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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 5 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 651 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), 236 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) 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:

   cond1(true, x, y) -> cond2(gr(x, 0), x, y)
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y)
   cond2(false, x, y) -> cond3(gr(y, 0), x, y)
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y))
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   or(false, false) -> false
   or(true, x) -> true
   or(x, true) -> true
   p(0) -> 0
   p(s(x)) -> x

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:

   cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1]
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
   cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1]
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [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:

   cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1]
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
   cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1]
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
or :: true:false -> true:false -> true:false
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
s :: 0:s -> 0:s

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:

   cond1(v0, v1, v2) -> null_cond1 [0]

And the following fresh constants:    null_cond1

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

(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:

   cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1]
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
   cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1]
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   cond1(v0, v1, v2) -> null_cond1 [0]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> null_cond1
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> null_cond1
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
or :: true:false -> true:false -> true:false
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> null_cond1
s :: 0:s -> 0:s
null_cond1 :: null_cond1

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 => 1
   0 => 0
   false => 0
   null_cond1 => 0

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

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

   cond1(z, z', z'') -{ 1 }-> cond2(gr(x, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   cond2(z, z', z'') -{ 1 }-> cond3(gr(y, 0), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   cond2(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond3(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   cond3(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0
   or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x
   or(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V2),0,[cond1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[cond2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[cond3(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[or(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
eq(cond1(V1, V, V2, Out),1,[gr(V4, 0, Ret0),cond2(Ret0, V4, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(cond2(V1, V, V2, Out),1,[gr(V5, 0, Ret00),gr(V6, 0, Ret01),or(Ret00, Ret01, Ret02),p(V5, Ret1),cond1(Ret02, Ret1, V6, Ret2)],[Out = Ret2,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]).
eq(cond2(V1, V, V2, Out),1,[gr(V7, 0, Ret03),cond3(Ret03, V8, V7, Ret3)],[Out = Ret3,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(cond3(V1, V, V2, Out),1,[gr(V9, 0, Ret001),gr(V10, 0, Ret011),or(Ret001, Ret011, Ret04),p(V10, Ret21),cond1(Ret04, V9, Ret21, Ret4)],[Out = Ret4,V = V9,V2 = V10,V1 = 1,V9 >= 0,V10 >= 0]).
eq(cond3(V1, V, V2, Out),1,[gr(V12, 0, Ret002),gr(V11, 0, Ret012),or(Ret002, Ret012, Ret05),cond1(Ret05, V12, V11, Ret5)],[Out = Ret5,V = V12,V2 = V11,V12 >= 0,V11 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 0,V = V13,V13 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]).
eq(gr(V1, V, Out),1,[gr(V16, V15, Ret6)],[Out = Ret6,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]).
eq(or(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(or(V1, V, Out),1,[],[Out = 1,V = V17,V1 = 1,V17 >= 0]).
eq(or(V1, V, Out),1,[],[Out = 1,V18 >= 0,V = 1,V1 = V18]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V19,V19 >= 0,V1 = 1 + V19]).
eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V21 >= 0,V2 = V22,V20 >= 0,V1 = V21,V = V20,V22 >= 0]).
input_output_vars(cond1(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(cond2(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(cond3(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(or(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [gr/3]
1. non_recursive  : [or/3]
2. non_recursive  : [p/2]
3. recursive  : [cond1/4,cond2/4,cond3/4]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into gr/3
1. SCC is partially evaluated into or/3
2. SCC is partially evaluated into p/2
3. SCC is partially evaluated into cond1/4
4. SCC is partially evaluated into start/3

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

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


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

### Ranking functions of CR gr(V1,V,Out) 
* RF of phase [16]: [V,V1]

#### Partial ranking functions of CR gr(V1,V,Out) 
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V
    V1


### Specialization of cost equations or/3 
* CE 15 is refined into CE [25] 
* CE 14 is refined into CE [26] 
* CE 13 is refined into CE [27] 


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

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

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


### Specialization of cost equations p/2 
* CE 17 is refined into CE [28] 
* CE 16 is refined into CE [29] 


### Cost equations --> "Loop" of p/2 
* CEs [28] --> Loop 20 
* CEs [29] --> Loop 21 

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations cond1/4 
* CE 21 is refined into CE [30] 
* CE 18 is refined into CE [31,32,33] 
* CE 19 is refined into CE [34] 
* CE 20 is refined into CE [35] 


### Cost equations --> "Loop" of cond1/4 
* CEs [32,33] --> Loop 22 
* CEs [31] --> Loop 23 
* CEs [34] --> Loop 24 
* CEs [35] --> Loop 25 
* CEs [30] --> Loop 26 

### Ranking functions of CR cond1(V1,V,V2,Out) 
* RF of phase [22]: [V]
* RF of phase [23]: [V]
* RF of phase [24]: [V2]

#### Partial ranking functions of CR cond1(V1,V,V2,Out) 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [36,37,38,39,40,41,42,43,44,45,46] 
* CE 5 is refined into CE [47,48,49,50,51,52,53,54,55,56,57] 
* CE 2 is refined into CE [58,59,60,61,62,63,64,65] 
* CE 3 is refined into CE [66,67,68] 
* CE 4 is refined into CE [69,70,71,72,73,74,75,76,77] 
* CE 6 is refined into CE [78,79,80,81] 
* CE 7 is refined into CE [82,83,84,85] 
* CE 8 is refined into CE [86,87,88] 
* CE 9 is refined into CE [89,90] 


### Cost equations --> "Loop" of start/3 
* CEs [53,56] --> Loop 27 
* CEs [39,40,50,51,78,80] --> Loop 28 
* CEs [42,45,88] --> Loop 29 
* CEs [37,38,48,49,79,83] --> Loop 30 
* CEs [36,41,43,44,46,47,52,54,55,57,81,84,85,87,90] --> Loop 31 
* CEs [58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,82,86,89] --> Loop 32 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of gr(V1,V,Out):
* Chain [[16],15]: 1*it(16)+1
  Such that:it(16) =< V1

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

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

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

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

* Chain [14]: 1
  with precondition: [V=0,Out=1,V1>=1] 


#### Cost of chains of or(V1,V,Out):
* Chain [19]: 1
  with precondition: [V1=0,V=0,Out=0] 

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

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


#### Cost of chains of p(V1,Out):
* Chain [21]: 1
  with precondition: [V1=0,Out=0] 

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


#### Cost of chains of cond1(V1,V,V2,Out):
* Chain [[24],26]: 9*it(24)+0
  Such that:it(24) =< V2

  with precondition: [V1=1,V=0,Out=0,V2>=1] 

* Chain [[24],25,26]: 9*it(24)+8
  Such that:it(24) =< V2

  with precondition: [V1=1,V=0,Out=0,V2>=1] 

* Chain [[23],26]: 7*it(23)+0
  Such that:it(23) =< V

  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [[23],25,26]: 7*it(23)+8
  Such that:it(23) =< V

  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [[22],[24],26]: 7*it(22)+9*it(24)+0
  Such that:it(22) =< V
it(24) =< V2

  with precondition: [V1=1,Out=0,V>=1,V2>=1] 

* Chain [[22],[24],25,26]: 7*it(22)+9*it(24)+8
  Such that:it(22) =< V
it(24) =< V2

  with precondition: [V1=1,Out=0,V>=1,V2>=1] 

* Chain [[22],26]: 7*it(22)+0
  Such that:it(22) =< V

  with precondition: [V1=1,Out=0,V>=1,V2>=1] 

* Chain [26]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [25,26]: 8
  with precondition: [V1=1,V=0,V2=0,Out=0] 


#### Cost of chains of start(V1,V,V2):
* Chain [32]: 108*s(11)+140*s(13)+15
  Such that:aux(5) =< V
aux(6) =< V2
s(13) =< aux(5)
s(11) =< aux(6)

  with precondition: [V1=0] 

* Chain [31]: 106*s(40)+90*s(41)+1*s(58)+13
  Such that:s(58) =< V1
aux(7) =< V
aux(8) =< V2
s(40) =< aux(7)
s(41) =< aux(8)

  with precondition: [V1>=1] 

* Chain [30]: 54*s(61)+13
  Such that:aux(9) =< V2
s(61) =< aux(9)

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

* Chain [29]: 36*s(67)+13
  Such that:aux(10) =< V2
s(67) =< aux(10)

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

* Chain [28]: 42*s(71)+13
  Such that:aux(11) =< V
s(71) =< aux(11)

  with precondition: [V1>=0,V>=0,V2>=0] 

* Chain [27]: 28*s(77)+13
  Such that:aux(12) =< V
s(77) =< aux(12)

  with precondition: [V1=1,V2=1,V>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [32] with precondition: [V1=0] 
    - Upper bound: nat(V)*140+15+nat(V2)*108 
    - Complexity: n 
* Chain [31] with precondition: [V1>=1] 
    - Upper bound: V1+13+nat(V)*106+nat(V2)*90 
    - Complexity: n 
* Chain [30] with precondition: [V=0,V1>=1] 
    - Upper bound: nat(V2)*54+13 
    - Complexity: n 
* Chain [29] with precondition: [V=1,V1>=0] 
    - Upper bound: nat(V2)*36+13 
    - Complexity: n 
* Chain [28] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 42*V+13 
    - Complexity: n 
* Chain [27] with precondition: [V1=1,V2=1,V>=1] 
    - Upper bound: 28*V+13 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([nat(V2)*54,nat(V2)*90+nat(V)*64+max([V1,nat(V)*34+2+nat(V2)*18])+nat(V)*14+nat(V)*28])+13 
Asymptotic class: n 
* Total analysis performed in 608 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:

   cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
   cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
   cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
   cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
   cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   or(false, false) -> false
   or(true, x) -> true
   or(x, true) -> true
   p(0') -> 0'
   p(s(x)) -> x

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

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

(14)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

(16)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
false

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:
cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
cond2, cond1, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3