WORST_CASE(Omega(n^1),O(n^1))
proof of input_qMy1r9MVhF.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) CompleteCoflocoProof [FINISHED, 135 ms]
    (20) BOUNDS(1, n^1)
(21) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 21 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 773 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 20 ms]
            (38) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   IS_EMPTY_1, HD_1, TL_1, APPEND_2, IFAPPEND_3

Compound Symbols:   c, c1, c2, c3, c4_2, c5, c6_1, c7_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   TL(cons(z0, z1)) -> c3
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_2, c7_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   hd(cons(z0, z1)) -> z0
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> z1
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:   is_empty_1, tl_1

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))

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

   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> z1

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

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

(12)
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:

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]

The TRS has the following type information:
APPEND :: nil:cons -> a -> c4
c4 :: c7 -> c4
IFAPPEND :: nil:cons -> a -> false:true -> c7
is_empty :: nil:cons -> false:true
false :: false:true
c7 :: c4 -> c7
tl :: nil:cons -> nil:cons
nil :: nil:cons
true :: false:true
cons :: b -> nil:cons -> nil:cons

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

(15) 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:

   is_empty(v0) -> null_is_empty [0]
   tl(v0) -> null_tl [0]
   IFAPPEND(v0, v1, v2) -> null_IFAPPEND [0]

And the following fresh constants:    null_is_empty, null_tl, null_IFAPPEND, const, const1, const2

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

(16)
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:

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]
   is_empty(v0) -> null_is_empty [0]
   tl(v0) -> null_tl [0]
   IFAPPEND(v0, v1, v2) -> null_IFAPPEND [0]

The TRS has the following type information:
APPEND :: nil:cons:null_tl -> a -> c4
c4 :: c7:null_IFAPPEND -> c4
IFAPPEND :: nil:cons:null_tl -> a -> false:true:null_is_empty -> c7:null_IFAPPEND
is_empty :: nil:cons:null_tl -> false:true:null_is_empty
false :: false:true:null_is_empty
c7 :: c4 -> c7:null_IFAPPEND
tl :: nil:cons:null_tl -> nil:cons:null_tl
nil :: nil:cons:null_tl
true :: false:true:null_is_empty
cons :: b -> nil:cons:null_tl -> nil:cons:null_tl
null_is_empty :: false:true:null_is_empty
null_tl :: nil:cons:null_tl
null_IFAPPEND :: c7:null_IFAPPEND
const :: c4
const1 :: a
const2 :: b

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

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

   false => 1
   nil => 0
   true => 2
   null_is_empty => 0
   null_tl => 0
   null_IFAPPEND => 0
   const => 0
   const1 => 0
   const2 => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z0, z1, is_empty(z0)) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   IFAPPEND(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(tl(z0), z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun1(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun2(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V4),0,[tl(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[fun2(V3, Ret12),fun1(V3, V2, Ret12, Ret1)],[Out = 1 + Ret1,V1 = V3,V2 >= 0,V = V2,V3 >= 0]).
eq(fun1(V1, V, V4, Out),1,[tl(V6, Ret10),fun(Ret10, V5, Ret11)],[Out = 1 + Ret11,V1 = V6,V5 >= 0,V = V5,V6 >= 0,V4 = 1]).
eq(fun2(V1, Out),0,[],[Out = 2,V1 = 0]).
eq(fun2(V1, Out),0,[],[Out = 1,V7 >= 0,V8 >= 0,V1 = 1 + V7 + V8]).
eq(tl(V1, Out),0,[],[Out = V10,V10 >= 0,V9 >= 0,V1 = 1 + V10 + V9]).
eq(fun2(V1, Out),0,[],[Out = 0,V11 >= 0,V1 = V11]).
eq(tl(V1, Out),0,[],[Out = 0,V12 >= 0,V1 = V12]).
eq(fun1(V1, V, V4, Out),0,[],[Out = 0,V14 >= 0,V4 = V15,V13 >= 0,V1 = V14,V = V13,V15 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun2(V1,Out),[V1],[Out]).
input_output_vars(tl(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [tl/2]
1. non_recursive  : [fun2/2]
2. recursive  : [fun/3,fun1/4]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into tl/2
1. SCC is partially evaluated into fun2/2
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations tl/2 
* CE 6 is refined into CE [13] 
* CE 7 is refined into CE [14] 


### Cost equations --> "Loop" of tl/2 
* CEs [13] --> Loop 9 
* CEs [14] --> Loop 10 

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

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


### Specialization of cost equations fun2/2 
* CE 11 is refined into CE [15] 
* CE 12 is refined into CE [16] 
* CE 10 is refined into CE [17] 


### Cost equations --> "Loop" of fun2/2 
* CEs [15] --> Loop 11 
* CEs [16] --> Loop 12 
* CEs [17] --> Loop 13 

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

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


### Specialization of cost equations fun/3 
* CE 9 is refined into CE [18,19] 
* CE 8 is refined into CE [20,21,22] 


### Cost equations --> "Loop" of fun/3 
* CEs [20,21,22] --> Loop 14 
* CEs [18,19] --> Loop 15 

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

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [23] 
* CE 2 is refined into CE [24,25,26] 
* CE 3 is refined into CE [27,28] 
* CE 4 is refined into CE [29,30,31] 
* CE 5 is refined into CE [32,33] 


### Cost equations --> "Loop" of start/3 
* CEs [24,25,26] --> Loop 16 
* CEs [23,27,28,29,30,31,32,33] --> Loop 17 

### Ranking functions of CR start(V1,V,V4) 

#### Partial ranking functions of CR start(V1,V,V4) 


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

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

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


#### Cost of chains of fun2(V1,Out):
* Chain [13]: 0
  with precondition: [V1=0,Out=2] 

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

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


#### Cost of chains of fun(V1,V,Out):
* Chain [[15],14]: 2*it(15)+1
  Such that:it(15) =< V1

  with precondition: [V>=0,Out>=3,2*V1+1>=Out] 

* Chain [14]: 1
  with precondition: [Out=1,V1>=0,V>=0] 


#### Cost of chains of start(V1,V,V4):
* Chain [17]: 2*s(1)+1
  Such that:s(1) =< V1

  with precondition: [V1>=0] 

* Chain [16]: 2*s(2)+2
  Such that:s(2) =< V1

  with precondition: [V4=1,V1>=0,V>=0] 


Closed-form bounds of start(V1,V,V4): 
-------------------------------------
* Chain [17] with precondition: [V1>=0] 
    - Upper bound: 2*V1+1 
    - Complexity: n 
* Chain [16] with precondition: [V4=1,V1>=0,V>=0] 
    - Upper bound: 2*V1+2 
    - Complexity: n 

### Maximum cost of start(V1,V,V4): 2*V1+2 
Asymptotic class: n 
* Total analysis performed in 94 ms.


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

(20)
BOUNDS(1, n^1)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   IS_EMPTY_1, HD_1, TL_1, APPEND_2, IFAPPEND_3

Compound Symbols:   c, c1, c2, c3, c4_2, c5, c6_1, c7_2


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

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

(24)
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:

   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))

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

   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

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

(26)
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:

   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))

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

   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

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

(28)
Obligation:
Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons

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

(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
APPEND, append
----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons10_8(0) <=> nil
gen_nil:cons10_8(+(x, 1)) <=> cons(hole_hd3_8, gen_nil:cons10_8(x))


The following defined symbols remain to be analysed:
APPEND, append
----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APPEND(gen_nil:cons10_8(n12_8), hole_a7_8) -> *11_8, rt in Omega(n12_8)

Induction Base:
APPEND(gen_nil:cons10_8(0), hole_a7_8)

Induction Step:
APPEND(gen_nil:cons10_8(+(n12_8, 1)), hole_a7_8) ->_R^Omega(1)
c4(IFAPPEND(gen_nil:cons10_8(+(n12_8, 1)), hole_a7_8, is_empty(gen_nil:cons10_8(+(n12_8, 1)))), IS_EMPTY(gen_nil:cons10_8(+(n12_8, 1)))) ->_R^Omega(0)
c4(IFAPPEND(gen_nil:cons10_8(+(1, n12_8)), hole_a7_8, false), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(1)
c4(c7(APPEND(tl(gen_nil:cons10_8(+(1, n12_8))), hole_a7_8), TL(gen_nil:cons10_8(+(1, n12_8)))), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(0)
c4(c7(APPEND(gen_nil:cons10_8(n12_8), hole_a7_8), TL(gen_nil:cons10_8(+(1, n12_8)))), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_IH
c4(c7(*11_8, TL(gen_nil:cons10_8(+(1, n12_8)))), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(1)
c4(c7(*11_8, c3), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(1)
c4(c7(*11_8, c3), c1)

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

(32)
Complex Obligation (BEST)

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

(33)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons10_8(0) <=> nil
gen_nil:cons10_8(+(x, 1)) <=> cons(hole_hd3_8, gen_nil:cons10_8(x))


The following defined symbols remain to be analysed:
APPEND, append
----------------------------------------

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons


Lemmas:
APPEND(gen_nil:cons10_8(n12_8), hole_a7_8) -> *11_8, rt in Omega(n12_8)


Generator Equations:
gen_nil:cons10_8(0) <=> nil
gen_nil:cons10_8(+(x, 1)) <=> cons(hole_hd3_8, gen_nil:cons10_8(x))


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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_nil:cons10_8(n1460_8), gen_nil:cons10_8(b)) -> gen_nil:cons10_8(+(n1460_8, b)), rt in Omega(0)

Induction Base:
append(gen_nil:cons10_8(0), gen_nil:cons10_8(b)) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(0), gen_nil:cons10_8(b), is_empty(gen_nil:cons10_8(0))) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(0), gen_nil:cons10_8(b), true) ->_R^Omega(0)
gen_nil:cons10_8(b)

Induction Step:
append(gen_nil:cons10_8(+(n1460_8, 1)), gen_nil:cons10_8(b)) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(+(n1460_8, 1)), gen_nil:cons10_8(b), is_empty(gen_nil:cons10_8(+(n1460_8, 1)))) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(+(1, n1460_8)), gen_nil:cons10_8(b), false) ->_R^Omega(0)
cons(hd(gen_nil:cons10_8(+(1, n1460_8))), append(tl(gen_nil:cons10_8(+(1, n1460_8))), gen_nil:cons10_8(b))) ->_R^Omega(0)
cons(hole_hd3_8, append(tl(gen_nil:cons10_8(+(1, n1460_8))), gen_nil:cons10_8(b))) ->_R^Omega(0)
cons(hole_hd3_8, append(gen_nil:cons10_8(n1460_8), gen_nil:cons10_8(b))) ->_IH
cons(hole_hd3_8, gen_nil:cons10_8(+(b, c1461_8)))

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

(38)
BOUNDS(1, INF)