WORST_CASE(Omega(n^1),O(n^2))
proof of input_E4id1KpZjK.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 426 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), 17 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 260 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 113 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (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:

   f(true, x, y) -> f(gt(x, y), s(x), s(s(y)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

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:
   f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(0, z0) -> c1
   GT(s(z0), 0) -> c2
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
S tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(0, z0) -> c1
   GT(s(z0), 0) -> c2
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, gt_2

Defined Pair Symbols:   F_3, GT_2

Compound Symbols:   c_2, c1, c2, c3_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   GT(0, z0) -> c1
   GT(s(z0), 0) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
S tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, gt_2

Defined Pair Symbols:   F_3, GT_2

Compound Symbols:   c_2, c3_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
S tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   gt_2

Defined Pair Symbols:   F_3, GT_2

Compound Symbols:   c_2, c3_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:

   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))

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

   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1)) [1]
   GT(s(z0), s(z1)) -> c3(GT(z0, z1)) [1]
   gt(0, z0) -> false [0]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1)) [1]
   GT(s(z0), s(z1)) -> c3(GT(z0, z1)) [1]
   gt(0, z0) -> false [0]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]

The TRS has the following type information:
F :: true:false -> s:0 -> s:0 -> c
true :: true:false
c :: c -> c3 -> c
gt :: s:0 -> s:0 -> true:false
s :: s:0 -> s:0
GT :: s:0 -> s:0 -> c3
c3 :: c3 -> c3
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:

   gt(v0, v1) -> null_gt [0]
   F(v0, v1, v2) -> null_F [0]
   GT(v0, v1) -> null_GT [0]

And the following fresh constants:    null_gt, null_F, null_GT

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

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1)) [1]
   GT(s(z0), s(z1)) -> c3(GT(z0, z1)) [1]
   gt(0, z0) -> false [0]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]
   gt(v0, v1) -> null_gt [0]
   F(v0, v1, v2) -> null_F [0]
   GT(v0, v1) -> null_GT [0]

The TRS has the following type information:
F :: true:false:null_gt -> s:0 -> s:0 -> c:null_F
true :: true:false:null_gt
c :: c:null_F -> c3:null_GT -> c:null_F
gt :: s:0 -> s:0 -> true:false:null_gt
s :: s:0 -> s:0
GT :: s:0 -> s:0 -> c3:null_GT
c3 :: c3:null_GT -> c3:null_GT
0 :: s:0
false :: true:false:null_gt
null_gt :: true:false:null_gt
null_F :: c:null_F
null_GT :: c3:null_GT

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_gt => 0
   null_F => 0
   null_GT => 0

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

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

   F(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   F(z, z', z'') -{ 1 }-> 1 + F(gt(z0, z1), 1 + z0, 1 + (1 + z1)) + GT(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   GT(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GT(z, z') -{ 1 }-> 1 + GT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   gt(z, z') -{ 0 }-> gt(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   gt(z, z') -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0
   gt(z, z') -{ 0 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   gt(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, V2),0,[fun(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, V2, Out),1,[gt(V4, V3, Ret010),fun(Ret010, 1 + V4, 1 + (1 + V3), Ret01),fun1(V4, V3, Ret1)],[Out = 1 + Ret01 + Ret1,V1 = 2,V3 >= 0,V4 >= 0,V = V4,V2 = V3]).
eq(fun1(V1, V, Out),1,[fun1(V6, V5, Ret11)],[Out = 1 + Ret11,V5 >= 0,V1 = 1 + V6,V6 >= 0,V = 1 + V5]).
eq(gt(V1, V, Out),0,[],[Out = 1,V7 >= 0,V1 = 0,V = V7]).
eq(gt(V1, V, Out),0,[],[Out = 2,V1 = 1 + V8,V8 >= 0,V = 0]).
eq(gt(V1, V, Out),0,[gt(V9, V10, Ret)],[Out = Ret,V10 >= 0,V1 = 1 + V9,V9 >= 0,V = 1 + V10]).
eq(gt(V1, V, Out),0,[],[Out = 0,V12 >= 0,V11 >= 0,V1 = V12,V = V11]).
eq(fun(V1, V, V2, Out),0,[],[Out = 0,V14 >= 0,V2 = V15,V13 >= 0,V1 = V14,V = V13,V15 >= 0]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V1 = V17,V = V16]).
input_output_vars(fun(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun1/3
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into fun/4
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations fun1/3 
* CE 7 is refined into CE [12] 
* CE 6 is refined into CE [13] 


### Cost equations --> "Loop" of fun1/3 
* CEs [13] --> Loop 10 
* CEs [12] --> Loop 11 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V
    V1


### Specialization of cost equations gt/3 
* CE 11 is refined into CE [14] 
* CE 9 is refined into CE [15] 
* CE 8 is refined into CE [16] 
* CE 10 is refined into CE [17] 


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

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

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


### Specialization of cost equations fun/4 
* CE 5 is refined into CE [18] 
* CE 4 is refined into CE [19,20,21,22,23,24,25,26] 


### Cost equations --> "Loop" of fun/4 
* CEs [25] --> Loop 16 
* CEs [26] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [22] --> Loop 20 
* CEs [21] --> Loop 21 
* CEs [20] --> Loop 22 
* CEs [19] --> Loop 23 
* CEs [18] --> Loop 24 

### Ranking functions of CR fun(V1,V,V2,Out) 
* RF of phase [16,17]: [V-V2]

#### Partial ranking functions of CR fun(V1,V,V2,Out) 
* Partial RF of phase [16,17]:
  - RF of loop [16:1,17:1]:
    V-V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [27,28,29,30,31,32,33,34] 
* CE 2 is refined into CE [35,36] 
* CE 3 is refined into CE [37,38,39,40,41] 


### Cost equations --> "Loop" of start/3 
* CEs [38] --> Loop 25 
* CEs [27,28,29,30,31,32,33] --> Loop 26 
* CEs [34,35,36,37,39,40,41] --> Loop 27 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of fun1(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 gt(V1,V,Out):
* Chain [[12],15]: 0
  with precondition: [Out=1,V1>=1,V>=V1] 

* Chain [[12],14]: 0
  with precondition: [Out=2,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=1,V>=0] 

* Chain [14]: 0
  with precondition: [V=0,Out=2,V1>=1] 

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


#### Cost of chains of fun(V1,V,V2,Out):
* Chain [[16,17],24]: 2*it(16)+1*s(3)+0
  Such that:aux(1) =< 2*V-V2
aux(4) =< V-V2
it(16) =< aux(4)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2>=1,Out>=1,V>=V2+1] 

* Chain [[16,17],21,24]: 2*it(16)+1*s(3)+1
  Such that:aux(1) =< 2*V-V2
aux(5) =< V-V2
it(16) =< aux(5)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2>=1,Out>=2,V>=V2+1] 

* Chain [[16,17],20,24]: 2*it(16)+1*s(3)+1*s(4)+1
  Such that:aux(1) =< 2*V-V2
s(4) =< 2*V-V2+2
aux(6) =< V-V2
it(16) =< aux(6)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2>=1,Out>=3,V>=V2+1] 

* Chain [[16,17],19,24]: 2*it(16)+1*s(3)+1
  Such that:aux(1) =< 2*V-V2
aux(7) =< V-V2
it(16) =< aux(7)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2>=1,V>=V2+1,Out+V2>=V+1] 

* Chain [[16,17],18,24]: 2*it(16)+1*s(3)+1*s(5)+1
  Such that:aux(1) =< 2*V-V2
s(5) =< 2*V-V2+1
aux(8) =< V-V2
it(16) =< aux(8)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2>=1,V>=V2+1,Out+V2>=V+2] 

* Chain [24]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [23,24]: 1
  with precondition: [V1=2,V=0,Out=1,V2>=0] 

* Chain [22,[16,17],24]: 2*it(16)+1*s(3)+1
  Such that:aux(4) =< V
aux(1) =< 2*V
it(16) =< aux(4)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2=0,V>=2,Out>=2] 

* Chain [22,[16,17],21,24]: 2*it(16)+1*s(3)+2
  Such that:aux(5) =< V
aux(1) =< 2*V
it(16) =< aux(5)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2=0,V>=2,Out>=3] 

* Chain [22,[16,17],20,24]: 2*it(16)+1*s(3)+1*s(4)+2
  Such that:aux(6) =< V
aux(1) =< 2*V
s(4) =< 2*V+2
it(16) =< aux(6)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2=0,V>=2,Out>=4] 

* Chain [22,[16,17],19,24]: 2*it(16)+1*s(3)+2
  Such that:aux(7) =< V
aux(1) =< 2*V
it(16) =< aux(7)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2=0,V>=2,Out>=V+1] 

* Chain [22,[16,17],18,24]: 2*it(16)+1*s(3)+1*s(5)+2
  Such that:aux(8) =< V
aux(1) =< 2*V
s(5) =< 2*V+1
it(16) =< aux(8)
s(3) =< it(16)*aux(1)

  with precondition: [V1=2,V2=0,V>=2,Out>=V+2] 

* Chain [22,24]: 1
  with precondition: [V1=2,V2=0,Out=1,V>=1] 

* Chain [22,21,24]: 2
  with precondition: [V1=2,V2=0,Out=2,V>=1] 

* Chain [22,20,24]: 1*s(4)+2
  Such that:s(4) =< 4

  with precondition: [V1=2,V2=0,4>=Out,V>=1,Out>=3] 

* Chain [22,19,24]: 2
  with precondition: [V1=2,V=1,V2=0,Out=2] 

* Chain [22,18,24]: 1*s(5)+2
  Such that:s(5) =< 3

  with precondition: [V1=2,V=1,V2=0,4>=Out,Out>=3] 

* Chain [21,24]: 1
  with precondition: [V1=2,Out=1,V>=0,V2>=0] 

* Chain [20,24]: 1*s(4)+1
  Such that:s(4) =< V2+2

  with precondition: [V1=2,Out>=2,V+1>=Out,V2+1>=Out] 

* Chain [19,24]: 1
  with precondition: [V1=2,Out=1,V>=1,V2>=V] 

* Chain [18,24]: 1*s(5)+1
  Such that:s(5) =< V+1

  with precondition: [V1=2,Out>=2,V2>=V,V+1>=Out] 


#### Cost of chains of start(V1,V,V2):
* Chain [27]: 1*s(54)+0
  Such that:s(54) =< V

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

* Chain [26]: 1*s(55)+1*s(56)+10*s(59)+5*s(60)+1*s(61)+1*s(62)+10*s(65)+5*s(66)+1*s(67)+1*s(68)+9
  Such that:s(57) =< V
s(67) =< V+1
s(63) =< V-V2
s(58) =< 2*V
s(55) =< 2*V+1
s(56) =< 2*V+2
s(64) =< 2*V-V2
s(61) =< 2*V-V2+1
s(62) =< 2*V-V2+2
s(68) =< V2+2
s(59) =< s(57)
s(60) =< s(59)*s(58)
s(65) =< s(63)
s(66) =< s(65)*s(64)

  with precondition: [V1=2,V>=0,V2>=0] 

* Chain [25]: 0
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [27] with precondition: [V1>=0,V>=0] 
    - Upper bound: V 
    - Complexity: n 
* Chain [26] with precondition: [V1=2,V>=0,V2>=0] 
    - Upper bound: 10*V+9+5*V*(2*V)+(V+1)+(V2+2)+(2*V+1)+(2*V+2)+nat(2*V-V2+1)+nat(2*V-V2+2)+nat(V-V2)*10+nat(V-V2)*5*nat(2*V-V2) 
    - Complexity: n^2 
* Chain [25] with precondition: [V=0,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V2): 9*V+9+5*V*(2*V)+(V+1)+nat(V2+2)+(2*V+1)+(2*V+2)+nat(2*V-V2+1)+nat(2*V-V2+2)+nat(V-V2)*10+nat(V-V2)*5*nat(2*V-V2)+V 
Asymptotic class: n^2 
* Total analysis performed in 442 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:
   f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(0, z0) -> c1
   GT(s(z0), 0) -> c2
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
S tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(0, z0) -> c1
   GT(s(z0), 0) -> c2
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, gt_2

Defined Pair Symbols:   F_3, GT_2

Compound Symbols:   c_2, c1, c2, c3_1


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

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(0, z0) -> c1
   GT(s(z0), 0) -> c2
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

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:

   F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
   GT(0', z0) -> c1
   GT(s(z0), 0') -> c2
   GT(s(z0), s(z1)) -> c3(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
   gt(0', z0) -> false
   gt(s(z0), 0') -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

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

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

(26)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
GT(0', z0) -> c1
GT(s(z0), 0') -> c2
GT(s(z0), s(z1)) -> c3(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c
true :: true:false
c :: c -> c1:c2:c3 -> c
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c1:c2:c3
0' :: s:0'
c1 :: c1:c2:c3
c2 :: c1:c2:c3
c3 :: c1:c2:c3 -> c1:c2:c3
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c1_4 :: c
hole_true:false2_4 :: true:false
hole_s:0'3_4 :: s:0'
hole_c1:c2:c34_4 :: c1:c2:c3
hole_f5_4 :: f
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_c1:c2:c38_4 :: Nat -> c1:c2:c3

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, gt, GT, f

They will be analysed ascendingly in the following order:
gt < F
GT < F
gt < f

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

(28)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
GT(0', z0) -> c1
GT(s(z0), 0') -> c2
GT(s(z0), s(z1)) -> c3(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c
true :: true:false
c :: c -> c1:c2:c3 -> c
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c1:c2:c3
0' :: s:0'
c1 :: c1:c2:c3
c2 :: c1:c2:c3
c3 :: c1:c2:c3 -> c1:c2:c3
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c1_4 :: c
hole_true:false2_4 :: true:false
hole_s:0'3_4 :: s:0'
hole_c1:c2:c34_4 :: c1:c2:c3
hole_f5_4 :: f
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_c1:c2:c38_4 :: Nat -> c1:c2:c3


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x), c1)
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_c1:c2:c38_4(0) <=> c1
gen_c1:c2:c38_4(+(x, 1)) <=> c3(gen_c1:c2:c38_4(x))


The following defined symbols remain to be analysed:
gt, F, GT, f

They will be analysed ascendingly in the following order:
gt < F
GT < F
gt < f

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'7_4(n10_4), gen_s:0'7_4(n10_4)) -> false, rt in Omega(0)

Induction Base:
gt(gen_s:0'7_4(0), gen_s:0'7_4(0)) ->_R^Omega(0)
false

Induction Step:
gt(gen_s:0'7_4(+(n10_4, 1)), gen_s:0'7_4(+(n10_4, 1))) ->_R^Omega(0)
gt(gen_s:0'7_4(n10_4), gen_s:0'7_4(n10_4)) ->_IH
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:
F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
GT(0', z0) -> c1
GT(s(z0), 0') -> c2
GT(s(z0), s(z1)) -> c3(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c
true :: true:false
c :: c -> c1:c2:c3 -> c
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c1:c2:c3
0' :: s:0'
c1 :: c1:c2:c3
c2 :: c1:c2:c3
c3 :: c1:c2:c3 -> c1:c2:c3
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c1_4 :: c
hole_true:false2_4 :: true:false
hole_s:0'3_4 :: s:0'
hole_c1:c2:c34_4 :: c1:c2:c3
hole_f5_4 :: f
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_c1:c2:c38_4 :: Nat -> c1:c2:c3


Lemmas:
gt(gen_s:0'7_4(n10_4), gen_s:0'7_4(n10_4)) -> false, rt in Omega(0)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x), c1)
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_c1:c2:c38_4(0) <=> c1
gen_c1:c2:c38_4(+(x, 1)) <=> c3(gen_c1:c2:c38_4(x))


The following defined symbols remain to be analysed:
GT, F, f

They will be analysed ascendingly in the following order:
GT < F

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GT(gen_s:0'7_4(n253_4), gen_s:0'7_4(n253_4)) -> gen_c1:c2:c38_4(n253_4), rt in Omega(1 + n253_4)

Induction Base:
GT(gen_s:0'7_4(0), gen_s:0'7_4(0)) ->_R^Omega(1)
c1

Induction Step:
GT(gen_s:0'7_4(+(n253_4, 1)), gen_s:0'7_4(+(n253_4, 1))) ->_R^Omega(1)
c3(GT(gen_s:0'7_4(n253_4), gen_s:0'7_4(n253_4))) ->_IH
c3(gen_c1:c2:c38_4(c254_4))

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:
F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
GT(0', z0) -> c1
GT(s(z0), 0') -> c2
GT(s(z0), s(z1)) -> c3(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c
true :: true:false
c :: c -> c1:c2:c3 -> c
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c1:c2:c3
0' :: s:0'
c1 :: c1:c2:c3
c2 :: c1:c2:c3
c3 :: c1:c2:c3 -> c1:c2:c3
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c1_4 :: c
hole_true:false2_4 :: true:false
hole_s:0'3_4 :: s:0'
hole_c1:c2:c34_4 :: c1:c2:c3
hole_f5_4 :: f
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_c1:c2:c38_4 :: Nat -> c1:c2:c3


Lemmas:
gt(gen_s:0'7_4(n10_4), gen_s:0'7_4(n10_4)) -> false, rt in Omega(0)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x), c1)
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_c1:c2:c38_4(0) <=> c1
gen_c1:c2:c38_4(+(x, 1)) <=> c3(gen_c1:c2:c38_4(x))


The following defined symbols remain to be analysed:
GT, F, f

They will be analysed ascendingly in the following order:
GT < F

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), s(z0), s(s(z1))), GT(z0, z1))
GT(0', z0) -> c1
GT(s(z0), 0') -> c2
GT(s(z0), s(z1)) -> c3(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), s(z0), s(s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c
true :: true:false
c :: c -> c1:c2:c3 -> c
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c1:c2:c3
0' :: s:0'
c1 :: c1:c2:c3
c2 :: c1:c2:c3
c3 :: c1:c2:c3 -> c1:c2:c3
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c1_4 :: c
hole_true:false2_4 :: true:false
hole_s:0'3_4 :: s:0'
hole_c1:c2:c34_4 :: c1:c2:c3
hole_f5_4 :: f
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_c1:c2:c38_4 :: Nat -> c1:c2:c3


Lemmas:
gt(gen_s:0'7_4(n10_4), gen_s:0'7_4(n10_4)) -> false, rt in Omega(0)
GT(gen_s:0'7_4(n253_4), gen_s:0'7_4(n253_4)) -> gen_c1:c2:c38_4(n253_4), rt in Omega(1 + n253_4)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x), c1)
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_c1:c2:c38_4(0) <=> c1
gen_c1:c2:c38_4(+(x, 1)) <=> c3(gen_c1:c2:c38_4(x))


The following defined symbols remain to be analysed:
F, f