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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 4 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 579 ms]
    (10) BOUNDS(1, n^1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 274 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 78 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(y, 0), x, y)
   cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y))
   cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   p(0) -> 0
   p(s(x)) -> x
   eq(0, 0) -> true
   eq(s(x), 0) -> false
   eq(0, s(x)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   and(true, true) -> true
   and(false, x) -> false
   and(x, false) -> false

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

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

(2)
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:

   cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1]
   cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) [1]
   cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   and(true, true) -> true [1]
   and(false, x) -> false [1]
   and(x, false) -> false [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1]
   cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) [1]
   cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   and(true, true) -> true [1]
   and(false, x) -> false [1]
   and(x, false) -> false [1]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0:s -> 0:s -> true:false
s :: 0:s -> 0:s

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

(5) 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:

   cond1(v0, v1, v2) -> null_cond1 [0]

And the following fresh constants:    null_cond1

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

(6)
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:

   cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1]
   cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) [1]
   cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   eq(0, 0) -> true [1]
   eq(s(x), 0) -> false [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   and(true, true) -> true [1]
   and(false, x) -> false [1]
   and(x, false) -> false [1]
   cond1(v0, v1, v2) -> null_cond1 [0]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> null_cond1
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> null_cond1
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0:s -> 0:s -> true:false
s :: 0:s -> 0:s
null_cond1 :: null_cond1

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

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

   true => 1
   0 => 0
   false => 0
   null_cond1 => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   cond1(z, z', z'') -{ 1 }-> cond2(gr(y, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   cond2(z, z', z'') -{ 1 }-> cond2(gr(y, 0), p(x), p(y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond2(z, z', z'') -{ 1 }-> cond1(and(eq(x, y), gr(x, 0)), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V2),0,[cond1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[cond2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[eq(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[and(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(cond1(V1, V, V2, Out),1,[gr(V3, 0, Ret0),cond2(Ret0, V4, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(cond2(V1, V, V2, Out),1,[gr(V6, 0, Ret01),p(V5, Ret1),p(V6, Ret2),cond2(Ret01, Ret1, Ret2, Ret3)],[Out = Ret3,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]).
eq(cond2(V1, V, V2, Out),1,[eq(V8, V7, Ret00),gr(V8, 0, Ret011),and(Ret00, Ret011, Ret02),cond1(Ret02, V8, V7, Ret4)],[Out = Ret4,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 0,V = V9,V9 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = 1 + V10,V = 0]).
eq(gr(V1, V, Out),1,[gr(V12, V11, Ret5)],[Out = Ret5,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V13,V13 >= 0,V1 = 1 + V13]).
eq(eq(V1, V, Out),1,[],[Out = 1,V1 = 0,V = 0]).
eq(eq(V1, V, Out),1,[],[Out = 0,V14 >= 0,V1 = 1 + V14,V = 0]).
eq(eq(V1, V, Out),1,[],[Out = 0,V = 1 + V15,V15 >= 0,V1 = 0]).
eq(eq(V1, V, Out),1,[eq(V17, V16, Ret6)],[Out = Ret6,V = 1 + V16,V17 >= 0,V16 >= 0,V1 = 1 + V17]).
eq(and(V1, V, Out),1,[],[Out = 1,V1 = 1,V = 1]).
eq(and(V1, V, Out),1,[],[Out = 0,V = V18,V18 >= 0,V1 = 0]).
eq(and(V1, V, Out),1,[],[Out = 0,V19 >= 0,V1 = V19,V = 0]).
eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V21 >= 0,V2 = V22,V20 >= 0,V1 = V21,V = V20,V22 >= 0]).
input_output_vars(cond1(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(cond2(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(eq(V1,V,Out),[V1,V],[Out]).
input_output_vars(and(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [and/3]
1. recursive  : [eq/3]
2. recursive  : [gr/3]
3. non_recursive  : [p/2]
4. recursive  : [cond1/4,cond2/4]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into and/3
1. SCC is partially evaluated into eq/3
2. SCC is partially evaluated into gr/3
3. SCC is partially evaluated into p/2
4. SCC is partially evaluated into cond2/4
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations and/3 
* CE 22 is refined into CE [23] 
* CE 20 is refined into CE [24] 
* CE 21 is refined into CE [25] 


### Cost equations --> "Loop" of and/3 
* CEs [23] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [25] --> Loop 19 

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

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


### Specialization of cost equations eq/3 
* CE 19 is refined into CE [26] 
* CE 17 is refined into CE [27] 
* CE 18 is refined into CE [28] 
* CE 16 is refined into CE [29] 


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

### Ranking functions of CR eq(V1,V,Out) 
* RF of phase [23]: [V,V1]

#### Partial ranking functions of CR eq(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V
    V1


### Specialization of cost equations gr/3 
* CE 10 is refined into CE [30] 
* CE 9 is refined into CE [31] 
* CE 8 is refined into CE [32] 


### Cost equations --> "Loop" of gr/3 
* CEs [31] --> Loop 24 
* CEs [32] --> Loop 25 
* CEs [30] --> Loop 26 

### Ranking functions of CR gr(V1,V,Out) 
* RF of phase [26]: [V,V1]

#### Partial ranking functions of CR gr(V1,V,Out) 
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V
    V1


### Specialization of cost equations p/2 
* CE 15 is refined into CE [33] 
* CE 14 is refined into CE [34] 


### Cost equations --> "Loop" of p/2 
* CEs [33] --> Loop 27 
* CEs [34] --> Loop 28 

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

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


### Specialization of cost equations cond2/4 
* CE 13 is refined into CE [35,36,37,38] 
* CE 12 is refined into CE [39] 
* CE 11 is refined into CE [40,41,42,43,44,45,46] 


### Cost equations --> "Loop" of cond2/4 
* CEs [45] --> Loop 29 
* CEs [44] --> Loop 30 
* CEs [46] --> Loop 31 
* CEs [43] --> Loop 32 
* CEs [41,42] --> Loop 33 
* CEs [40] --> Loop 34 
* CEs [38] --> Loop 35 
* CEs [36] --> Loop 36 
* CEs [37] --> Loop 37 
* CEs [35] --> Loop 38 
* CEs [39] --> Loop 39 

### Ranking functions of CR cond2(V1,V,V2,Out) 
* RF of phase [35]: [V,V2]
* RF of phase [37]: [V2]

#### Partial ranking functions of CR cond2(V1,V,V2,Out) 
* Partial RF of phase [35]:
  - RF of loop [35:1]:
    V
    V2
* Partial RF of phase [37]:
  - RF of loop [37:1]:
    V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [47] 
* CE 2 is refined into CE [48,49,50,51,52,53,54] 
* CE 3 is refined into CE [55,56,57,58,59,60,61,62,63,64,65,66,67,68] 
* CE 4 is refined into CE [69,70,71,72] 
* CE 5 is refined into CE [73,74] 
* CE 6 is refined into CE [75,76,77,78,79,80] 
* CE 7 is refined into CE [81,82,83] 


### Cost equations --> "Loop" of start/3 
* CEs [54,68] --> Loop 40 
* CEs [53,67] --> Loop 41 
* CEs [52,66] --> Loop 42 
* CEs [51,65] --> Loop 43 
* CEs [49,64] --> Loop 44 
* CEs [63,71,72,74,78,79,80,82] --> Loop 45 
* CEs [47,50,62] --> Loop 46 
* CEs [48,61,70,77,83] --> Loop 47 
* CEs [55,56,57,58,59,60,69,73,75,76,81] --> Loop 48 

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

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


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

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

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

* Chain [17]: 1
  with precondition: [V=0,Out=0,V1>=0] 


#### Cost of chains of eq(V1,V,Out):
* Chain [[23],22]: 1*it(23)+1
  Such that:it(23) =< V1

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

* Chain [[23],21]: 1*it(23)+1
  Such that:it(23) =< V1

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

* Chain [[23],20]: 1*it(23)+1
  Such that:it(23) =< V

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

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

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

* Chain [20]: 1
  with precondition: [V=0,Out=0,V1>=1] 


#### Cost of chains of gr(V1,V,Out):
* Chain [[26],25]: 1*it(26)+1
  Such that:it(26) =< V1

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

* Chain [[26],24]: 1*it(26)+1
  Such that:it(26) =< V

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

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

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


#### Cost of chains of p(V1,Out):
* Chain [28]: 1
  with precondition: [V1=0,Out=0] 

* Chain [27]: 1
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of cond2(V1,V,V2,Out):
* Chain [[37],38,34]: 4*it(37)+8
  Such that:it(37) =< V2

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

* Chain [[35],[37],38,34]: 4*it(35)+4*it(37)+8
  Such that:it(37) =< -V+V2
it(35) =< V

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

* Chain [[35],38,34]: 4*it(35)+8
  Such that:it(35) =< V

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

* Chain [[35],36,34]: 4*it(35)+8
  Such that:it(35) =< V

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

* Chain [[35],36,32]: 4*it(35)+8
  Such that:it(35) =< V2

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

* Chain [39,[35],38,34]: 5*it(35)+14
  Such that:aux(1) =< V2
it(35) =< aux(1)

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

* Chain [38,34]: 8
  with precondition: [V1=1,V=0,V2=0,Out=0] 

* Chain [36,34]: 8
  with precondition: [V1=1,V=1,V2=0,Out=0] 

* Chain [36,32]: 8
  with precondition: [V1=1,V2=0,Out=0,V>=2] 

* Chain [34]: 4
  with precondition: [V1=0,V=0,V2=0,Out=0] 

* Chain [33]: 4
  with precondition: [V1=0,V=0,Out=0,V2>=1] 

* Chain [32]: 4
  with precondition: [V1=0,V2=0,Out=0,V>=1] 

* Chain [31]: 1*s(2)+4
  Such that:s(2) =< V2

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

* Chain [30]: 1*s(3)+4
  Such that:s(3) =< V

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

* Chain [29]: 1*s(4)+4
  Such that:s(4) =< V2

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


#### Cost of chains of start(V1,V,V2):
* Chain [48]: 7*s(9)+1*s(10)+14
  Such that:s(10) =< V
aux(3) =< V2
s(9) =< aux(3)

  with precondition: [V1=0] 

* Chain [47]: 8
  with precondition: [V=0,V1>=0] 

* Chain [46]: 8*s(12)+10
  Such that:aux(4) =< V2
s(12) =< aux(4)

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

* Chain [45]: 2*s(14)+3*s(15)+8
  Such that:aux(5) =< V1
aux(6) =< V
s(14) =< aux(5)
s(15) =< aux(6)

  with precondition: [V1>=1] 

* Chain [44]: 8
  with precondition: [V1=1,V2=0,V>=1] 

* Chain [43]: 8*s(19)+10
  Such that:aux(7) =< V2
s(19) =< aux(7)

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

* Chain [42]: 8*s(21)+10
  Such that:aux(8) =< V2+1
s(21) =< aux(8)

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

* Chain [41]: 8*s(23)+8*s(24)+10
  Such that:aux(9) =< -V+V2
aux(10) =< V
s(23) =< aux(9)
s(24) =< aux(10)

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

* Chain [40]: 8*s(27)+10
  Such that:aux(11) =< V2
s(27) =< aux(11)

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


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [48] with precondition: [V1=0] 
    - Upper bound: nat(V)+14+nat(V2)*7 
    - Complexity: n 
* Chain [47] with precondition: [V=0,V1>=0] 
    - Upper bound: 8 
    - Complexity: constant 
* Chain [46] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 8*V2+10 
    - Complexity: n 
* Chain [45] with precondition: [V1>=1] 
    - Upper bound: 2*V1+8+nat(V)*3 
    - Complexity: n 
* Chain [44] with precondition: [V1=1,V2=0,V>=1] 
    - Upper bound: 8 
    - Complexity: constant 
* Chain [43] with precondition: [V1=1,V=V2,V>=1] 
    - Upper bound: 8*V2+10 
    - Complexity: n 
* Chain [42] with precondition: [V1=1,V=V2+1,V>=2] 
    - Upper bound: 8*V2+18 
    - Complexity: n 
* Chain [41] with precondition: [V1=1,V>=1,V2>=V+1] 
    - Upper bound: 8*V2+10 
    - Complexity: n 
* Chain [40] with precondition: [V1=1,V2>=1,V>=V2+2] 
    - Upper bound: 8*V2+10 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([max([nat(V2)*8+2,nat(V2+1)*8+2]),nat(V)+max([nat(V)*2+max([2*V1,nat(V)*5+2+nat(-V+V2)*8]),nat(V2)*7+6])])+8 
Asymptotic class: n 
* Total analysis performed in 484 ms.


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

(10)
BOUNDS(1, n^1)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
   cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y))
   cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   p(0') -> 0'
   p(s(x)) -> x
   eq(0', 0') -> true
   eq(s(x), 0') -> false
   eq(0', s(x)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   and(true, true) -> true
   and(false, x) -> false
   and(x, false) -> false

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

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

(14)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y))
cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(x)) -> false
eq(s(x), s(y)) -> eq(x, y)
and(true, true) -> true
and(false, x) -> false
and(x, false) -> false

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr, eq

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
gr < cond2
eq < cond2

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

(16)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y))
cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(x)) -> false
eq(s(x), s(y)) -> eq(x, y)
and(true, true) -> true
and(false, x) -> false
and(x, false) -> false

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2, eq

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
gr < cond2
eq < cond2

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
false

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

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y))
cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(x)) -> false
eq(s(x), s(y)) -> eq(x, y)
and(true, true) -> true
and(false, x) -> false
and(x, false) -> false

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2, eq

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
gr < cond2
eq < cond2

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y))
cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(x)) -> false
eq(s(x), s(y)) -> eq(x, y)
and(true, true) -> true
and(false, x) -> false
and(x, false) -> false

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
eq, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
eq < cond2

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s4_0(n295_0), gen_0':s4_0(n295_0)) -> true, rt in Omega(1 + n295_0)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
true

Induction Step:
eq(gen_0':s4_0(+(n295_0, 1)), gen_0':s4_0(+(n295_0, 1))) ->_R^Omega(1)
eq(gen_0':s4_0(n295_0), gen_0':s4_0(n295_0)) ->_IH
true

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

(24)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y))
cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(x)) -> false
eq(s(x), s(y)) -> eq(x, y)
and(true, true) -> true
and(false, x) -> false
and(x, false) -> false

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
false :: true:false
and :: true:false -> true:false -> true:false
eq :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
eq(gen_0':s4_0(n295_0), gen_0':s4_0(n295_0)) -> true, rt in Omega(1 + n295_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2