WORST_CASE(Omega(n^1),O(n^1))
proof of input_Vfa5qfCile.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 209 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 694 ms]
    (12) BOUNDS(1, n^1)
(13) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
    (15) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 289 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 126 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 446 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 29 ms]
            (34) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (UPPER BOUND(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^1).


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

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:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

The TRS has the following type information:
a :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z

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:

   and(v0, v1) -> null_and [0]
   second(v0) -> null_second [0]
   first(v0) -> null_first [0]
   a(v0, v1) -> null_a [0]
   eqZList(v0, v1) -> null_eqZList [0]

And the following fresh constants:    null_and, null_second, null_first, null_a, null_eqZList

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

(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:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]
   second(v0) -> null_second [0]
   first(v0) -> null_first [0]
   a(v0, v1) -> null_a [0]
   eqZList(v0, v1) -> null_eqZList [0]

The TRS has the following type information:
a :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
C :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
Z :: C:Z:null_second:null_first:null_a
eqZList :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> False:True:null_and:null_eqZList
and :: False:True:null_and:null_eqZList -> False:True:null_and:null_eqZList -> False:True:null_and:null_eqZList
False :: False:True:null_and:null_eqZList
True :: False:True:null_and:null_eqZList
second :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
first :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
null_and :: False:True:null_and:null_eqZList
null_second :: C:Z:null_second:null_first:null_a
null_first :: C:Z:null_second:null_first:null_a
null_a :: C:Z:null_second:null_first:null_a
null_eqZList :: False:True:null_and:null_eqZList

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:

   Z => 0
   False => 1
   True => 2
   null_and => 0
   null_second => 0
   null_first => 0
   null_a => 0
   null_eqZList => 0

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

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

   a(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   a(z, z') -{ 1 }-> 1 + a(x1, y) + a(x2, 1 + x1 + x2) :|: x1 >= 0, y >= 0, z = 1 + x1 + x2, z' = y, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   eqZList(z, z') -{ 1 }-> and(eqZList(x1, y1), eqZList(x2, y2)) :|: y1 >= 0, x1 >= 0, z' = 1 + y1 + y2, z = 1 + x1 + x2, y2 >= 0, x2 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   first(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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),0,[a(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[eqZList(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[second(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[first(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[and(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(a(V1, V, Out),1,[a(V4, V2, Ret01),a(V3, 1 + V4 + V3, Ret1)],[Out = 1 + Ret01 + Ret1,V4 >= 0,V2 >= 0,V1 = 1 + V3 + V4,V = V2,V3 >= 0]).
eq(a(V1, V, Out),1,[],[Out = 0,V5 >= 0,V1 = 0,V = V5]).
eq(eqZList(V1, V, Out),1,[eqZList(V8, V9, Ret0),eqZList(V6, V7, Ret11),and(Ret0, Ret11, Ret)],[Out = Ret,V9 >= 0,V8 >= 0,V = 1 + V7 + V9,V1 = 1 + V6 + V8,V7 >= 0,V6 >= 0]).
eq(eqZList(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = 1 + V10 + V11,V11 >= 0,V = 0]).
eq(eqZList(V1, V, Out),1,[],[Out = 1,V12 >= 0,V = 1 + V12 + V13,V13 >= 0,V1 = 0]).
eq(eqZList(V1, V, Out),1,[],[Out = 2,V1 = 0,V = 0]).
eq(second(V1, Out),1,[],[Out = V14,V15 >= 0,V1 = 1 + V14 + V15,V14 >= 0]).
eq(first(V1, Out),1,[],[Out = V17,V17 >= 0,V1 = 1 + V16 + V17,V16 >= 0]).
eq(and(V1, V, Out),0,[],[Out = 1,V1 = 1,V = 1]).
eq(and(V1, V, Out),0,[],[Out = 1,V1 = 2,V = 1]).
eq(and(V1, V, Out),0,[],[Out = 1,V = 2,V1 = 1]).
eq(and(V1, V, Out),0,[],[Out = 2,V1 = 2,V = 2]).
eq(and(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
eq(second(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(first(V1, Out),0,[],[Out = 0,V21 >= 0,V1 = V21]).
eq(a(V1, V, Out),0,[],[Out = 0,V22 >= 0,V23 >= 0,V1 = V22,V = V23]).
eq(eqZList(V1, V, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V24,V = V25]).
input_output_vars(a(V1,V,Out),[V1,V],[Out]).
input_output_vars(eqZList(V1,V,Out),[V1,V],[Out]).
input_output_vars(second(V1,Out),[V1],[Out]).
input_output_vars(first(V1,Out),[V1],[Out]).
input_output_vars(and(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [a/3]
1. non_recursive  : [and/3]
2. recursive [non_tail,multiple] : [eqZList/3]
3. non_recursive  : [first/2]
4. non_recursive  : [second/2]
5. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into a/3
1. SCC is partially evaluated into and/3
2. SCC is partially evaluated into eqZList/3
3. SCC is partially evaluated into first/2
4. SCC is partially evaluated into second/2
5. SCC is partially evaluated into start/2

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

### Specialization of cost equations a/3 
* CE 7 is refined into CE [23] 
* CE 8 is refined into CE [24] 
* CE 6 is refined into CE [25] 


### Cost equations --> "Loop" of a/3 
* CEs [25] --> Loop 18 
* CEs [23,24] --> Loop 19 

### Ranking functions of CR a(V1,V,Out) 
* RF of phase [18]: [V1]

#### Partial ranking functions of CR a(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1,18:2]:
    V1


### Specialization of cost equations and/3 
* CE 22 is refined into CE [26] 
* CE 21 is refined into CE [27] 
* CE 19 is refined into CE [28] 
* CE 20 is refined into CE [29] 
* CE 18 is refined into CE [30] 


### Cost equations --> "Loop" of and/3 
* CEs [26] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [30] --> Loop 24 

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

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


### Specialization of cost equations eqZList/3 
* CE 13 is refined into CE [31] 
* CE 10 is refined into CE [32] 
* CE 11 is refined into CE [33] 
* CE 12 is refined into CE [34] 
* CE 9 is refined into CE [35,36,37,38,39] 


### Cost equations --> "Loop" of eqZList/3 
* CEs [38] --> Loop 25 
* CEs [37] --> Loop 26 
* CEs [36] --> Loop 27 
* CEs [35] --> Loop 28 
* CEs [39] --> Loop 29 
* CEs [31] --> Loop 30 
* CEs [32] --> Loop 31 
* CEs [33] --> Loop 32 
* CEs [34] --> Loop 33 

### Ranking functions of CR eqZList(V1,V,Out) 
* RF of phase [25]: [V,V1]
* RF of phase [26,27,28]: [V,V1]
* RF of phase [29]: [V,V1]

#### Partial ranking functions of CR eqZList(V1,V,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1,25:2]:
    V
    V1
* Partial RF of phase [26,27,28]:
  - RF of loop [26:1,26:2,27:1,27:2,28:1,28:2]:
    V
    V1
* Partial RF of phase [29]:
  - RF of loop [29:1,29:2]:
    V
    V1


### Specialization of cost equations first/2 
* CE 16 is refined into CE [40] 
* CE 17 is refined into CE [41] 


### Cost equations --> "Loop" of first/2 
* CEs [40] --> Loop 34 
* CEs [41] --> Loop 35 

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

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


### Specialization of cost equations second/2 
* CE 14 is refined into CE [42] 
* CE 15 is refined into CE [43] 


### Cost equations --> "Loop" of second/2 
* CEs [42] --> Loop 36 
* CEs [43] --> Loop 37 

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [44,45] 
* CE 2 is refined into CE [46,47,48,49,50,51] 
* CE 3 is refined into CE [52,53] 
* CE 4 is refined into CE [54,55] 
* CE 5 is refined into CE [56,57,58,59,60] 


### Cost equations --> "Loop" of start/2 
* CEs [48] --> Loop 38 
* CEs [59] --> Loop 39 
* CEs [58] --> Loop 40 
* CEs [57] --> Loop 41 
* CEs [51,56] --> Loop 42 
* CEs [47] --> Loop 43 
* CEs [44,45,46,49,50,52,53,54,55,60] --> Loop 44 

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

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


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

#### Cost of chains of a(V1,V,Out):
* Chain [19]: 1
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [multiple([18],[[19]])]: 1*it(18)+1*it([19])+0
  Such that:it(18) =< V1
it([19]) =< V1+1

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


#### Cost of chains of and(V1,V,Out):
* Chain [24]: 0
  with precondition: [V1=1,V=1,Out=1] 

* Chain [23]: 0
  with precondition: [V1=1,V=2,Out=1] 

* Chain [22]: 0
  with precondition: [V1=2,V=1,Out=1] 

* Chain [21]: 0
  with precondition: [V1=2,V=2,Out=2] 

* Chain [20]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of eqZList(V1,V,Out):
* Chain [33]: 1
  with precondition: [V1=0,V=0,Out=2] 

* Chain [32]: 1
  with precondition: [V1=0,Out=1,V>=1] 

* Chain [31]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [30]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [multiple([29],[[multiple([26,27,28],[[multiple([25],[[33]])],[33],[32],[31]])],[multiple([25],[[33]])],[33],[32],[31],[30]])]: 1*it(29)+1*it([31])+1*it([32])+1*it([33])+5*s(1)+1*s(3)+1*s(4)+1*s(5)+1*s(6)+1*s(7)+0
  Such that:aux(11) =< V1
aux(12) =< V1+1
aux(13) =< V1/2+1/2
aux(14) =< V1/3+V/3+2/3
aux(15) =< V+1
aux(16) =< V/2+1/2
it(29) =< aux(11)
it([31]) =< aux(11)
it([31]) =< aux(12)
it([32]) =< aux(12)
it([33]) =< aux(12)
it([31]) =< aux(13)
it([33]) =< aux(15)
s(1) =< aux(15)
it([32]) =< aux(16)
s(11) =< aux(15)*(1/2)
s(9) =< aux(14)*(6/5)
s(6) =< aux(13)
s(5) =< aux(14)
s(3) =< s(9)
s(4) =< s(9)
s(5) =< s(9)
s(4) =< aux(15)
s(5) =< aux(15)
s(7) =< aux(15)
s(7) =< s(11)
s(5) =< s(1)*(1/3)+aux(14)
s(3) =< s(1)*(1/5)+s(9)
s(4) =< s(1)*(1/5)+s(9)
s(5) =< s(1)*(1/5)+s(9)

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

* Chain [multiple([26,27,28],[[multiple([25],[[33]])],[33],[32],[31]])]: 1*it(26)+1*it(27)+1*it(28)+1*it([31])+1*it([32])+3*it([33])+0
  Such that:aux(3) =< V1/2+1/2
aux(4) =< V1/3+V/3
aux(5) =< 2/5*V1+2/5*V
aux(6) =< V
aux(7) =< V+1
aux(8) =< V/2+1/2
it([31]) =< aux(3)
it(28) =< aux(4)
it(26) =< aux(5)
it(27) =< aux(5)
it(28) =< aux(5)
it(27) =< aux(6)
it(28) =< aux(6)
it([32]) =< aux(6)
it([33]) =< aux(7)
it([32]) =< aux(8)
it(28) =< it([33])*(1/3)+aux(4)
it(26) =< it([33])*(1/5)+aux(5)
it(27) =< it([33])*(1/5)+aux(5)
it(28) =< it([33])*(1/5)+aux(5)

  with precondition: [Out=1,V1>=1,V>=1,V+V1>=3] 

* Chain [multiple([25],[[33]])]: 1*it(25)+1*it([33])+0
  Such that:it(25) =< V
it([33]) =< V+1

  with precondition: [Out=2,V1=V,V1>=1] 


#### Cost of chains of first(V1,Out):
* Chain [35]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [34]: 1
  with precondition: [Out>=0,V1>=Out+1] 


#### Cost of chains of second(V1,Out):
* Chain [37]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [36]: 1
  with precondition: [Out>=0,V1>=Out+1] 


#### Cost of chains of start(V1,V):
* Chain [44]: 2*s(33)+1*s(34)+1*s(42)+1*s(43)+1*s(44)+8*s(45)+2*s(48)+1*s(49)+1*s(50)+1*s(51)+1*s(52)+1*s(60)+1*s(61)+1*s(62)+1*s(63)+1
  Such that:s(54) =< V1/3+V/3
s(38) =< V1/3+V/3+2/3
s(55) =< 2/5*V1+2/5*V
s(56) =< V
aux(17) =< V1
aux(18) =< V1+1
aux(19) =< V1/2+1/2
aux(20) =< V+1
aux(21) =< V/2+1/2
s(33) =< aux(17)
s(34) =< aux(18)
s(42) =< aux(17)
s(42) =< aux(18)
s(43) =< aux(18)
s(44) =< aux(18)
s(42) =< aux(19)
s(44) =< aux(20)
s(45) =< aux(20)
s(43) =< aux(21)
s(46) =< aux(20)*(1/2)
s(47) =< s(38)*(6/5)
s(48) =< aux(19)
s(49) =< s(38)
s(50) =< s(47)
s(51) =< s(47)
s(49) =< s(47)
s(51) =< aux(20)
s(49) =< aux(20)
s(52) =< aux(20)
s(52) =< s(46)
s(49) =< s(45)*(1/3)+s(38)
s(50) =< s(45)*(1/5)+s(47)
s(51) =< s(45)*(1/5)+s(47)
s(49) =< s(45)*(1/5)+s(47)
s(60) =< s(54)
s(61) =< s(55)
s(62) =< s(55)
s(60) =< s(55)
s(62) =< s(56)
s(60) =< s(56)
s(63) =< s(56)
s(63) =< aux(21)
s(60) =< s(45)*(1/3)+s(54)
s(61) =< s(45)*(1/5)+s(55)
s(62) =< s(45)*(1/5)+s(55)
s(60) =< s(45)*(1/5)+s(55)

  with precondition: [V1>=0] 

* Chain [43]: 1
  with precondition: [V1=0,V>=1] 

* Chain [42]: 1*s(65)+1*s(66)+0
  Such that:s(65) =< V
s(66) =< V+1

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

* Chain [41]: 0
  with precondition: [V1=1,V=2] 

* Chain [40]: 0
  with precondition: [V1=2,V=1] 

* Chain [39]: 0
  with precondition: [V1=2,V=2] 

* Chain [38]: 1
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [44] with precondition: [V1>=0] 
    - Upper bound: 3*V1+1+nat(V)+(3*V1+3)+nat(V+1)*9+nat(2/5*V1+2/5*V)*2+17/5*nat(V1/3+V/3+2/3)+(V1+1)+nat(V1/3+V/3) 
    - Complexity: n 
* Chain [43] with precondition: [V1=0,V>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [42] with precondition: [V1=V,V1>=1] 
    - Upper bound: 2*V+1 
    - Complexity: n 
* Chain [41] with precondition: [V1=1,V=2] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [40] with precondition: [V1=2,V=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [39] with precondition: [V1=2,V=2] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [38] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V): max([1,6*V1+4+nat(V+1)*8+nat(2/5*V1+2/5*V)*2+17/5*nat(V1/3+V/3+2/3)+(V1+1)+nat(V1/3+V/3)+(nat(V+1)+nat(V))]) 
Asymptotic class: n 
* Total analysis performed in 549 ms.


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

(12)
BOUNDS(1, n^1)

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

(13) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0
Tuples:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13
K tuples:none
Defined Rule Symbols:   a_2, eqZList_2, second_1, first_1, and_2

Defined Pair Symbols:   AND_2, A_2, EQZLIST_2, SECOND_1, FIRST_1

Compound Symbols:   c, c1, c2, c3, c4_1, c5_1, c6, c7_2, c8_2, c9, c10, c11, c12, c13


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

(15) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

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

(17) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

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

(19) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
A(C(z0, z1), z2) -> c4(A(z0, z2))
A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
A(Z, z0) -> c6
EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
EQZLIST(C(z0, z1), Z) -> c9
EQZLIST(Z, C(z0, z1)) -> c10
EQZLIST(Z, Z) -> c11
SECOND(C(z0, z1)) -> c12
FIRST(C(z0, z1)) -> c13
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
a(Z, z0) -> Z
eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
eqZList(C(z0, z1), Z) -> False
eqZList(Z, C(z0, z1)) -> False
eqZList(Z, Z) -> True
second(C(z0, z1)) -> z1
first(C(z0, z1)) -> z0

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
A, EQZLIST, eqZList, a

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

(22)
Obligation:
Innermost TRS:
Rules:
A(C(z0, z1), z2) -> c4(A(z0, z2))
A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
A(Z, z0) -> c6
EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
EQZLIST(C(z0, z1), Z) -> c9
EQZLIST(Z, C(z0, z1)) -> c10
EQZLIST(Z, Z) -> c11
SECOND(C(z0, z1)) -> c12
FIRST(C(z0, z1)) -> c13
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
a(Z, z0) -> Z
eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
eqZList(C(z0, z1), Z) -> False
eqZList(Z, C(z0, z1)) -> False
eqZList(Z, Z) -> True
second(C(z0, z1)) -> z1
first(C(z0, z1)) -> z0

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


The following defined symbols remain to be analysed:
A, EQZLIST, eqZList, a

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)

Induction Base:
A(gen_C:Z9_14(0), gen_C:Z9_14(b)) ->_R^Omega(1)
c6

Induction Step:
A(gen_C:Z9_14(+(n12_14, 1)), gen_C:Z9_14(b)) ->_R^Omega(1)
c4(A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b))) ->_IH
c4(gen_c4:c5:c68_14(c13_14))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
A(C(z0, z1), z2) -> c4(A(z0, z2))
A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
A(Z, z0) -> c6
EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
EQZLIST(C(z0, z1), Z) -> c9
EQZLIST(Z, C(z0, z1)) -> c10
EQZLIST(Z, Z) -> c11
SECOND(C(z0, z1)) -> c12
FIRST(C(z0, z1)) -> c13
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
a(Z, z0) -> Z
eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
eqZList(C(z0, z1), Z) -> False
eqZList(Z, C(z0, z1)) -> False
eqZList(Z, Z) -> True
second(C(z0, z1)) -> z1
first(C(z0, z1)) -> z0

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


The following defined symbols remain to be analysed:
A, EQZLIST, eqZList, a

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
A(C(z0, z1), z2) -> c4(A(z0, z2))
A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
A(Z, z0) -> c6
EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
EQZLIST(C(z0, z1), Z) -> c9
EQZLIST(Z, C(z0, z1)) -> c10
EQZLIST(Z, Z) -> c11
SECOND(C(z0, z1)) -> c12
FIRST(C(z0, z1)) -> c13
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
a(Z, z0) -> Z
eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
eqZList(C(z0, z1), Z) -> False
eqZList(Z, C(z0, z1)) -> False
eqZList(Z, Z) -> True
second(C(z0, z1)) -> z1
first(C(z0, z1)) -> z0

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Lemmas:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


The following defined symbols remain to be analysed:
eqZList, EQZLIST, a

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)) -> False, rt in Omega(0)

Induction Base:
eqZList(gen_C:Z9_14(+(1, 0)), gen_C:Z9_14(0)) ->_R^Omega(0)
False

Induction Step:
eqZList(gen_C:Z9_14(+(1, +(n675_14, 1))), gen_C:Z9_14(+(n675_14, 1))) ->_R^Omega(0)
and(eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)), eqZList(Z, Z)) ->_IH
and(False, eqZList(Z, Z)) ->_R^Omega(0)
and(False, True) ->_R^Omega(0)
False

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
A(C(z0, z1), z2) -> c4(A(z0, z2))
A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
A(Z, z0) -> c6
EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
EQZLIST(C(z0, z1), Z) -> c9
EQZLIST(Z, C(z0, z1)) -> c10
EQZLIST(Z, Z) -> c11
SECOND(C(z0, z1)) -> c12
FIRST(C(z0, z1)) -> c13
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
a(Z, z0) -> Z
eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
eqZList(C(z0, z1), Z) -> False
eqZList(Z, C(z0, z1)) -> False
eqZList(Z, Z) -> True
second(C(z0, z1)) -> z1
first(C(z0, z1)) -> z0

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Lemmas:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)
eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)) -> False, rt in Omega(0)


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


The following defined symbols remain to be analysed:
EQZLIST, a
----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14))) -> *11_14, rt in Omega(n2887_14)

Induction Base:
EQZLIST(gen_C:Z9_14(+(2, 0)), gen_C:Z9_14(+(1, 0)))

Induction Step:
EQZLIST(gen_C:Z9_14(+(2, +(n2887_14, 1))), gen_C:Z9_14(+(1, +(n2887_14, 1)))) ->_R^Omega(1)
c7(AND(eqZList(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14))), eqZList(Z, Z)), EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_L^Omega(0)
c7(AND(False, eqZList(Z, Z)), EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_R^Omega(0)
c7(AND(False, True), EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_R^Omega(0)
c7(c2, EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_IH
c7(c2, *11_14)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
A(C(z0, z1), z2) -> c4(A(z0, z2))
A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
A(Z, z0) -> c6
EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
EQZLIST(C(z0, z1), Z) -> c9
EQZLIST(Z, C(z0, z1)) -> c10
EQZLIST(Z, Z) -> c11
SECOND(C(z0, z1)) -> c12
FIRST(C(z0, z1)) -> c13
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
a(Z, z0) -> Z
eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
eqZList(C(z0, z1), Z) -> False
eqZList(Z, C(z0, z1)) -> False
eqZList(Z, Z) -> True
second(C(z0, z1)) -> z1
first(C(z0, z1)) -> z0

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Lemmas:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)
eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)) -> False, rt in Omega(0)
EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14))) -> *11_14, rt in Omega(n2887_14)


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


The following defined symbols remain to be analysed:
a
----------------------------------------

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_C:Z9_14(n5491_14), gen_C:Z9_14(b)) -> gen_C:Z9_14(n5491_14), rt in Omega(0)

Induction Base:
a(gen_C:Z9_14(0), gen_C:Z9_14(b)) ->_R^Omega(0)
Z

Induction Step:
a(gen_C:Z9_14(+(n5491_14, 1)), gen_C:Z9_14(b)) ->_R^Omega(0)
C(a(gen_C:Z9_14(n5491_14), gen_C:Z9_14(b)), a(Z, C(gen_C:Z9_14(n5491_14), Z))) ->_IH
C(gen_C:Z9_14(c5492_14), a(Z, C(gen_C:Z9_14(n5491_14), Z))) ->_R^Omega(0)
C(gen_C:Z9_14(n5491_14), Z)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(34)
BOUNDS(1, INF)