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


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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), 1 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 390 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), 267 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), 19 ms]
            (24) proven lower bound
            (25) LowerBoundPropagationProof [FINISHED, 0 ms]
            (26) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(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^2).


The TRS R consists of the following rules:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]

The TRS has the following type information:
nonZero :: 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
p :: 0:s -> 0:s
id_inc :: 0:s -> 0:s
random :: 0:s -> 0:s
rand :: 0:s -> 0:s -> 0:s
if :: false:true -> 0: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:

   p(v0) -> null_p [0]
   nonZero(v0) -> null_nonZero [0]
   if(v0, v1, v2) -> null_if [0]

And the following fresh constants:    null_p, null_nonZero, 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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]
   p(v0) -> null_p [0]
   nonZero(v0) -> null_nonZero [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
nonZero :: 0:s:null_p:null_if -> false:true:null_nonZero
0 :: 0:s:null_p:null_if
false :: false:true:null_nonZero
s :: 0:s:null_p:null_if -> 0:s:null_p:null_if
true :: false:true:null_nonZero
p :: 0:s:null_p:null_if -> 0:s:null_p:null_if
id_inc :: 0:s:null_p:null_if -> 0:s:null_p:null_if
random :: 0:s:null_p:null_if -> 0:s:null_p:null_if
rand :: 0:s:null_p:null_if -> 0:s:null_p:null_if -> 0:s:null_p:null_if
if :: false:true:null_nonZero -> 0:s:null_p:null_if -> 0:s:null_p:null_if -> 0:s:null_p:null_if
null_p :: 0:s:null_p:null_if
null_nonZero :: false:true:null_nonZero
null_if :: 0:s:null_p: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
   false => 1
   true => 2
   null_p => 0
   null_nonZero => 0
   null_if => 0

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

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

   id_inc(z) -{ 1 }-> x :|: x >= 0, z = x
   id_inc(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 1 }-> rand(p(x), id_inc(y)) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   nonZero(z) -{ 1 }-> 2 :|: x >= 0, z = 1 + x
   nonZero(z) -{ 1 }-> 1 :|: z = 0
   nonZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)
   rand(z, z') -{ 1 }-> if(nonZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   random(z) -{ 1 }-> rand(x, 0) :|: x >= 0, z = x

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, V6, V9),0,[nonZero(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[p(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[random(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[rand(V, V6, Out)],[V >= 0,V6 >= 0]).
eq(start(V, V6, V9),0,[if(V, V6, V9, Out)],[V >= 0,V6 >= 0,V9 >= 0]).
eq(nonZero(V, Out),1,[],[Out = 1,V = 0]).
eq(nonZero(V, Out),1,[],[Out = 2,V1 >= 0,V = 1 + V1]).
eq(p(V, Out),1,[],[Out = 0,V = 1]).
eq(p(V, Out),1,[p(1 + V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V = 2 + V2]).
eq(fun(V, Out),1,[],[Out = V3,V3 >= 0,V = V3]).
eq(fun(V, Out),1,[],[Out = 1 + V4,V4 >= 0,V = V4]).
eq(random(V, Out),1,[rand(V5, 0, Ret)],[Out = Ret,V5 >= 0,V = V5]).
eq(rand(V, V6, Out),1,[nonZero(V7, Ret0),if(Ret0, V7, V8, Ret2)],[Out = Ret2,V7 >= 0,V8 >= 0,V = V7,V6 = V8]).
eq(if(V, V6, V9, Out),1,[],[Out = V11,V6 = V10,V9 = V11,V = 1,V10 >= 0,V11 >= 0]).
eq(if(V, V6, V9, Out),1,[p(V13, Ret01),fun(V12, Ret11),rand(Ret01, Ret11, Ret3)],[Out = Ret3,V = 2,V6 = V13,V9 = V12,V13 >= 0,V12 >= 0]).
eq(p(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
eq(nonZero(V, Out),0,[],[Out = 0,V15 >= 0,V = V15]).
eq(if(V, V6, V9, Out),0,[],[Out = 0,V17 >= 0,V9 = V18,V16 >= 0,V = V17,V6 = V16,V18 >= 0]).
input_output_vars(nonZero(V,Out),[V],[Out]).
input_output_vars(p(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(random(V,Out),[V],[Out]).
input_output_vars(rand(V,V6,Out),[V,V6],[Out]).
input_output_vars(if(V,V6,V9,Out),[V,V6,V9],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [fun/2]
1. recursive  : [p/2]
2. non_recursive  : [nonZero/2]
3. recursive  : [if/4,rand/3]
4. non_recursive  : [random/2]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into nonZero/2
3. SCC is partially evaluated into rand/3
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations fun/2 
* CE 12 is refined into CE [20] 
* CE 13 is refined into CE [21] 


### Cost equations --> "Loop" of fun/2 
* CEs [20] --> Loop 13 
* CEs [21] --> Loop 14 

### Ranking functions of CR fun(V,Out) 

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


### Specialization of cost equations p/2 
* CE 9 is refined into CE [22] 
* CE 11 is refined into CE [23] 
* CE 10 is refined into CE [24] 


### Cost equations --> "Loop" of p/2 
* CEs [24] --> Loop 15 
* CEs [22,23] --> Loop 16 

### Ranking functions of CR p(V,Out) 
* RF of phase [15]: [V-1]

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


### Specialization of cost equations nonZero/2 
* CE 18 is refined into CE [25] 
* CE 19 is refined into CE [26] 
* CE 17 is refined into CE [27] 


### Cost equations --> "Loop" of nonZero/2 
* CEs [25] --> Loop 17 
* CEs [26] --> Loop 18 
* CEs [27] --> Loop 19 

### Ranking functions of CR nonZero(V,Out) 

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


### Specialization of cost equations rand/3 
* CE 16 is refined into CE [28] 
* CE 14 is refined into CE [29,30,31] 
* CE 15 is refined into CE [32,33,34,35] 


### Cost equations --> "Loop" of rand/3 
* CEs [35] --> Loop 20 
* CEs [34] --> Loop 21 
* CEs [33] --> Loop 22 
* CEs [32] --> Loop 23 
* CEs [28] --> Loop 24 
* CEs [29,30,31] --> Loop 25 

### Ranking functions of CR rand(V,V6,Out) 
* RF of phase [20,21]: [V-1]

#### Partial ranking functions of CR rand(V,V6,Out) 
* Partial RF of phase [20,21]:
  - RF of loop [20:1,21:1]:
    V-1


### Specialization of cost equations start/3 
* CE 2 is refined into CE [36,37,38,39,40,41,42,43,44,45,46,47,48,49] 
* CE 1 is refined into CE [50] 
* CE 3 is refined into CE [51] 
* CE 4 is refined into CE [52,53,54] 
* CE 5 is refined into CE [55,56] 
* CE 6 is refined into CE [57,58] 
* CE 7 is refined into CE [59,60,61,62,63,64] 
* CE 8 is refined into CE [65,66,67,68,69,70] 


### Cost equations --> "Loop" of start/3 
* CEs [36,37,38,39,40,41,42,43,44,45,46,47,48,49] --> Loop 26 
* CEs [51] --> Loop 27 
* CEs [50,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70] --> Loop 28 

### Ranking functions of CR start(V,V6,V9) 

#### Partial ranking functions of CR start(V,V6,V9) 


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

#### Cost of chains of fun(V,Out):
* Chain [14]: 1
  with precondition: [V+1=Out,V>=0] 

* Chain [13]: 1
  with precondition: [V=Out,V>=0] 


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

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

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


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

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

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


#### Cost of chains of rand(V,V6,Out):
* Chain [[20,21],25]: 10*it(20)+1*s(5)+1*s(6)+2
  Such that:aux(5) =< V
it(20) =< aux(5)
aux(2) =< aux(5)
s(5) =< it(20)*aux(5)
s(6) =< it(20)*aux(2)

  with precondition: [Out=0,V>=2,V6>=0] 

* Chain [[20,21],23,25]: 10*it(20)+1*s(5)+1*s(6)+7
  Such that:aux(6) =< V
it(20) =< aux(6)
aux(2) =< aux(6)
s(5) =< it(20)*aux(6)
s(6) =< it(20)*aux(2)

  with precondition: [Out=0,V>=2,V6>=0] 

* Chain [[20,21],23,24]: 10*it(20)+1*s(5)+1*s(6)+8
  Such that:aux(7) =< V
it(20) =< aux(7)
aux(2) =< aux(7)
s(5) =< it(20)*aux(7)
s(6) =< it(20)*aux(2)

  with precondition: [V>=2,V6>=0,Out>=V6+1,V+V6>=Out] 

* Chain [[20,21],22,25]: 10*it(20)+1*s(5)+1*s(6)+7
  Such that:aux(8) =< V
it(20) =< aux(8)
aux(2) =< aux(8)
s(5) =< it(20)*aux(8)
s(6) =< it(20)*aux(2)

  with precondition: [Out=0,V>=2,V6>=0] 

* Chain [[20,21],22,24]: 10*it(20)+1*s(5)+1*s(6)+8
  Such that:aux(9) =< V
it(20) =< aux(9)
aux(2) =< aux(9)
s(5) =< it(20)*aux(9)
s(6) =< it(20)*aux(2)

  with precondition: [V>=2,V6>=0,Out>=V6,V+V6>=Out+1] 

* Chain [25]: 2
  with precondition: [Out=0,V>=0,V6>=0] 

* Chain [24]: 3
  with precondition: [V=0,V6=Out,V6>=0] 

* Chain [23,25]: 7
  with precondition: [Out=0,V>=1,V6>=0] 

* Chain [23,24]: 8
  with precondition: [Out=V6+1,V>=1,Out>=1] 

* Chain [22,25]: 7
  with precondition: [Out=0,V>=1,V6>=0] 

* Chain [22,24]: 8
  with precondition: [V6=Out,V>=1,V6>=0] 


#### Cost of chains of start(V,V6,V9):
* Chain [28]: 101*s(22)+10*s(26)+10*s(27)+9
  Such that:aux(11) =< V
s(22) =< aux(11)
s(25) =< aux(11)
s(26) =< s(22)*aux(11)
s(27) =< s(22)*s(25)

  with precondition: [V>=0] 

* Chain [27]: 1
  with precondition: [V=1,V6>=0,V9>=0] 

* Chain [26]: 110*s(63)+10*s(67)+10*s(68)+11
  Such that:aux(18) =< V6
s(63) =< aux(18)
s(66) =< aux(18)
s(67) =< s(63)*aux(18)
s(68) =< s(63)*s(66)

  with precondition: [V=2,V6>=0,V9>=0] 


Closed-form bounds of start(V,V6,V9): 
-------------------------------------
* Chain [28] with precondition: [V>=0] 
    - Upper bound: 101*V+9+20*V*V 
    - Complexity: n^2 
* Chain [27] with precondition: [V=1,V6>=0,V9>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [26] with precondition: [V=2,V6>=0,V9>=0] 
    - Upper bound: 110*V6+11+20*V6*V6 
    - Complexity: n^2 

### Maximum cost of start(V,V6,V9): max([101*V+8+20*V*V,nat(V6)*110+10+nat(V6)*20*nat(V6)])+1 
Asymptotic class: n^2 
* Total analysis performed in 306 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^2, INF).


The TRS R consists of the following rules:

   nonZero(0') -> false
   nonZero(s(x)) -> true
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0')
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(y))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
nonZero(0') -> false
nonZero(s(x)) -> true
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
p, rand

They will be analysed ascendingly in the following order:
p < rand

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

(16)
Obligation:
Innermost TRS:
Rules:
nonZero(0') -> false
nonZero(s(x)) -> true
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
p, rand

They will be analysed ascendingly in the following order:
p < rand

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':s3_0(+(1, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
p(gen_0':s3_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
s(p(s(gen_0':s3_0(n5_0)))) ->_IH
s(gen_0':s3_0(c6_0))

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:
nonZero(0') -> false
nonZero(s(x)) -> true
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
p, rand

They will be analysed ascendingly in the following order:
p < rand

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
nonZero(0') -> false
nonZero(s(x)) -> true
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s3_0(+(1, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
rand
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rand(gen_0':s3_0(n230_0), gen_0':s3_0(b)) -> gen_0':s3_0(b), rt in Omega(1 + n230_0 + n230_0^2)

Induction Base:
rand(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
if(nonZero(gen_0':s3_0(0)), gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
if(false, gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
gen_0':s3_0(b)

Induction Step:
rand(gen_0':s3_0(+(n230_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
if(nonZero(gen_0':s3_0(+(n230_0, 1))), gen_0':s3_0(+(n230_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
if(true, gen_0':s3_0(+(1, n230_0)), gen_0':s3_0(b)) ->_R^Omega(1)
rand(p(gen_0':s3_0(+(1, n230_0))), id_inc(gen_0':s3_0(b))) ->_L^Omega(1 + n230_0)
rand(gen_0':s3_0(n230_0), id_inc(gen_0':s3_0(b))) ->_R^Omega(1)
rand(gen_0':s3_0(n230_0), gen_0':s3_0(b)) ->_IH
gen_0':s3_0(b)

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(24)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
nonZero(0') -> false
nonZero(s(x)) -> true
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s3_0(+(1, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
rand
----------------------------------------

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

(26)
BOUNDS(n^2, INF)