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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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), 4 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 2753 ms]
    (10) BOUNDS(1, n^2)
(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), 258 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^2).


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
   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^2).


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) [1]
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) [1]
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) [1]
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) [1]
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) [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, z) -> cond2(gr(x, 0), x, y, z) [1]
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) [1]
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) [1]
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) [1]
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) [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 -> 0:s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0:s -> 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 -> 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, v3) -> 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, z) -> cond2(gr(x, 0), x, y, z) [1]
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) [1]
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) [1]
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) [1]
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) [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, v3) -> null_cond1 [0]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> 0:s -> null_cond1
true :: true:false
cond2 :: true:false -> 0:s -> 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 -> 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'', z1, z2) -{ 1 }-> cond2(gr(x, 0), x, y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   cond1(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   cond2(z', z'', z1, z2) -{ 1 }-> cond3(gr(y, 0), x, y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0
   cond2(z', z'', z1, z2) -{ 1 }-> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   cond3(z', z'', z1, z2) -{ 1 }-> cond1(or(gr(x, z), gr(y, z)), x, y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0
   cond3(z', z'', z1, z2) -{ 1 }-> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   gr(z', z'') -{ 1 }-> gr(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   gr(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
   or(z', z'') -{ 1 }-> 1 :|: x >= 0, z'' = x, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1
   or(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   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(V, V1, V6, V2),0,[cond1(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]).
eq(start(V, V1, V6, V2),0,[cond2(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]).
eq(start(V, V1, V6, V2),0,[cond3(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]).
eq(start(V, V1, V6, V2),0,[gr(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V6, V2),0,[or(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V6, V2),0,[p(V, Out)],[V >= 0]).
eq(cond1(V, V1, V6, V2, Out),1,[gr(V4, 0, Ret0),cond2(Ret0, V4, V3, V5, Ret)],[Out = Ret,V6 = V3,V5 >= 0,V2 = V5,V4 >= 0,V3 >= 0,V1 = V4,V = 1]).
eq(cond2(V, V1, V6, V2, Out),1,[gr(V7, V8, Ret00),gr(V9, V8, Ret01),or(Ret00, Ret01, Ret02),p(V7, Ret1),cond1(Ret02, Ret1, V9, V8, Ret2)],[Out = Ret2,V6 = V9,V8 >= 0,V2 = V8,V7 >= 0,V9 >= 0,V1 = V7,V = 1]).
eq(cond2(V, V1, V6, V2, Out),1,[gr(V10, 0, Ret03),cond3(Ret03, V11, V10, V12, Ret3)],[Out = Ret3,V6 = V10,V12 >= 0,V2 = V12,V11 >= 0,V10 >= 0,V1 = V11,V = 0]).
eq(cond3(V, V1, V6, V2, Out),1,[gr(V13, V14, Ret001),gr(V15, V14, Ret011),or(Ret001, Ret011, Ret04),p(V15, Ret21),cond1(Ret04, V13, Ret21, V14, Ret4)],[Out = Ret4,V6 = V15,V14 >= 0,V2 = V14,V13 >= 0,V15 >= 0,V1 = V13,V = 1]).
eq(cond3(V, V1, V6, V2, Out),1,[gr(V17, V18, Ret002),gr(V16, V18, Ret012),or(Ret002, Ret012, Ret05),cond1(Ret05, V17, V16, V18, Ret5)],[Out = Ret5,V6 = V16,V18 >= 0,V2 = V18,V17 >= 0,V16 >= 0,V1 = V17,V = 0]).
eq(gr(V, V1, Out),1,[],[Out = 0,V19 >= 0,V1 = V19,V = 0]).
eq(gr(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V20,V20 >= 0]).
eq(gr(V, V1, Out),1,[gr(V22, V21, Ret6)],[Out = Ret6,V = 1 + V22,V22 >= 0,V21 >= 0,V1 = 1 + V21]).
eq(or(V, V1, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(or(V, V1, Out),1,[],[Out = 1,V23 >= 0,V1 = V23,V = 1]).
eq(or(V, V1, Out),1,[],[Out = 1,V = V24,V24 >= 0,V1 = 1]).
eq(p(V, Out),1,[],[Out = 0,V = 0]).
eq(p(V, Out),1,[],[Out = V25,V = 1 + V25,V25 >= 0]).
eq(cond1(V, V1, V6, V2, Out),0,[],[Out = 0,V2 = V28,V27 >= 0,V6 = V29,V26 >= 0,V1 = V26,V29 >= 0,V28 >= 0,V = V27]).
input_output_vars(cond1(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]).
input_output_vars(cond2(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]).
input_output_vars(cond3(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]).
input_output_vars(gr(V,V1,Out),[V,V1],[Out]).
input_output_vars(or(V,V1,Out),[V,V1],[Out]).
input_output_vars(p(V,Out),[V],[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  : [cond1/5,cond2/5,cond3/5]
4. non_recursive  : [start/4]

#### 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 cond1/5
4. SCC is partially evaluated into start/4

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(V,V1,Out) 
* RF of phase [16]: [V,V1]

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

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

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


### Specialization of cost equations cond1/5 
* CE 21 is refined into CE [30] 
* CE 18 is refined into CE [31,32,33,34,35,36,37,38,39,40] 
* CE 19 is refined into CE [41,42,43] 
* CE 20 is refined into CE [44] 


### Cost equations --> "Loop" of cond1/5 
* CEs [39,40] --> Loop 22 
* CEs [38] --> Loop 23 
* CEs [36] --> Loop 24 
* CEs [35] --> Loop 25 
* CEs [32,33] --> Loop 26 
* CEs [37] --> Loop 27 
* CEs [34] --> Loop 28 
* CEs [31] --> Loop 29 
* CEs [43] --> Loop 30 
* CEs [42] --> Loop 31 
* CEs [41] --> Loop 32 
* CEs [44] --> Loop 33 
* CEs [30] --> Loop 34 

### Ranking functions of CR cond1(V,V1,V6,V2,Out) 
* RF of phase [22]: [V1-1,V1-V2]
* RF of phase [23]: [V1-1,V1-V2,V1-V6]
* RF of phase [24]: [V1]
* RF of phase [26]: [V1]
* RF of phase [27]: [V1-1,V1-V2]
* RF of phase [29]: [V1]
* RF of phase [30]: [V6-1,V6-V2]
* RF of phase [32]: [V6]

#### Partial ranking functions of CR cond1(V,V1,V6,V2,Out) 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V1-1
    V1-V2
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V1-1
    V1-V2
    V1-V6
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V1
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V1-1
    V1-V2
* Partial RF of phase [29]:
  - RF of loop [29:1]:
    V1
* Partial RF of phase [30]:
  - RF of loop [30:1]:
    V6-1
    V6-V2
* Partial RF of phase [32]:
  - RF of loop [32:1]:
    V6


### Specialization of cost equations start/4 
* CE 1 is refined into CE [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75] 
* CE 5 is refined into CE [76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106] 
* CE 2 is refined into CE [107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131] 
* CE 3 is refined into CE [132,133,134,135,136,137] 
* CE 4 is refined into CE [138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161] 
* CE 6 is refined into CE [162,163,164,165,166,167,168,169,170,171,172,173] 
* CE 7 is refined into CE [174,175,176,177] 
* CE 8 is refined into CE [178,179,180] 
* CE 9 is refined into CE [181,182] 


### Cost equations --> "Loop" of start/4 
* CEs [94] --> Loop 35 
* CEs [68] --> Loop 36 
* CEs [71,74] --> Loop 37 
* CEs [102,105] --> Loop 38 
* CEs [99] --> Loop 39 
* CEs [86,89] --> Loop 40 
* CEs [59,91,167] --> Loop 41 
* CEs [64,65,66,96,97,168] --> Loop 42 
* CEs [51,52,83,84,166] --> Loop 43 
* CEs [62,180] --> Loop 44 
* CEs [53,54,55,56,57,58,85,87,88,90,169] --> Loop 45 
* CEs [48,79,164] --> Loop 46 
* CEs [49,50,80,81,82,162,165] --> Loop 47 
* CEs [46,47,77,78,163,175] --> Loop 48 
* CEs [45,60,61,63,67,69,70,72,73,75,76,92,93,95,98,100,101,103,104,106,170,171,172,173,176,177,179,182] --> Loop 49 
* CEs [107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,174,178,181] --> Loop 50 

### Ranking functions of CR start(V,V1,V6,V2) 

#### Partial ranking functions of CR start(V,V1,V6,V2) 


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

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

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

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

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

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

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


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

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

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


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

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


#### Cost of chains of cond1(V,V1,V6,V2,Out):
* Chain [[32],34]: 9*it(32)+0
  Such that:it(32) =< V6

  with precondition: [V=1,V1=0,V2=0,Out=0,V6>=1] 

* Chain [[32],33,34]: 9*it(32)+8
  Such that:it(32) =< V6

  with precondition: [V=1,V1=0,V2=0,Out=0,V6>=1] 

* Chain [[30],34]: 9*it(30)+1*s(3)+0
  Such that:it(30) =< V6-V2
aux(1) =< V2
s(3) =< it(30)*aux(1)

  with precondition: [V=1,V1=0,Out=0,V2>=1,V6>=V2+1] 

* Chain [[30],31,34]: 9*it(30)+1*s(3)+1*s(4)+9
  Such that:it(30) =< V6-V2
aux(2) =< V2
s(4) =< aux(2)
s(3) =< it(30)*aux(2)

  with precondition: [V=1,V1=0,Out=0,V2>=1,V6>=V2+1] 

* Chain [[29],34]: 7*it(29)+0
  Such that:it(29) =< V1

  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] 

* Chain [[29],33,34]: 7*it(29)+8
  Such that:it(29) =< V1

  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] 

* Chain [[27],34]: 7*it(27)+1*s(7)+0
  Such that:it(27) =< V1-V2
aux(3) =< V2
s(7) =< it(27)*aux(3)

  with precondition: [V=1,V6=0,Out=0,V2>=1,V1>=V2+1] 

* Chain [[27],28,34]: 7*it(27)+1*s(7)+1*s(8)+7
  Such that:it(27) =< V1-V2
aux(4) =< V2
s(8) =< aux(4)
s(7) =< it(27)*aux(4)

  with precondition: [V=1,V6=0,Out=0,V2>=1,V1>=V2+1] 

* Chain [[26],[32],34]: 7*it(26)+9*it(32)+0
  Such that:it(26) =< V1
it(32) =< V6

  with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=1] 

* Chain [[26],[32],33,34]: 7*it(26)+9*it(32)+8
  Such that:it(26) =< V1
it(32) =< V6

  with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=1] 

* Chain [[26],34]: 7*it(26)+0
  Such that:it(26) =< V1

  with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=1] 

* Chain [[24],[30],34]: 7*it(24)+9*it(30)+1*s(3)+1*s(13)+1*s(14)+0
  Such that:it(30) =< V6-V2
aux(8) =< V1
aux(9) =< V2
it(24) =< aux(8)
s(3) =< it(30)*aux(9)
s(14) =< it(24)*aux(9)
s(13) =< it(24)*aux(8)

  with precondition: [V=1,Out=0,V1>=1,V2>=V1,V6>=V2+1] 

* Chain [[24],[30],31,34]: 7*it(24)+9*it(30)+1*s(3)+1*s(4)+1*s(13)+1*s(14)+9
  Such that:it(30) =< V6-V2
aux(10) =< V1
aux(11) =< V2
it(24) =< aux(10)
s(4) =< aux(11)
s(3) =< it(30)*aux(11)
s(14) =< it(24)*aux(11)
s(13) =< it(24)*aux(10)

  with precondition: [V=1,Out=0,V1>=1,V2>=V1,V6>=V2+1] 

* Chain [[24],34]: 7*it(24)+1*s(13)+1*s(14)+0
  Such that:aux(6) =< V2
aux(12) =< V1
it(24) =< aux(12)
s(14) =< it(24)*aux(6)
s(13) =< it(24)*aux(12)

  with precondition: [V=1,Out=0,V1>=1,V2>=V1,V6>=V2+1] 

* Chain [[23],34]: 7*it(23)+1*s(19)+1*s(20)+0
  Such that:it(23) =< V1-V2
aux(13) =< V6
aux(14) =< V2
s(19) =< it(23)*aux(14)
s(20) =< it(23)*aux(13)

  with precondition: [V=1,Out=0,V6>=1,V2>=V6,V1>=V2+1] 

* Chain [[23],25,34]: 7*it(23)+1*s(19)+1*s(20)+1*s(21)+1*s(22)+7
  Such that:it(23) =< V1-V2
aux(15) =< V6
aux(16) =< V2
s(22) =< aux(15)
s(21) =< aux(16)
s(19) =< it(23)*aux(16)
s(20) =< it(23)*aux(15)

  with precondition: [V=1,Out=0,V6>=1,V2>=V6,V1>=V2+1] 

* Chain [[22],[24],[30],34]: 7*it(22)+7*it(24)+9*it(30)+1*s(3)+2*s(13)+4*s(29)+0
  Such that:it(22) =< V1-V2
it(30) =< V6-V2
aux(21) =< V2
it(24) =< aux(21)
s(3) =< it(30)*aux(21)
s(13) =< it(24)*aux(21)
s(30) =< it(22)*aux(21)
s(29) =< s(30)

  with precondition: [V=1,Out=0,V2>=1,V1>=V2+1,V6>=V2+1] 

* Chain [[22],[24],[30],31,34]: 7*it(22)+8*it(24)+9*it(30)+1*s(3)+2*s(13)+4*s(29)+9
  Such that:it(22) =< V1-V2
it(30) =< V6-V2
aux(22) =< V2
it(24) =< aux(22)
s(3) =< it(30)*aux(22)
s(13) =< it(24)*aux(22)
s(30) =< it(22)*aux(22)
s(29) =< s(30)

  with precondition: [V=1,Out=0,V2>=1,V1>=V2+1,V6>=V2+1] 

* Chain [[22],[24],34]: 7*it(22)+7*it(24)+2*s(13)+4*s(29)+0
  Such that:it(22) =< V1-V2
aux(23) =< V2
it(24) =< aux(23)
s(13) =< it(24)*aux(23)
s(30) =< it(22)*aux(23)
s(29) =< s(30)

  with precondition: [V=1,Out=0,V2>=1,V1>=V2+1,V6>=V2+1] 

* Chain [[22],34]: 7*it(22)+4*s(29)+0
  Such that:it(22) =< V1-V2
aux(20) =< V2
s(30) =< it(22)*aux(20)
s(29) =< s(30)

  with precondition: [V=1,Out=0,V2>=1,V1>=V2+1,V6>=V2+1] 

* Chain [34]: 0
  with precondition: [Out=0,V>=0,V1>=0,V6>=0,V2>=0] 

* Chain [33,34]: 8
  with precondition: [V=1,V1=0,V6=0,Out=0,V2>=0] 

* Chain [31,34]: 1*s(4)+9
  Such that:s(4) =< V6

  with precondition: [V=1,V1=0,Out=0,V6>=1,V2>=V6] 

* Chain [28,34]: 1*s(8)+7
  Such that:s(8) =< V1

  with precondition: [V=1,V6=0,Out=0,V1>=1,V2>=V1] 

* Chain [25,34]: 1*s(21)+1*s(22)+7
  Such that:s(21) =< V1
s(22) =< V6

  with precondition: [V=1,Out=0,V1>=1,V6>=1,V2>=V1,V2>=V6] 


#### Cost of chains of start(V,V1,V6,V2):
* Chain [50]: 118*s(113)+142*s(115)+144*s(121)+16*s(123)+192*s(125)+6*s(153)+6*s(154)+1*s(158)+210*s(161)+18*s(163)+4*s(173)+64*s(194)+24*s(196)+16
  Such that:s(158) =< 1
aux(67) =< V1
aux(68) =< V1-V2
aux(69) =< V6
aux(70) =< V6-V2
aux(71) =< V2
s(125) =< aux(67)
s(113) =< aux(69)
s(115) =< aux(71)
s(161) =< aux(68)
s(163) =< s(161)*aux(71)
s(173) =< s(161)*aux(69)
s(121) =< aux(70)
s(193) =< s(161)*aux(71)
s(194) =< s(193)
s(196) =< s(115)*aux(71)
s(123) =< s(121)*aux(71)
s(153) =< s(125)*aux(71)
s(154) =< s(125)*aux(67)

  with precondition: [V=0] 

* Chain [49]: 71*s(315)+10*s(316)+140*s(318)+144*s(324)+16*s(327)+9*s(328)+9*s(329)+182*s(337)+6*s(340)+6*s(341)+80*s(352)+30*s(354)+1*s(454)+14
  Such that:s(454) =< V
aux(88) =< V1
aux(89) =< V1-V2
aux(90) =< V6
aux(91) =< V6-V2
aux(92) =< V2
s(315) =< aux(88)
s(316) =< aux(90)
s(318) =< aux(92)
s(324) =< aux(91)
s(327) =< s(324)*aux(92)
s(328) =< s(315)*aux(92)
s(329) =< s(315)*aux(88)
s(337) =< aux(89)
s(340) =< s(337)*aux(92)
s(341) =< s(337)*aux(90)
s(351) =< s(337)*aux(92)
s(352) =< s(351)
s(354) =< s(318)*aux(92)

  with precondition: [V>=1] 

* Chain [48]: 54*s(457)+13
  Such that:aux(93) =< V6
s(457) =< aux(93)

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

* Chain [47]: 9*s(462)+54*s(466)+6*s(468)+14
  Such that:aux(97) =< V6-V2
aux(98) =< V2
s(462) =< aux(98)
s(466) =< aux(97)
s(468) =< s(466)*aux(98)

  with precondition: [V>=0,V1>=0,V6>=0,V2>=0] 

* Chain [46]: 3*s(483)+13
  Such that:aux(99) =< V6
s(483) =< aux(99)

  with precondition: [V=1,V1=0,V6>=1,V2>=V6] 

* Chain [45]: 126*s(487)+105*s(490)+13
  Such that:aux(100) =< V1
aux(101) =< V6
s(490) =< aux(100)
s(487) =< aux(101)

  with precondition: [V=1,V2=0,V1>=1,V6>=1] 

* Chain [44]: 1*s(510)+2*s(511)+18*s(514)+2*s(516)+14
  Such that:s(510) =< 1
s(512) =< V6-V2
aux(102) =< V2
s(511) =< aux(102)
s(514) =< s(512)
s(516) =< s(514)*aux(102)

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

* Chain [43]: 42*s(518)+13
  Such that:aux(103) =< V1
s(518) =< aux(103)

  with precondition: [V=1,V6=0,V2=0,V1>=1] 

* Chain [42]: 9*s(523)+42*s(529)+6*s(531)+13
  Such that:aux(107) =< V1-V2
aux(108) =< V2
s(523) =< aux(108)
s(529) =< aux(107)
s(531) =< s(529)*aux(108)

  with precondition: [V=1,V6=0,V2>=1,V1>=V2+1] 

* Chain [41]: 3*s(544)+13
  Such that:aux(109) =< V1
s(544) =< aux(109)

  with precondition: [V=1,V6=0,V1>=1,V2>=V1] 

* Chain [40]: 28*s(548)+13
  Such that:aux(110) =< V1
s(548) =< aux(110)

  with precondition: [V=1,V6=1,V2=0,V1>=1] 

* Chain [39]: 2*s(551)+1*s(552)+14*s(555)+2*s(557)+12
  Such that:s(552) =< 1
s(553) =< V1-V2
aux(111) =< V2
s(551) =< aux(111)
s(555) =< s(553)
s(557) =< s(555)*aux(111)

  with precondition: [V=1,V6=1,V2>=1,V1>=V2+1] 

* Chain [38]: 8*s(558)+28*s(563)+8*s(566)+12
  Such that:aux(114) =< V1-V2
aux(115) =< V2
s(558) =< aux(115)
s(563) =< aux(114)
s(566) =< s(563)*aux(115)

  with precondition: [V=1,V6=V2+1,V6>=2,V1>=V6] 

* Chain [37]: 48*s(578)+36*s(583)+4*s(586)+12*s(587)+14
  Such that:aux(118) =< V6-V2
aux(119) =< V2
s(578) =< aux(119)
s(583) =< aux(118)
s(586) =< s(583)*aux(119)
s(587) =< s(578)*aux(119)

  with precondition: [V=1,V1=V2+1,V1>=2,V6>=V1] 

* Chain [36]: 2*s(600)+2*s(601)+12
  Such that:aux(120) =< V6
aux(121) =< V2
s(601) =< aux(120)
s(600) =< aux(121)

  with precondition: [V=1,V1=V2+1,V6>=1,V1>=V6+1] 

* Chain [35]: 2*s(604)+2*s(605)+12
  Such that:aux(122) =< V1
aux(123) =< V2
s(604) =< aux(122)
s(605) =< aux(123)

  with precondition: [V=1,V6=V2+1,V1>=1,V6>=V1+1] 


Closed-form bounds of start(V,V1,V6,V2): 
-------------------------------------
* Chain [50] with precondition: [V=0] 
    - Upper bound: nat(V1)*192+17+nat(V1)*6*nat(V1)+nat(V1)*6*nat(V2)+nat(V6)*118+nat(V6)*4*nat(V1-V2)+nat(V2)*142+nat(V2)*24*nat(V2)+nat(V2)*82*nat(V1-V2)+nat(V2)*16*nat(V6-V2)+nat(V1-V2)*210+nat(V6-V2)*144 
    - Complexity: n^2 
* Chain [49] with precondition: [V>=1] 
    - Upper bound: V+14+nat(V1)*71+nat(V1)*9*nat(V1)+nat(V1)*9*nat(V2)+nat(V6)*10+nat(V6)*6*nat(V1-V2)+nat(V2)*140+nat(V2)*30*nat(V2)+nat(V2)*86*nat(V1-V2)+nat(V2)*16*nat(V6-V2)+nat(V1-V2)*182+nat(V6-V2)*144 
    - Complexity: n^2 
* Chain [48] with precondition: [V1=0,V>=1] 
    - Upper bound: nat(V6)*54+13 
    - Complexity: n 
* Chain [47] with precondition: [V>=0,V1>=0,V6>=0,V2>=0] 
    - Upper bound: 9*V2+14+6*V2*nat(V6-V2)+nat(V6-V2)*54 
    - Complexity: n^2 
* Chain [46] with precondition: [V=1,V1=0,V6>=1,V2>=V6] 
    - Upper bound: 3*V6+13 
    - Complexity: n 
* Chain [45] with precondition: [V=1,V2=0,V1>=1,V6>=1] 
    - Upper bound: 105*V1+126*V6+13 
    - Complexity: n 
* Chain [44] with precondition: [V1=1,V>=0] 
    - Upper bound: nat(V2)*2+15+nat(V2)*2*nat(V6-V2)+nat(V6-V2)*18 
    - Complexity: n^2 
* Chain [43] with precondition: [V=1,V6=0,V2=0,V1>=1] 
    - Upper bound: 42*V1+13 
    - Complexity: n 
* Chain [42] with precondition: [V=1,V6=0,V2>=1,V1>=V2+1] 
    - Upper bound: 42*V1-42*V2+(9*V2+13+(V1-V2)*(6*V2)) 
    - Complexity: n^2 
* Chain [41] with precondition: [V=1,V6=0,V1>=1,V2>=V1] 
    - Upper bound: 3*V1+13 
    - Complexity: n 
* Chain [40] with precondition: [V=1,V6=1,V2=0,V1>=1] 
    - Upper bound: 28*V1+13 
    - Complexity: n 
* Chain [39] with precondition: [V=1,V6=1,V2>=1,V1>=V2+1] 
    - Upper bound: 14*V1-14*V2+(2*V2+13+(V1-V2)*(2*V2)) 
    - Complexity: n^2 
* Chain [38] with precondition: [V=1,V6=V2+1,V6>=2,V1>=V6] 
    - Upper bound: 28*V1-28*V2+(8*V2+12+(V1-V2)*(8*V2)) 
    - Complexity: n^2 
* Chain [37] with precondition: [V=1,V1=V2+1,V1>=2,V6>=V1] 
    - Upper bound: 36*V6-36*V2+(48*V2+14+12*V2*V2+(V6-V2)*(4*V2)) 
    - Complexity: n^2 
* Chain [36] with precondition: [V=1,V1=V2+1,V6>=1,V1>=V6+1] 
    - Upper bound: 2*V6+2*V2+12 
    - Complexity: n 
* Chain [35] with precondition: [V=1,V6=V2+1,V1>=1,V6>=V1+1] 
    - Upper bound: 2*V1+2*V2+12 
    - Complexity: n 

### Maximum cost of start(V,V1,V6,V2): max([max([nat(V6)*54,nat(V6)*126+nat(V1)*63+nat(V1)*14+nat(V1)*25+nat(V1)*3])+1,nat(V2)*2+max([max([nat(V1)*2,nat(V6)*2,nat(V2)*2*nat(V1-V2)+1+nat(V1-V2)*14,nat(V2)*2*nat(V6-V2)+3+nat(V6-V2)*18]),nat(V2)*6+max([nat(V2)*8*nat(V1-V2)+nat(V1-V2)*28,nat(V2)+1+max([nat(V2)*6*nat(V1-V2)+nat(V1-V2)*42,nat(V2)*4*nat(V6-V2)+1+nat(V6-V2)*36+max([nat(V2)*2*nat(V6-V2)+nat(V6-V2)*18,nat(V1)*6*nat(V1)+nat(V1)*71+nat(V1)*6*nat(V2)+nat(V6)*10+nat(V6)*4*nat(V1-V2)+nat(V2)*92+nat(V2)*12*nat(V2)+nat(V2)*82*nat(V1-V2)+nat(V2)*12*nat(V6-V2)+nat(V1-V2)*182+nat(V6-V2)*108+max([nat(V1)*121+3+nat(V6)*108+nat(V2)*2+nat(V1-V2)*28,nat(V1)*3*nat(V1)+V+nat(V1)*3*nat(V2)+nat(V6)*2*nat(V1-V2)+nat(V2)*6*nat(V2)+nat(V2)*4*nat(V1-V2)])+(nat(V2)*12*nat(V2)+nat(V2)*39)])])])])])+12 
Asymptotic class: n^2 
* Total analysis performed in 2561 ms.


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

(10)
BOUNDS(1, n^2)

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

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