/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 53 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 1329 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 232 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   quot(x, 0) -> quotZeroErro
   quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0)
   quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
   if(true, x, y, z, u, v) -> v
   if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
   if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
   if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
quotIter([], s(y), z, u, v)
quotIter(x, s(y), [], u, v)
if(false, x, [], z, u, v)
if(false, x, y, z, [], v)


The defined contexts are:
if([], x1, s(x2), x3, x4, x5)
if2([], x1, x2, s(x3), s(x4), x5)


[] just represents basic- or constructor-terms in the following defined contexts:
if([], x1, s(x2), x3, x4, x5)
if2([], x1, x2, s(x3), s(x4), x5)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   quot(x, 0) -> quotZeroErro
   quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0)
   quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
   if(true, x, y, z, u, v) -> v
   if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
   if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
   if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   quot(x, 0) -> quotZeroErro [1]
   quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) [1]
   quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) [1]
   if(true, x, y, z, u, v) -> v [1]
   if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) [1]
   if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) [1]
   if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   quot(x, 0) -> quotZeroErro [1]
   quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) [1]
   quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) [1]
   if(true, x, y, z, u, v) -> v [1]
   if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) [1]
   if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) [1]
   if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) [1]

The TRS has the following type information:
le :: 0:s:quotZeroErro -> 0:s:quotZeroErro -> true:false
0 :: 0:s:quotZeroErro
true :: true:false
s :: 0:s:quotZeroErro -> 0:s:quotZeroErro
false :: true:false
quot :: 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro
quotZeroErro :: 0:s:quotZeroErro
quotIter :: 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro
if :: true:false -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro
if2 :: true:false -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro

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

(7) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   le(v0, v1) -> null_le [0]
   quot(v0, v1) -> null_quot [0]
   quotIter(v0, v1, v2, v3, v4) -> null_quotIter [0]
   if(v0, v1, v2, v3, v4, v5) -> null_if [0]
   if2(v0, v1, v2, v3, v4, v5) -> null_if2 [0]

And the following fresh constants:    null_le, null_quot, null_quotIter, null_if, null_if2

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

(8)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   quot(x, 0) -> quotZeroErro [1]
   quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) [1]
   quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) [1]
   if(true, x, y, z, u, v) -> v [1]
   if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) [1]
   if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) [1]
   if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) [1]
   le(v0, v1) -> null_le [0]
   quot(v0, v1) -> null_quot [0]
   quotIter(v0, v1, v2, v3, v4) -> null_quotIter [0]
   if(v0, v1, v2, v3, v4, v5) -> null_if [0]
   if2(v0, v1, v2, v3, v4, v5) -> null_if2 [0]

The TRS has the following type information:
le :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> true:false:null_le
0 :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
true :: true:false:null_le
s :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
false :: true:false:null_le
quot :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
quotZeroErro :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
quotIter :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
if :: true:false:null_le -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
if2 :: true:false:null_le -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
null_le :: true:false:null_le
null_quot :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
null_quotIter :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
null_if :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2
null_if2 :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2

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

(9) 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
   quotZeroErro => 1
   null_le => 0
   null_quot => 0
   null_quotIter => 0
   null_if => 0
   null_if2 => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   if(z', z'', z1, z2, z3, z4) -{ 1 }-> v :|: z1 = y, z4 = v, z >= 0, v >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0
   if(z', z'', z1, z2, z3, z4) -{ 1 }-> if2(le(y, 1 + u), x, y, 1 + z, 1 + u, v) :|: z1 = y, z4 = v, z >= 0, v >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if(z', z'', z1, z2, z3, z4) -{ 0 }-> 0 :|: z2 = v3, z4 = v5, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, v5 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0
   if2(z', z'', z1, z2, z3, z4) -{ 1 }-> quotIter(x, y, z, u, v) :|: z1 = y, z4 = v, z >= 0, v >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if2(z', z'', z1, z2, z3, z4) -{ 1 }-> quotIter(x, y, z, 0, 1 + v) :|: z1 = y, z4 = v, z >= 0, v >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0
   if2(z', z'', z1, z2, z3, z4) -{ 0 }-> 0 :|: z2 = v3, z4 = v5, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, v5 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   quot(z', z'') -{ 1 }-> quotIter(x, 1 + y, 0, 0, 0) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   quot(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   quotIter(z', z'', z1, z2, z3) -{ 1 }-> if(le(x, z), x, 1 + y, z, u, v) :|: z1 = z, z2 = u, z >= 0, v >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y, z3 = v, u >= 0
   quotIter(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

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V1, V16, V14, V15, V20),0,[le(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V16, V14, V15, V20),0,[quot(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V16, V14, V15, V20),0,[quotIter(V, V1, V16, V14, V15, Out)],[V >= 0,V1 >= 0,V16 >= 0,V14 >= 0,V15 >= 0]).
eq(start(V, V1, V16, V14, V15, V20),0,[if(V, V1, V16, V14, V15, V20, Out)],[V >= 0,V1 >= 0,V16 >= 0,V14 >= 0,V15 >= 0,V20 >= 0]).
eq(start(V, V1, V16, V14, V15, V20),0,[if2(V, V1, V16, V14, V15, V20, Out)],[V >= 0,V1 >= 0,V16 >= 0,V14 >= 0,V15 >= 0,V20 >= 0]).
eq(le(V, V1, Out),1,[],[Out = 2,V1 = V2,V2 >= 0,V = 0]).
eq(le(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V3,V3 >= 0]).
eq(le(V, V1, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0,V5 >= 0,V1 = 1 + V5]).
eq(quot(V, V1, Out),1,[],[Out = 1,V1 = 0,V = V6,V6 >= 0]).
eq(quot(V, V1, Out),1,[quotIter(V7, 1 + V8, 0, 0, 0, Ret1)],[Out = Ret1,V = V7,V7 >= 0,V8 >= 0,V1 = 1 + V8]).
eq(quotIter(V, V1, V16, V14, V15, Out),1,[le(V10, V11, Ret0),if(Ret0, V10, 1 + V12, V11, V9, V13, Ret2)],[Out = Ret2,V16 = V11,V14 = V9,V11 >= 0,V13 >= 0,V = V10,V10 >= 0,V12 >= 0,V1 = 1 + V12,V15 = V13,V9 >= 0]).
eq(if(V, V1, V16, V14, V15, V20, Out),1,[],[Out = V21,V16 = V17,V20 = V21,V18 >= 0,V21 >= 0,V = 2,V14 = V18,V22 >= 0,V17 >= 0,V1 = V22,V15 = V19,V19 >= 0]).
eq(if(V, V1, V16, V14, V15, V20, Out),1,[le(V23, 1 + V25, Ret01),if2(Ret01, V26, V23, 1 + V24, 1 + V25, V27, Ret3)],[Out = Ret3,V16 = V23,V20 = V27,V24 >= 0,V27 >= 0,V14 = V24,V26 >= 0,V23 >= 0,V1 = V26,V = 1,V15 = V25,V25 >= 0]).
eq(if2(V, V1, V16, V14, V15, V20, Out),1,[quotIter(V28, V29, V30, V31, V32, Ret4)],[Out = Ret4,V16 = V29,V20 = V32,V30 >= 0,V32 >= 0,V14 = V30,V28 >= 0,V29 >= 0,V1 = V28,V = 1,V15 = V31,V31 >= 0]).
eq(if2(V, V1, V16, V14, V15, V20, Out),1,[quotIter(V34, V35, V36, 0, 1 + V37, Ret5)],[Out = Ret5,V16 = V35,V20 = V37,V36 >= 0,V37 >= 0,V = 2,V14 = V36,V34 >= 0,V35 >= 0,V1 = V34,V15 = V33,V33 >= 0]).
eq(le(V, V1, Out),0,[],[Out = 0,V39 >= 0,V38 >= 0,V1 = V38,V = V39]).
eq(quot(V, V1, Out),0,[],[Out = 0,V41 >= 0,V40 >= 0,V1 = V40,V = V41]).
eq(quotIter(V, V1, V16, V14, V15, Out),0,[],[Out = 0,V14 = V45,V43 >= 0,V44 >= 0,V16 = V46,V42 >= 0,V1 = V42,V15 = V44,V46 >= 0,V45 >= 0,V = V43]).
eq(if(V, V1, V16, V14, V15, V20, Out),0,[],[Out = 0,V14 = V52,V20 = V50,V47 >= 0,V51 >= 0,V16 = V48,V49 >= 0,V50 >= 0,V1 = V49,V15 = V51,V48 >= 0,V52 >= 0,V = V47]).
eq(if2(V, V1, V16, V14, V15, V20, Out),0,[],[Out = 0,V14 = V55,V20 = V57,V54 >= 0,V58 >= 0,V16 = V56,V53 >= 0,V57 >= 0,V1 = V53,V15 = V58,V56 >= 0,V55 >= 0,V = V54]).
input_output_vars(le(V,V1,Out),[V,V1],[Out]).
input_output_vars(quot(V,V1,Out),[V,V1],[Out]).
input_output_vars(quotIter(V,V1,V16,V14,V15,Out),[V,V1,V16,V14,V15],[Out]).
input_output_vars(if(V,V1,V16,V14,V15,V20,Out),[V,V1,V16,V14,V15,V20],[Out]).
input_output_vars(if2(V,V1,V16,V14,V15,V20,Out),[V,V1,V16,V14,V15,V20],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [le/3]
1. recursive  : [if/7,if2/7,quotIter/6]
2. non_recursive  : [quot/3]
3. non_recursive  : [start/6]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into le/3
1. SCC is partially evaluated into quotIter/6
2. SCC is partially evaluated into quot/3
3. SCC is partially evaluated into start/6

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

### Specialization of cost equations le/3 
* CE 14 is refined into CE [24] 
* CE 12 is refined into CE [25] 
* CE 11 is refined into CE [26] 
* CE 13 is refined into CE [27] 


### Cost equations --> "Loop" of le/3 
* CEs [27] --> Loop 14 
* CEs [24] --> Loop 15 
* CEs [25] --> Loop 16 
* CEs [26] --> Loop 17 

### Ranking functions of CR le(V,V1,Out) 
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR le(V,V1,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V1


### Specialization of cost equations quotIter/6 
* CE 19 is refined into CE [28,29] 
* CE 15 is refined into CE [30,31,32,33,34,35] 
* CE 18 is refined into CE [36,37,38,39,40] 
* CE 20 is refined into CE [41] 
* CE 17 is refined into CE [42,43] 
* CE 16 is refined into CE [44,45] 


### Cost equations --> "Loop" of quotIter/6 
* CEs [43] --> Loop 18 
* CEs [45] --> Loop 19 
* CEs [42] --> Loop 20 
* CEs [44] --> Loop 21 
* CEs [29] --> Loop 22 
* CEs [30,31,32,37] --> Loop 23 
* CEs [28] --> Loop 24 
* CEs [33,34,35,36,38,39,40,41] --> Loop 25 

### Ranking functions of CR quotIter(V,V1,V16,V14,V15,Out) 
* RF of phase [18,19]: [V-V16]

#### Partial ranking functions of CR quotIter(V,V1,V16,V14,V15,Out) 
* Partial RF of phase [18,19]:
  - RF of loop [18:1]:
    V1-V14-1 depends on loops [19:1] 
  - RF of loop [18:1,19:1]:
    V-V16


### Specialization of cost equations quot/3 
* CE 22 is refined into CE [46,47,48,49,50,51] 
* CE 23 is refined into CE [52] 
* CE 21 is refined into CE [53] 


### Cost equations --> "Loop" of quot/3 
* CEs [51] --> Loop 26 
* CEs [50] --> Loop 27 
* CEs [53] --> Loop 28 
* CEs [48] --> Loop 29 
* CEs [47] --> Loop 30 
* CEs [46,49,52] --> Loop 31 

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

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


### Specialization of cost equations start/6 
* CE 4 is refined into CE [54,55,56,57,58,59,60,61] 
* CE 7 is refined into CE [62] 
* CE 1 is refined into CE [63,64,65,66] 
* CE 2 is refined into CE [67] 
* CE 3 is refined into CE [68,69,70,71,72] 
* CE 5 is refined into CE [73,74,75,76] 
* CE 6 is refined into CE [77,78,79,80,81,82,83,84] 
* CE 8 is refined into CE [85,86,87,88,89] 
* CE 9 is refined into CE [90,91,92,93,94] 
* CE 10 is refined into CE [95,96,97,98,99,100,101,102] 


### Cost equations --> "Loop" of start/6 
* CEs [100] --> Loop 32 
* CEs [99] --> Loop 33 
* CEs [86,92] --> Loop 34 
* CEs [59] --> Loop 35 
* CEs [58] --> Loop 36 
* CEs [56] --> Loop 37 
* CEs [55,93] --> Loop 38 
* CEs [54,57,60,61,62] --> Loop 39 
* CEs [82] --> Loop 40 
* CEs [81] --> Loop 41 
* CEs [97] --> Loop 42 
* CEs [96] --> Loop 43 
* CEs [63,68] --> Loop 44 
* CEs [78,79,90] --> Loop 45 
* CEs [64,65,66,69,70,71,72,73,74,75,76,77,80,83,84] --> Loop 46 
* CEs [67,85,87,88,89,91,94,95,98,101,102] --> Loop 47 

### Ranking functions of CR start(V,V1,V16,V14,V15,V20) 

#### Partial ranking functions of CR start(V,V1,V16,V14,V15,V20) 


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

#### Cost of chains of le(V,V1,Out):
* Chain [[14],17]: 1*it(14)+1
  Such that:it(14) =< V

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

* Chain [[14],16]: 1*it(14)+1
  Such that:it(14) =< V1

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

* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< V1

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

* Chain [17]: 1
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [16]: 1
  with precondition: [V1=0,Out=1,V>=1] 

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


#### Cost of chains of quotIter(V,V1,V16,V14,V15,Out):
* Chain [[18,19],25]: 10*it(18)+6*s(2)+3*s(3)+1*s(7)+1*s(19)+1*s(20)+1*s(21)+4
  Such that:s(7) =< V1
aux(13) =< V1+V14
aux(14) =< V
aux(15) =< V-V16
aux(16) =< V-V16+V14+1
s(2) =< aux(14)
s(3) =< aux(16)
it(18) =< aux(15)
aux(7) =< aux(14)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(13)
s(19) =< it(18)*aux(14)

  with precondition: [Out=0,V1>=1,V16>=1,V14>=0,V15>=0,V>=V16+1] 

* Chain [[18,19],22]: 10*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(22)+1*s(23)+3
  Such that:s(22) =< V-V16+V14
aux(13) =< V1+V14
aux(17) =< V
aux(18) =< V-V16
s(23) =< aux(17)
it(18) =< aux(18)
aux(7) =< aux(17)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(13)
s(19) =< it(18)*aux(17)

  with precondition: [V1>=1,V16>=1,V14>=0,V15>=0,V>=V16+1,Out>=V15,Out+V1>=V15+2,V+V15>=Out+V16] 

* Chain [25]: 5*s(2)+2*s(3)+1*s(7)+1*s(10)+4
  Such that:s(10) =< V
s(7) =< V1
aux(1) =< V16
aux(2) =< V14+1
s(2) =< aux(1)
s(3) =< aux(2)

  with precondition: [Out=0,V>=0,V1>=0,V16>=0,V14>=0,V15>=0] 

* Chain [24]: 3
  with precondition: [V=0,V15=Out,V1>=1,V16>=0,V14>=0,V15>=0] 

* Chain [23]: 2*s(24)+1*s(26)+4
  Such that:s(26) =< V1
aux(19) =< V14+1
s(24) =< aux(19)

  with precondition: [V16=0,Out=0,V>=1,V1>=1,V14>=0,V15>=0] 

* Chain [22]: 1*s(23)+3
  Such that:s(23) =< V

  with precondition: [V15=Out,V>=1,V1>=1,V14>=0,V15>=0,V16>=V] 

* Chain [21,[18,19],25]: 19*it(18)+2*s(7)+1*s(19)+1*s(20)+1*s(21)+9
  Such that:aux(20) =< V
aux(21) =< V1
s(7) =< aux(21)
it(18) =< aux(20)
aux(7) =< aux(20)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(21)
s(19) =< it(18)*aux(20)

  with precondition: [V16=0,Out=0,V>=2,V1>=1,V15>=0,V14+1>=V1] 

* Chain [21,[18,19],22]: 12*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(27)+8
  Such that:aux(22) =< V
aux(23) =< V1
it(18) =< aux(22)
s(27) =< aux(23)
aux(7) =< aux(22)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(23)
s(19) =< it(18)*aux(22)

  with precondition: [V16=0,V>=2,V1>=1,V15>=0,V14+1>=V1,Out>=V15+1,Out+V1>=V15+3,V+V15>=Out] 

* Chain [21,25]: 7*s(2)+2*s(7)+1*s(10)+9
  Such that:s(10) =< V
aux(24) =< 1
aux(25) =< V1
s(7) =< aux(25)
s(2) =< aux(24)

  with precondition: [V16=0,Out=0,V>=1,V1>=1,V15>=0,V14+1>=V1] 

* Chain [21,22]: 1*s(23)+1*s(27)+8
  Such that:s(23) =< 1
s(27) =< V1

  with precondition: [V=1,V16=0,Out=V15+1,V1>=1,Out>=1,V14+1>=V1] 

* Chain [20,[18,19],25]: 16*it(18)+3*s(3)+1*s(7)+1*s(19)+1*s(20)+1*s(21)+1*s(28)+9
  Such that:aux(16) =< V+V14+1
s(7) =< V1
aux(13) =< V1+V14+1
s(28) =< V14+1
aux(26) =< V
it(18) =< aux(26)
s(3) =< aux(16)
aux(7) =< aux(26)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(13)
s(19) =< it(18)*aux(26)

  with precondition: [V16=0,Out=0,V>=2,V14>=0,V15>=0,V1>=V14+2] 

* Chain [20,[18,19],22]: 11*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(22)+1*s(28)+8
  Such that:s(22) =< V+V14
aux(13) =< V1+V14+1
s(28) =< V14+1
aux(27) =< V
it(18) =< aux(27)
aux(7) =< aux(27)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(13)
s(19) =< it(18)*aux(27)

  with precondition: [V16=0,V>=2,V14>=0,V15>=0,V1>=V14+2,Out>=V15,V+V15>=Out+1] 

* Chain [20,25]: 5*s(2)+2*s(3)+1*s(7)+1*s(10)+1*s(28)+9
  Such that:aux(1) =< 1
s(10) =< V
s(7) =< V1
s(28) =< V14+1
aux(2) =< V14+2
s(2) =< aux(1)
s(3) =< aux(2)

  with precondition: [V16=0,Out=0,V>=1,V14>=0,V15>=0,V1>=V14+2] 

* Chain [20,22]: 1*s(23)+1*s(28)+8
  Such that:s(23) =< 1
s(28) =< V14+1

  with precondition: [V=1,V16=0,V15=Out,V14>=0,V15>=0,V1>=V14+2] 


#### Cost of chains of quot(V,V1,Out):
* Chain [31]: 54*s(92)+9*s(93)+18*s(94)+2*s(96)+3*s(98)+2*s(99)+3*s(100)+6*s(102)+1*s(103)+10
  Such that:aux(32) =< 1
s(87) =< 2
aux(33) =< V
aux(34) =< V+1
aux(35) =< V1
s(85) =< V1+1
s(92) =< aux(33)
s(93) =< aux(35)
s(94) =< aux(32)
s(96) =< s(87)
s(97) =< aux(33)
s(98) =< s(92)*s(97)
s(99) =< s(92)*aux(35)
s(100) =< s(92)*aux(33)
s(102) =< aux(34)
s(103) =< s(92)*s(85)

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

* Chain [30]: 2*s(109)+9
  Such that:aux(36) =< 1
s(109) =< aux(36)

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

* Chain [29]: 2*s(111)+9
  Such that:aux(37) =< 1
s(111) =< aux(37)

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

* Chain [28]: 1
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [27]: 12*s(115)+1*s(116)+1*s(118)+1*s(119)+1*s(120)+9
  Such that:s(114) =< 1
s(113) =< V
s(115) =< s(113)
s(116) =< s(114)
s(117) =< s(113)
s(118) =< s(115)*s(117)
s(119) =< s(115)*s(114)
s(120) =< s(115)*s(113)

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

* Chain [26]: 12*s(121)+1*s(123)+1*s(127)+1*s(128)+1*s(129)+9
  Such that:s(123) =< 1
s(122) =< V1+1
aux(38) =< V
s(121) =< aux(38)
s(126) =< aux(38)
s(127) =< s(121)*s(126)
s(128) =< s(121)*s(122)
s(129) =< s(121)*aux(38)

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


#### Cost of chains of start(V,V1,V16,V14,V15,V20):
* Chain [47]: 20*s(148)+113*s(150)+33*s(157)+2*s(160)+6*s(162)+3*s(163)+6*s(164)+6*s(165)+2*s(166)+6*s(188)+2*s(190)+5*s(195)+3*s(196)+1*s(197)+3*s(198)+20*s(199)+2*s(200)+2*s(201)+2*s(202)+1*s(204)+10
  Such that:s(151) =< 2
s(153) =< V+1
s(204) =< V-V16+V14
s(176) =< V-V16+V14+1
s(177) =< V+V14+1
s(179) =< V1+V14+1
s(180) =< V16
s(185) =< V14+1
s(181) =< V14+2
aux(40) =< 1
aux(41) =< V
aux(42) =< V-V16
aux(43) =< V1
aux(44) =< V1+1
aux(45) =< V1+V14
s(157) =< aux(40)
s(150) =< aux(41)
s(148) =< aux(43)
s(160) =< s(151)
s(161) =< aux(41)
s(162) =< s(150)*s(161)
s(163) =< s(150)*aux(43)
s(164) =< s(150)*aux(41)
s(165) =< s(153)
s(166) =< s(150)*aux(44)
s(188) =< s(185)
s(190) =< s(181)
s(195) =< s(180)
s(196) =< s(177)
s(197) =< s(150)*s(179)
s(198) =< s(176)
s(199) =< aux(42)
s(200) =< s(199)*s(161)
s(201) =< s(199)*aux(45)
s(202) =< s(199)*aux(41)

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

* Chain [46]: 12*s(214)+32*s(216)+138*s(230)+42*s(232)+2*s(234)+6*s(236)+3*s(237)+6*s(238)+10*s(239)+3*s(240)+1*s(241)+64*s(242)+6*s(244)+2*s(245)+6*s(246)+8*s(275)+2*s(277)+3*s(283)+1*s(284)+6*s(285)+2*s(288)+2*s(293)+5*s(323)+3*s(324)+1*s(325)+2*s(329)+12
  Such that:s(225) =< 2
s(221) =< V1+1
s(305) =< V1+V15+1
s(264) =< V1+V15+2
s(223) =< V16+1
s(266) =< V16+V15+2
s(308) =< V14
s(268) =< V15+3
aux(51) =< 1
aux(52) =< V1
aux(53) =< V1-V14
aux(54) =< V1-V14+V15
aux(55) =< V1-V14+V15+1
aux(56) =< V16
aux(57) =< V16+V15
aux(58) =< V16+V15+1
aux(59) =< V14+1
aux(60) =< V15+1
aux(61) =< V15+2
s(230) =< aux(52)
s(293) =< aux(54)
s(216) =< aux(56)
s(214) =< aux(60)
s(275) =< aux(61)
s(232) =< aux(51)
s(277) =< s(268)
s(235) =< aux(52)
s(236) =< s(230)*s(235)
s(237) =< s(230)*aux(56)
s(238) =< s(230)*aux(52)
s(239) =< aux(59)
s(283) =< s(264)
s(284) =< s(230)*s(266)
s(285) =< aux(55)
s(242) =< aux(53)
s(244) =< s(242)*s(235)
s(288) =< s(242)*aux(58)
s(246) =< s(242)*aux(52)
s(323) =< s(308)
s(324) =< s(305)
s(325) =< s(230)*aux(58)
s(329) =< s(242)*aux(57)
s(234) =< s(225)
s(240) =< s(221)
s(241) =< s(230)*s(223)
s(245) =< s(242)*aux(56)

  with precondition: [V=1,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 

* Chain [45]: 4*s(342)+1*s(343)+1*s(345)+9
  Such that:s(343) =< V16
s(345) =< V15+1
aux(62) =< 1
s(342) =< aux(62)

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

* Chain [44]: 44*s(359)+18*s(361)+2*s(363)+2*s(365)+2*s(367)+5*s(368)+3*s(369)+1*s(370)+13*s(371)+1*s(373)+1*s(375)+12
  Such that:aux(64) =< 1
s(354) =< 2
s(356) =< V1
s(350) =< V1+1
aux(65) =< V1-V14
s(353) =< V14+1
s(359) =< s(356)
s(361) =< aux(64)
s(363) =< s(354)
s(364) =< s(356)
s(365) =< s(359)*s(364)
s(367) =< s(359)*s(356)
s(368) =< s(353)
s(369) =< s(350)
s(370) =< s(359)*aux(64)
s(371) =< aux(65)
s(373) =< s(371)*s(364)
s(375) =< s(371)*s(356)

  with precondition: [V=1,V16=0,V1>=0,V14>=0,V15>=0,V20>=0] 

* Chain [43]: 1*s(376)+1*s(377)+8
  Such that:s(376) =< 1
s(377) =< V1

  with precondition: [V=1,V16=0,V1>=1,V15>=0,V14+1>=V1] 

* Chain [42]: 1*s(378)+1*s(379)+8
  Such that:s(378) =< 1
s(379) =< V14+1

  with precondition: [V=1,V16=0,V14>=0,V15>=0,V1>=V14+2] 

* Chain [41]: 12*s(382)+1*s(383)+1*s(385)+1*s(386)+1*s(387)+9
  Such that:s(380) =< V1
s(381) =< V16
s(382) =< s(380)
s(383) =< s(381)
s(384) =< s(380)
s(385) =< s(382)*s(384)
s(386) =< s(382)*s(381)
s(387) =< s(382)*s(380)

  with precondition: [V=1,V14=0,V1>=2,V16>=1,V20>=0,V15+1>=V16] 

* Chain [40]: 1*s(388)+1*s(390)+11*s(392)+1*s(394)+1*s(395)+1*s(396)+9
  Such that:s(391) =< V1
s(388) =< V1+V15
s(389) =< V16+V15+1
s(390) =< V15+1
s(392) =< s(391)
s(393) =< s(391)
s(394) =< s(392)*s(393)
s(395) =< s(392)*s(389)
s(396) =< s(392)*s(391)

  with precondition: [V=1,V14=0,V1>=2,V15>=0,V20>=0,V16>=V15+2] 

* Chain [39]: 46*s(408)+9*s(409)+18*s(410)+2*s(412)+2*s(414)+1*s(415)+2*s(416)+5*s(417)+3*s(418)+1*s(419)+3*s(420)+21*s(421)+2*s(422)+2*s(423)+2*s(424)+10
  Such that:aux(66) =< 1
s(403) =< 2
s(399) =< V1+1
s(398) =< V1-V14+1
s(401) =< V16+1
s(402) =< V14
aux(69) =< V1
aux(70) =< V1-V14
aux(71) =< V16
s(408) =< aux(69)
s(409) =< aux(71)
s(410) =< aux(66)
s(412) =< s(403)
s(413) =< aux(69)
s(414) =< s(408)*s(413)
s(415) =< s(408)*aux(71)
s(416) =< s(408)*aux(69)
s(417) =< s(402)
s(418) =< s(399)
s(419) =< s(408)*s(401)
s(420) =< s(398)
s(421) =< aux(70)
s(422) =< s(421)*s(413)
s(423) =< s(421)*aux(71)
s(424) =< s(421)*aux(69)

  with precondition: [V=2,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 

* Chain [38]: 3*s(436)+12*s(440)+1*s(443)+1*s(444)+1*s(445)+9
  Such that:s(439) =< V
aux(73) =< 1
s(436) =< aux(73)
s(440) =< s(439)
s(442) =< s(439)
s(443) =< s(440)*s(442)
s(444) =< s(440)*aux(73)
s(445) =< s(440)*s(439)

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

* Chain [37]: 2*s(446)+9
  Such that:aux(74) =< 1
s(446) =< aux(74)

  with precondition: [V=2,V1=1,V14=0,V16>=2,V15>=0,V20>=0] 

* Chain [36]: 12*s(450)+1*s(451)+1*s(453)+1*s(454)+1*s(455)+9
  Such that:s(449) =< 1
s(448) =< V1
s(450) =< s(448)
s(451) =< s(449)
s(452) =< s(448)
s(453) =< s(450)*s(452)
s(454) =< s(450)*s(449)
s(455) =< s(450)*s(448)

  with precondition: [V=2,V16=1,V14=0,V1>=2,V15>=0,V20>=0] 

* Chain [35]: 12*s(456)+1*s(458)+1*s(462)+1*s(463)+1*s(464)+9
  Such that:s(458) =< 1
s(457) =< V16+1
aux(75) =< V1
s(456) =< aux(75)
s(461) =< aux(75)
s(462) =< s(456)*s(461)
s(463) =< s(456)*s(457)
s(464) =< s(456)*aux(75)

  with precondition: [V=2,V14=0,V1>=2,V16>=2,V15>=0,V20>=0] 

* Chain [34]: 1
  with precondition: [V1=0,V>=0] 

* Chain [33]: 12*s(467)+1*s(468)+1*s(470)+1*s(471)+1*s(472)+8
  Such that:s(465) =< V
s(466) =< V1
s(467) =< s(465)
s(468) =< s(466)
s(469) =< s(465)
s(470) =< s(467)*s(469)
s(471) =< s(467)*s(466)
s(472) =< s(467)*s(465)

  with precondition: [V16=0,V>=2,V1>=1,V15>=0,V14+1>=V1] 

* Chain [32]: 1*s(473)+1*s(475)+11*s(477)+1*s(479)+1*s(480)+1*s(481)+8
  Such that:s(476) =< V
s(473) =< V+V14
s(474) =< V1+V14+1
s(475) =< V14+1
s(477) =< s(476)
s(478) =< s(476)
s(479) =< s(477)*s(478)
s(480) =< s(477)*s(474)
s(481) =< s(477)*s(476)

  with precondition: [V16=0,V>=2,V14>=0,V15>=0,V1>=V14+2] 


Closed-form bounds of start(V,V1,V16,V14,V15,V20): 
-------------------------------------
* Chain [47] with precondition: [V>=0,V1>=0] 
    - Upper bound: 113*V+47+12*V*V+3*V*V1+(V1+1)*(2*V)+nat(V1+V14+1)*V+4*V*nat(V-V16)+20*V1+nat(V16)*5+(6*V+6)+nat(V1+V14)*2*nat(V-V16)+nat(V14+1)*6+nat(V14+2)*2+nat(V+V14+1)*3+nat(V-V16+V14+1)*3+nat(V-V16+V14)+nat(V-V16)*20 
    - Complexity: n^2 
* Chain [46] with precondition: [V=1,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 
    - Upper bound: 138*V1+58+12*V1*V1+3*V1*V16+(V16+1)*V1+(V16+V15+1)*V1+(V16+V15+2)*V1+12*V1*nat(V1-V14)+32*V16+2*V16*nat(V1-V14)+5*V14+(3*V1+3)+(2*V16+2*V15)*nat(V1-V14)+(10*V14+10)+(12*V15+12)+(8*V15+16)+(2*V15+6)+(3*V1+3*V15+3)+(3*V1+3*V15+6)+(2*V16+2*V15+2)*nat(V1-V14)+nat(V1-V14+V15+1)*6+nat(V1-V14+V15)*2+nat(V1-V14)*64 
    - Complexity: n^2 
* Chain [45] with precondition: [V=1,V1=1] 
    - Upper bound: nat(V16)+13+nat(V15+1) 
    - Complexity: n 
* Chain [44] with precondition: [V=1,V16=0,V1>=0,V14>=0,V15>=0,V20>=0] 
    - Upper bound: 45*V1+34+4*V1*V1+2*V1*nat(V1-V14)+(3*V1+3)+(5*V14+5)+nat(V1-V14)*13 
    - Complexity: n^2 
* Chain [43] with precondition: [V=1,V16=0,V1>=1,V15>=0,V14+1>=V1] 
    - Upper bound: V1+9 
    - Complexity: n 
* Chain [42] with precondition: [V=1,V16=0,V14>=0,V15>=0,V1>=V14+2] 
    - Upper bound: V14+10 
    - Complexity: n 
* Chain [41] with precondition: [V=1,V14=0,V1>=2,V16>=1,V20>=0,V15+1>=V16] 
    - Upper bound: 12*V1+9+2*V1*V1+V16*V1+V16 
    - Complexity: n^2 
* Chain [40] with precondition: [V=1,V14=0,V1>=2,V15>=0,V20>=0,V16>=V15+2] 
    - Upper bound: 11*V1+9+2*V1*V1+(V16+V15+1)*V1+(V1+V15)+(V15+1) 
    - Complexity: n^2 
* Chain [39] with precondition: [V=2,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 
    - Upper bound: 46*V1+32+4*V1*V1+V16*V1+(V16+1)*V1+4*V1*nat(V1-V14)+9*V16+2*V16*nat(V1-V14)+5*V14+(3*V1+3)+nat(V1-V14+1)*3+nat(V1-V14)*21 
    - Complexity: n^2 
* Chain [38] with precondition: [V1=1,V>=2] 
    - Upper bound: 13*V+12+2*V*V 
    - Complexity: n^2 
* Chain [37] with precondition: [V=2,V1=1,V14=0,V16>=2,V15>=0,V20>=0] 
    - Upper bound: 11 
    - Complexity: constant 
* Chain [36] with precondition: [V=2,V16=1,V14=0,V1>=2,V15>=0,V20>=0] 
    - Upper bound: 13*V1+10+2*V1*V1 
    - Complexity: n^2 
* Chain [35] with precondition: [V=2,V14=0,V1>=2,V16>=2,V15>=0,V20>=0] 
    - Upper bound: 12*V1+10+2*V1*V1+(V16+1)*V1 
    - Complexity: n^2 
* Chain [34] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [33] with precondition: [V16=0,V>=2,V1>=1,V15>=0,V14+1>=V1] 
    - Upper bound: 12*V+8+2*V*V+V1*V+V1 
    - Complexity: n^2 
* Chain [32] with precondition: [V16=0,V>=2,V14>=0,V15>=0,V1>=V14+2] 
    - Upper bound: 11*V+8+2*V*V+(V1+V14+1)*V+(V+V14)+(V14+1) 
    - Complexity: n^2 

### Maximum cost of start(V,V1,V16,V14,V15,V20): max([max([max([10,nat(V14+1)+8,nat(V16)+12+nat(V15+1)]),11*V+7+2*V*V+max([2*V+4,nat(V1+V14+1)*V+nat(V+V14)+nat(V14+1)])]),V1+7+max([max([1,2*V*V+12*V+V1*V]),10*V1+1+max([max([2*V1*V1+max([max([nat(V16)*V1+nat(V16),nat(V16+1)*V1+1]),32*V1+22+2*V1*V1+2*V1*nat(V1-V14)+(3*V1+3)+nat(V1-V14)*13+max([nat(V14+1)*5+2,nat(V16)*V1+V1+nat(V16+1)*V1+2*V1*nat(V1-V14)+nat(V16)*9+nat(V16)*2*nat(V1-V14)+nat(V14)*5+nat(V1-V14)*8+max([nat(V1-V14+1)*3,92*V1+26+8*V1*V1+2*V1*nat(V16)+nat(V16+V15+1)*V1+nat(V16+V15+2)*V1+8*V1*nat(V1-V14)+nat(V16)*23+nat(V16+V15)*2*nat(V1-V14)+nat(V14+1)*10+nat(V15+1)*12+nat(V15+2)*8+nat(V15+3)*2+nat(V1+V15+1)*3+nat(V1+V15+2)*3+nat(V16+V15+1)*2*nat(V1-V14)+nat(V1-V14+V15+1)*6+nat(V1-V14+V15)*2+nat(V1-V14)*43])])+(V1+1)]),113*V+38+12*V*V+3*V*V1+(V1+1)*(2*V)+nat(V1+V14+1)*V+4*V*nat(V-V16)+8*V1+nat(V16)*5+(6*V+6)+nat(V1+V14)*2*nat(V-V16)+nat(V14+1)*6+nat(V14+2)*2+nat(V+V14+1)*3+nat(V-V16+V14+1)*3+nat(V-V16+V14)+nat(V-V16)*20])+V1,2*V1*V1+nat(V16+V15+1)*V1+nat(V1+V15)+nat(V15+1)])])])+1 
Asymptotic class: n^2 
* Total analysis performed in 1175 ms.


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

(12)
BOUNDS(1, n^2)

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

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

(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   quot(x, 0') -> quotZeroErro
   quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0')
   quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
   if(true, x, y, z, u, v) -> v
   if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
   if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
   if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v))

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

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

(16)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, 0') -> quotZeroErro
quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) -> v
if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro -> 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, quotIter

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

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

(18)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, 0') -> quotZeroErro
quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) -> v
if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro -> 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro


Generator Equations:
gen_0':s:quotZeroErro3_0(0) <=> 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) <=> s(gen_0':s:quotZeroErro3_0(x))


The following defined symbols remain to be analysed:
le, quotIter

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_0':s:quotZeroErro3_0(0), gen_0':s:quotZeroErro3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s:quotZeroErro3_0(+(n5_0, 1)), gen_0':s:quotZeroErro3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_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).
----------------------------------------

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, 0') -> quotZeroErro
quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) -> v
if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro -> 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro


Generator Equations:
gen_0':s:quotZeroErro3_0(0) <=> 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) <=> s(gen_0':s:quotZeroErro3_0(x))


The following defined symbols remain to be analysed:
le, quotIter

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

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, 0') -> quotZeroErro
quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) -> v
if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro -> 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro


Lemmas:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:quotZeroErro3_0(0) <=> 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) <=> s(gen_0':s:quotZeroErro3_0(x))


The following defined symbols remain to be analysed:
quotIter