13)
s(15) =< aux(13)*s(14)
s(16) =< s(12)*aux(13)

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

* Chain [20]: 1*s(55)+2*s(56)+1*s(58)+1*s(59)+1
  Such that:s(53) =< 2/3*V
aux(15) =< V
s(56) =< aux(15)
s(55) =< aux(15)
s(55) =< s(53)
s(57) =< aux(15)
s(58) =< aux(15)*s(57)
s(59) =< s(55)*aux(15)

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

* Chain [19]: 2*s(64)+2*s(65)+2*s(67)+2*s(68)+1*s(80)+1*s(81)+1*s(83)+1*s(84)+1
  Such that:s(77) =< V
s(78) =< 2/3*V
aux(16) =< V1
aux(17) =< 2/3*V1
s(64) =< aux(16)
s(65) =< aux(16)
s(64) =< aux(17)
s(66) =< aux(16)
s(67) =< aux(16)*s(66)
s(68) =< s(64)*aux(16)
s(80) =< s(77)
s(81) =< s(77)
s(80) =< s(78)
s(82) =< s(77)
s(83) =< s(77)*s(82)
s(84) =< s(80)*s(77)

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

* Chain [18]: 1*s(88)+1*s(89)+1*s(91)+1*s(92)+1*s(96)+2*s(97)+1*s(99)+1*s(100)+1
  Such that:s(85) =< V1
s(86) =< 2/3*V1
s(94) =< 2/3*V
aux(18) =< V
s(97) =< aux(18)
s(96) =< aux(18)
s(96) =< s(94)
s(98) =< aux(18)
s(99) =< aux(18)*s(98)
s(100) =< s(96)*aux(18)
s(88) =< s(85)
s(89) =< s(85)
s(88) =< s(86)
s(90) =< s(85)
s(91) =< s(85)*s(90)
s(92) =< s(88)*s(85)

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


#### Cost of chains of fun2(V1,Out):
* Chain [24]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [23]: 0
  with precondition: [Out=1,V1>=0] 

* Chain [22]: 1*s(105)+1*s(106)+1*s(108)+1*s(109)+0
  Such that:s(102) =< V1
s(103) =< 2/3*V1
s(105) =< s(102)
s(106) =< s(102)
s(105) =< s(103)
s(107) =< s(102)
s(108) =< s(102)*s(107)
s(109) =< s(105)*s(102)

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


#### Cost of chains of start(V1,V):
* Chain [25]: 12*s(110)+7*s(115)+7*s(116)+7*s(118)+7*s(119)+6*s(130)+6*s(132)+6*s(133)+1
  Such that:aux(19) =< V1
aux(20) =< 2/3*V1
aux(21) =< V
aux(22) =< 2/3*V
s(110) =< aux(21)
s(115) =< aux(19)
s(116) =< aux(19)
s(115) =< aux(20)
s(117) =< aux(19)
s(118) =< aux(19)*s(117)
s(119) =< s(115)*aux(19)
s(130) =< aux(21)
s(130) =< aux(22)
s(131) =< aux(21)
s(132) =< aux(21)*s(131)
s(133) =< s(130)*aux(21)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [25] with precondition: [] 
    - Upper bound: nat(V1)*14+1+nat(V1)*14*nat(V1)+nat(V)*18+nat(V)*12*nat(V) 
    - Complexity: n^2 

### Maximum cost of start(V1,V): nat(V1)*14+1+nat(V1)*14*nat(V1)+nat(V)*18+nat(V)*12*nat(V) 
Asymptotic class: n^2 
* Total analysis performed in 310 ms.


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

(14)
BOUNDS(1, n^2)

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

(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
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(17) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

f(x, g(y)) ->^+ f(g(x), y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [y / g(y)].

The result substitution is [x / g(x)].




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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST