/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 2 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 1207 ms]
    (12) BOUNDS(1, n^2)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) TRS for Loop Detection


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

(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)
   mod(x, 0) -> modZeroErro
   mod(x, s(y)) -> modIter(x, s(y), 0, 0)
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u)
   if(true, x, y, z, u) -> u
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u))
   if2(false, x, y, z, u) -> modIter(x, y, z, u)
   if2(true, x, y, z, u) -> modIter(x, y, z, 0)

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

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

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


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


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


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)
   mod(x, 0) -> modZeroErro
   mod(x, s(y)) -> modIter(x, s(y), 0, 0)
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u)
   if(true, x, y, z, u) -> u
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u))
   if2(false, x, y, z, u) -> modIter(x, y, z, u)
   if2(true, x, y, z, u) -> modIter(x, y, z, 0)

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]
   mod(x, 0) -> modZeroErro [1]
   mod(x, s(y)) -> modIter(x, s(y), 0, 0) [1]
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u) [1]
   if(true, x, y, z, u) -> u [1]
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u)) [1]
   if2(false, x, y, z, u) -> modIter(x, y, z, u) [1]
   if2(true, x, y, z, u) -> modIter(x, y, z, 0) [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]
   mod(x, 0) -> modZeroErro [1]
   mod(x, s(y)) -> modIter(x, s(y), 0, 0) [1]
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u) [1]
   if(true, x, y, z, u) -> u [1]
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u)) [1]
   if2(false, x, y, z, u) -> modIter(x, y, z, u) [1]
   if2(true, x, y, z, u) -> modIter(x, y, z, 0) [1]

The TRS has the following type information:
le :: 0:s:modZeroErro -> 0:s:modZeroErro -> true:false
0 :: 0:s:modZeroErro
true :: true:false
s :: 0:s:modZeroErro -> 0:s:modZeroErro
false :: true:false
mod :: 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro
modZeroErro :: 0:s:modZeroErro
modIter :: 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro
if :: true:false -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro
if2 :: true:false -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro -> 0:s:modZeroErro

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]
   mod(v0, v1) -> null_mod [0]
   modIter(v0, v1, v2, v3) -> null_modIter [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]
   if2(v0, v1, v2, v3, v4) -> null_if2 [0]

And the following fresh constants:    null_le, null_mod, null_modIter, 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]
   mod(x, 0) -> modZeroErro [1]
   mod(x, s(y)) -> modIter(x, s(y), 0, 0) [1]
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u) [1]
   if(true, x, y, z, u) -> u [1]
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u)) [1]
   if2(false, x, y, z, u) -> modIter(x, y, z, u) [1]
   if2(true, x, y, z, u) -> modIter(x, y, z, 0) [1]
   le(v0, v1) -> null_le [0]
   mod(v0, v1) -> null_mod [0]
   modIter(v0, v1, v2, v3) -> null_modIter [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]
   if2(v0, v1, v2, v3, v4) -> null_if2 [0]

The TRS has the following type information:
le :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> true:false:null_le
0 :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
true :: true:false:null_le
s :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
false :: true:false:null_le
mod :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
modZeroErro :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
modIter :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
if :: true:false:null_le -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
if2 :: true:false:null_le -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2 -> 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
null_le :: true:false:null_le
null_mod :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
null_modIter :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
null_if :: 0:s:modZeroErro:null_mod:null_modIter:null_if:null_if2
null_if2 :: 0:s:modZeroErro:null_mod:null_modIter: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
   modZeroErro => 1
   null_le => 0
   null_mod => 0
   null_modIter => 0
   null_if => 0
   null_if2 => 0

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> u :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(le(y, 1 + u), x, y, 1 + z, 1 + u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   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
   if2(z', z'', z1, z2, z3) -{ 1 }-> modIter(x, y, z, u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> modIter(x, y, z, 0) :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0
   if2(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
   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
   mod(z', z'') -{ 1 }-> modIter(x, 1 + y, 0, 0) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   mod(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   mod(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   modIter(z', z'', z1, z2) -{ 1 }-> if(le(x, z), x, 1 + y, z, u) :|: z1 = z, z2 = u, z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y, u >= 0
   modIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, 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, V14, V13, V19),0,[le(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V14, V13, V19),0,[mod(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V14, V13, V19),0,[modIter(V, V1, V14, V13, Out)],[V >= 0,V1 >= 0,V14 >= 0,V13 >= 0]).
eq(start(V, V1, V14, V13, V19),0,[if(V, V1, V14, V13, V19, Out)],[V >= 0,V1 >= 0,V14 >= 0,V13 >= 0,V19 >= 0]).
eq(start(V, V1, V14, V13, V19),0,[if2(V, V1, V14, V13, V19, Out)],[V >= 0,V1 >= 0,V14 >= 0,V13 >= 0,V19 >= 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(mod(V, V1, Out),1,[],[Out = 1,V1 = 0,V = V6,V6 >= 0]).
eq(mod(V, V1, Out),1,[modIter(V7, 1 + V8, 0, 0, Ret1)],[Out = Ret1,V = V7,V7 >= 0,V8 >= 0,V1 = 1 + V8]).
eq(modIter(V, V1, V14, V13, Out),1,[le(V10, V11, Ret0),if(Ret0, V10, 1 + V12, V11, V9, Ret2)],[Out = Ret2,V14 = V11,V13 = V9,V11 >= 0,V = V10,V10 >= 0,V12 >= 0,V1 = 1 + V12,V9 >= 0]).
eq(if(V, V1, V14, V13, V19, Out),1,[],[Out = V16,V14 = V15,V17 >= 0,V = 2,V13 = V17,V18 >= 0,V15 >= 0,V1 = V18,V19 = V16,V16 >= 0]).
eq(if(V, V1, V14, V13, V19, Out),1,[le(V21, 1 + V22, Ret01),if2(Ret01, V23, V21, 1 + V20, 1 + V22, Ret3)],[Out = Ret3,V14 = V21,V20 >= 0,V13 = V20,V23 >= 0,V21 >= 0,V1 = V23,V = 1,V19 = V22,V22 >= 0]).
eq(if2(V, V1, V14, V13, V19, Out),1,[modIter(V25, V26, V24, V27, Ret4)],[Out = Ret4,V14 = V26,V24 >= 0,V13 = V24,V25 >= 0,V26 >= 0,V1 = V25,V = 1,V19 = V27,V27 >= 0]).
eq(if2(V, V1, V14, V13, V19, Out),1,[modIter(V29, V31, V30, 0, Ret5)],[Out = Ret5,V14 = V31,V30 >= 0,V = 2,V13 = V30,V29 >= 0,V31 >= 0,V1 = V29,V19 = V28,V28 >= 0]).
eq(le(V, V1, Out),0,[],[Out = 0,V33 >= 0,V32 >= 0,V1 = V32,V = V33]).
eq(mod(V, V1, Out),0,[],[Out = 0,V35 >= 0,V34 >= 0,V1 = V34,V = V35]).
eq(modIter(V, V1, V14, V13, Out),0,[],[Out = 0,V13 = V38,V37 >= 0,V14 = V39,V36 >= 0,V1 = V36,V39 >= 0,V38 >= 0,V = V37]).
eq(if(V, V1, V14, V13, V19, Out),0,[],[Out = 0,V13 = V44,V40 >= 0,V43 >= 0,V14 = V41,V42 >= 0,V1 = V42,V19 = V43,V41 >= 0,V44 >= 0,V = V40]).
eq(if2(V, V1, V14, V13, V19, Out),0,[],[Out = 0,V13 = V47,V46 >= 0,V49 >= 0,V14 = V48,V45 >= 0,V1 = V45,V19 = V49,V48 >= 0,V47 >= 0,V = V46]).
input_output_vars(le(V,V1,Out),[V,V1],[Out]).
input_output_vars(mod(V,V1,Out),[V,V1],[Out]).
input_output_vars(modIter(V,V1,V14,V13,Out),[V,V1,V14,V13],[Out]).
input_output_vars(if(V,V1,V14,V13,V19,Out),[V,V1,V14,V13,V19],[Out]).
input_output_vars(if2(V,V1,V14,V13,V19,Out),[V,V1,V14,V13,V19],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [le/3]
1. recursive  : [if/6,if2/6,modIter/5]
2. non_recursive  : [(mod)/3]
3. non_recursive  : [start/5]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into le/3
1. SCC is partially evaluated into modIter/5
2. SCC is partially evaluated into (mod)/3
3. SCC is partially evaluated into start/5

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 modIter/5 
* 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 modIter/5 
* 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 modIter(V,V1,V14,V13,Out) 
* RF of phase [18,19]: [V-V14]

#### Partial ranking functions of CR modIter(V,V1,V14,V13,Out) 
* Partial RF of phase [18,19]:
  - RF of loop [18:1]:
    V1-V13-1 depends on loops [19:1] 
  - RF of loop [18:1,19:1]:
    V-V14


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


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

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

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


### Specialization of cost equations start/5 
* CE 4 is refined into CE [53,54,55,56,57,58,59] 
* CE 7 is refined into CE [60] 
* CE 1 is refined into CE [61,62,63,64] 
* CE 2 is refined into CE [65] 
* CE 3 is refined into CE [66,67,68,69,70] 
* CE 5 is refined into CE [71,72,73,74] 
* CE 6 is refined into CE [75,76,77,78,79,80,81] 
* CE 8 is refined into CE [82,83,84,85,86] 
* CE 9 is refined into CE [87,88,89,90] 
* CE 10 is refined into CE [91,92,93,94,95,96,97] 


### Cost equations --> "Loop" of start/5 
* CEs [94] --> Loop 31 
* CEs [95] --> Loop 32 
* CEs [83,88] --> Loop 33 
* CEs [56] --> Loop 34 
* CEs [57] --> Loop 35 
* CEs [55] --> Loop 36 
* CEs [53,54,58,59,60] --> Loop 37 
* CEs [78] --> Loop 38 
* CEs [79] --> Loop 39 
* CEs [87,93] --> Loop 40 
* CEs [61,66] --> Loop 41 
* CEs [77] --> Loop 42 
* CEs [62,63,64,67,68,69,70,71,72,73,74,75,76,80,81] --> Loop 43 
* CEs [65,82,84,85,86,89,90,91,92,96,97] --> Loop 44 

### Ranking functions of CR start(V,V1,V14,V13,V19) 

#### Partial ranking functions of CR start(V,V1,V14,V13,V19) 


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 modIter(V,V1,V14,V13,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+V13
aux(14) =< V
aux(15) =< V-V14
aux(16) =< V-V14+V13+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,V14>=1,V13>=0,V>=V14+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-V14+V13-Out
aux(13) =< V1+V13
aux(17) =< V
aux(18) =< V-V14
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: [V14>=1,V13>=0,Out>=0,V>=V14+1,V1>=Out+1,V+V13>=Out+V14] 

* 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) =< V14
aux(2) =< V13+1
s(2) =< aux(1)
s(3) =< aux(2)

  with precondition: [Out=0,V>=0,V1>=0,V14>=0,V13>=0] 

* Chain [24]: 3
  with precondition: [V=0,V13=Out,V1>=1,V14>=0,V13>=0] 

* Chain [23]: 2*s(24)+1*s(26)+4
  Such that:s(26) =< V1
aux(19) =< V13+1
s(24) =< aux(19)

  with precondition: [V14=0,Out=0,V>=1,V1>=1,V13>=0] 

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

  with precondition: [V13=Out,V>=1,V1>=1,V13>=0,V14>=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: [V14=0,Out=0,V>=2,V1>=1,V13+1>=V1] 

* Chain [21,[18,19],22]: 11*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(22)+1*s(27)+8
  Such that:s(22) =< V-Out
aux(22) =< V
aux(23) =< V1
s(27) =< aux(23)
it(18) =< aux(22)
aux(7) =< aux(22)
s(21) =< it(18)*aux(7)
s(20) =< it(18)*aux(23)
s(19) =< it(18)*aux(22)

  with precondition: [V14=0,V>=2,Out>=0,V13+1>=V1,V>=Out+1,V1>=Out+1] 

* 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: [V14=0,Out=0,V>=1,V1>=1,V13+1>=V1] 

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

  with precondition: [V=1,V14=0,Out=0,V1>=1,V13+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+V13+1
s(7) =< V1
aux(13) =< V1+V13+1
s(28) =< V13+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: [V14=0,Out=0,V>=2,V13>=0,V1>=V13+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+V13-Out
aux(13) =< V1+V13+1
s(28) =< V13+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: [V14=0,V>=2,V13>=0,Out>=0,V1>=V13+2,V1>=Out+1,V+V13>=Out] 

* 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) =< V13+1
aux(2) =< V13+2
s(2) =< aux(1)
s(3) =< aux(2)

  with precondition: [V14=0,Out=0,V>=1,V13>=0,V1>=V13+2] 

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

  with precondition: [V=1,V14=0,V13+1=Out,V13>=0,V1>=V13+2] 


#### Cost of chains of mod(V,V1,Out):
* Chain [30]: 19*s(94)+54*s(95)+10*s(96)+2*s(98)+3*s(100)+2*s(101)+3*s(102)+6*s(104)+1*s(105)+10
  Such that:aux(32) =< 1
s(89) =< 2
aux(33) =< V
aux(34) =< V+1
aux(35) =< V1
s(87) =< V1+1
s(94) =< aux(32)
s(95) =< aux(33)
s(96) =< aux(35)
s(98) =< s(89)
s(99) =< aux(33)
s(100) =< s(95)*s(99)
s(101) =< s(95)*aux(35)
s(102) =< s(95)*aux(33)
s(104) =< aux(34)
s(105) =< s(95)*s(87)

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

* Chain [29]: 2*s(111)+9
  Such that:aux(36) =< 1
s(111) =< aux(36)

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

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

* Chain [27]: 12*s(113)+1*s(116)+1*s(119)+1*s(120)+1*s(121)+9
  Such that:s(115) =< 1
aux(37) =< V
s(113) =< aux(37)
s(116) =< s(115)
s(118) =< aux(37)
s(119) =< s(113)*s(118)
s(120) =< s(113)*s(115)
s(121) =< s(113)*aux(37)

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

* Chain [26]: 12*s(122)+1*s(124)+1*s(128)+1*s(129)+1*s(130)+9
  Such that:s(124) =< 1
s(123) =< V1+1
aux(38) =< V
s(122) =< aux(38)
s(127) =< aux(38)
s(128) =< s(122)*s(127)
s(129) =< s(122)*s(123)
s(130) =< s(122)*aux(38)

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


#### Cost of chains of start(V,V1,V14,V13,V19):
* Chain [44]: 22*s(155)+125*s(157)+34*s(165)+7*s(167)+1*s(168)+7*s(169)+2*s(171)+3*s(172)+6*s(173)+2*s(174)+6*s(197)+2*s(198)+5*s(203)+3*s(204)+1*s(205)+3*s(206)+20*s(207)+2*s(208)+2*s(209)+2*s(210)+1*s(212)+10
  Such that:s(158) =< 2
s(159) =< V+1
s(212) =< V-V14+V13
s(184) =< V-V14+V13+1
s(185) =< V+V13+1
s(187) =< V1+V13+1
s(188) =< V14
s(193) =< V13+1
s(189) =< V13+2
aux(41) =< 1
aux(42) =< V
aux(43) =< V-V14
aux(44) =< V1
aux(45) =< V1+1
aux(46) =< V1+V13
s(165) =< aux(41)
s(157) =< aux(42)
s(155) =< aux(44)
s(166) =< aux(42)
s(167) =< s(157)*s(166)
s(168) =< s(157)*aux(41)
s(169) =< s(157)*aux(42)
s(171) =< s(158)
s(172) =< s(157)*aux(44)
s(173) =< s(159)
s(174) =< s(157)*aux(45)
s(197) =< s(193)
s(198) =< s(189)
s(203) =< s(188)
s(204) =< s(185)
s(205) =< s(157)*s(187)
s(206) =< s(184)
s(207) =< aux(43)
s(208) =< s(207)*s(166)
s(209) =< s(207)*aux(46)
s(210) =< s(207)*aux(42)

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

* Chain [43]: 12*s(222)+35*s(224)+45*s(238)+138*s(239)+2*s(242)+6*s(244)+3*s(245)+6*s(246)+10*s(247)+3*s(248)+1*s(249)+64*s(250)+6*s(252)+2*s(253)+6*s(254)+8*s(284)+2*s(285)+3*s(291)+1*s(292)+6*s(293)+2*s(296)+2*s(301)+5*s(331)+3*s(332)+1*s(333)+2*s(337)+12
  Such that:s(233) =< 2
s(229) =< V1+1
s(313) =< V1+V19+1
s(272) =< V1+V19+2
s(231) =< V14+1
s(274) =< V14+V19+2
s(316) =< V13
s(276) =< V19+3
aux(52) =< 1
aux(53) =< V1
aux(54) =< V1-V13
aux(55) =< V1-V13+V19
aux(56) =< V1-V13+V19+1
aux(57) =< V14
aux(58) =< V14+V19
aux(59) =< V14+V19+1
aux(60) =< V13+1
aux(61) =< V19+1
aux(62) =< V19+2
s(239) =< aux(53)
s(301) =< aux(55)
s(224) =< aux(57)
s(222) =< aux(61)
s(238) =< aux(52)
s(284) =< aux(62)
s(285) =< s(276)
s(243) =< aux(53)
s(244) =< s(239)*s(243)
s(245) =< s(239)*aux(57)
s(246) =< s(239)*aux(53)
s(247) =< aux(60)
s(291) =< s(272)
s(292) =< s(239)*s(274)
s(293) =< aux(56)
s(250) =< aux(54)
s(252) =< s(250)*s(243)
s(296) =< s(250)*aux(59)
s(254) =< s(250)*aux(53)
s(331) =< s(316)
s(332) =< s(313)
s(333) =< s(239)*aux(59)
s(337) =< s(250)*aux(58)
s(242) =< s(233)
s(248) =< s(229)
s(249) =< s(239)*s(231)
s(253) =< s(250)*aux(57)

  with precondition: [V=1,V1>=0,V14>=0,V13>=0,V19>=0] 

* Chain [42]: 1*s(350)+1*s(351)+9
  Such that:s(350) =< 1
s(351) =< V19+1

  with precondition: [V=1,V1=1,V13=0,V19>=0,V14>=V19+2] 

* Chain [41]: 19*s(363)+44*s(364)+2*s(367)+2*s(369)+2*s(371)+5*s(372)+3*s(373)+1*s(374)+13*s(375)+1*s(377)+1*s(379)+12
  Such that:aux(64) =< 1
s(358) =< 2
s(360) =< V1
s(354) =< V1+1
aux(65) =< V1-V13
s(357) =< V13+1
s(363) =< aux(64)
s(364) =< s(360)
s(367) =< s(358)
s(368) =< s(360)
s(369) =< s(364)*s(368)
s(371) =< s(364)*s(360)
s(372) =< s(357)
s(373) =< s(354)
s(374) =< s(364)*aux(64)
s(375) =< aux(65)
s(377) =< s(375)*s(368)
s(379) =< s(375)*s(360)

  with precondition: [V=1,V14=0,V1>=0,V13>=0,V19>=0] 

* Chain [40]: 3*s(381)+1*s(383)+9
  Such that:s(383) =< V13+1
aux(66) =< 1
s(381) =< aux(66)

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

* Chain [39]: 12*s(384)+1*s(387)+1*s(390)+1*s(391)+1*s(392)+9
  Such that:s(386) =< V14
aux(67) =< V1
s(384) =< aux(67)
s(387) =< s(386)
s(389) =< aux(67)
s(390) =< s(384)*s(389)
s(391) =< s(384)*s(386)
s(392) =< s(384)*aux(67)

  with precondition: [V=1,V13=0,V1>=2,V14>=1,V19+1>=V14] 

* Chain [38]: 1*s(393)+1*s(395)+11*s(397)+1*s(399)+1*s(400)+1*s(401)+9
  Such that:s(396) =< V1
s(393) =< V1+V19
s(394) =< V14+V19+1
s(395) =< V19+1
s(397) =< s(396)
s(398) =< s(396)
s(399) =< s(397)*s(398)
s(400) =< s(397)*s(394)
s(401) =< s(397)*s(396)

  with precondition: [V=1,V13=0,V1>=2,V19>=0,V14>=V19+2] 

* Chain [37]: 19*s(413)+46*s(414)+10*s(415)+2*s(417)+2*s(419)+1*s(420)+2*s(421)+5*s(422)+3*s(423)+1*s(424)+3*s(425)+21*s(426)+2*s(427)+2*s(428)+2*s(429)+10
  Such that:aux(68) =< 1
s(408) =< 2
s(404) =< V1+1
s(403) =< V1-V13+1
s(406) =< V14+1
s(407) =< V13
aux(71) =< V1
aux(72) =< V1-V13
aux(73) =< V14
s(414) =< aux(71)
s(413) =< aux(68)
s(415) =< aux(73)
s(417) =< s(408)
s(418) =< aux(71)
s(419) =< s(414)*s(418)
s(420) =< s(414)*aux(73)
s(421) =< s(414)*aux(71)
s(422) =< s(407)
s(423) =< s(404)
s(424) =< s(414)*s(406)
s(425) =< s(403)
s(426) =< aux(72)
s(427) =< s(426)*s(418)
s(428) =< s(426)*aux(73)
s(429) =< s(426)*aux(71)

  with precondition: [V=2,V1>=0,V14>=0,V13>=0,V19>=0] 

* Chain [36]: 2*s(441)+9
  Such that:aux(74) =< 1
s(441) =< aux(74)

  with precondition: [V=2,V1=1,V13=0,V14>=2,V19>=0] 

* Chain [35]: 12*s(443)+1*s(446)+1*s(449)+1*s(450)+1*s(451)+9
  Such that:s(445) =< 1
aux(75) =< V1
s(443) =< aux(75)
s(446) =< s(445)
s(448) =< aux(75)
s(449) =< s(443)*s(448)
s(450) =< s(443)*s(445)
s(451) =< s(443)*aux(75)

  with precondition: [V=2,V14=1,V13=0,V1>=2,V19>=0] 

* Chain [34]: 12*s(452)+1*s(454)+1*s(458)+1*s(459)+1*s(460)+9
  Such that:s(454) =< 1
s(453) =< V14+1
aux(76) =< V1
s(452) =< aux(76)
s(457) =< aux(76)
s(458) =< s(452)*s(457)
s(459) =< s(452)*s(453)
s(460) =< s(452)*aux(76)

  with precondition: [V=2,V13=0,V1>=2,V14>=2,V19>=0] 

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

* Chain [32]: 12*s(461)+1*s(464)+1*s(467)+1*s(468)+1*s(469)+8
  Such that:s(463) =< V1
aux(77) =< V
s(461) =< aux(77)
s(464) =< s(463)
s(466) =< aux(77)
s(467) =< s(461)*s(466)
s(468) =< s(461)*s(463)
s(469) =< s(461)*aux(77)

  with precondition: [V14=0,V>=2,V1>=1,V13+1>=V1] 

* Chain [31]: 1*s(470)+1*s(472)+11*s(474)+1*s(476)+1*s(477)+1*s(478)+8
  Such that:s(473) =< V
s(470) =< V+V13
s(471) =< V1+V13+1
s(472) =< V13+1
s(474) =< s(473)
s(475) =< s(473)
s(476) =< s(474)*s(475)
s(477) =< s(474)*s(471)
s(478) =< s(474)*s(473)

  with precondition: [V14=0,V>=2,V13>=0,V1>=V13+2] 


Closed-form bounds of start(V,V1,V14,V13,V19): 
-------------------------------------
* Chain [44] with precondition: [V>=0,V1>=0] 
    - Upper bound: 126*V+48+14*V*V+3*V*V1+(V1+1)*(2*V)+nat(V1+V13+1)*V+4*V*nat(V-V14)+22*V1+nat(V14)*5+(6*V+6)+nat(V1+V13)*2*nat(V-V14)+nat(V13+1)*6+nat(V13+2)*2+nat(V+V13+1)*3+nat(V-V14+V13+1)*3+nat(V-V14+V13)+nat(V-V14)*20 
    - Complexity: n^2 
* Chain [43] with precondition: [V=1,V1>=0,V14>=0,V13>=0,V19>=0] 
    - Upper bound: 138*V1+61+12*V1*V1+3*V1*V14+(V14+1)*V1+(V14+V19+1)*V1+(V14+V19+2)*V1+12*V1*nat(V1-V13)+35*V14+2*V14*nat(V1-V13)+5*V13+(3*V1+3)+(2*V14+2*V19)*nat(V1-V13)+(10*V13+10)+(12*V19+12)+(8*V19+16)+(2*V19+6)+(3*V1+3*V19+3)+(3*V1+3*V19+6)+(2*V14+2*V19+2)*nat(V1-V13)+nat(V1-V13+V19+1)*6+nat(V1-V13+V19)*2+nat(V1-V13)*64 
    - Complexity: n^2 
* Chain [42] with precondition: [V=1,V1=1,V13=0,V19>=0,V14>=V19+2] 
    - Upper bound: V19+11 
    - Complexity: n 
* Chain [41] with precondition: [V=1,V14=0,V1>=0,V13>=0,V19>=0] 
    - Upper bound: 45*V1+35+4*V1*V1+2*V1*nat(V1-V13)+(3*V1+3)+(5*V13+5)+nat(V1-V13)*13 
    - Complexity: n^2 
* Chain [40] with precondition: [V=1,V1>=2] 
    - Upper bound: nat(V13+1)+12 
    - Complexity: n 
* Chain [39] with precondition: [V=1,V13=0,V1>=2,V14>=1,V19+1>=V14] 
    - Upper bound: 12*V1+9+2*V1*V1+V14*V1+V14 
    - Complexity: n^2 
* Chain [38] with precondition: [V=1,V13=0,V1>=2,V19>=0,V14>=V19+2] 
    - Upper bound: 11*V1+9+2*V1*V1+(V14+V19+1)*V1+(V1+V19)+(V19+1) 
    - Complexity: n^2 
* Chain [37] with precondition: [V=2,V1>=0,V14>=0,V13>=0,V19>=0] 
    - Upper bound: 46*V1+33+4*V1*V1+V14*V1+(V14+1)*V1+4*V1*nat(V1-V13)+10*V14+2*V14*nat(V1-V13)+5*V13+(3*V1+3)+nat(V1-V13+1)*3+nat(V1-V13)*21 
    - Complexity: n^2 
* Chain [36] with precondition: [V=2,V1=1,V13=0,V14>=2,V19>=0] 
    - Upper bound: 11 
    - Complexity: constant 
* Chain [35] with precondition: [V=2,V14=1,V13=0,V1>=2,V19>=0] 
    - Upper bound: 13*V1+10+2*V1*V1 
    - Complexity: n^2 
* Chain [34] with precondition: [V=2,V13=0,V1>=2,V14>=2,V19>=0] 
    - Upper bound: 12*V1+10+2*V1*V1+(V14+1)*V1 
    - Complexity: n^2 
* Chain [33] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [32] with precondition: [V14=0,V>=2,V1>=1,V13+1>=V1] 
    - Upper bound: 12*V+8+2*V*V+V1*V+V1 
    - Complexity: n^2 
* Chain [31] with precondition: [V14=0,V>=2,V13>=0,V1>=V13+2] 
    - Upper bound: 11*V+8+2*V*V+(V1+V13+1)*V+(V+V13)+(V13+1) 
    - Complexity: n^2 

### Maximum cost of start(V,V1,V14,V13,V19): max([max([max([10,nat(V19+1)+9]),nat(V13+1)+7+max([4,2*V*V+11*V+nat(V1+V13+1)*V+nat(V+V13)])]),V1+7+max([10*V1+1+max([max([2*V1*V1+max([max([nat(V14)*V1+nat(V14),nat(V14+1)*V1+1]),32*V1+23+2*V1*V1+2*V1*nat(V1-V13)+(3*V1+3)+nat(V1-V13)*13+max([nat(V13+1)*5+2,nat(V14)*V1+V1+nat(V14+1)*V1+2*V1*nat(V1-V13)+nat(V14)*10+nat(V14)*2*nat(V1-V13)+nat(V13)*5+nat(V1-V13)*8+max([nat(V1-V13+1)*3,92*V1+28+8*V1*V1+2*V1*nat(V14)+nat(V14+V19+1)*V1+nat(V14+V19+2)*V1+8*V1*nat(V1-V13)+nat(V14)*25+nat(V14+V19)*2*nat(V1-V13)+nat(V13+1)*10+nat(V19+1)*12+nat(V19+2)*8+nat(V19+3)*2+nat(V1+V19+1)*3+nat(V1+V19+2)*3+nat(V14+V19+1)*2*nat(V1-V13)+nat(V1-V13+V19+1)*6+nat(V1-V13+V19)*2+nat(V1-V13)*43])])+(V1+1)]),126*V+39+14*V*V+3*V*V1+(V1+1)*(2*V)+nat(V1+V13+1)*V+4*V*nat(V-V14)+10*V1+nat(V14)*5+(6*V+6)+nat(V1+V13)*2*nat(V-V14)+nat(V13+1)*6+nat(V13+2)*2+nat(V+V13+1)*3+nat(V-V14+V13+1)*3+nat(V-V14+V13)+nat(V-V14)*20])+V1,2*V1*V1+nat(V14+V19+1)*V1+nat(V1+V19)+nat(V19+1)]),2*V*V+12*V+V1*V])])+1 
Asymptotic class: n^2 
* Total analysis performed in 1048 ms.


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(12)
BOUNDS(1, n^2)

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(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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)
   mod(x, 0) -> modZeroErro
   mod(x, s(y)) -> modIter(x, s(y), 0, 0)
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u)
   if(true, x, y, z, u) -> u
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u))
   if2(false, x, y, z, u) -> modIter(x, y, z, u)
   if2(true, x, y, z, u) -> modIter(x, y, z, 0)

S is empty.
Rewrite Strategy: FULL
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(15) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

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

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

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

The result substitution is [ ].




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(16)
Complex Obligation (BEST)

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(17)
Obligation:
Proved the lower bound n^1 for the following 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)
   mod(x, 0) -> modZeroErro
   mod(x, s(y)) -> modIter(x, s(y), 0, 0)
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u)
   if(true, x, y, z, u) -> u
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u))
   if2(false, x, y, z, u) -> modIter(x, y, z, u)
   if2(true, x, y, z, u) -> modIter(x, y, z, 0)

S is empty.
Rewrite Strategy: FULL
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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^1, INF)

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(20)
Obligation:
Analyzing the following TRS for decreasing loops:

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)
   mod(x, 0) -> modZeroErro
   mod(x, s(y)) -> modIter(x, s(y), 0, 0)
   modIter(x, s(y), z, u) -> if(le(x, z), x, s(y), z, u)
   if(true, x, y, z, u) -> u
   if(false, x, y, z, u) -> if2(le(y, s(u)), x, y, s(z), s(u))
   if2(false, x, y, z, u) -> modIter(x, y, z, u)
   if2(true, x, y, z, u) -> modIter(x, y, z, 0)

S is empty.
Rewrite Strategy: FULL