WORST_CASE(Omega(n^1),O(n^3))
proof of input_u22tRbllXD.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 691 ms]
    (10) BOUNDS(1, n^3)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 10 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 306 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 123 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 105 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 12 ms]
            (32) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   div(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> 0
   if2(true, x, y) -> s(div(minus(x, y), y))
   if2(false, x, y) -> 0

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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^3).


The TRS R consists of the following rules:

   minus(s(x), y) -> if(gt(s(x), y), x, y) [1]
   if(true, x, y) -> s(minus(x, y)) [1]
   if(false, x, y) -> 0 [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   div(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> 0 [1]
   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
   if2(false, x, y) -> 0 [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:

   minus(s(x), y) -> if(gt(s(x), y), x, y) [1]
   if(true, x, y) -> s(minus(x, y)) [1]
   if(false, x, y) -> 0 [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   div(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> 0 [1]
   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
   if2(false, x, y) -> 0 [1]

The TRS has the following type information:
minus :: s:0 -> s:0 -> s:0
s :: s:0 -> s:0
if :: true:false -> s:0 -> s:0 -> s:0
gt :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
0 :: s:0
ge :: s:0 -> s:0 -> true:false
div :: s:0 -> s:0 -> s:0
if1 :: true:false -> s:0 -> s:0 -> s:0
if2 :: true:false -> s:0 -> s:0 -> s:0

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:

   minus(v0, v1) -> null_minus [0]
   ge(v0, v1) -> null_ge [0]
   gt(v0, v1) -> null_gt [0]
   if(v0, v1, v2) -> null_if [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]

And the following fresh constants:    null_minus, null_ge, null_gt, null_if, null_if1, null_if2

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

(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:

   minus(s(x), y) -> if(gt(s(x), y), x, y) [1]
   if(true, x, y) -> s(minus(x, y)) [1]
   if(false, x, y) -> 0 [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   div(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> 0 [1]
   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
   if2(false, x, y) -> 0 [1]
   minus(v0, v1) -> null_minus [0]
   ge(v0, v1) -> null_ge [0]
   gt(v0, v1) -> null_gt [0]
   if(v0, v1, v2) -> null_if [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]

The TRS has the following type information:
minus :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
s :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
if :: true:false:null_ge:null_gt -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
gt :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> true:false:null_ge:null_gt
true :: true:false:null_ge:null_gt
false :: true:false:null_ge:null_gt
0 :: s:0:null_minus:null_if:null_if1:null_if2
ge :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> true:false:null_ge:null_gt
div :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
if1 :: true:false:null_ge:null_gt -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
if2 :: true:false:null_ge:null_gt -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
null_minus :: s:0:null_minus:null_if:null_if1:null_if2
null_ge :: true:false:null_ge:null_gt
null_gt :: true:false:null_ge:null_gt
null_if :: s:0:null_minus:null_if:null_if1:null_if2
null_if1 :: s:0:null_minus:null_if:null_if1:null_if2
null_if2 :: s:0:null_minus:null_if:null_if1:null_if2

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

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

   true => 2
   false => 1
   0 => 0
   null_minus => 0
   null_ge => 0
   null_gt => 0
   null_if => 0
   null_if1 => 0
   null_if2 => 0

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

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

   div(z, z') -{ 1 }-> if1(ge(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   gt(z, z') -{ 1 }-> gt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gt(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   gt(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   gt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> if2(gt(y, 0), x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 + div(minus(x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   minus(z, z') -{ 1 }-> if(gt(1 + x, y), x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[if(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[if1(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[if2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(minus(V1, V, Out),1,[gt(1 + V3, V2, Ret0),if(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = 1 + V3,V = V2]).
eq(if(V1, V, V5, Out),1,[minus(V4, V6, Ret1)],[Out = 1 + Ret1,V1 = 2,V = V4,V5 = V6,V4 >= 0,V6 >= 0]).
eq(if(V1, V, V5, Out),1,[],[Out = 0,V = V8,V5 = V7,V1 = 1,V8 >= 0,V7 >= 0]).
eq(ge(V1, V, Out),1,[],[Out = 2,V9 >= 0,V1 = V9,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V10,V10 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V12, V11, Ret2)],[Out = Ret2,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]).
eq(gt(V1, V, Out),1,[],[Out = 1,V13 >= 0,V1 = 0,V = V13]).
eq(gt(V1, V, Out),1,[],[Out = 2,V14 >= 0,V1 = 1 + V14,V = 0]).
eq(gt(V1, V, Out),1,[gt(V16, V15, Ret3)],[Out = Ret3,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]).
eq(div(V1, V, Out),1,[ge(V17, V18, Ret01),if1(Ret01, V17, V18, Ret4)],[Out = Ret4,V17 >= 0,V18 >= 0,V1 = V17,V = V18]).
eq(if1(V1, V, V5, Out),1,[gt(V19, 0, Ret02),if2(Ret02, V20, V19, Ret5)],[Out = Ret5,V1 = 2,V = V20,V5 = V19,V20 >= 0,V19 >= 0]).
eq(if1(V1, V, V5, Out),1,[],[Out = 0,V = V22,V5 = V21,V1 = 1,V22 >= 0,V21 >= 0]).
eq(if2(V1, V, V5, Out),1,[minus(V24, V23, Ret10),div(Ret10, V23, Ret11)],[Out = 1 + Ret11,V1 = 2,V = V24,V5 = V23,V24 >= 0,V23 >= 0]).
eq(if2(V1, V, V5, Out),1,[],[Out = 0,V = V25,V5 = V26,V1 = 1,V25 >= 0,V26 >= 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V28 >= 0,V27 >= 0,V1 = V28,V = V27]).
eq(ge(V1, V, Out),0,[],[Out = 0,V30 >= 0,V29 >= 0,V1 = V30,V = V29]).
eq(gt(V1, V, Out),0,[],[Out = 0,V32 >= 0,V31 >= 0,V1 = V32,V = V31]).
eq(if(V1, V, V5, Out),0,[],[Out = 0,V33 >= 0,V5 = V35,V34 >= 0,V1 = V33,V = V34,V35 >= 0]).
eq(if1(V1, V, V5, Out),0,[],[Out = 0,V37 >= 0,V5 = V38,V36 >= 0,V1 = V37,V = V36,V38 >= 0]).
eq(if2(V1, V, V5, Out),0,[],[Out = 0,V40 >= 0,V5 = V41,V39 >= 0,V1 = V40,V = V39,V41 >= 0]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).
input_output_vars(if1(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(if2(V1,V,V5,Out),[V1,V,V5],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [ge/3]
1. recursive  : [gt/3]
2. recursive  : [if/4,minus/3]
3. recursive  : [(div)/3,if1/4,if2/4]
4. non_recursive  : [start/3]

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

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

### Specialization of cost equations ge/3 
* CE 28 is refined into CE [29] 
* CE 25 is refined into CE [30] 
* CE 26 is refined into CE [31] 
* CE 27 is refined into CE [32] 


### Cost equations --> "Loop" of ge/3 
* CEs [32] --> Loop 15 
* CEs [29] --> Loop 16 
* CEs [30] --> Loop 17 
* CEs [31] --> Loop 18 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
    V1


### Specialization of cost equations gt/3 
* CE 15 is refined into CE [33] 
* CE 13 is refined into CE [34] 
* CE 12 is refined into CE [35] 
* CE 14 is refined into CE [36] 


### Cost equations --> "Loop" of gt/3 
* CEs [36] --> Loop 19 
* CEs [33] --> Loop 20 
* CEs [34] --> Loop 21 
* CEs [35] --> Loop 22 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [19]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 16 is refined into CE [37,38,39,40] 
* CE 17 is refined into CE [41] 
* CE 19 is refined into CE [42] 
* CE 18 is refined into CE [43,44] 


### Cost equations --> "Loop" of minus/3 
* CEs [44] --> Loop 23 
* CEs [43] --> Loop 24 
* CEs [37,38,39,40,41,42] --> Loop 25 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [23]: [V1-1,V1-V]
* RF of phase [24]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V1-1
    V1-V
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V1


### Specialization of cost equations (div)/3 
* CE 20 is refined into CE [45,46,47,48] 
* CE 21 is refined into CE [49] 
* CE 23 is refined into CE [50,51,52,53,54] 
* CE 24 is refined into CE [55,56] 
* CE 22 is refined into CE [57,58] 


### Cost equations --> "Loop" of (div)/3 
* CEs [58] --> Loop 26 
* CEs [57] --> Loop 27 
* CEs [45,46,49,51] --> Loop 28 
* CEs [47,48,50,52,53,54,55,56] --> Loop 29 

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

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [59,60,61] 
* CE 3 is refined into CE [62] 
* CE 5 is refined into CE [63,64,65,66,67] 
* CE 6 is refined into CE [68,69,70,71,72,73] 
* CE 7 is refined into CE [74,75,76] 
* CE 2 is refined into CE [77] 
* CE 4 is refined into CE [78] 
* CE 8 is refined into CE [79,80,81] 
* CE 9 is refined into CE [82,83,84,85,86] 
* CE 10 is refined into CE [87,88,89,90,91] 
* CE 11 is refined into CE [92,93,94,95] 


### Cost equations --> "Loop" of start/3 
* CEs [79,83,88] --> Loop 30 
* CEs [59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76] --> Loop 31 
* CEs [78] --> Loop 32 
* CEs [77,80,81,82,84,85,86,87,89,90,91,92,93,94,95] --> Loop 33 

### Ranking functions of CR start(V1,V,V5) 

#### Partial ranking functions of CR start(V1,V,V5) 


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

#### Cost of chains of ge(V1,V,Out):
* Chain [[15],18]: 1*it(15)+1
  Such that:it(15) =< V1

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

* Chain [[15],17]: 1*it(15)+1
  Such that:it(15) =< V

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

* Chain [[15],16]: 1*it(15)+0
  Such that:it(15) =< V

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

* Chain [18]: 1
  with precondition: [V1=0,Out=1,V>=1] 

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

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


#### Cost of chains of gt(V1,V,Out):
* Chain [[19],22]: 1*it(19)+1
  Such that:it(19) =< V1

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

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

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

* Chain [[19],20]: 1*it(19)+0
  Such that:it(19) =< V

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

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

* Chain [21]: 1
  with precondition: [V=0,Out=2,V1>=1] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[24],25]: 3*it(24)+2*s(4)+3
  Such that:aux(1) =< V1-Out
it(24) =< Out
s(4) =< aux(1)

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

* Chain [[23],25]: 3*it(23)+2*s(3)+2*s(4)+1*s(9)+3
  Such that:aux(1) =< V1-Out
it(23) =< Out
aux(4) =< V
s(4) =< aux(1)
s(3) =< aux(4)
s(9) =< it(23)*aux(4)

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

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

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


#### Cost of chains of div(V1,V,Out):
* Chain [[26],29]: 8*it(26)+4*s(10)+7*s(14)+3*s(31)+1*s(33)+4
  Such that:aux(10) =< V1-V
aux(12) =< V1
aux(13) =< V
it(26) =< aux(12)
s(14) =< aux(12)
s(10) =< aux(13)
it(26) =< aux(10)
s(31) =< it(26)*aux(10)
s(33) =< s(31)*aux(13)

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

* Chain [[26],27,29]: 8*it(26)+7*s(10)+7*s(30)+3*s(31)+1*s(33)+12
  Such that:aux(10) =< V1-V
aux(16) =< V1
aux(17) =< V
it(26) =< aux(16)
s(10) =< aux(17)
s(30) =< aux(16)
it(26) =< aux(10)
s(31) =< it(26)*aux(10)
s(33) =< s(31)*aux(17)

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

* Chain [29]: 4*s(10)+2*s(14)+4
  Such that:aux(5) =< V1
aux(6) =< V
s(14) =< aux(5)
s(10) =< aux(6)

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

* Chain [28]: 5
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [27,29]: 7*s(10)+2*s(39)+12
  Such that:s(37) =< V1
aux(15) =< V
s(10) =< aux(15)
s(39) =< s(37)

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


#### Cost of chains of start(V1,V,V5):
* Chain [33]: 27*s(48)+30*s(49)+1*s(55)+16*s(73)+6*s(76)+2*s(77)+12
  Such that:aux(19) =< V1
aux(20) =< V1-V
aux(21) =< V
s(48) =< aux(19)
s(49) =< aux(21)
s(73) =< aux(19)
s(73) =< aux(20)
s(76) =< s(73)*aux(20)
s(77) =< s(76)*aux(21)
s(55) =< s(48)*aux(21)

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

* Chain [32]: 1
  with precondition: [V1=1,V>=0,V5>=0] 

* Chain [31]: 99*s(89)+76*s(90)+9*s(100)+32*s(124)+12*s(127)+4*s(128)+18
  Such that:aux(43) =< V
aux(44) =< V-V5
aux(45) =< V5
s(89) =< aux(43)
s(90) =< aux(45)
s(100) =< s(89)*aux(45)
s(124) =< aux(43)
s(124) =< aux(44)
s(127) =< s(124)*aux(44)
s(128) =< s(127)*aux(45)

  with precondition: [V1=2,V>=0,V5>=0] 

* Chain [30]: 5*s(220)+3
  Such that:aux(46) =< V1
s(220) =< aux(46)

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


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [33] with precondition: [V1>=0,V>=0] 
    - Upper bound: 43*V1+12+V*V1+2*V1*V*nat(V1-V)+6*V1*nat(V1-V)+30*V 
    - Complexity: n^3 
* Chain [32] with precondition: [V1=1,V>=0,V5>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [31] with precondition: [V1=2,V>=0,V5>=0] 
    - Upper bound: 131*V+18+9*V*V5+4*V*V5*nat(V-V5)+12*V*nat(V-V5)+76*V5 
    - Complexity: n^3 
* Chain [30] with precondition: [V=0,V1>=0] 
    - Upper bound: 5*V1+3 
    - Complexity: n 

### Maximum cost of start(V1,V,V5): max([131*V+17+9*V*nat(V5)+4*V*nat(V5)*nat(V-V5)+12*V*nat(V-V5)+nat(V5)*76,38*V1+9+V*V1+2*V1*V*nat(V1-V)+6*V1*nat(V1-V)+30*V+(5*V1+2)])+1 
Asymptotic class: n^3 
* Total analysis performed in 583 ms.


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

(10)
BOUNDS(1, n^3)

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

(11) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   div(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> 0
   if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
   if2(false, z0, z1) -> 0
Tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
   GT(0, z0) -> c6
   GT(s(z0), 0) -> c7
   GT(s(z0), s(z1)) -> c8(GT(z0, z1))
   DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c10(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c11
   IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c13
S tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
   GT(0, z0) -> c6
   GT(s(z0), 0) -> c7
   GT(s(z0), s(z1)) -> c8(GT(z0, z1))
   DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c10(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c11
   IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c13
K tuples:none
Defined Rule Symbols:   minus_2, if_3, ge_2, gt_2, div_2, if1_3, if2_3

Defined Pair Symbols:   MINUS_2, IF_3, GE_2, GT_2, DIV_2, IF1_3, IF2_3

Compound Symbols:   c_2, c1_1, c2, c3, c4, c5_1, c6, c7, c8_1, c9_2, c10_2, c11, c12_2, c13


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

(13) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

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


The TRS R consists of the following rules:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
   GT(0, z0) -> c6
   GT(s(z0), 0) -> c7
   GT(s(z0), s(z1)) -> c8(GT(z0, z1))
   DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c10(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c11
   IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c13

The (relative) TRS S consists of the following rules:

   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   div(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> 0
   if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
   if2(false, z0, z1) -> 0

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GE(z0, 0') -> c3
   GE(0', s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
   GT(0', z0) -> c6
   GT(s(z0), 0') -> c7
   GT(s(z0), s(z1)) -> c8(GT(z0, z1))
   DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
   IF1(false, z0, z1) -> c11
   IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c13

The (relative) TRS S consists of the following rules:

   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0'
   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gt(0', z0) -> false
   gt(s(z0), 0') -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   div(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
   if1(false, z0, z1) -> 0'
   if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
   if2(false, z0, z1) -> 0'

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

(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, gt, GT, GE, DIV, ge, minus, div

They will be analysed ascendingly in the following order:
gt < MINUS
GT < MINUS
MINUS < DIV
gt < DIV
gt < minus
gt < div
GT < DIV
GE < DIV
ge < DIV
minus < DIV
ge < div
minus < div

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

(20)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5


Generator Equations:
gen_s:0'10_14(0) <=> 0'
gen_s:0'10_14(+(x, 1)) <=> s(gen_s:0'10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c3:c4:c512_14(0) <=> c3
gen_c3:c4:c512_14(+(x, 1)) <=> c5(gen_c3:c4:c512_14(x))


The following defined symbols remain to be analysed:
gt, MINUS, GT, GE, DIV, ge, minus, div

They will be analysed ascendingly in the following order:
gt < MINUS
GT < MINUS
MINUS < DIV
gt < DIV
gt < minus
gt < div
GT < DIV
GE < DIV
ge < DIV
minus < DIV
ge < div
minus < div

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) -> false, rt in Omega(0)

Induction Base:
gt(gen_s:0'10_14(0), gen_s:0'10_14(0)) ->_R^Omega(0)
false

Induction Step:
gt(gen_s:0'10_14(+(n14_14, 1)), gen_s:0'10_14(+(n14_14, 1))) ->_R^Omega(0)
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) ->_IH
false

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'10_14(0) <=> 0'
gen_s:0'10_14(+(x, 1)) <=> s(gen_s:0'10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c3:c4:c512_14(0) <=> c3
gen_c3:c4:c512_14(+(x, 1)) <=> c5(gen_c3:c4:c512_14(x))


The following defined symbols remain to be analysed:
GT, MINUS, GE, DIV, ge, minus, div

They will be analysed ascendingly in the following order:
GT < MINUS
MINUS < DIV
GT < DIV
GE < DIV
ge < DIV
minus < DIV
ge < div
minus < div

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GT(gen_s:0'10_14(n425_14), gen_s:0'10_14(n425_14)) -> gen_c6:c7:c811_14(n425_14), rt in Omega(1 + n425_14)

Induction Base:
GT(gen_s:0'10_14(0), gen_s:0'10_14(0)) ->_R^Omega(1)
c6

Induction Step:
GT(gen_s:0'10_14(+(n425_14, 1)), gen_s:0'10_14(+(n425_14, 1))) ->_R^Omega(1)
c8(GT(gen_s:0'10_14(n425_14), gen_s:0'10_14(n425_14))) ->_IH
c8(gen_c6:c7:c811_14(c426_14))

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'10_14(0) <=> 0'
gen_s:0'10_14(+(x, 1)) <=> s(gen_s:0'10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c3:c4:c512_14(0) <=> c3
gen_c3:c4:c512_14(+(x, 1)) <=> c5(gen_c3:c4:c512_14(x))


The following defined symbols remain to be analysed:
GT, MINUS, GE, DIV, ge, minus, div

They will be analysed ascendingly in the following order:
GT < MINUS
MINUS < DIV
GT < DIV
GE < DIV
ge < DIV
minus < DIV
ge < div
minus < div

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) -> false, rt in Omega(0)
GT(gen_s:0'10_14(n425_14), gen_s:0'10_14(n425_14)) -> gen_c6:c7:c811_14(n425_14), rt in Omega(1 + n425_14)


Generator Equations:
gen_s:0'10_14(0) <=> 0'
gen_s:0'10_14(+(x, 1)) <=> s(gen_s:0'10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c3:c4:c512_14(0) <=> c3
gen_c3:c4:c512_14(+(x, 1)) <=> c5(gen_c3:c4:c512_14(x))


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

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

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_s:0'10_14(n1507_14), gen_s:0'10_14(n1507_14)) -> gen_c3:c4:c512_14(n1507_14), rt in Omega(1 + n1507_14)

Induction Base:
GE(gen_s:0'10_14(0), gen_s:0'10_14(0)) ->_R^Omega(1)
c3

Induction Step:
GE(gen_s:0'10_14(+(n1507_14, 1)), gen_s:0'10_14(+(n1507_14, 1))) ->_R^Omega(1)
c5(GE(gen_s:0'10_14(n1507_14), gen_s:0'10_14(n1507_14))) ->_IH
c5(gen_c3:c4:c512_14(c1508_14))

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

(30)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) -> false, rt in Omega(0)
GT(gen_s:0'10_14(n425_14), gen_s:0'10_14(n425_14)) -> gen_c6:c7:c811_14(n425_14), rt in Omega(1 + n425_14)
GE(gen_s:0'10_14(n1507_14), gen_s:0'10_14(n1507_14)) -> gen_c3:c4:c512_14(n1507_14), rt in Omega(1 + n1507_14)


Generator Equations:
gen_s:0'10_14(0) <=> 0'
gen_s:0'10_14(+(x, 1)) <=> s(gen_s:0'10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c3:c4:c512_14(0) <=> c3
gen_c3:c4:c512_14(+(x, 1)) <=> c5(gen_c3:c4:c512_14(x))


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

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'10_14(n2228_14), gen_s:0'10_14(n2228_14)) -> true, rt in Omega(0)

Induction Base:
ge(gen_s:0'10_14(0), gen_s:0'10_14(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_s:0'10_14(+(n2228_14, 1)), gen_s:0'10_14(+(n2228_14, 1))) ->_R^Omega(0)
ge(gen_s:0'10_14(n2228_14), gen_s:0'10_14(n2228_14)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
GT(0', z0) -> c6
GT(s(z0), 0') -> c7
GT(s(z0), s(z1)) -> c8(GT(z0, z1))
DIV(z0, z1) -> c9(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c10(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c11
IF2(true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c13
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
div(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> 0'
if2(true, z0, z1) -> s(div(minus(z0, z1), z1))
if2(false, z0, z1) -> 0'

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c6:c7:c8 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c6:c7:c8
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GE :: s:0' -> s:0' -> c3:c4:c5
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
DIV :: s:0' -> s:0' -> c9
c9 :: c10:c11 -> c3:c4:c5 -> c9
IF1 :: true:false -> s:0' -> s:0' -> c10:c11
ge :: s:0' -> s:0' -> true:false
c10 :: c12:c13 -> c6:c7:c8 -> c10:c11
IF2 :: true:false -> s:0' -> s:0' -> c12:c13
c11 :: c10:c11
c12 :: c9 -> c -> c12:c13
minus :: s:0' -> s:0' -> s:0'
c13 :: c12:c13
if :: true:false -> s:0' -> s:0' -> s:0'
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_14 :: c
hole_s:0'2_14 :: s:0'
hole_c1:c23_14 :: c1:c2
hole_c6:c7:c84_14 :: c6:c7:c8
hole_true:false5_14 :: true:false
hole_c3:c4:c56_14 :: c3:c4:c5
hole_c97_14 :: c9
hole_c10:c118_14 :: c10:c11
hole_c12:c139_14 :: c12:c13
gen_s:0'10_14 :: Nat -> s:0'
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c3:c4:c512_14 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'10_14(n14_14), gen_s:0'10_14(n14_14)) -> false, rt in Omega(0)
GT(gen_s:0'10_14(n425_14), gen_s:0'10_14(n425_14)) -> gen_c6:c7:c811_14(n425_14), rt in Omega(1 + n425_14)
GE(gen_s:0'10_14(n1507_14), gen_s:0'10_14(n1507_14)) -> gen_c3:c4:c512_14(n1507_14), rt in Omega(1 + n1507_14)
ge(gen_s:0'10_14(n2228_14), gen_s:0'10_14(n2228_14)) -> true, rt in Omega(0)


Generator Equations:
gen_s:0'10_14(0) <=> 0'
gen_s:0'10_14(+(x, 1)) <=> s(gen_s:0'10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c3:c4:c512_14(0) <=> c3
gen_c3:c4:c512_14(+(x, 1)) <=> c5(gen_c3:c4:c512_14(x))


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

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