WORST_CASE(Omega(n^1),O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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), 6 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 326 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), 17 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 290 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), 1263 ms]
            (24) BOUNDS(1, INF)


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

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

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

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

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

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

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

   cond1(z, z') -{ 1 }-> cond2(gr(1 + x, y), 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   cond1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cond2(z, z', z'') -{ 1 }-> cond1(y, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond2(z, z', z'') -{ 1 }-> cond1(p(x), y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 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
   neq(z, z') -{ 1 }-> neq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   neq(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   neq(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   neq(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, V5),0,[cond1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[cond2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[neq(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[p(V1, Out)],[V1 >= 0]).
eq(cond1(V1, V, Out),1,[gr(1 + V3, V2, Ret0),cond2(Ret0, 1 + V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = 1 + V3,V = V2]).
eq(cond2(V1, V, V5, Out),1,[cond1(V6, V6, Ret1)],[Out = Ret1,V = V4,V5 = V6,V1 = 1,V4 >= 0,V6 >= 0]).
eq(cond2(V1, V, V5, Out),1,[p(V8, Ret01),cond1(Ret01, V7, Ret2)],[Out = Ret2,V = V8,V5 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 0,V = V9,V9 >= 0,V1 = 0]).
eq(gr(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = 1 + V10,V = 0]).
eq(gr(V1, V, Out),1,[gr(V12, V11, Ret3)],[Out = Ret3,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]).
eq(neq(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(neq(V1, V, Out),1,[],[Out = 1,V = 1 + V13,V13 >= 0,V1 = 0]).
eq(neq(V1, V, Out),1,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]).
eq(neq(V1, V, Out),1,[neq(V16, V15, Ret4)],[Out = Ret4,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V17,V17 >= 0,V1 = 1 + V17]).
eq(cond1(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
input_output_vars(cond1(V1,V,Out),[V1,V],[Out]).
input_output_vars(cond2(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(neq(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. non_recursive  : [p/2]
1. recursive  : [gr/3]
2. recursive  : [cond1/3,cond2/4]
3. recursive  : [neq/3]
4. non_recursive  : [start/3]

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

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

### Specialization of cost equations p/2 
* CE 8 is refined into CE [19] 
* CE 7 is refined into CE [20] 


### Cost equations --> "Loop" of p/2 
* CEs [19] --> Loop 14 
* CEs [20] --> Loop 15 

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

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


### Specialization of cost equations gr/3 
* CE 14 is refined into CE [21] 
* CE 13 is refined into CE [22] 
* CE 12 is refined into CE [23] 


### Cost equations --> "Loop" of gr/3 
* CEs [22] --> Loop 16 
* CEs [23] --> Loop 17 
* CEs [21] --> Loop 18 

### Ranking functions of CR gr(V1,V,Out) 
* RF of phase [18]: [V,V1]

#### Partial ranking functions of CR gr(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V
    V1


### Specialization of cost equations cond1/3 
* CE 11 is refined into CE [24] 
* CE 9 is refined into CE [25] 
* CE 10 is refined into CE [26,27] 


### Cost equations --> "Loop" of cond1/3 
* CEs [27] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [26] --> Loop 21 
* CEs [24] --> Loop 22 

### Ranking functions of CR cond1(V1,V,Out) 
* RF of phase [20]: [V1]

#### Partial ranking functions of CR cond1(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V1


### Specialization of cost equations neq/3 
* CE 18 is refined into CE [28] 
* CE 17 is refined into CE [29] 
* CE 16 is refined into CE [30] 
* CE 15 is refined into CE [31] 


### Cost equations --> "Loop" of neq/3 
* CEs [29] --> Loop 23 
* CEs [30] --> Loop 24 
* CEs [31] --> Loop 25 
* CEs [28] --> Loop 26 

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

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


### Specialization of cost equations start/3 
* CE 2 is refined into CE [32] 
* CE 1 is refined into CE [33,34] 
* CE 3 is refined into CE [35] 
* CE 4 is refined into CE [36,37,38,39] 
* CE 5 is refined into CE [40,41,42,43,44,45] 
* CE 6 is refined into CE [46,47] 


### Cost equations --> "Loop" of start/3 
* CEs [43] --> Loop 27 
* CEs [37,42,47] --> Loop 28 
* CEs [32,35,38,39,44,45] --> Loop 29 
* CEs [33,34,36,40,41,46] --> Loop 30 

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

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


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

#### Cost of chains of p(V1,Out):
* Chain [15]: 1
  with precondition: [V1=0,Out=0] 

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


#### Cost of chains of gr(V1,V,Out):
* Chain [[18],17]: 1*it(18)+1
  Such that:it(18) =< V1

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

* Chain [[18],16]: 1*it(18)+1
  Such that:it(18) =< V

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

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

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


#### Cost of chains of cond1(V1,V,Out):
* Chain [[20],22]: 4*it(20)+1*s(3)+0
  Such that:aux(3) =< V1
it(20) =< aux(3)
s(3) =< it(20)*aux(3)

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

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

* Chain [21,22]: 3
  with precondition: [V=0,Out=0,V1>=1] 

* Chain [19,[20],22]: 5*it(20)+1*s(3)+3
  Such that:aux(4) =< V
it(20) =< aux(4)
s(3) =< it(20)*aux(4)

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

* Chain [19,22]: 1*s(4)+3
  Such that:s(4) =< V

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


#### Cost of chains of neq(V1,V,Out):
* Chain [[26],25]: 1*it(26)+1
  Such that:it(26) =< V1

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

* Chain [[26],24]: 1*it(26)+1
  Such that:it(26) =< V1

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

* Chain [[26],23]: 1*it(26)+1
  Such that:it(26) =< V

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

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

* Chain [24]: 1
  with precondition: [V1=0,Out=1,V>=1] 

* Chain [23]: 1
  with precondition: [V=0,Out=1,V1>=1] 


#### Cost of chains of start(V1,V,V5):
* Chain [30]: 12*s(14)+2*s(17)+4*s(21)+1*s(22)+5
  Such that:s(18) =< V
aux(6) =< V5
s(14) =< aux(6)
s(21) =< s(18)
s(22) =< s(21)*s(18)
s(17) =< s(14)*aux(6)

  with precondition: [V1=0] 

* Chain [29]: 10*s(26)+2*s(28)+8*s(32)+6*s(33)+1*s(34)+1*s(35)+4
  Such that:aux(7) =< V5
aux(8) =< V1
aux(9) =< V
s(33) =< aux(8)
s(32) =< aux(9)
s(34) =< s(33)*aux(8)
s(35) =< s(32)*aux(9)
s(26) =< aux(7)
s(28) =< s(26)*aux(7)

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

* Chain [28]: 1
  with precondition: [V1>=1] 

* Chain [27]: 1*s(40)+1
  Such that:s(40) =< V

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


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [30] with precondition: [V1=0] 
    - Upper bound: nat(V)*4+5+nat(V)*nat(V)+nat(V5)*12+nat(V5)*2*nat(V5) 
    - Complexity: n^2 
* Chain [29] with precondition: [V1>=0,V>=0] 
    - Upper bound: 6*V1+4+V1*V1+8*V+V*V+nat(V5)*10+nat(V5)*2*nat(V5) 
    - Complexity: n^2 
* Chain [28] with precondition: [V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [27] with precondition: [V1=V,V1>=1] 
    - Upper bound: V+1 
    - Complexity: n 

### Maximum cost of start(V1,V,V5): nat(V)*3+3+nat(V)*nat(V)+nat(V5)*10+nat(V5)*2*nat(V5)+max([nat(V5)*2+1,6*V1+V1*V1+nat(V)*4])+nat(V)+1 
Asymptotic class: n^2 
* Total analysis performed in 244 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(s(x), y) -> cond2(gr(s(x), y), s(x), y)
   cond2(true, x, y) -> cond1(y, y)
   cond2(false, x, y) -> cond1(p(x), y)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   neq(0', 0') -> false
   neq(0', s(x)) -> true
   neq(s(x), 0') -> true
   neq(s(x), s(y)) -> neq(x, y)
   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(s(x), y) -> cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) -> cond1(y, y)
cond2(false, x, y) -> cond1(p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: s:0' -> s:0' -> cond1:cond2
s :: s:0' -> s:0'
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
gr :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
p :: s:0' -> s:0'
0' :: s:0'
neq :: s:0' -> s:0' -> true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat -> s:0'

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

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

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

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

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

Types:
cond1 :: s:0' -> s:0' -> cond1:cond2
s :: s:0' -> s:0'
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
gr :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
p :: s:0' -> s:0'
0' :: s:0'
neq :: s:0' -> s:0' -> true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gr(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(gen_s:0'4_0(n6_0), gen_s:0'4_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(s(x), y) -> cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) -> cond1(y, y)
cond2(false, x, y) -> cond1(p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: s:0' -> s:0' -> cond1:cond2
s :: s:0' -> s:0'
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
gr :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
p :: s:0' -> s:0'
0' :: s:0'
neq :: s:0' -> s:0' -> true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


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

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

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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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

Types:
cond1 :: s:0' -> s:0' -> cond1:cond2
s :: s:0' -> s:0'
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
gr :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
p :: s:0' -> s:0'
0' :: s:0'
neq :: s:0' -> s:0' -> true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
cond1, neq
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
neq(gen_s:0'4_0(n16104_0), gen_s:0'4_0(n16104_0)) -> false, rt in Omega(1 + n16104_0)

Induction Base:
neq(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
false

Induction Step:
neq(gen_s:0'4_0(+(n16104_0, 1)), gen_s:0'4_0(+(n16104_0, 1))) ->_R^Omega(1)
neq(gen_s:0'4_0(n16104_0), gen_s:0'4_0(n16104_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
BOUNDS(1, INF)