/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^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 1125 ms]
    (10) BOUNDS(1, n^2)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond2(false, x, y) -> cond3(eq(x, y), x, y)
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   p(0) -> 0
   p(s(x)) -> x

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^2).


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
   cond2(false, x, y) -> cond3(eq(x, y), x, y) [1]
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   add(0, x) -> x [1]
   add(s(x), y) -> s(add(x, y)) [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [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(x, y), x, y) [1]
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
   cond2(false, x, y) -> cond3(eq(x, y), x, y) [1]
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   add(0, x) -> x [1]
   add(s(x), y) -> s(add(x, y)) [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
gr :: 0:s -> 0:s -> true:false
add :: 0:s -> 0:s -> 0:s
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
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(x, y), x, y) [1]
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
   cond2(false, x, y) -> cond3(eq(x, y), x, y) [1]
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   add(0, x) -> x [1]
   add(s(x), y) -> s(add(x, y)) [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [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
add :: 0:s -> 0:s -> 0:s
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> null_cond1
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:

   add(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   cond1(z, z', z'') -{ 1 }-> cond2(gr(x, y), 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 }-> cond3(eq(x, y), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   cond2(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond3(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), x, p(y)) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   cond3(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 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 :|: z' = 1 + x, x >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, 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,[cond3(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,[add(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[eq(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
eq(cond1(V1, V, V2, Out),1,[gr(V4, V3, 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,[add(V5, V6, Ret00),gr(Ret00, 0, Ret01),p(V5, Ret1),cond1(Ret01, Ret1, V6, Ret2)],[Out = Ret2,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]).
eq(cond2(V1, V, V2, Out),1,[eq(V8, V7, Ret02),cond3(Ret02, V8, V7, Ret3)],[Out = Ret3,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(cond3(V1, V, V2, Out),1,[add(V9, V10, Ret001),gr(Ret001, 0, Ret03),p(V9, Ret11),cond1(Ret03, Ret11, V10, Ret4)],[Out = Ret4,V = V9,V2 = V10,V1 = 1,V9 >= 0,V10 >= 0]).
eq(cond3(V1, V, V2, Out),1,[add(V12, V11, Ret002),gr(Ret002, 0, Ret04),p(V11, Ret21),cond1(Ret04, V12, Ret21, Ret5)],[Out = Ret5,V = V12,V2 = V11,V12 >= 0,V11 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 0,V = V13,V13 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]).
eq(gr(V1, V, Out),1,[gr(V16, V15, Ret6)],[Out = Ret6,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]).
eq(add(V1, V, Out),1,[],[Out = V17,V = V17,V17 >= 0,V1 = 0]).
eq(add(V1, V, Out),1,[add(V19, V18, Ret12)],[Out = 1 + Ret12,V19 >= 0,V18 >= 0,V1 = 1 + V19,V = V18]).
eq(eq(V1, V, Out),1,[],[Out = 1,V1 = 0,V = 0]).
eq(eq(V1, V, Out),1,[],[Out = 0,V = 1 + V20,V20 >= 0,V1 = 0]).
eq(eq(V1, V, Out),1,[],[Out = 0,V21 >= 0,V1 = 1 + V21,V = 0]).
eq(eq(V1, V, Out),1,[eq(V22, V23, Ret7)],[Out = Ret7,V = 1 + V23,V22 >= 0,V23 >= 0,V1 = 1 + V22]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V24,V24 >= 0,V1 = 1 + V24]).
eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V26 >= 0,V2 = V27,V25 >= 0,V1 = V26,V = V25,V27 >= 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(cond3(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(add(V1,V,Out),[V1,V],[Out]).
input_output_vars(eq(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [add/3]
1. recursive  : [gr/3]
2. non_recursive  : [p/2]
3. recursive  : [eq/3]
4. recursive  : [cond1/4,cond2/4,cond3/4]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into add/3
1. SCC is partially evaluated into gr/3
2. SCC is partially evaluated into p/2
3. SCC is partially evaluated into eq/3
4. SCC is partially evaluated into cond1/4
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations add/3 
* CE 11 is refined into CE [25] 
* CE 10 is refined into CE [26] 


### Cost equations --> "Loop" of add/3 
* CEs [26] --> Loop 17 
* CEs [25] --> Loop 18 

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

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


### Specialization of cost equations gr/3 
* CE 14 is refined into CE [27] 
* CE 13 is refined into CE [28] 
* CE 12 is refined into CE [29] 


### Cost equations --> "Loop" of gr/3 
* CEs [28] --> Loop 19 
* CEs [29] --> Loop 20 
* CEs [27] --> Loop 21 

### Ranking functions of CR gr(V1,V,Out) 
* RF of phase [21]: [V,V1]

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


### Specialization of cost equations p/2 
* CE 16 is refined into CE [30] 
* CE 15 is refined into CE [31] 


### Cost equations --> "Loop" of p/2 
* CEs [30] --> Loop 22 
* CEs [31] --> Loop 23 

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

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


### Specialization of cost equations eq/3 
* CE 24 is refined into CE [32] 
* CE 23 is refined into CE [33] 
* CE 22 is refined into CE [34] 
* CE 21 is refined into CE [35] 


### Cost equations --> "Loop" of eq/3 
* CEs [33] --> Loop 24 
* CEs [34] --> Loop 25 
* CEs [35] --> Loop 26 
* CEs [32] --> Loop 27 

### Ranking functions of CR eq(V1,V,Out) 
* RF of phase [27]: [V,V1]

#### Partial ranking functions of CR eq(V1,V,Out) 
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V
    V1


### Specialization of cost equations cond1/4 
* CE 20 is refined into CE [36] 
* CE 17 is refined into CE [37,38] 
* CE 18 is refined into CE [39,40] 
* CE 19 is refined into CE [41,42] 


### Cost equations --> "Loop" of cond1/4 
* CEs [42] --> Loop 28 
* CEs [38] --> Loop 29 
* CEs [40] --> Loop 30 
* CEs [37] --> Loop 31 
* CEs [41] --> Loop 32 
* CEs [39] --> Loop 33 
* CEs [36] --> Loop 34 

### Ranking functions of CR cond1(V1,V,V2,Out) 
* RF of phase [28,30]: [V+V2-1]
* RF of phase [29]: [V-1,V-V2]
* RF of phase [31]: [V]
* RF of phase [32]: [V2]

#### Partial ranking functions of CR cond1(V1,V,V2,Out) 
* Partial RF of phase [28,30]:
  - RF of loop [28:1]:
    -V+V2 depends on loops [30:1] 
    V2-1
  - RF of loop [30:1]:
    V
    V-V2+1 depends on loops [28:1] 
* Partial RF of phase [29]:
  - RF of loop [29:1]:
    V-1
    V-V2
* Partial RF of phase [31]:
  - RF of loop [31:1]:
    V
* Partial RF of phase [32]:
  - RF of loop [32:1]:
    V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [43,44,45,46,47,48,49,50] 
* CE 2 is refined into CE [51,52,53,54] 
* CE 3 is refined into CE [55,56,57,58,59,60,61,62,63] 
* CE 4 is refined into CE [64,65,66,67,68,69,70,71,72] 
* CE 5 is refined into CE [73,74,75,76,77] 
* CE 6 is refined into CE [78,79,80,81] 
* CE 7 is refined into CE [82,83] 
* CE 8 is refined into CE [84,85,86,87,88,89] 
* CE 9 is refined into CE [90,91] 


### Cost equations --> "Loop" of start/3 
* CEs [89] --> Loop 35 
* CEs [48,75] --> Loop 36 
* CEs [46,47,49,50,76,77] --> Loop 37 
* CEs [44,45,73,74] --> Loop 38 
* CEs [43,79,80,81,83,86,87,88,91] --> Loop 39 
* CEs [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,78,82,84,85,90] --> Loop 40 

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

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


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

#### Cost of chains of add(V1,V,Out):
* Chain [[18],17]: 1*it(18)+1
  Such that:it(18) =< -V+Out

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

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


#### Cost of chains of gr(V1,V,Out):
* Chain [[21],20]: 1*it(21)+1
  Such that:it(21) =< V1

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

* Chain [[21],19]: 1*it(21)+1
  Such that:it(21) =< V

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

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

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


#### Cost of chains of p(V1,Out):
* Chain [23]: 1
  with precondition: [V1=0,Out=0] 

* Chain [22]: 1
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of eq(V1,V,Out):
* Chain [[27],26]: 1*it(27)+1
  Such that:it(27) =< V1

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

* Chain [[27],25]: 1*it(27)+1
  Such that:it(27) =< V1

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

* Chain [[27],24]: 1*it(27)+1
  Such that:it(27) =< V

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

* Chain [26]: 1
  with precondition: [V1=0,V=0,Out=1] 

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

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


#### Cost of chains of cond1(V1,V,V2,Out):
* Chain [[32],34]: 8*it(32)+0
  Such that:it(32) =< V2

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

* Chain [[32],33,34]: 8*it(32)+8
  Such that:it(32) =< V2

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

* Chain [[31],34]: 6*it(31)+1*s(3)+0
  Such that:aux(3) =< V
it(31) =< aux(3)
s(3) =< it(31)*aux(3)

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

* Chain [[31],33,34]: 6*it(31)+1*s(3)+8
  Such that:aux(4) =< V
it(31) =< aux(4)
s(3) =< it(31)*aux(4)

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

* Chain [[29],[28,30],[32],34]: 16*it(28)+6*it(29)+8*it(32)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+0
  Such that:aux(20) =< V
it(29) =< V-V2
aux(17) =< 2*V2
aux(21) =< V2
it(28) =< aux(21)
it(32) =< aux(17)
it(28) =< aux(17)
s(17) =< it(28)*aux(17)
s(15) =< it(28)*aux(21)
s(16) =< s(17)
s(14) =< s(15)
it(29) =< aux(20)
s(22) =< it(29)*aux(21)
s(23) =< it(29)*aux(20)

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

* Chain [[29],[28,30],[32],33,34]: 16*it(28)+6*it(29)+8*it(32)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+8
  Such that:aux(20) =< V
it(29) =< V-V2
aux(23) =< 2*V2
aux(24) =< V2
it(28) =< aux(24)
it(32) =< aux(23)
it(28) =< aux(23)
s(17) =< it(28)*aux(23)
s(15) =< it(28)*aux(24)
s(16) =< s(17)
s(14) =< s(15)
it(29) =< aux(20)
s(22) =< it(29)*aux(24)
s(23) =< it(29)*aux(20)

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

* Chain [[29],[28,30],34]: 16*it(28)+6*it(29)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+0
  Such that:aux(20) =< V
it(29) =< V-V2
aux(26) =< 2*V2
aux(27) =< V2
it(28) =< aux(27)
it(28) =< aux(26)
s(17) =< it(28)*aux(26)
s(15) =< it(28)*aux(27)
s(16) =< s(17)
s(14) =< s(15)
it(29) =< aux(20)
s(22) =< it(29)*aux(27)
s(23) =< it(29)*aux(20)

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

* Chain [[29],34]: 6*it(29)+1*s(22)+1*s(23)+0
  Such that:aux(20) =< V
it(29) =< V-V2
aux(19) =< V2
it(29) =< aux(20)
s(22) =< it(29)*aux(19)
s(23) =< it(29)*aux(20)

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

* Chain [[28,30],[32],34]: 8*it(28)+8*it(30)+8*it(32)+3*s(14)+3*s(16)+0
  Such that:it(28) =< V2
aux(16) =< V
aux(17) =< V+V2
it(30) =< aux(16)
it(32) =< aux(17)
it(28) =< aux(17)
it(30) =< aux(17)
s(17) =< it(30)*aux(17)
s(15) =< it(28)*aux(16)
s(16) =< s(17)
s(14) =< s(15)

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

* Chain [[28,30],[32],33,34]: 8*it(28)+8*it(30)+8*it(32)+3*s(14)+3*s(16)+8
  Such that:it(28) =< V2
aux(22) =< V
aux(23) =< V+V2
it(30) =< aux(22)
it(32) =< aux(23)
it(28) =< aux(23)
it(30) =< aux(23)
s(17) =< it(30)*aux(23)
s(15) =< it(28)*aux(22)
s(16) =< s(17)
s(14) =< s(15)

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

* Chain [[28,30],34]: 8*it(28)+8*it(30)+3*s(14)+3*s(16)+0
  Such that:it(28) =< V2
aux(25) =< V
aux(26) =< V+V2
it(30) =< aux(25)
it(28) =< aux(26)
it(30) =< aux(26)
s(17) =< it(30)*aux(26)
s(15) =< it(28)*aux(25)
s(16) =< s(17)
s(14) =< s(15)

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

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

* Chain [33,34]: 8
  with precondition: [V1=1,V=0,V2=0,Out=0] 


#### Cost of chains of start(V1,V,V2):
* Chain [40]: 38*s(98)+19*s(100)+144*s(109)+27*s(113)+27*s(114)+48*s(115)+63*s(118)+8*s(122)+48*s(130)+48*s(131)+18*s(134)+18*s(135)+32*s(136)+48*s(150)+8*s(151)+8*s(152)+14
  Such that:aux(50) =< 1
aux(51) =< V
aux(52) =< V-V2+1
aux(53) =< V+V2
aux(54) =< V2
aux(55) =< 2*V2
s(100) =< aux(50)
s(118) =< aux(51)
s(98) =< aux(54)
s(122) =< s(118)*aux(51)
s(150) =< aux(52)
s(150) =< aux(51)
s(151) =< s(150)*aux(54)
s(152) =< s(150)*aux(51)
s(109) =< aux(54)
s(109) =< aux(55)
s(111) =< s(109)*aux(55)
s(112) =< s(109)*aux(54)
s(113) =< s(111)
s(114) =< s(112)
s(115) =< aux(55)
s(130) =< aux(54)
s(131) =< aux(51)
s(130) =< aux(53)
s(131) =< aux(53)
s(132) =< s(131)*aux(53)
s(133) =< s(130)*aux(51)
s(134) =< s(132)
s(135) =< s(133)
s(136) =< aux(53)

  with precondition: [V1=0] 

* Chain [39]: 3*s(196)+2*s(197)+12
  Such that:aux(56) =< V1
aux(57) =< V
s(196) =< aux(56)
s(197) =< aux(57)

  with precondition: [V1>=1] 

* Chain [38]: 32*s(202)+12
  Such that:aux(58) =< V2
s(202) =< aux(58)

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

* Chain [37]: 3*s(205)+1*s(206)+16*s(208)+48*s(213)+48*s(214)+18*s(217)+18*s(218)+32*s(219)+48*s(225)+8*s(226)+8*s(227)+96*s(228)+18*s(231)+18*s(232)+32*s(233)+12
  Such that:s(206) =< 1
aux(61) =< V
aux(62) =< V-V2
aux(63) =< V+V2
aux(64) =< V2
aux(65) =< 2*V2
s(205) =< aux(61)
s(208) =< aux(64)
s(213) =< aux(64)
s(214) =< aux(61)
s(213) =< aux(63)
s(214) =< aux(63)
s(215) =< s(214)*aux(63)
s(216) =< s(213)*aux(61)
s(217) =< s(215)
s(218) =< s(216)
s(219) =< aux(63)
s(225) =< aux(62)
s(225) =< aux(61)
s(226) =< s(225)*aux(64)
s(227) =< s(225)*aux(61)
s(228) =< aux(64)
s(228) =< aux(65)
s(229) =< s(228)*aux(65)
s(230) =< s(228)*aux(64)
s(231) =< s(229)
s(232) =< s(230)
s(233) =< aux(65)

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

* Chain [36]: 25*s(257)+4*s(260)+12
  Such that:aux(67) =< V
s(257) =< aux(67)
s(260) =< s(257)*aux(67)

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

* Chain [35]: 1*s(264)+1
  Such that:s(264) =< V

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


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [40] with precondition: [V1=0] 
    - Upper bound: nat(V)*111+33+nat(V)*8*nat(V)+nat(V)*18*nat(V2)+nat(V)*18*nat(V+V2)+nat(V)*8*nat(V-V2+1)+nat(V2)*230+nat(V2)*27*nat(V2)+nat(V2)*27*nat(2*V2)+nat(V2)*8*nat(V-V2+1)+nat(2*V2)*48+nat(V+V2)*32+nat(V-V2+1)*48 
    - Complexity: n^2 
* Chain [39] with precondition: [V1>=1] 
    - Upper bound: 3*V1+12+nat(V)*2 
    - Complexity: n 
* Chain [38] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 32*V2+12 
    - Complexity: n 
* Chain [37] with precondition: [V1=1,V>=1,V2>=0] 
    - Upper bound: 51*V+13+18*V*V2+(V+V2)*(18*V)+8*V*nat(V-V2)+160*V2+18*V2*V2+18*V2*(2*V2)+8*V2*nat(V-V2)+64*V2+(32*V+32*V2)+nat(V-V2)*48 
    - Complexity: n^2 
* Chain [36] with precondition: [V1=1,V2=0,V>=1] 
    - Upper bound: 25*V+12+4*V*V 
    - Complexity: n^2 
* Chain [35] with precondition: [V1=V,V1>=1] 
    - Upper bound: V+1 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([nat(V2)*32+11,nat(V)+11+max([3*V1,nat(V)*23+max([nat(V)*4*nat(V),nat(V)*26+1+nat(V)*18*nat(V2)+nat(V)*18*nat(V+V2)+nat(V2)*160+nat(V2)*18*nat(V2)+nat(V2)*18*nat(2*V2)+nat(2*V2)*32+nat(V+V2)*32+max([nat(V2)*8*nat(V-V2)+nat(V)*8*nat(V-V2)+nat(V-V2)*48,nat(V)*60+20+nat(V)*8*nat(V)+nat(V)*8*nat(V-V2+1)+nat(V2)*70+nat(V2)*9*nat(V2)+nat(V2)*9*nat(2*V2)+nat(V2)*8*nat(V-V2+1)+nat(2*V2)*16+nat(V-V2+1)*48])])])+nat(V)])+1 
Asymptotic class: n^2 
* Total analysis performed in 975 ms.


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(10)
BOUNDS(1, n^2)

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(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond2(false, x, y) -> cond3(eq(x, y), x, y)
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   p(0) -> 0
   p(s(x)) -> x

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

(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

add(s(x), y) ->^+ s(add(x, y))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / s(x)].

The result substitution is [ ].




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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond2(false, x, y) -> cond3(eq(x, y), x, y)
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   p(0) -> 0
   p(s(x)) -> x

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond2(false, x, y) -> cond3(eq(x, y), x, y)
   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST