/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^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 512 ms]
    (10) BOUNDS(1, n^1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 298 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   div(x, y) -> ify(ge(y, s(0)), x, y)
   ify(false, x, y) -> divByZeroError
   ify(true, x, y) -> if(ge(x, y), x, y)
   if(false, x, y) -> 0
   if(true, x, y) -> s(div(minus(x, y), y))

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

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

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


The TRS R consists of the following rules:

   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   div(x, y) -> ify(ge(y, s(0)), x, y) [1]
   ify(false, x, y) -> divByZeroError [1]
   ify(true, x, y) -> if(ge(x, y), x, y) [1]
   if(false, x, y) -> 0 [1]
   if(true, x, y) -> s(div(minus(x, y), y)) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   div(x, y) -> ify(ge(y, s(0)), x, y) [1]
   ify(false, x, y) -> divByZeroError [1]
   ify(true, x, y) -> if(ge(x, y), x, y) [1]
   if(false, x, y) -> 0 [1]
   if(true, x, y) -> s(div(minus(x, y), y)) [1]

The TRS has the following type information:
ge :: 0:s:divByZeroError -> 0:s:divByZeroError -> true:false
0 :: 0:s:divByZeroError
true :: true:false
s :: 0:s:divByZeroError -> 0:s:divByZeroError
false :: true:false
minus :: 0:s:divByZeroError -> 0:s:divByZeroError -> 0:s:divByZeroError
div :: 0:s:divByZeroError -> 0:s:divByZeroError -> 0:s:divByZeroError
ify :: true:false -> 0:s:divByZeroError -> 0:s:divByZeroError -> 0:s:divByZeroError
divByZeroError :: 0:s:divByZeroError
if :: true:false -> 0:s:divByZeroError -> 0:s:divByZeroError -> 0:s:divByZeroError

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

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

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

And the following fresh constants:    null_ge, null_minus, null_ify, null_if

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

(6)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   div(x, y) -> ify(ge(y, s(0)), x, y) [1]
   ify(false, x, y) -> divByZeroError [1]
   ify(true, x, y) -> if(ge(x, y), x, y) [1]
   if(false, x, y) -> 0 [1]
   if(true, x, y) -> s(div(minus(x, y), y)) [1]
   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   ify(v0, v1, v2) -> null_ify [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
ge :: 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if -> true:false:null_ge
0 :: 0:s:divByZeroError:null_minus:null_ify:null_if
true :: true:false:null_ge
s :: 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if
false :: true:false:null_ge
minus :: 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if
div :: 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if
ify :: true:false:null_ge -> 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if
divByZeroError :: 0:s:divByZeroError:null_minus:null_ify:null_if
if :: true:false:null_ge -> 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if -> 0:s:divByZeroError:null_minus:null_ify:null_if
null_ge :: true:false:null_ge
null_minus :: 0:s:divByZeroError:null_minus:null_ify:null_if
null_ify :: 0:s:divByZeroError:null_minus:null_ify:null_if
null_if :: 0:s:divByZeroError:null_minus:null_ify:null_if

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

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

   0 => 0
   true => 2
   false => 1
   divByZeroError => 1
   null_ge => 0
   null_minus => 0
   null_ify => 0
   null_if => 0

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

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

   div(z, z') -{ 1 }-> ify(ge(y, 1 + 0), 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
   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 + div(minus(x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   ify(z, z', z'') -{ 1 }-> if(ge(x, y), x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   ify(z, z', z'') -{ 1 }-> 1 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   ify(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   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, V12),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[ify(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(start(V1, V, V12),0,[if(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(ge(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = V2,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V3,V3 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]).
eq(minus(V1, V, Out),1,[minus(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
eq(div(V1, V, Out),1,[ge(V9, 1 + 0, Ret0),ify(Ret0, V10, V9, Ret2)],[Out = Ret2,V10 >= 0,V9 >= 0,V1 = V10,V = V9]).
eq(ify(V1, V, V12, Out),1,[],[Out = 1,V = V13,V12 = V11,V1 = 1,V13 >= 0,V11 >= 0]).
eq(ify(V1, V, V12, Out),1,[ge(V15, V14, Ret01),if(Ret01, V15, V14, Ret3)],[Out = Ret3,V1 = 2,V = V15,V12 = V14,V15 >= 0,V14 >= 0]).
eq(if(V1, V, V12, Out),1,[],[Out = 0,V = V17,V12 = V16,V1 = 1,V17 >= 0,V16 >= 0]).
eq(if(V1, V, V12, Out),1,[minus(V19, V18, Ret10),div(Ret10, V18, Ret11)],[Out = 1 + Ret11,V1 = 2,V = V19,V12 = V18,V19 >= 0,V18 >= 0]).
eq(ge(V1, V, Out),0,[],[Out = 0,V21 >= 0,V20 >= 0,V1 = V21,V = V20]).
eq(minus(V1, V, Out),0,[],[Out = 0,V23 >= 0,V22 >= 0,V1 = V23,V = V22]).
eq(ify(V1, V, V12, Out),0,[],[Out = 0,V25 >= 0,V12 = V26,V24 >= 0,V1 = V25,V = V24,V26 >= 0]).
eq(if(V1, V, V12, Out),0,[],[Out = 0,V27 >= 0,V12 = V28,V29 >= 0,V1 = V27,V = V29,V28 >= 0]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).
input_output_vars(ify(V1,V,V12,Out),[V1,V,V12],[Out]).
input_output_vars(if(V1,V,V12,Out),[V1,V,V12],[Out]).


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

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

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

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

### Specialization of cost equations ge/3 
* CE 13 is refined into CE [22] 
* CE 10 is refined into CE [23] 
* CE 11 is refined into CE [24] 
* CE 12 is refined into CE [25] 


### Cost equations --> "Loop" of ge/3 
* CEs [25] --> Loop 13 
* CEs [22] --> Loop 14 
* CEs [23] --> Loop 15 
* CEs [24] --> Loop 16 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [13]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 16 is refined into CE [26] 
* CE 14 is refined into CE [27] 
* CE 15 is refined into CE [28] 


### Cost equations --> "Loop" of minus/3 
* CEs [28] --> Loop 17 
* CEs [26] --> Loop 18 
* CEs [27] --> Loop 19 

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

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


### Specialization of cost equations (div)/3 
* CE 18 is refined into CE [29] 
* CE 17 is refined into CE [30,31,32] 
* CE 19 is refined into CE [33,34] 
* CE 21 is refined into CE [35,36,37,38] 
* CE 20 is refined into CE [39,40] 


### Cost equations --> "Loop" of (div)/3 
* CEs [40] --> Loop 20 
* CEs [39] --> Loop 21 
* CEs [29] --> Loop 22 
* CEs [30] --> Loop 23 
* CEs [31,32,33,34,35,36,37,38] --> Loop 24 

### Ranking functions of CR div(V1,V,Out) 
* RF of phase [20]: [V1,V1-V+1]

#### Partial ranking functions of CR div(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V1
    V1-V+1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [41,42] 
* CE 4 is refined into CE [43,44,45,46,47,48,49] 
* CE 5 is refined into CE [50,51,52,53,54] 
* CE 6 is refined into CE [55,56,57,58,59,60] 
* CE 1 is refined into CE [61] 
* CE 2 is refined into CE [62] 
* CE 7 is refined into CE [63,64,65,66,67] 
* CE 8 is refined into CE [68,69,70] 
* CE 9 is refined into CE [71,72,73] 


### Cost equations --> "Loop" of start/3 
* CEs [64,68,72] --> Loop 25 
* CEs [43,44,45,46,51,55,56,58] --> Loop 26 
* CEs [41,42,47,48,49,50,52,53,54,57,59,60] --> Loop 27 
* CEs [62] --> Loop 28 
* CEs [61,63,65,66,67,69,70,71,73] --> Loop 29 

### Ranking functions of CR start(V1,V,V12) 

#### Partial ranking functions of CR start(V1,V,V12) 


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

#### Cost of chains of ge(V1,V,Out):
* Chain [[13],16]: 1*it(13)+1
  Such that:it(13) =< V1

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

* Chain [[13],15]: 1*it(13)+1
  Such that:it(13) =< V

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

* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< V

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

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

* Chain [15]: 1
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [14]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[17],19]: 1*it(17)+1
  Such that:it(17) =< V

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

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

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

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

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


#### Cost of chains of div(V1,V,Out):
* Chain [[20],24]: 6*it(20)+8*s(3)+4*s(7)+2*s(10)+1*s(21)+5
  Such that:aux(1) =< 1
aux(7) =< V1-V+1
aux(3) =< V
aux(10) =< V1
s(3) =< aux(1)
s(7) =< aux(10)
s(10) =< aux(3)
aux(5) =< aux(10)
it(20) =< aux(10)
aux(5) =< aux(7)
it(20) =< aux(7)
s(21) =< aux(5)

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

* Chain [[20],21,24]: 6*it(20)+9*s(3)+4*s(10)+1*s(21)+2*s(22)+10
  Such that:aux(12) =< 1
aux(6) =< V1
aux(7) =< V1-V+1
aux(13) =< V
aux(14) =< V1-V
s(3) =< aux(12)
s(10) =< aux(13)
aux(5) =< aux(6)
it(20) =< aux(6)
s(23) =< aux(6)
aux(5) =< aux(7)
it(20) =< aux(7)
aux(5) =< aux(14)
it(20) =< aux(14)
s(23) =< aux(14)
s(21) =< aux(5)
s(22) =< s(23)

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

* Chain [24]: 8*s(3)+2*s(7)+2*s(10)+5
  Such that:aux(1) =< 1
aux(2) =< V1
aux(3) =< V
s(3) =< aux(1)
s(7) =< aux(2)
s(10) =< aux(3)

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

* Chain [23]: 2
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [22]: 3
  with precondition: [V=0,Out=1,V1>=0] 

* Chain [21,24]: 9*s(3)+4*s(10)+10
  Such that:aux(12) =< 1
aux(13) =< V
s(3) =< aux(12)
s(10) =< aux(13)

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


#### Cost of chains of start(V1,V,V12):
* Chain [29]: 16*s(59)+7*s(60)+34*s(67)+6*s(78)+1*s(80)+2*s(81)+6*s(84)+1*s(85)+10
  Such that:s(70) =< V1-V
s(73) =< V1-V+1
aux(19) =< 1
aux(20) =< V1
aux(21) =< V
s(60) =< aux(20)
s(59) =< aux(21)
s(67) =< aux(19)
s(77) =< aux(20)
s(78) =< aux(20)
s(79) =< aux(20)
s(77) =< s(73)
s(78) =< s(73)
s(77) =< s(70)
s(78) =< s(70)
s(79) =< s(70)
s(80) =< s(77)
s(81) =< s(79)
s(83) =< aux(20)
s(84) =< aux(20)
s(83) =< s(73)
s(84) =< s(73)
s(85) =< s(83)

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

* Chain [28]: 1
  with precondition: [V1=1,V>=0,V12>=0] 

* Chain [27]: 2*s(86)+39*s(87)+84*s(92)+12*s(101)+12*s(113)+2*s(115)+4*s(116)+12*s(119)+2*s(120)+14
  Such that:aux(28) =< 1
aux(29) =< V
aux(30) =< V-2*V12
aux(31) =< V-2*V12+1
aux(32) =< V-V12
aux(33) =< V12
s(86) =< aux(29)
s(87) =< aux(33)
s(92) =< aux(28)
s(101) =< aux(32)
s(112) =< aux(32)
s(113) =< aux(32)
s(114) =< aux(32)
s(112) =< aux(31)
s(113) =< aux(31)
s(112) =< aux(30)
s(113) =< aux(30)
s(114) =< aux(30)
s(115) =< s(112)
s(116) =< s(114)
s(118) =< aux(32)
s(119) =< aux(32)
s(118) =< aux(31)
s(119) =< aux(31)
s(120) =< s(118)

  with precondition: [V1=2,V>=0,V12>=0] 

* Chain [26]: 24*s(158)+4*s(159)+9
  Such that:aux(35) =< 1
aux(36) =< V
s(158) =< aux(35)
s(159) =< aux(36)

  with precondition: [V1=2,V12=0,V>=0] 

* Chain [25]: 3
  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V,V12): 
-------------------------------------
* Chain [29] with precondition: [V1>=0,V>=0] 
    - Upper bound: 23*V1+16*V+44 
    - Complexity: n 
* Chain [28] with precondition: [V1=1,V>=0,V12>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [27] with precondition: [V1=2,V>=0,V12>=0] 
    - Upper bound: 2*V+39*V12+98+nat(V-V12)*44 
    - Complexity: n 
* Chain [26] with precondition: [V1=2,V12=0,V>=0] 
    - Upper bound: 4*V+33 
    - Complexity: n 
* Chain [25] with precondition: [V=0,V1>=0] 
    - Upper bound: 3 
    - Complexity: constant 

### Maximum cost of start(V1,V,V12): 2*V+30+max([23*V1+14*V+11,nat(V12)*39+65+nat(V-V12)*44])+3 
Asymptotic class: n 
* Total analysis performed in 424 ms.


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

(10)
BOUNDS(1, n^1)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   div(x, y) -> ify(ge(y, s(0')), x, y)
   ify(false, x, y) -> divByZeroError
   ify(true, x, y) -> if(ge(x, y), x, y)
   if(false, x, y) -> 0'
   if(true, x, y) -> s(div(minus(x, y), y))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError

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

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

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

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

(16)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
ge(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s:divByZeroError3_0(+(n5_0, 1)), gen_0':s:divByZeroError3_0(+(n5_0, 1))) ->_R^Omega(1)
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) ->_IH
true

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


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

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

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:divByZeroError3_0(n293_0), gen_0':s:divByZeroError3_0(n293_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n293_0)

Induction Base:
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
gen_0':s:divByZeroError3_0(0)

Induction Step:
minus(gen_0':s:divByZeroError3_0(+(n293_0, 1)), gen_0':s:divByZeroError3_0(+(n293_0, 1))) ->_R^Omega(1)
minus(gen_0':s:divByZeroError3_0(n293_0), gen_0':s:divByZeroError3_0(n293_0)) ->_IH
gen_0':s:divByZeroError3_0(0)

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

(24)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s:divByZeroError3_0(n293_0), gen_0':s:divByZeroError3_0(n293_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n293_0)


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


The following defined symbols remain to be analysed:
div