WORST_CASE(Omega(n^1),O(n^2))
proof of input_dO4Nc3Ny6y.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 365 ms]
    (18) BOUNDS(1, n^2)
(19) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
    (21) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRelTRS
    (23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 21 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 390 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
            (36) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   int(x, y) -> if(le(x, y), x, y)
   if(true, x, y) -> cons(x, int(s(x), y))
   if(false, x, y) -> nil

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:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   int(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> cons(z0, int(s(z0), z1))
   if(false, z0, z1) -> nil
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
   IF(false, z0, z1) -> c5
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
   IF(false, z0, z1) -> c5
K tuples:none
Defined Rule Symbols:   le_2, int_2, if_3

Defined Pair Symbols:   LE_2, INT_2, IF_3

Compound Symbols:   c, c1, c2_1, c3_2, c4_1, c5


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   IF(false, z0, z1) -> c5
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   int(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> cons(z0, int(s(z0), z1))
   if(false, z0, z1) -> nil
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
K tuples:none
Defined Rule Symbols:   le_2, int_2, if_3

Defined Pair Symbols:   LE_2, INT_2, IF_3

Compound Symbols:   c2_1, c3_2, c4_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   int(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> cons(z0, int(s(z0), z1))
   if(false, z0, z1) -> nil

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
K tuples:none
Defined Rule Symbols:   le_2

Defined Pair Symbols:   LE_2, INT_2, IF_3

Compound Symbols:   c2_1, c3_2, c4_1


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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)

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

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

(10)
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:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1)) [1]
   IF(true, z0, z1) -> c4(INT(s(z0), z1)) [1]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1)) [1]
   IF(true, z0, z1) -> c4(INT(s(z0), z1)) [1]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]

The TRS has the following type information:
LE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
INT :: s:0 -> s:0 -> c3
c3 :: c4 -> c2 -> c3
IF :: true:false -> s:0 -> s:0 -> c4
le :: s:0 -> s:0 -> true:false
true :: true:false
c4 :: c3 -> c4
0 :: s:0
false :: true:false

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

(13) 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:

   le(v0, v1) -> null_le [0]
   LE(v0, v1) -> null_LE [0]
   IF(v0, v1, v2) -> null_IF [0]

And the following fresh constants:    null_le, null_LE, null_IF, const

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

(14)
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:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1)) [1]
   IF(true, z0, z1) -> c4(INT(s(z0), z1)) [1]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   le(v0, v1) -> null_le [0]
   LE(v0, v1) -> null_LE [0]
   IF(v0, v1, v2) -> null_IF [0]

The TRS has the following type information:
LE :: s:0 -> s:0 -> c2:null_LE
s :: s:0 -> s:0
c2 :: c2:null_LE -> c2:null_LE
INT :: s:0 -> s:0 -> c3
c3 :: c4:null_IF -> c2:null_LE -> c3
IF :: true:false:null_le -> s:0 -> s:0 -> c4:null_IF
le :: s:0 -> s:0 -> true:false:null_le
true :: true:false:null_le
c4 :: c3 -> c4:null_IF
0 :: s:0
false :: true:false:null_le
null_le :: true:false:null_le
null_LE :: c2:null_LE
null_IF :: c4:null_IF
const :: c3

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

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

   true => 2
   0 => 0
   false => 1
   null_le => 0
   null_LE => 0
   null_IF => 0
   const => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   IF(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + INT(1 + z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   INT(z, z') -{ 1 }-> 1 + IF(le(z0, z1), z0, z1) + LE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   LE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0
   le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V6),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun2(V1, V, V6, Out)],[V1 >= 0,V >= 0,V6 >= 0]).
eq(start(V1, V, V6),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[le(V5, V4, Ret010),fun2(Ret010, V5, V4, Ret01),fun(V5, V4, Ret11)],[Out = 1 + Ret01 + Ret11,V1 = V5,V4 >= 0,V = V4,V5 >= 0]).
eq(fun2(V1, V, V6, Out),1,[fun1(1 + V8, V7, Ret12)],[Out = 1 + Ret12,V1 = 2,V7 >= 0,V8 >= 0,V = V8,V6 = V7]).
eq(le(V1, V, Out),0,[],[Out = 2,V9 >= 0,V1 = 0,V = V9]).
eq(le(V1, V, Out),0,[],[Out = 1,V1 = 1 + V10,V10 >= 0,V = 0]).
eq(le(V1, V, Out),0,[le(V11, V12, Ret)],[Out = Ret,V12 >= 0,V1 = 1 + V11,V11 >= 0,V = 1 + V12]).
eq(le(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun2(V1, V, V6, Out),0,[],[Out = 0,V18 >= 0,V6 = V19,V17 >= 0,V1 = V18,V = V17,V19 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,V6,Out),[V1,V,V6],[Out]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into le/3
2. SCC is partially evaluated into fun1/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations fun/3 
* CE 9 is refined into CE [14] 
* CE 8 is refined into CE [15] 


### Cost equations --> "Loop" of fun/3 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 

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

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


### Specialization of cost equations le/3 
* CE 13 is refined into CE [16] 
* CE 11 is refined into CE [17] 
* CE 10 is refined into CE [18] 
* CE 12 is refined into CE [19] 


### Cost equations --> "Loop" of le/3 
* CEs [19] --> Loop 12 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR le(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations fun1/3 
* CE 7 is refined into CE [20,21,22] 
* CE 6 is refined into CE [23,24,25,26,27,28,29,30] 


### Cost equations --> "Loop" of fun1/3 
* CEs [28] --> Loop 16 
* CEs [26,30] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [23,25,27,29] --> Loop 19 
* CEs [22] --> Loop 20 
* CEs [21] --> Loop 21 
* CEs [20] --> Loop 22 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [20,21]: [-V1+V+1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [20,21]:
  - RF of loop [20:1,21:1]:
    -V1+V+1


### Specialization of cost equations start/3 
* CE 1 is refined into CE [31] 
* CE 2 is refined into CE [32,33,34] 
* CE 3 is refined into CE [35,36] 
* CE 4 is refined into CE [37,38,39,40,41,42] 
* CE 5 is refined into CE [43,44,45,46,47] 


### Cost equations --> "Loop" of start/3 
* CEs [44] --> Loop 23 
* CEs [32,33,34] --> Loop 24 
* CEs [31,35,36,37,38,39,40,41,42,43,45,46,47] --> Loop 25 

### Ranking functions of CR start(V1,V,V6) 

#### Partial ranking functions of CR start(V1,V,V6) 


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< Out

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

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


#### Cost of chains of le(V1,V,Out):
* Chain [[12],15]: 0
  with precondition: [Out=2,V1>=1,V>=V1] 

* Chain [[12],14]: 0
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[12],13]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [15]: 0
  with precondition: [V1=0,Out=2,V>=0] 

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

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,V,Out):
* Chain [[20,21],19]: 4*it(20)+1*s(3)+1
  Such that:aux(1) =< V+1
aux(4) =< -V1+V+1
it(20) =< aux(4)
s(3) =< it(20)*aux(1)

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

* Chain [[20,21],17]: 4*it(20)+1*s(3)+1*s(4)+1*s(5)+1
  Such that:s(4) =< V
aux(5) =< -V1+V+1
aux(6) =< V+1
s(5) =< aux(6)
it(20) =< aux(5)
s(3) =< it(20)*aux(6)

  with precondition: [V1>=1,Out>=4,V>=V1] 

* Chain [[20,21],16]: 4*it(20)+1*s(3)+1*s(6)+1
  Such that:s(6) =< V
aux(1) =< V+1
aux(7) =< -V1+V+1
it(20) =< aux(7)
s(3) =< it(20)*aux(1)

  with precondition: [V1>=1,V>=V1,Out+2*V1>=2*V+4] 

* Chain [22,[20,21],19]: 4*it(20)+1*s(3)+3
  Such that:aux(4) =< V
aux(1) =< V+1
it(20) =< aux(4)
s(3) =< it(20)*aux(1)

  with precondition: [V1=0,V>=1,Out>=5] 

* Chain [22,[20,21],17]: 5*it(20)+1*s(3)+1*s(5)+3
  Such that:aux(6) =< V+1
aux(8) =< V
it(20) =< aux(8)
s(5) =< aux(6)
s(3) =< it(20)*aux(6)

  with precondition: [V1=0,V>=1,Out>=6] 

* Chain [22,[20,21],16]: 5*it(20)+1*s(3)+3
  Such that:aux(1) =< V+1
aux(9) =< V
it(20) =< aux(9)
s(3) =< it(20)*aux(1)

  with precondition: [V1=0,V>=1,Out>=2*V+4] 

* Chain [22,19]: 3
  with precondition: [V1=0,Out=3,V>=0] 

* Chain [22,18]: 3
  with precondition: [V1=0,V=0,Out=3] 

* Chain [22,17]: 1*s(4)+1*s(5)+3
  Such that:s(5) =< 1
s(4) =< V

  with precondition: [V1=0,Out=4,V>=1] 

* Chain [19]: 1
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [18]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [17]: 1*s(4)+1*s(5)+1
  Such that:s(5) =< V1
s(4) =< V

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

* Chain [16]: 1*s(6)+1
  Such that:s(6) =< V

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


#### Cost of chains of start(V1,V,V6):
* Chain [25]: 20*s(38)+1*s(39)+3*s(44)+2*s(45)+12*s(50)+3*s(51)+1*s(53)+3
  Such that:s(39) =< 1
s(46) =< -V1+V+1
s(53) =< V1
aux(16) =< V
aux(17) =< V+1
s(38) =< aux(16)
s(50) =< s(46)
s(51) =< s(50)*aux(17)
s(45) =< aux(17)
s(44) =< s(38)*aux(17)

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

* Chain [24]: 4*s(59)+12*s(60)+3*s(61)+1*s(62)+1*s(63)+2
  Such that:s(56) =< -V+V6
s(63) =< V+1
s(58) =< V6+1
aux(18) =< V6
s(59) =< aux(18)
s(60) =< s(56)
s(61) =< s(60)*s(58)
s(62) =< s(58)

  with precondition: [V1=2,V>=0,V6>=0] 

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


Closed-form bounds of start(V1,V,V6): 
-------------------------------------
* Chain [25] with precondition: [V1>=0,V>=0] 
    - Upper bound: V1+20*V+4+(V+1)*(3*V)+(2*V+2)+(3*V+3)*nat(-V1+V+1)+nat(-V1+V+1)*12 
    - Complexity: n^2 
* Chain [24] with precondition: [V1=2,V>=0,V6>=0] 
    - Upper bound: V+5*V6+4+(3*V6+3)*nat(-V+V6)+nat(-V+V6)*12 
    - Complexity: n^2 
* Chain [23] with precondition: [V=0,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V6): V+3+max([nat(V6)*4+nat(V6+1)+nat(V6+1)*3*nat(-V+V6)+nat(-V+V6)*12,V1+20*V+2+(V+1)*(3*V)+(V+1)+(3*V+3)*nat(-V1+V+1)+nat(-V1+V+1)*12]) 
Asymptotic class: n^2 
* Total analysis performed in 315 ms.


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

(18)
BOUNDS(1, n^2)

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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   int(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> cons(z0, int(s(z0), z1))
   if(false, z0, z1) -> nil
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
   IF(false, z0, z1) -> c5
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
   IF(false, z0, z1) -> c5
K tuples:none
Defined Rule Symbols:   le_2, int_2, if_3

Defined Pair Symbols:   LE_2, INT_2, IF_3

Compound Symbols:   c, c1, c2_1, c3_2, c4_1, c5


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

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

(22)
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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
   IF(false, z0, z1) -> c5

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   int(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> cons(z0, int(s(z0), z1))
   if(false, z0, z1) -> nil

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

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

(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:

   LE(0', z0) -> c
   LE(s(z0), 0') -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c4(INT(s(z0), z1))
   IF(false, z0, z1) -> c5

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

   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   int(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> cons(z0, int(s(z0), z1))
   if(false, z0, z1) -> nil

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

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

(26)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c4(INT(s(z0), z1))
IF(false, z0, z1) -> c5
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
int(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> cons(z0, int(s(z0), z1))
if(false, z0, z1) -> nil

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INT :: 0':s -> 0':s -> c3
c3 :: c4:c5 -> c:c1:c2 -> c3
IF :: true:false -> 0':s -> 0':s -> c4:c5
le :: 0':s -> 0':s -> true:false
true :: true:false
c4 :: c3 -> c4:c5
false :: true:false
c5 :: c4:c5
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c:c1:c21_6 :: c:c1:c2
hole_0':s2_6 :: 0':s
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_true:false5_6 :: true:false
hole_cons:nil6_6 :: cons:nil
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_0':s8_6 :: Nat -> 0':s
gen_cons:nil9_6 :: Nat -> cons:nil

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, INT, le, int

They will be analysed ascendingly in the following order:
LE < INT
le < INT
le < int

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

(28)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c4(INT(s(z0), z1))
IF(false, z0, z1) -> c5
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
int(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> cons(z0, int(s(z0), z1))
if(false, z0, z1) -> nil

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INT :: 0':s -> 0':s -> c3
c3 :: c4:c5 -> c:c1:c2 -> c3
IF :: true:false -> 0':s -> 0':s -> c4:c5
le :: 0':s -> 0':s -> true:false
true :: true:false
c4 :: c3 -> c4:c5
false :: true:false
c5 :: c4:c5
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c:c1:c21_6 :: c:c1:c2
hole_0':s2_6 :: 0':s
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_true:false5_6 :: true:false
hole_cons:nil6_6 :: cons:nil
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_0':s8_6 :: Nat -> 0':s
gen_cons:nil9_6 :: Nat -> cons:nil


Generator Equations:
gen_c:c1:c27_6(0) <=> c
gen_c:c1:c27_6(+(x, 1)) <=> c2(gen_c:c1:c27_6(x))
gen_0':s8_6(0) <=> 0'
gen_0':s8_6(+(x, 1)) <=> s(gen_0':s8_6(x))
gen_cons:nil9_6(0) <=> nil
gen_cons:nil9_6(+(x, 1)) <=> cons(0', gen_cons:nil9_6(x))


The following defined symbols remain to be analysed:
LE, INT, le, int

They will be analysed ascendingly in the following order:
LE < INT
le < INT
le < int

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s8_6(n11_6), gen_0':s8_6(n11_6)) -> gen_c:c1:c27_6(n11_6), rt in Omega(1 + n11_6)

Induction Base:
LE(gen_0':s8_6(0), gen_0':s8_6(0)) ->_R^Omega(1)
c

Induction Step:
LE(gen_0':s8_6(+(n11_6, 1)), gen_0':s8_6(+(n11_6, 1))) ->_R^Omega(1)
c2(LE(gen_0':s8_6(n11_6), gen_0':s8_6(n11_6))) ->_IH
c2(gen_c:c1:c27_6(c12_6))

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

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c4(INT(s(z0), z1))
IF(false, z0, z1) -> c5
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
int(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> cons(z0, int(s(z0), z1))
if(false, z0, z1) -> nil

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INT :: 0':s -> 0':s -> c3
c3 :: c4:c5 -> c:c1:c2 -> c3
IF :: true:false -> 0':s -> 0':s -> c4:c5
le :: 0':s -> 0':s -> true:false
true :: true:false
c4 :: c3 -> c4:c5
false :: true:false
c5 :: c4:c5
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c:c1:c21_6 :: c:c1:c2
hole_0':s2_6 :: 0':s
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_true:false5_6 :: true:false
hole_cons:nil6_6 :: cons:nil
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_0':s8_6 :: Nat -> 0':s
gen_cons:nil9_6 :: Nat -> cons:nil


Generator Equations:
gen_c:c1:c27_6(0) <=> c
gen_c:c1:c27_6(+(x, 1)) <=> c2(gen_c:c1:c27_6(x))
gen_0':s8_6(0) <=> 0'
gen_0':s8_6(+(x, 1)) <=> s(gen_0':s8_6(x))
gen_cons:nil9_6(0) <=> nil
gen_cons:nil9_6(+(x, 1)) <=> cons(0', gen_cons:nil9_6(x))


The following defined symbols remain to be analysed:
LE, INT, le, int

They will be analysed ascendingly in the following order:
LE < INT
le < INT
le < int

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

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

(33)
BOUNDS(n^1, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c4(INT(s(z0), z1))
IF(false, z0, z1) -> c5
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
int(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> cons(z0, int(s(z0), z1))
if(false, z0, z1) -> nil

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INT :: 0':s -> 0':s -> c3
c3 :: c4:c5 -> c:c1:c2 -> c3
IF :: true:false -> 0':s -> 0':s -> c4:c5
le :: 0':s -> 0':s -> true:false
true :: true:false
c4 :: c3 -> c4:c5
false :: true:false
c5 :: c4:c5
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c:c1:c21_6 :: c:c1:c2
hole_0':s2_6 :: 0':s
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_true:false5_6 :: true:false
hole_cons:nil6_6 :: cons:nil
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_0':s8_6 :: Nat -> 0':s
gen_cons:nil9_6 :: Nat -> cons:nil


Lemmas:
LE(gen_0':s8_6(n11_6), gen_0':s8_6(n11_6)) -> gen_c:c1:c27_6(n11_6), rt in Omega(1 + n11_6)


Generator Equations:
gen_c:c1:c27_6(0) <=> c
gen_c:c1:c27_6(+(x, 1)) <=> c2(gen_c:c1:c27_6(x))
gen_0':s8_6(0) <=> 0'
gen_0':s8_6(+(x, 1)) <=> s(gen_0':s8_6(x))
gen_cons:nil9_6(0) <=> nil
gen_cons:nil9_6(+(x, 1)) <=> cons(0', gen_cons:nil9_6(x))


The following defined symbols remain to be analysed:
le, INT, int

They will be analysed ascendingly in the following order:
le < INT
le < int

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s8_6(n491_6), gen_0':s8_6(n491_6)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s8_6(0), gen_0':s8_6(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s8_6(+(n491_6, 1)), gen_0':s8_6(+(n491_6, 1))) ->_R^Omega(0)
le(gen_0':s8_6(n491_6), gen_0':s8_6(n491_6)) ->_IH
true

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

(36)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
INT(z0, z1) -> c3(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c4(INT(s(z0), z1))
IF(false, z0, z1) -> c5
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
int(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> cons(z0, int(s(z0), z1))
if(false, z0, z1) -> nil

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INT :: 0':s -> 0':s -> c3
c3 :: c4:c5 -> c:c1:c2 -> c3
IF :: true:false -> 0':s -> 0':s -> c4:c5
le :: 0':s -> 0':s -> true:false
true :: true:false
c4 :: c3 -> c4:c5
false :: true:false
c5 :: c4:c5
int :: 0':s -> 0':s -> cons:nil
if :: true:false -> 0':s -> 0':s -> cons:nil
cons :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c:c1:c21_6 :: c:c1:c2
hole_0':s2_6 :: 0':s
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_true:false5_6 :: true:false
hole_cons:nil6_6 :: cons:nil
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_0':s8_6 :: Nat -> 0':s
gen_cons:nil9_6 :: Nat -> cons:nil


Lemmas:
LE(gen_0':s8_6(n11_6), gen_0':s8_6(n11_6)) -> gen_c:c1:c27_6(n11_6), rt in Omega(1 + n11_6)
le(gen_0':s8_6(n491_6), gen_0':s8_6(n491_6)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c27_6(0) <=> c
gen_c:c1:c27_6(+(x, 1)) <=> c2(gen_c:c1:c27_6(x))
gen_0':s8_6(0) <=> 0'
gen_0':s8_6(+(x, 1)) <=> s(gen_0':s8_6(x))
gen_cons:nil9_6(0) <=> nil
gen_cons:nil9_6(+(x, 1)) <=> cons(0', gen_cons:nil9_6(x))


The following defined symbols remain to be analysed:
INT, int