/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 1114 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), 260 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), 55 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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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:

   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   quot(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) [1]
   if(false, b, x, y, z) -> div_by_zero [1]
   if(true, false, x, y, z) -> z [1]
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) [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:

   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   quot(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) [1]
   if(false, b, x, y, z) -> div_by_zero [1]
   if(true, false, x, y, z) -> z [1]
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) [1]

The TRS has the following type information:
ge :: 0:s:div_by_zero -> 0:s:div_by_zero -> true:false
0 :: 0:s:div_by_zero
true :: true:false
s :: 0:s:div_by_zero -> 0:s:div_by_zero
false :: true:false
minus :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
id_inc :: 0:s:div_by_zero -> 0:s:div_by_zero
quot :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
div :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
if :: true:false -> true:false -> 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
div_by_zero :: 0:s:div_by_zero

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:

   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]

And the following fresh constants:    null_ge, null_minus, null_if

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

(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:

   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   quot(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) [1]
   if(false, b, x, y, z) -> div_by_zero [1]
   if(true, false, x, y, z) -> z [1]
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) [1]
   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]

The TRS has the following type information:
ge :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> true:false:null_ge
0 :: 0:s:div_by_zero:null_minus:null_if
true :: true:false:null_ge
s :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
false :: true:false:null_ge
minus :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
id_inc :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
quot :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
div :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
if :: true:false:null_ge -> true:false:null_ge -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
div_by_zero :: 0:s:div_by_zero:null_minus:null_if
null_ge :: true:false:null_ge
null_minus :: 0:s:div_by_zero:null_minus:null_if
null_if :: 0:s:div_by_zero:null_minus:null_if

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:

   0 => 0
   true => 2
   false => 1
   div_by_zero => 1
   null_ge => 0
   null_minus => 0
   null_if => 0

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

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

   div(z', z'', z1) -{ 1 }-> if(ge(y, 1 + 0), ge(x, y), x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   id_inc(z') -{ 1 }-> x :|: z' = x, x >= 0
   id_inc(z') -{ 1 }-> 1 + x :|: z' = x, x >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> z :|: z2 = y, z >= 0, z' = 2, z3 = z, x >= 0, y >= 0, z'' = 1, z1 = x
   if(z', z'', z1, z2, z3) -{ 1 }-> div(minus(x, y), y, id_inc(z)) :|: z2 = y, z >= 0, z' = 2, z3 = z, x >= 0, y >= 0, z'' = 2, z1 = x
   if(z', z'', z1, z2, z3) -{ 1 }-> 1 :|: b >= 0, z2 = y, z >= 0, z'' = b, z3 = z, x >= 0, y >= 0, z' = 1, z1 = x
   if(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0
   minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   minus(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   quot(z', z'') -{ 1 }-> div(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 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(V, V1, V17, V22, V23),0,[ge(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V17, V22, V23),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V17, V22, V23),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V1, V17, V22, V23),0,[quot(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V17, V22, V23),0,[div(V, V1, V17, Out)],[V >= 0,V1 >= 0,V17 >= 0]).
eq(start(V, V1, V17, V22, V23),0,[if(V, V1, V17, V22, V23, Out)],[V >= 0,V1 >= 0,V17 >= 0,V22 >= 0,V23 >= 0]).
eq(ge(V, V1, Out),1,[],[Out = 2,V1 = 0,V = V2,V2 >= 0]).
eq(ge(V, V1, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]).
eq(ge(V, V1, Out),1,[ge(V4, V5, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0,V5 >= 0,V1 = 1 + V5]).
eq(minus(V, V1, Out),1,[],[Out = V6,V1 = 0,V = V6,V6 >= 0]).
eq(minus(V, V1, Out),1,[],[Out = 0,V1 = V7,V7 >= 0,V = 0]).
eq(minus(V, V1, Out),1,[minus(V8, V9, Ret1)],[Out = Ret1,V = 1 + V8,V8 >= 0,V9 >= 0,V1 = 1 + V9]).
eq(fun(V, Out),1,[],[Out = V10,V = V10,V10 >= 0]).
eq(fun(V, Out),1,[],[Out = 1 + V11,V = V11,V11 >= 0]).
eq(quot(V, V1, Out),1,[div(V13, V12, 0, Ret2)],[Out = Ret2,V = V13,V1 = V12,V13 >= 0,V12 >= 0]).
eq(div(V, V1, V17, Out),1,[ge(V14, 1 + 0, Ret0),ge(V16, V14, Ret11),if(Ret0, Ret11, V16, V14, V15, Ret3)],[Out = Ret3,V17 = V15,V15 >= 0,V = V16,V1 = V14,V16 >= 0,V14 >= 0]).
eq(if(V, V1, V17, V22, V23, Out),1,[],[Out = 1,V21 >= 0,V22 = V20,V18 >= 0,V1 = V21,V23 = V18,V19 >= 0,V20 >= 0,V = 1,V17 = V19]).
eq(if(V, V1, V17, V22, V23, Out),1,[],[Out = V24,V22 = V25,V24 >= 0,V = 2,V23 = V24,V26 >= 0,V25 >= 0,V1 = 1,V17 = V26]).
eq(if(V, V1, V17, V22, V23, Out),1,[minus(V29, V28, Ret01),fun(V27, Ret21),div(Ret01, V28, Ret21, Ret4)],[Out = Ret4,V22 = V28,V27 >= 0,V = 2,V23 = V27,V29 >= 0,V28 >= 0,V1 = 2,V17 = V29]).
eq(ge(V, V1, Out),0,[],[Out = 0,V31 >= 0,V30 >= 0,V1 = V30,V = V31]).
eq(minus(V, V1, Out),0,[],[Out = 0,V33 >= 0,V32 >= 0,V1 = V32,V = V33]).
eq(if(V, V1, V17, V22, V23, Out),0,[],[Out = 0,V22 = V37,V35 >= 0,V36 >= 0,V17 = V38,V34 >= 0,V1 = V34,V23 = V36,V38 >= 0,V37 >= 0,V = V35]).
input_output_vars(ge(V,V1,Out),[V,V1],[Out]).
input_output_vars(minus(V,V1,Out),[V,V1],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(quot(V,V1,Out),[V,V1],[Out]).
input_output_vars(div(V,V1,V17,Out),[V,V1,V17],[Out]).
input_output_vars(if(V,V1,V17,V22,V23,Out),[V,V1,V17,V22,V23],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [ge/3]
1. non_recursive  : [fun/2]
2. recursive  : [minus/3]
3. recursive  : [(div)/4,if/6]
4. non_recursive  : [quot/3]
5. non_recursive  : [start/5]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into ge/3
1. SCC is partially evaluated into fun/2
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into (div)/4
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into start/5

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

### Specialization of cost equations ge/3 
* CE 23 is refined into CE [24] 
* CE 20 is refined into CE [25] 
* CE 21 is refined into CE [26] 
* CE 22 is refined into CE [27] 


### Cost equations --> "Loop" of ge/3 
* CEs [27] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [25] --> Loop 19 
* CEs [26] --> Loop 20 

### Ranking functions of CR ge(V,V1,Out) 
* RF of phase [17]: [V,V1]

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


### Specialization of cost equations fun/2 
* CE 14 is refined into CE [28] 
* CE 15 is refined into CE [29] 


### Cost equations --> "Loop" of fun/2 
* CEs [28] --> Loop 21 
* CEs [29] --> Loop 22 

### Ranking functions of CR fun(V,Out) 

#### Partial ranking functions of CR fun(V,Out) 


### Specialization of cost equations minus/3 
* CE 10 is refined into CE [30] 
* CE 11 is refined into CE [31] 
* CE 13 is refined into CE [32] 
* CE 12 is refined into CE [33] 


### Cost equations --> "Loop" of minus/3 
* CEs [33] --> Loop 23 
* CEs [30] --> Loop 24 
* CEs [31,32] --> Loop 25 

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

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


### Specialization of cost equations (div)/4 
* CE 18 is refined into CE [34,35] 
* CE 19 is refined into CE [36,37] 
* CE 16 is refined into CE [38,39,40,41,42,43,44,45,46,47,48] 
* CE 17 is refined into CE [49,50,51,52] 


### Cost equations --> "Loop" of (div)/4 
* CEs [52] --> Loop 26 
* CEs [51] --> Loop 27 
* CEs [50] --> Loop 28 
* CEs [49] --> Loop 29 
* CEs [35] --> Loop 30 
* CEs [36,37] --> Loop 31 
* CEs [38,39,41] --> Loop 32 
* CEs [34] --> Loop 33 
* CEs [40,42,43,44,45,46,47,48] --> Loop 34 

### Ranking functions of CR div(V,V1,V17,Out) 
* RF of phase [26,27]: [V,V-V1+1]

#### Partial ranking functions of CR div(V,V1,V17,Out) 
* Partial RF of phase [26,27]:
  - RF of loop [26:1,27:1]:
    V
    V-V1+1


### Specialization of cost equations start/5 
* CE 2 is refined into CE [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76] 
* CE 3 is refined into CE [77] 
* CE 1 is refined into CE [78] 
* CE 4 is refined into CE [79] 
* CE 5 is refined into CE [80,81,82,83,84] 
* CE 6 is refined into CE [85,86,87] 
* CE 7 is refined into CE [88,89] 
* CE 8 is refined into CE [90,91,92,93,94,95,96,97] 
* CE 9 is refined into CE [98,99,100,101,102,103,104,105] 


### Cost equations --> "Loop" of start/5 
* CEs [81,85,92,100] --> Loop 35 
* CEs [63,70] --> Loop 36 
* CEs [53,54,55,56,57,58,59,60,61,62,64,65,66,67,68,69,71,72,73,74,75,76] --> Loop 37 
* CEs [77] --> Loop 38 
* CEs [79] --> Loop 39 
* CEs [78,80,82,83,84,86,87,88,89,90,91,93,94,95,96,97,98,99,101,102,103,104,105] --> Loop 40 

### Ranking functions of CR start(V,V1,V17,V22,V23) 

#### Partial ranking functions of CR start(V,V1,V17,V22,V23) 


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

#### Cost of chains of ge(V,V1,Out):
* Chain [[17],20]: 1*it(17)+1
  Such that:it(17) =< V

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

* Chain [[17],19]: 1*it(17)+1
  Such that:it(17) =< V1

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

* Chain [[17],18]: 1*it(17)+0
  Such that:it(17) =< V1

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

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

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

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


#### Cost of chains of fun(V,Out):
* Chain [22]: 1
  with precondition: [V+1=Out,V>=0] 

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


#### Cost of chains of minus(V,V1,Out):
* Chain [[23],25]: 1*it(23)+1
  Such that:it(23) =< V1

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

* Chain [[23],24]: 1*it(23)+1
  Such that:it(23) =< V1

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

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

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


#### Cost of chains of div(V,V1,V17,Out):
* Chain [[26,27],34]: 12*it(26)+8*s(3)+4*s(5)+6*s(7)+2*s(29)+3
  Such that:aux(1) =< 1
aux(9) =< V-V1+1
aux(3) =< V1
aux(12) =< V
s(3) =< aux(1)
s(7) =< aux(12)
s(5) =< aux(3)
aux(6) =< aux(12)
it(26) =< aux(12)
aux(6) =< aux(9)
it(26) =< aux(9)
s(29) =< aux(6)

  with precondition: [Out=0,V1>=1,V17>=0,V>=V1] 

* Chain [[26,27],33]: 12*it(26)+2*s(29)+2*s(30)+2*s(33)+1*s(35)+4
  Such that:s(35) =< 1
aux(9) =< V-V1+1
s(31) =< V+V17-Out
aux(13) =< V
aux(6) =< aux(13)
it(26) =< aux(13)
aux(6) =< aux(9)
it(26) =< aux(9)
s(29) =< aux(6)
s(33) =< aux(13)
s(30) =< s(31)

  with precondition: [V1>=1,V17>=0,V>=V1,Out>=V17,V+V17+1>=Out+V1] 

* Chain [[26,27],30]: 12*it(26)+2*s(29)+2*s(30)+3*s(33)+1*s(36)+4
  Such that:s(36) =< 1
aux(9) =< V-V1+1
s(31) =< V+V17-Out
aux(14) =< V
s(33) =< aux(14)
aux(6) =< aux(14)
it(26) =< aux(14)
aux(6) =< aux(9)
it(26) =< aux(9)
s(29) =< aux(6)
s(30) =< s(31)

  with precondition: [V1>=2,V17>=0,V>=V1+1,Out>=V17,V+2*V17+1>=2*Out+V1] 

* Chain [[26,27],29,34]: 12*it(26)+9*s(3)+6*s(5)+2*s(29)+2*s(30)+2*s(33)+9
  Such that:aux(16) =< 1
aux(8) =< V
aux(9) =< V-V1+1
aux(17) =< V1
aux(18) =< V-V1
s(3) =< aux(16)
s(5) =< aux(17)
aux(6) =< aux(8)
it(26) =< aux(8)
s(34) =< aux(8)
aux(6) =< aux(9)
it(26) =< aux(9)
aux(6) =< aux(18)
it(26) =< aux(18)
s(34) =< aux(18)
s(29) =< aux(6)
s(33) =< s(34)
s(30) =< aux(18)

  with precondition: [Out=0,V1>=1,V17>=0,V>=2*V1] 

* Chain [[26,27],29,33]: 12*it(26)+2*s(29)+2*s(30)+2*s(33)+2*s(35)+2*s(39)+10
  Such that:aux(19) =< 1
aux(8) =< V
aux(9) =< V-V1+1
s(31) =< V-V1+V17-Out+1
aux(15) =< V1
aux(20) =< V-V1
s(35) =< aux(19)
s(39) =< aux(15)
aux(6) =< aux(8)
it(26) =< aux(8)
s(34) =< aux(8)
aux(6) =< aux(9)
it(26) =< aux(9)
aux(6) =< aux(20)
it(26) =< aux(20)
s(34) =< aux(20)
s(29) =< aux(6)
s(33) =< s(34)
s(30) =< s(31)

  with precondition: [V1>=1,V17>=0,V>=2*V1,Out>=V17+1,V+V17+2>=2*V1+Out] 

* Chain [[26,27],28,34]: 12*it(26)+9*s(3)+6*s(5)+2*s(29)+2*s(30)+2*s(33)+9
  Such that:aux(22) =< 1
aux(8) =< V
aux(9) =< V-V1+1
aux(23) =< V1
aux(24) =< V-V1
s(3) =< aux(22)
s(5) =< aux(23)
aux(6) =< aux(8)
it(26) =< aux(8)
s(34) =< aux(8)
aux(6) =< aux(9)
it(26) =< aux(9)
aux(6) =< aux(24)
it(26) =< aux(24)
s(34) =< aux(24)
s(29) =< aux(6)
s(33) =< s(34)
s(30) =< aux(24)

  with precondition: [Out=0,V1>=1,V17>=0,V>=2*V1] 

* Chain [[26,27],28,33]: 12*it(26)+2*s(29)+2*s(30)+2*s(33)+2*s(35)+2*s(42)+10
  Such that:aux(25) =< 1
aux(8) =< V
aux(9) =< V-V1+1
s(31) =< V-V1+V17-Out
aux(21) =< V1
aux(26) =< V-V1
s(35) =< aux(25)
s(42) =< aux(21)
aux(6) =< aux(8)
it(26) =< aux(8)
s(34) =< aux(8)
aux(6) =< aux(9)
it(26) =< aux(9)
aux(6) =< aux(26)
it(26) =< aux(26)
s(34) =< aux(26)
s(29) =< aux(6)
s(33) =< s(34)
s(30) =< s(31)

  with precondition: [V1>=1,V17>=0,V>=2*V1,Out>=V17,V+V17+1>=2*V1+Out] 

* Chain [34]: 8*s(3)+4*s(5)+2*s(7)+3
  Such that:aux(1) =< 1
aux(2) =< V
aux(3) =< V1
s(3) =< aux(1)
s(7) =< aux(2)
s(5) =< aux(3)

  with precondition: [Out=0,V>=0,V1>=0,V17>=0] 

* Chain [33]: 1*s(35)+4
  Such that:s(35) =< 1

  with precondition: [V=0,V17=Out,V1>=1,V17>=0] 

* Chain [32]: 4
  with precondition: [V1=0,Out=0,V>=0,V17>=0] 

* Chain [31]: 4
  with precondition: [V1=0,Out=1,V>=0,V17>=0] 

* Chain [30]: 1*s(36)+1*s(37)+4
  Such that:s(36) =< 1
s(37) =< V

  with precondition: [V17=Out,V>=1,V17>=0,V1>=V+1] 

* Chain [29,34]: 9*s(3)+6*s(5)+9
  Such that:aux(16) =< 1
aux(17) =< V1
s(3) =< aux(16)
s(5) =< aux(17)

  with precondition: [Out=0,V1>=1,V17>=0,V>=V1] 

* Chain [29,33]: 2*s(35)+2*s(39)+10
  Such that:aux(15) =< V1
aux(19) =< 1
s(35) =< aux(19)
s(39) =< aux(15)

  with precondition: [Out=V17+1,V1>=1,Out>=1,V>=V1] 

* Chain [28,34]: 9*s(3)+6*s(5)+9
  Such that:aux(22) =< 1
aux(23) =< V1
s(3) =< aux(22)
s(5) =< aux(23)

  with precondition: [Out=0,V1>=1,V17>=0,V>=V1] 

* Chain [28,33]: 2*s(35)+2*s(42)+10
  Such that:aux(21) =< V1
aux(25) =< 1
s(35) =< aux(25)
s(42) =< aux(21)

  with precondition: [V17=Out,V1>=1,V17>=0,V>=V1] 


#### Cost of chains of start(V,V1,V17,V22,V23):
* Chain [40]: 84*s(119)+37*s(120)+128*s(124)+96*s(134)+16*s(136)+16*s(137)+16*s(138)+72*s(140)+12*s(141)+11
  Such that:aux(42) =< 1
aux(43) =< V
aux(44) =< V-V1
aux(45) =< V-V1+1
aux(46) =< V1
s(124) =< aux(42)
s(120) =< aux(43)
s(119) =< aux(46)
s(133) =< aux(43)
s(134) =< aux(43)
s(135) =< aux(43)
s(133) =< aux(45)
s(134) =< aux(45)
s(133) =< aux(44)
s(134) =< aux(44)
s(135) =< aux(44)
s(136) =< s(133)
s(137) =< s(135)
s(138) =< aux(44)
s(139) =< aux(43)
s(140) =< aux(43)
s(139) =< aux(45)
s(140) =< aux(45)
s(141) =< s(139)

  with precondition: [V>=0] 

* Chain [39]: 1
  with precondition: [V=1,V1>=0,V17>=0,V22>=0,V23>=0] 

* Chain [38]: 1
  with precondition: [V=2,V1=1,V17>=0,V22>=0,V23>=0] 

* Chain [37]: 336*s(257)+32*s(259)+72*s(261)+12*s(263)+160*s(286)+8*s(302)+36*s(336)+96*s(338)+16*s(340)+16*s(341)+16*s(342)+72*s(344)+12*s(345)+13
  Such that:aux(67) =< 1
aux(68) =< V17
aux(69) =< V17+1
aux(70) =< V17-2*V22
aux(71) =< V17-2*V22+1
aux(72) =< V17-V22
aux(73) =< -V22
aux(74) =< V22
s(257) =< aux(67)
s(336) =< aux(72)
s(286) =< aux(74)
s(259) =< aux(68)
s(260) =< aux(68)
s(261) =< aux(68)
s(260) =< aux(69)
s(261) =< aux(69)
s(263) =< s(260)
s(302) =< aux(73)
s(337) =< aux(72)
s(338) =< aux(72)
s(339) =< aux(72)
s(337) =< aux(71)
s(338) =< aux(71)
s(337) =< aux(70)
s(338) =< aux(70)
s(339) =< aux(70)
s(340) =< s(337)
s(341) =< s(339)
s(342) =< aux(70)
s(343) =< aux(72)
s(344) =< aux(72)
s(343) =< aux(71)
s(344) =< aux(71)
s(345) =< s(343)

  with precondition: [V=2,V1=2,V17>=0,V22>=0,V23>=0] 

* Chain [36]: 2*s(466)+2*s(467)+7
  Such that:aux(75) =< 1
aux(76) =< V22
s(467) =< aux(75)
s(466) =< aux(76)

  with precondition: [V=2,V1=2,V17=V22,V17>=1,V23>=0] 

* Chain [35]: 5
  with precondition: [V1=0,V>=0] 


Closed-form bounds of start(V,V1,V17,V22,V23): 
-------------------------------------
* Chain [40] with precondition: [V>=0] 
    - Upper bound: 249*V+139+nat(V1)*84+nat(V-V1)*16 
    - Complexity: n 
* Chain [39] with precondition: [V=1,V1>=0,V17>=0,V22>=0,V23>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [38] with precondition: [V=2,V1=1,V17>=0,V22>=0,V23>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [37] with precondition: [V=2,V1=2,V17>=0,V22>=0,V23>=0] 
    - Upper bound: 116*V17+160*V22+349+nat(V17-V22)*248+nat(V17-2*V22)*16 
    - Complexity: n 
* Chain [36] with precondition: [V=2,V1=2,V17=V22,V17>=1,V23>=0] 
    - Upper bound: 2*V22+9 
    - Complexity: n 
* Chain [35] with precondition: [V1=0,V>=0] 
    - Upper bound: 5 
    - Complexity: constant 

### Maximum cost of start(V,V1,V17,V22,V23): max([max([4,249*V+138+nat(V1)*84+nat(V-V1)*16]),nat(V17)*116+340+nat(V22)*158+nat(V17-V22)*248+nat(V17-2*V22)*16+(nat(V22)*2+8)])+1 
Asymptotic class: n 
* Total analysis performed in 963 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:

   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0') -> x
   minus(0', y) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0')
   div(x, y, z) -> if(ge(y, s(0')), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
quot(x, y) -> div(x, y, 0')
div(x, y, z) -> if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) -> div_by_zero
if(true, false, x, y, z) -> z
if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
quot :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ge, minus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

(16)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
quot(x, y) -> div(x, y, 0')
div(x, y, z) -> if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) -> div_by_zero
if(true, false, x, y, z) -> z
if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
quot :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


The following defined symbols remain to be analysed:
ge, minus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
ge(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s:div_by_zero3_0(+(n5_0, 1)), gen_0':s:div_by_zero3_0(+(n5_0, 1))) ->_R^Omega(1)
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) ->_IH
true

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:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
quot(x, y) -> div(x, y, 0')
div(x, y, z) -> if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) -> div_by_zero
if(true, false, x, y, z) -> z
if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
quot :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


The following defined symbols remain to be analysed:
ge, minus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
quot(x, y) -> div(x, y, 0')
div(x, y, z) -> if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) -> div_by_zero
if(true, false, x, y, z) -> z
if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
quot :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


The following defined symbols remain to be analysed:
minus, div

They will be analysed ascendingly in the following order:
minus < div

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:div_by_zero3_0(n329_0), gen_0':s:div_by_zero3_0(n329_0)) -> gen_0':s:div_by_zero3_0(0), rt in Omega(1 + n329_0)

Induction Base:
minus(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) ->_R^Omega(1)
gen_0':s:div_by_zero3_0(0)

Induction Step:
minus(gen_0':s:div_by_zero3_0(+(n329_0, 1)), gen_0':s:div_by_zero3_0(+(n329_0, 1))) ->_R^Omega(1)
minus(gen_0':s:div_by_zero3_0(n329_0), gen_0':s:div_by_zero3_0(n329_0)) ->_IH
gen_0':s:div_by_zero3_0(0)

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:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
quot(x, y) -> div(x, y, 0')
div(x, y, z) -> if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) -> div_by_zero
if(true, false, x, y, z) -> z
if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
quot :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s:div_by_zero3_0(n329_0), gen_0':s:div_by_zero3_0(n329_0)) -> gen_0':s:div_by_zero3_0(0), rt in Omega(1 + n329_0)


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


The following defined symbols remain to be analysed:
div